Geometric Complexity Theory IV: quantum group for the Kronecker ...
arXiv:cs/0703110v2 [cs.CC] 29 Mar 2007
Geometric Complexity Theory IV: quantum group for the Kronecker problem Dedicated to Sri Ramakrishna Ketan D. Mulmuley The University of Chicago and I.I.T., Mumbai∗ Milind Sohoni I.I.T., Mumbai February 1, 2008
Abstract A fundamental problem in representation theory is to find an explicit positive rule, akin to the Littlewood-Richardson rule, for decomposing the tensor product of two irreducible representations of the symmetric group (Kronecker problem). In this paper a generalization of the Drinfeld-Jimbo quantum group, with a compact real form, is constructed, and also an associated semisimple algebra that has conjecturally the same relationship with the generalized quantum group that the Hecke algebra has with the Drinfeld-Jimbo quantum group. In the sequel [24] it is observed that an explicit positive decomposition rule for the Kronecker problem exists assuming that the coordinate ring of the generalized quantum group has a basis analogous to the canonical basis for the coordinate ring of the Drinfeld-Jimbo quantum group, as per Kashiwara and Lusztig [12, 17], or in the dual setting–the associated algebra has a basis analogous to the Kazhdan-Lusztig basis for the Hecke algebra [13], as suggested by the experimental and theoretical results therein. In the other sequel [23], similar quantum group and algebra are constructed for the generalized plethysm problem, of which the Kronecker problem studied here is a special case. These problems play a central role in geometric complexity theory– an approach to the P vs. N P and related problems. ∗
Visiting faculty member
1
1
Introduction
This is a continuation of the series of articles [26, 27, 28] on geometric complexity theory, an approach to the P vs. N P and related problems. A basic philosophy of this approach is called the flip, which was proposed in [25], with a detailed exposition to appear in [22]. The flip, in essence, reduces the negative lower bound problems in complexity theory to positive problems in mathematics. One central positive problem that arises in the flip turns out to be the following fundamental problem in representation theory of the symmetric group Sn ; cf. [5, 20, 33] for its history and significance. Problem 1.1 (Kronecker problem) Find an explicit positive decomposition rule for the tensor product of two irreducible representations (Specht modules) of Sn over C. Specifically, given partitions (Young diagrams) λ, µ, π, find an explicit positive formula π , which is the multiplicity of the irreducible for the Kronecker coefficient kλ,µ representation (Specht module) Sπ of the symmetric group Sn in the tensor product Sλ ⊗ Sµ . By an explicit positive formula, we mean a formula involving no alternating signs, as in the Littlewood-Richardson rule. This problem is a special case of the following slightly more general problem [5]. Given a partition λ of height at most n, let Vλ (G) denote the corresponding irreducible Weyl module of G. Let V = C n , W = Cm , X = V ⊗ W , and consider the natural homomorphism H = GL(V ) × GL(W ) → G = GL(V ⊗ W ) = GL(X).
(1)
Any irreducible representation Vλ (G) of G decomposes as an H-module: Vλ (G) = ⊕α,β mλα,β Vα (GL(V )) ⊗ Vβ (GL(W )),
(2)
where α and β range over Young diagrams of height at most n and m, respectively, and Vα (GL(V )) and Vβ (GL(W )) denote the corresponding irreducible representations of GL(V ) and GL(W ), respectively. Problem 1.2 (Kronecker problem: second version) Find an explicit positive rule for the decomposition (2). Specifically, given partitions α, β, λ, find an explicit positive formula for the (generalized) Kronecker coefficient mλα,β , which is the multiplicity of the H-module Vα (GL(V )) ⊗ Vβ (GL(W )) in the G-module Vλ (G). 2
1.1
The basic idea
Our approach to this problem is motivated by a transparent proof for the positive Littlewood-Richardson rule based on the theory of quantum groups, which works for arbitrary symmetrizable Kac-Moody algebras [10, 11, 16, 18, 17, 29]. To recall the basic idea of this proof, we need some notation. Given Z = Cn , G = GL(Z), let G be the Lie algebra of G, U (G) its enveloping algebra, and O(G) its coordinate algebra. There are two equivalent ways of defining a standard quantum group associated with G: either by defining (1) a quantization of the enveloping algebra U (G), namely, the Drinfeld-Jimbo quantized enveloping algebra Uq (G) [4, 7], or (2) a quantization Oq (G) of the coordinate algebra O(G), namely, the FRT-algebra [32] that is dual to Uq (G). Let GLq (Z) denote the standard quantum group deformation of G = GL(Z)–which is only a virtual object–whose coordinate ring is Oq (G). Now a proof of the generalized Littlewood-Richardson rule mentioned above roughly goes as follows. Let H = GLn (C). Given partitions α and β of height at most n, the tensor product of the Weyl modules Vα (H) and Vβ (H) decomposes as: Vα (H) ⊗ Vβ (H) = ⊕λ cλα,β Vλ (H), where λ ranges over Young diagrams of height at most n and cλα,β is the Littlewood-Richardson coefficient. The problem is to find an explicit positive formula for this coefficient. Equivalently, letting G = H × H, the problem is to find an explicit positive decomposition of a given G–module Vα (H) ⊗ Vβ (H), when considered as an H-module via the diagonal embedding: H → G = H × H. Let H be the Lie algebra of H, and U (H) its enveloping algebra. The infinitesimal version the diagonal map above is the map U (H) →∆ U (H) ⊗ U (H), where ∆(x) = 1 ⊗ x + x ⊗ 1 for x ∈ H. Drinfeld and Jimbo [4, 7] have given a quantum deformation of this map: Uq (H) →∆ Uq (H) ⊗ Uq (H), where Uq (H) is the Drinfeld-Jimbo algebra, and such a deformation works for arbitrary symmetrizable Kac-Moody algebras. The (generalized) LittlewoodRichardson rule then comes out of the properties of canonical (local/global 3
crystal) bases of Kashiwara and Lusztig [10, 11, 16, 17, 29] for representations of Uq (H). This suggests the following analogous strategy to address the Kronecker problem (Problem 1.2). (1) Find a quantization of the homomorphism (1) H = GL(V ) × GL(W ) → G = GL(V ⊗ W ) = GL(X). Instead of working in the setting of Drinfeld-Jimbo quantized algebra, we shall work in the dual setting FRT-coordinate algebras [32]. So what we seek is a quantization of the form: ¯ GLq (V ) × GLq (W ) → GLq (X),
(3)
¯ where the quantum groups GLq (V ) and GLq (W ) are standard, and GLq (X) is a newly sought quantum group deformation of GL(V ⊗ W ). Furthermore, ¯ should have a compact real form in the sense like GLq (X), the new GLq (X) of Woronowicz [36]. This is required in the present context for two reasons. First, we want representation theory of the new quantum group to be rich like that of the standard quantum group. In particular, we want quantum analogues of the classical complete reducibility theorem, and Peter-Weyl theorem to hold, for which compactness is crucial. Second, we want representations of the new quantum group to have bases akin to the canonical bases for representations of the standard quantum group. Compactness is important for existence of such bases. Indeed, Kashiwara’s theory of local crystal bases [10], which was later globalized in [12], came out of an analysis of q-orthonormal Gelfand-Tsetlin bases for representations of the standard quantum group, whose construction depends on compactness of its real form in the sense of Woronowicz. (2) Develop a theory of canonical (local/global crystal) bases for represen¯ akin to the canonical bases for representations of the tations of GLq (X) standard quantum group GLq (X), as per Kashiwara and Lusztig [10, 11, 18]. (3) The required explicit positive decomposition rule should then follow from the properties of canonical bases. This is the route that we shall adopt. 4
1.2
Quantization
As expected, the theory of Drinfeld-Jimbo quantum group does not work for the Kronecker problem, because it can be shown (as in [8]) that the homomorphism (1) can not be quantized in the category of Drinfeld-Jimbo quantum groups. What is needed an analogue of the Drinfeld-Jimbo quantum group in the present setting. This is provided by the following result, which addresses the first step. Theorem 1.3 (a) The homomorphism H = GL(V ) × GL(W ) → G = GL(V ⊗ W ) = GL(X).
(4)
can be quantized in the form ¯ GLq (V ) × GLq (W ) → GLq (X),
(5)
¯ where the quantum groups GLq (V ) and GLq (W ) are standard, and GLq (X) is a new quantum group deformation of GL(V ⊗ W ) defined in this paper. ¯ has a real form Uq (X), ¯ which is a compact (b) The quantum group GLq (X) quantum group in the sense of Woronowicz [36]. This implies that ¯ is completely re1. Every finite dimensional representation of GLq (X) ducible as a direct sum of irreducible representations. 2. Quantum analogue of the Peter-Weyl theorem holds. This is proved in Sections 3-8. When W is trivial, the new quantum group specializes to the Drinfeld-Jimbo quantum group GLq (V ). Furthermore, let C[Sr ] be the group algebra of the symmetric group Sr , and C[Sr ] → C[Sr ] × C[Sr ] the embedding corresponding to the diagonal homomorphism Sr → Sr × Sr . Then Proposition 1.4 The embedding C[Sr ] ֒→ C[Sr ] × C[Sr ] can be similarly quantized. That is, there is a quantized semisimple algebra Br = Br (q) with an embedding Br → Hr ⊗ Hr , where Hr = Hr (q) denotes the Hecke algebra associated with the symmetric group Sr –it is a quantization of the group algebra C[Sr ]. This is proved in Sections 10-10.3. The relationship between Br (q) and ¯ here is conjecturally similar to that between the Hecke algebra GLq (X) Hr (q) and the standard quantum group GLq (X) (Section 10.2). 5
1.3
Explicit positive decomposition
The following addresses the second step. ¯ has a basis that Conjecture 1.5 (cf.[24]) The coordinate ring of GLq (X) is akin to the canonical basis of the coordinate ring of GLq (X) as per Kashiwara and Lusztig [12, 19, 17]. The algebra Br (q) has a basis that is akin to the canonical (KazhdanLusztig) basis of Hr (q) [13, 14]. The precise meaning of “akin to” will be made clear in [24], along with theoretical and experimental evidence in support of the second statement. See also Section 12 for an example of a canonical basis for B3 (q). The first and the second statements here are closely related, in view of the close relationship between analogous two statements for the standard quantum group and Hecke algebra [6]. Assuming this conjecture, it follows [24] that there is an explicit positive rule for the Kronecker problem. This addresses the third step.
1.4
Comparison
¯ and the algebra Br (q) are qualitatively similar, The quantum group GLq (X) but at the same time fundamentally different, in comparison to the standard quantum group GLq (X) and the Hecke algebra Hr (q), respectively. We now compare their properties. ¯ shares (1) Compactness Compactness is a crucial property that GLq (X) with the standard quantum group GLq (X). This means Woronowicz’ theory ¯ [36] of compact quantum groups is applicable to GLq (X). In particular, compactness implies existence of orthonormal bases for representations of ¯ GLq (X). We can also expect nice orthonormal bases for representations ¯ that yield local crystal bases, akin to the orthonormal Gelfandof GLq (X) Tsetlin bases for representations of GLq (X), which, after renormalization, yield local crystal bases [10]. It may be remarked that among the several quantum deformations of GL(Cn ) known by now, besides the one due to Drinfeld and Jimbo–e.g., Reshetikhin, Takhtadzhyan, Faddeev [32, 31], Manin [21], Sudbery [34], Artin, Schelter and Tate [2]–the standard Drinfeld-Jimbo deformation GLq (Cn ) is the only one that has a compact real form in the sense of Woronowicz [36]. 6
Not surprisingly, it is the only deformation of GL(Cn ) that has been studied in depth in the literature. (2) Determinants and minors The quantum determinant and minors of the standard quantum group have simple formulae, very similar to the classical ones. In contrast, explicit formulae for the quantum determinant ¯ turn out to be nonelementary; cf. Propoand quantum minors of GLq (X) sition 6.1. These involve Clebsch-Gordon coefficients and certain q-special functions that arise in the theory of the standard quantum group, which have been intensively studied (cf. the surveys [15]), but not yet completely understood. This difference in the complexity of the determinants and minors is the source of main problems that arise in the new setting. (3) Representations The irreducible representations of the standard quantum group GLq (X) are q-deformations of the irreducible representations of ¯ Because the poincare series of GL(X). This is no longer so for GLq (X). ¯ GLq (X) is different from the Poincare series of GL(X) (cf. Proposition 5.4). Thus, unlike in the standard case, the representation theory of the new quantum group does not run parallel to the representation theory of the classical general linear group. But conjecturally each irreducible representation Vα of G = GL(X) has a unique q-analogue that is a possibly reducible repre¯ cf. Section 9 for a basic example which illustrates this sentation of GL(X); phenomenon, and [24] for a detailed treatment. Thus the representations of GL(X) are not lost in a virtual sense in the transition from GL(X) to ¯ GLq (X). (4) Standard bases The main reason for the one-to-one correspondence between representations of GL(X) and of the standard quantum group GLq (X) is that the formulae for the determinants and minors of GLq (X) are very similar to those for GL(X). Hence, the standard monomial basis in terms of minors of X can be quantized [30] to get standard monomial bases for irreducible representations of GLq (X) [15]. Since, the formulae for the ¯ are no longer elementary, construction determinants and minors of GLq (X) ¯ and their standard bases turns of the irreducible representations of GLq (X) out to be a challenge that we have not been able to meet. (5) Explicit presentation: The Hecke algebra Hr (q) has an explicit presentation in terms of generators and relations. These relations are quantizations of the usual relations among the generators of C[Sr ], the degree three relations being the braid relations. This explicit presentation is necessary for the construction of irreducible representations of Hr (q) and their special bases as in [13]. Analogous explicit presentation for Br (q) turns out to be
7
a challenging problem, as an example for r = 4 in Section 13 illustrates. Hence, explicit construction of irreducible representations of Br (q) and their bases turns out to be challenge that we have not been able to meet. The algebraic structure of Br (q) and its combinatorial properties, which may shed light on this, will be studied in more detail in [1].
1.5
Plethysm and subgroup restriction problem
The Kronecker problem is a special case of the following generalized plethysm problem with H = GL(V ) × GL(W ) and G = GL(V ⊗ W ) = GL(X). Problem 1.6 (The plethysm problem) Given partitions λ, µ, π, give an explicit positive formula for the the plethysm constant aπλ,µ . This is the multiplicity of the irreducible representation Vπ (H) of H = GLn (C) in the irreducible representation Vλ (G) of G = GL(Vµ ), where Vµ = Vµ (H) is an irreducible representation H. Here Vλ (G) is considered an H-module via the representation map ρ : H → G = GL(Vµ ). (The generalized plethysm problem) As above, allowing H to be any connected reductive group, and letting µ and π be dominant weights of H. This problem will be addressed in [23]. Specifically, a quantization of the map H → G in Problem 1.6 is constructed there, which specializes to the one in Theorem 1.3, when H = GL(V ) × GL(W ) and G = GL(X). Analogue of Br (q) in this context will also be constructed. Assuming analogue of Conjecture 1.5 for this quantization, it then follows that there exists an explicit positive formula as sought in Problem 1.6. Also addressed in [23] is a more general subgroup restriction problem, where one is explicitly given a polynomial homomorphism H → G, H and G being arbitrary connected reductive groups.
1.6
Organization
The rest of this paper is organized as follows. In Section 2, we recall basic results and notions concerning the standard (Drinfeld-Jimbo) quantum group in the setting of FRT-algebras [32]. In Sections 3-8 we prove Theorem 1.3. In Sections 10-10.3, we prove Proposition 1.4. Sections 9,11,13 describe concrete examples. Section 12 gives a canonical basis of B3 . 8
2
The standard quantum group
In this section, we briefly recall basic notions concerning the standard quantum group; cf. [15, 32, 21]. It can be defined by specifying either its enveloping algebra, as in Drinfeld and Jimbo [4, 7], or its coordinate FRT-algebra, as in [32]. In this paper, we shall adopt the latter view. We shall mostly follow the terminology in [15]. Let V be vector space of dimension n, R = RV,V a linear mapping of ˆ V,V = R ˆ = τ ◦ R, where τ is the flip of V ⊗ V . Let V ⊗ V to itself, and R u be an n × n variable matrix, and Chui the free algebra over the variable entries of u. Then the defining relations of the FRT bialgebra A(R) [32] in the matrix form is ˆ 1 u2 = u1 u2 R, ˆ Ru (6) i.e.,
ˆ ⊗ u) = (u ⊗ u)R, ˆ R(u
(7)
where u1 = u ⊗ I and u2 = I ⊗ u, and I denotes the identity matrix. This is equivalent to: Ru1 u2 = u2 u1 R. (8) Thus,
ˆ 1 u2 − u1 u2 Ri. ˆ A(R) = Chui/hRu
(9)
This is a bialgebra with comultiplication and counit given by X ∆(uij ) = uik ⊗ ukj , ǫ(uij ) = δij , k
or in the matrix form ˙ ∆(u) = u⊗u, and ǫ(u) = I, where · denotes matrix multiplication. The coordinate algebra O(Mq (V )) of the standard quantum matrix space Mq (V ) is the FRT-algebra A(R) for the specific R given by: ji Rmn = q δij δin δjm + (q − q −1 )δim δjn θ(j − i),
where θ(k) is 1 if k > 0 and 0 otherwise. It is known that the corresponding ˆ satisfies the quadratic equation R ˆ − qI)(R ˆ + q −1 I) = 0, (R 9
(10)
and has the spectral decomposition ˆ = qP+ − q −1 P, R
(11)
where the projections P+ = P+V and P− = P−V are P+ =
ˆ + q −1 I R , q + q −1
P− =
ˆ + qI −R , q + q −1
(12)
so that I = P+ P−
(13)
is the spectral decomposition of the identity. These projections are quantum analogues of the symmetrization and antisymmetrization operators on Cl ⊗ Cl , respectively. Specifically, let the symmetric subspace Sq (V ⊗ V ) be the image of P+ , and the antisymmetric space Aq (V ⊗V ) the image of P− . Thus Sq (V ⊗ V ) is defined by the equation P− v1 v2 = 0,
(14)
where v1 = v ⊗ I and v2 = I ⊗ v. In terms of the entries vi ’s of v, this becomes vi vj = qvj vi , i < j. (15) The antisymmetric space Aq (V ⊗ V ) is defined by the equation P+ v1 v2 = 0,
(16)
or equivalently, vi2 = 0, and vi vj = −q −1 vj vi ,
i < j.
(17)
Let Cq [V ] be the algebra over the entries vi ’s of v subject to the relation (15). It will called the quantum symmetric algebra of V . Let ∧q [V ] be the algebra over the entries vi ’s of v subject to the relation (17). It will called the quantum exterior algebra of V . Let Crq [V ] and ∧rq [V ] be the degree r-components of Cq [V ] and ∧q [V ], respectively. Both Cq [V ] and ∧q [V ] are left and right corepresentations of O(Mq (V )). By (12), the defining relation (6) of O(Mq (V )) is equivalent to P+ u1 u2 = u1 u2 P+ ,
(18)
P− u1 u2 = u1 u2 P− .
(19)
or equivalently,
10
These can also be rewritten as: P+ (u ⊗ u) = (u ⊗ u)P+ ,
(20)
P− (u ⊗ u) = (u ⊗ u)P− .
(21)
or equivalently, It can be shown [15] that (20) is equivalent to saying that the defining relation (15) of the quantum symmetric algebra Cq [V ] is preserved by the left and right actions of u on v given by v → uv and vt → vt u. Similarly, (21) is equivalent to saying that the defining relation (17) of the quantum antisymmetric algebra ∧q [V ] is also preserved by the left and right actions of u on v. We think of Cq [V ] as the coordinate algebra of a virtual symmetric quantum space Vsym , isomorphic to V as a vector space, with commuting coordinates (in the quantum sense), and ∧q [V ] as the coordinate algebra of a virtual antisymmetric quantum space V∧ , isomorphic to V as a vector space, with anti-commuting coordinates (in the quantum sense). Thus Mq (V ) is the set of linear transformations of the symmetric quantum space Vsym or the antisymmetric quantum space V∧ , on which each transformation acts from the left and as well as the right. This view of the standard quantum group, emphasized by Manin [21], will be a starting point for the definition ¯ of the new quantum group GLq (X). Let φR,r be the right coaction of O(Mq (V )) on ∧rq [V ]: φR,r : ∧rq [V ] → ∧rq [V ] ⊗ O(Mq (V )), and φL,r the left coaction. Let Ωr be the set of subsets of {1, . . . , n} of size r. For a subset I ∈ Ωr , with I = {i1 , . . . , ir }, i1 < i2 < · · · , let vI = vi1 · · · vir . The left quantum r-minors of O(Mq (V )) are defined to be the matrix coefficients of the left corepresentation map φL,r . Specifically, for I, J ∈ Ωr , the left quantum r-minors DJL,I (V ) are such that φL,r (vI ) =
X
DJL,I (V ) ⊗ vJ .
J
The right quantum r-minors DIR,J (V ) are such that φR,r (vI ) =
X
vJ ⊗ DIR,J (V ).
J
11
Then
DJI = DJL,I (V ) = DJR,I (V ) 6= 0.
(22)
The quantum determinant Dq = Dq (V ) of u is defined to be DJL,I (V ) = DJR,I (V ), with I = J = [1, n]. Explicitly: DJI =
X
i
i
, (−q)l(σ) ujσ(1) · · · ujσ(n) n 1
(23)
σ∈Sn
where l(σ) is the number of inversions in the permutation σ. The coordinate algebra O(GLq (V )) of the quantum group GLq (V ) is obtained by adjoining the inverse Dq (V )−1 to O(Mq (V )). We have a nondegenerate pairing ∧qn−1 [V ] × ∧1q [V ] → ∧nq [V ], where ∧1q [V ] = V is the fundamental vector representation. Its matrix form ˜ so that lets us define the cofactor matrix u ˜ u = u˜ u u = Dq (V )I. Then we can formally define u−1 = Dq (V )−1 u. This gives the following Hopf structure on Oq (GLq (V )): ·
1. ∆(u) = u ⊗ u. 2. ǫ(u) = I. 3. S(uij ) = u ˜ij Dq−1 , S(Dq−1 ) = Dq , where uij are the entries of u and u ˜ij ˜. are the entries of u The Poincare series of O(Mq (V )) coincides with the Poincare series of the commutative algebra C[U ] = C[uij ]. Because, just as in the classical case, O(Mq (V )) has a basis consisting of the standard monomials (u11 )k11 (u12 )k12 · · · (unn )knn , kij being nonnegative integers. To show this [30, 2], the monomials are ordered lexicographically, and the defining equations (20) of O(Mq (V )) are recast in the form of a reduction system: ujk uik ukj uki ujk uil ujl uik
→ → → →
q −1 uik ujk (i < j) q −1 uki ukj (i < j) uil ujk (i < j, k < l) uik ujl − (q − q −1 )uil ujk
12
(24) (i < j, k < l).
Then, by the diamond lemma [15], it suffices to show that all ambiguities in this reduction system are resolvable. This means any term of the form uij ukl urs , when reduced in any way, leads to the same result. This has to be checked for 24 different types of configurations of the three indices (i, j), (k, l), (r, s); see [2, 15, 30] for details.
2.1
Compactness, unitary transformations
What sets the standard quantum group apart from other known deformations [2, 21, 32, 31, 34] of GL(V ) is that it has a real form that is compact. To see what this means, we have to recall the notion of compactness due to Woronowicz in the quantum setting; cf. [36] or Chapter 11 in [15] for details. Let A be the coordinate Hopf algebra of a quantum group Gq . Suppose there is an involution ∗ on A so that it is a Hopf ∗-algebra [15]. We say that ∗ defines a real form of the quantum group Gq . A finite dimensional corepresentation of A on a vector space V with a Hermitian form is called unitary if the matrix v = (vij ) of this corepresentation with respect to an a orthonormal basis {ei } of V satisfies v ∗ v = vv ∗ = I. The algebra A is called a compact matrix group algebra (CMQG) if (1) it is the linear span of all matrix elements of finite-dimensional corepresentations of A, (2) it is generated as an algebra by finitely many elements. Then Theorem 2.1 (Woronowicz [36]; cf. chapter 11 in [15]) (a) A Hopf ∗algebra A is a CMQG algebra iff there is a finite dimensional unitary corepresentation of A whose matrix elements generate A as an algebra. (b) If A is a CMQG algebra then the quantum analogue of the Peter-Weyl theorem holds, and any finite dimensional corepresentation of A is unitarizable, and hence, a direct sum of irreducible corepresentations. There is a unique involution ∗ on the algebra O(GLq (V ) such that = S(uji ). Equipped with this involution, O(GLq (V )) becomes a Hopf ∗-algebra, called denoted by O(Uq (V )), and called the coordinate algebra of the quantum unitary group Uq (V )–which is, again, a virtual object. Furthermore, O(GLq (V )) is a CMQG algebra. (uij )∗
Woronowicz [36] has shown that the usual results for real compact groups, such as Harmonic analysis, existence of orthonormal bases, and so on, generalize to CMQG algebras. 13
2.2
Gelfand-Tsetlin bases and Clebsch-Gordon coefficients
In particular, standard results for the unitary group U (V ) have their analogues for Uq (V ). In this section, we describe results of this kind that we need; cf. [15, 35] for their detailed description. Given a partition λ, let Vλ (V ) denote the corresponding Weyl module of GL(V ), and Vq,λ (V ) denote the corresponding irreducible representation (qWeyl module) of the standard quantum group GLq (V ). Let {|M i} denote the orthonormal Gelfand-Tsetlin basis for Vq,λ (V ), where M ranges over Gelfand-Tsetlin tableau of shape λ. The tensor product of two irreducible representations of GLq (V ) decomposes as Vq,α (V ) ⊗ Vq,β (V ) = ⊕γ,r Vq,γ,r (V ), (25) where r labels different copies of Vq,γ (V )–the number of these copies is the Littlewood-Richardson coefficient cα,β,γ . The Clebsch-Gordon (Wigner) coefficients (CGCs) of this tensor product are defined by the formula X α,β,γ |M ir = CN KM,r |N i ⊗ |Ki, (26) N,K
where N and K range over Gelfand-Tsetlin tableau of shapes α and β, respectively, and |M ir denotes the Gelfand-Tsetlin basis element of Vq,γ,r (V ) in (25) labelled by the Gelfand-Tsetlin tableau M of shape γ. We denote α,β,γ CN KM,r by simply CN KM,r if the shapes are understood in a context. Furthermore, if the multiplicity of Vq,γ (V ) is one in (25), we denote CN KM,r by CN KM . These coefficients have been intensively studied in the literature; cf. [15, 35] and the references therein. An explicit formula for them is known when either Vq,α (V ) or Vq,β (V ) is a fundamental vector representation, or more generally, a symmetric representation. In the presence of multiplicities, the Clebsch-Gordon coefficients are not uniquely determined, and do not have explicit formulae in general. We now briefly recall explicit formulae for the fundamental ClebschGordon coeffficients–i.e., when Vq,α (V ) or Vq,β (V ) is a fundamental representation. Suppose the multiplicity of each irreducible representation in the tensor product decomposition (25) is one, as in the fundamental case. Then each CGC CN KM is a product of so called reduced CGC’s, which depend on the consecutive two rows of N, K, M . The s-reduced CGC is denoted by the 14
symbol
Ns Ks Ms Ns−1 Ks−1 Ms−1
,
where Ns denote the s-th row of the Gelfand-Tsetlin tableau N . Let us assume that Vq,β (V ) is a fundamental representation. Then each (valid) reduced CGC is either of the form:
Ns (1, 0) Ns + ei Ns−1 (0, 0) Ns−1
,
where 0 = (0, . . . , 0) and ei denotes a unit-row with i-th entry one, or of the form
Ns (1, 0) Ns + ei Ns−1 (1, 0) Ns−1 + ej
.
The first CGC is equal to
q
−1/2(i+1+
P
j Nj,s−1 −
P
j6=i Nj,s )
×
Qs−1
j=1 [Nj,s−1
Q
− Ni,s − j + i − 1]
j6=i [Nj,s
− Ni,s − j + i]
!1/2
,
(27)
and the second CGC is equal to ν(j − i)q −(Nj,s−1 −Ni,s −j+i)/2 × Q [Nk,s −Nj,s−1 −k+j] Q k6=i
[Nk,s −Ni,s −k+i]
[Nk,s−1 −Ni,s −k+i−1] 1/2 , k6=j [Nk,s−1 −Nj,s−1 −k+j−1
(28)
where [m] = [m]q = (q m − q −m )/(q − q −1 ) is the q-number, Ni,k are the components of Ni , and ν(j − i) := 1 if j − i ≥ 0 and −1 otherwise.
3
Quantization of GL(V ) × GL(W ) ֒→ GL(V ⊗ W )
Now we turn to Theorem 1.3. Let H = GL(V ) × GL(W ), where V and W are vector spaces of dimension n and m. Let X = V ⊗ W be the fundamental representation of H, and ρ : H ֒→ G = GL(X) = GL(V ⊗ W ) the corresponding homomorphism. The goal is to quantize this homomorphism in the form: ¯ GLq (V ) × GLq (W ) → GLq (X), 15
(29)
¯ where the quantum groups GLq (V ) and GLq (W ) are standard, and GLq (X) is a newly sought quantum group deformation of GL(V ⊗W ) such that (1) it has compact real form and (2) its dimension is the same as that of GL(X). This then can be considered to be a correct quantization of ρ. We shall prove (1), and conjecture (2) (cf. Conjecture 5.5). When we think of the vector space X as a fundamental representation of ¯ defined GLq (V ) × GLq (W )–or as a fundamental representation of GLq (X) ¯ as in eq.(29). later–we denote it by X, ¯ is motivated by the view of the standard The construction of GLq (X) quantum group described in Section 2: 1. First we construct (Section 4) symmetric and antisymmetric quantum ¯ Cq (X) ¯ and ∧q (X), ¯ which can be thought of as coordialgebras of X, ¯ sym and X ¯∧ , respectively. nate algebras of (virtual) quantum spaces X ∗ ¯ When W = V , Xsym is isomorphic to the standard quantum space ¯ sym are the same as (24). Mq (V ); i.e., the defining equations of X ¯ is defined (Section 5) as the space 2. The quantum matrix space Mq (X) ¯ ¯ ∧ ; i.e., so that of linear transformations of Xsym , or equivalently, of X ¯ on X ¯ sym or X ¯ ∧ preserve their the left and right actions of Mq (X) defining equations. 3. It is shown that there is a natural bialgebra homomorphism from ¯ to O(Mq (V )) ⊗ O(Mq (W )); cf. Section 5.2. O(Mq (X)) ¯ r be the degree r component of ∧q [X]. ¯ Then the left and 4. Let ∧q [X] right quantum r-minors are defined as the matrix coefficients of the left ¯ on ∧rq [X], ¯ and their basic properties and right coactions of O(Mq (X)) are proved; cf. Section 6. ¯ are de5. The cofactor and the inverse of a generic matrix u ∈ Mq (X) ¯ fined using determinants and quantum minors, and GLq (X) is defined ¯ as the subset of nonsingular transformations in Mq (X). Formally, ¯ GLq (X) is defined by putting a Hopf structure on the coordinate al¯ gebra O(Mq (X)); cf. Section 7. ¯ 6. A natural ∗-structure is put on O(GLq (X)), and using Woronowicz’ ¯ results [36], it is shown that GLq (X) has a compact real form, which ¯ we shall denote by Uq (X)–this is the analogue of the unitary group in this setting; cf. Section 8.
16
7. Finally, it is shown that Theorem 1.3 follows from these results in conjunction with Woronowicz’s results [36]. In the subsequent sections we address these steps one at a time.
4
Quantum symmetric and antisymmetric algebras
ˆ V,V ⊗ R ˆ W,W be the R-matrix ˆ ¯ = V ⊗ W as ˆ¯ ¯ =R associated with X Let R X,X ˆ a representation of GLq (V ) × GLq (W ). This is different from the R-matrix ˆ X,X obtained by thinking of X = V ⊗ W as a vector representation of the R standard quantum group GLq (X). ˆ V,V and R ˆ W,W are diagonalizable with eigenvalues q and −q −1 . Both R ˆ V,V ⊗ R ˆ W,W is diagonalizable with eigenvalues q 2 , −1, q −2 . ˆ¯ ¯ =R Hence, R X,X In what follows, we assume that q is positive and transcendental. The ¯ ⊗ X) ¯ ⊆X ¯ ⊗X ¯ is defined to be the span quantum symmetric subspace Sq (X 2 of the eigenspaces for the positive eigenvalues q , q −2 (cf. Chapter 8,[15]). ¯ ⊗ X) ¯ ⊆X ¯ ⊗X ¯ is defined to be The quantum antisymmetric subspace Aq (X the eigenspace for the negative eigenvalue −1. Let ˆ ¯ ¯) ˆ ¯ ¯ ) + P− (R I = P+ (R X,X X,X be the spectral decomposition of the identity corresponding to the spectral ˆ ¯ ¯ . In what follows, we let decomposition of the diagonalizable R X,X ¯
P−X ¯ P+X
ˆ ¯ ¯) = P− (R X,X ˆ ¯ ¯ ). = P+ (R X,X
(30)
¯ ¯ ˆ ¯ ¯ . The symmetric subspace Both P+X and P−X are polynomials in R X,X ¯ ¯ ⊗ X) ¯ is the image of P+X and the antisymmetric space Aq (X ¯ ⊗ X) ¯ is Sq (X ¯ ¯ ⊗ X) ¯ is defined by the equation the image of P−X . In other words, Sq (X ¯
P−X x1 x2 = 0,
(31)
where x1 = x ⊗ I and x2 = I ⊗ x, and Aq (X ⊗ X) is defined by the equation ¯
P+X x1 x2 = 0,
(32)
¯ of X ¯ be the algebra over the Let the quantum symmetric algebra Cq [X] entries xi ’s of x subject to relation (31). It will be called the coordinate ring 17
¯ sym , which is only a virtual object. Let the quantum of the quantum space X ¯ ¯ be the algebra over the entries xi ’s of x subject exterior algebra ∧q [X] of X to relation (32). It will called the coordinate ring of the virtual quantum ¯ ∧ . We shall sometimes refer to X ¯ sym and X ¯ ∧ as symmetric and space X antisymmetric quantum spaces (though the meaning of the term symmetric ¯ quantum space here is different from the one in the literature). Let Crq [X] r ¯ ¯ ¯ and ∧q [X] be the degree r components of Cq [X] and ∧q [X], respectively. ˆ V,V , and Let AV be an eigenbasis of Sq (V ⊗ V ) for the eigenvalue q of R −1 BV an eigenbasis for the eigenvalue −q ; AW , BW are defined similarly. ¯ ⊗X ¯ Given c ∈ V ⊗ V and W ⊗ W , we define their c∗d ∈ X Pd ∈ P restitution 1 2 k 1 2 k as follows. Let c = i ci ⊗ ci , ci ∈ V , and d = j dj ⊗ dj , dj ∈ W . Then X (33) c∗d= (c1i ⊗ d1j ) ⊗ (c2i ⊗ d2j ). i,j
¯ ⊗ X) ¯ and Aq (X ¯ ⊗ X): ¯ Then the following are eigenbases of Sq (X ¯ ⊗ X) ¯ : Sq (X
{a ∗ a′ | a ∈ AV , a′ ∈ AW } ∪ {b ∗ b′ | b ∈ BV , b′ ∈ BW }.
¯ ⊗ X) ¯ : {a ∗ b′ | a ∈ AV , b′ ∈ BW } ∪ {b ∗ a′ | b ∈ BV , a′ ∈ AW }. Aq (X (34) An explicit form of the defining relation (31) for the symmetric quantum ¯ is obtained by setting the eigenbasis of Aq (X ¯ ⊗ X) ¯ to zero: algebra Cq [X] a ∗ b′ = 0 a ∈ AV , b′ ∈ BW , b ∗ a′ = 0 b ∈ BV , a′ ∈ AW .
(35)
An explicit form of the defining relation (32) for the antisymmetric quantum ¯ is obtained by setting the eigenbasis of Sq (X ¯ ⊗ X) ¯ to zero: algebra ∧q [X] a ∗ a′ = 0 a ∈ AV , a′ ∈ aW , b ∗ b′ = 0 b ∈ BV , b′ ∈ BW .
(36)
Example Let dim(V ) = dim(W ) = 2, V = hv1 , v2 i, W = hw1 , w2 i. Then AV = {A11, A22, B12} and BV = {B12}, where, deleting the ⊗ sign and letting p = 1/q, A11(V ) = v1 v1 A22(V ) = v2 v2 (37) A12(V ) = v1 v2 + pv2 v1 B12(V ) = v1 v2 − qv2 v1 18
¯ The following is a basis AW and BW are similar. Let xij = vi ⊗ wj ∈ X. ¯ ¯ of the symmetric space Sq (X ⊗ X): a11 := A11(V ) ∗ A11(W ) a22 := A11(V ) ∗ A22(W ) a12 := A11(V ) ∗ A12(W ) a33 := A22(V ) ∗ A11(W ) a44 := A22(V ) ∗ A22(W ) a34 := A22(V ) ∗ A12(W ) a13 := A12(V ) ∗ A11(W ) a24 := A12(V ) ∗ A22(W ) a14 := A12(V ) ∗ A12(W ) a23 := B12(V ) ∗ B12(W )
= x11 x11 = x12 x12 = x11 x12 + px12 x11 = x21 x21 = x22 x22 = x21 x22 + px22 x21 = x11 x21 + px21 x11 = x12 x22 + px22 x12 = x11 x22 + px12 x21 + px21 x12 + p2 x22 x11 = x11 x22 − qx12 x21 − qx21 x12 + q 2 x22 x11 (38) ¯ ¯ The following is a basis of the antisymmetric space Aq (X ⊗ X):
b12 := A11(V ) ∗ B12(W ) b34 := A22(V ) ∗ B12(W ) b13 := B12(V ) ∗ A11(W ) b24 := B12(V ) ∗ A22(W ) b14 := A12(V ) ∗ B12(W ) b23 := B12(V ) ∗ A12(W ) Proposition 4.1
= x11 x12 − qx12 x11 = x21 x22 − qx22 x21 = x11 x21 − qx21 x11 = x12 x22 − qx22 x12 = x11 x22 − qx12 x21 + px21 x12 − x22 x11 = x11 x22 + px12 x21 − qx21 x12 − x22 x11
(39)
Q kij ¯ is { 1. A basis of Cq [X] i,j xij | ki,j ∈ Z≥0 }.
¯ is {Q xkij | ki,j ∈ {0, 1}}. 2. A basis of ∧q [X] i,j ij
¯ and ∧q [X]. ¯ These bases will be called standard monomial bases of Cq [X]
Proof: (1) The relations (35) (cf. (39)) can be reformulated in the form of the following reduction system: xjk xik xkj xki xjk xil xjl xik
→ → → →
q −1 xik xjk (i < j) q −1 xki xkj (i < j) xil xjk (i < j, k < l) xik xjl − (q − q −1 )xil xjk
(40) (i < j, k < l).
When dim(V ) = dim(W ), these coincide with the defining relations (24) for the standard quantum matrix space Mq (V ) after the change of variables 19
¯ sym ∼ xij → uij . In other words, X = Mq (V ). In this case, the ambiguities in this reduction system can be resolved just as in the case of the reduction system for O(Mq (V )) [30, 2]; cf. Section 2. This is also so when dim(V ) 6= dim(W ). Hence the result follows from the diamond lemma [15]. (2) The relations (36) can be reformulated in the form of the following reduction system: x2ij → 0 xjk xik xkj xki xjl xik xjk xil
→ → → →
−qxik xjk (i < j) −qxki xkj (i < j) −xik xjl (i < j, k < l) −xil xjk + (q −1 − q)xik xjl
(41) (i < j, k < l).
Ambiguities in this reduction system can also be resolved just as in (1); we omit the details. So the result again follows from the diamond lemma [15]. Q.E.D. For future reference, we note down a corollary of the proof above. Let ˆ Y,Y = R ˆ V ∗ ,V ∗ ⊗ R ˆ V,V be an R ˆ matrix Y = End(V, V ) = V ∗ ⊗ V . Let R associated with Y . It is diagonalizable. Let I = P−Y + P+Y be the associated spectral decomposition of the identity, where P−Y and P+Y ˆ Y,Y for the eigenvalues with denote the projections onto the eigenspaces of R sign − and +, respectively. Proposition 4.2 Eq.(18) or eq.(19) defining O(Mq (V )) is equivalent to the relation P−Y (u ⊗ u) = 0. (42) At q = 1, this relation simply says that the entries of u commute with each other. So this is a quantized version of commutativity. Proof: See the proof of Proposition 4.1 (1), and the remark therein. Q.E.D. Proposition 4.3 (1) As a GLq (V ) × GLq (W )-module X ¯ = Symd (V ⊗ W ) ∼ Vq,λ (V ) ⊗ Vq,λ (W ), Cdq [X] = q
(43)
λ
where λ ranges over all Young diagrams of size d, with at most dim(V ) or dim(W ) rows. 20
(2) Similarly, ¯ = ∧dq (V ⊗ W ) ∼ ∧dq [X] = ⊕λ Vq,λ (V ) ⊗ Vq,λ′ (W ),
(44)
where λ ranges over all Young diagrams of size d, with at most dim(V ) rows and at most dim(W ) columns, and λ′ denotes the conjugate of λ obtained by interchanging rows and columns. ¯ as a GLq (V ) × Proof: (1) By Proposition 4.1, the character of Cdq [X] GLq (W )-module coincides with the character of Cd [X] as a GL(V )×GL(W )module. Since, finite dimensional representations of the GLq (V ) × GLq (W ) are completely reducible, the irreducible representations of GLq (V )×GLq (W ) are in one-to-one correspondence with the those of GL(V ) × GL(W ), and the characters of the corresponding representations coincide [15], (1) follows from the the classical result Cd [X] = Symd (V ⊗ W ) ∼ = ⊕λ Vλ (V ) ⊗ Vλ (W ).
(45)
Similarly (2) is a q-analogue of the classical result ∧d (V ⊗ W ) ∼ = ⊕λ Vλ (V ) ⊗ Vλ′ (W ),
(46)
Q.E.D. ¯ is isomorphic to the coordinate algeWe have already noted that Cdq [X] bra O(Mq (V )) of the standard quantum matrix space, when W = V ∗ . In this case, Proposition 4.3 (1) is the q-analogue of the Peter-Weyl theorem for the standard quantum group GLq (V ), and Proposition 4.3 (2) is the q-analogue of the antisymmetric form of the Peter-Weyl theorem.
4.1
Explicit product formulae
We wish to give explicit formulae for products in the symmetric and anti¯ and ∧q [X]. ¯ symmetric algebras Cq [X] Let
˜ s¯ = ∪λ {|Mλ i ⊗ |Nλ i} B X
(47)
¯ as per the decomposition in Propobe the Gelfand-Tsetlin basis for Cdq [X] sition 4.3 (1) and ˜ ∧¯ = ∪λ {|Mλ i ⊗ |Nλ′ i} (48) B X ¯ as per Proposition 4.3 (2). that for ∧dq [X] 21
When W = V ∗ , the basis element |M i ⊗ |N i ∈ Vq,λ (V ) ⊗ Vq,λ (V ∗ ) ⊆ ¯ stands for the matrix coefficient uλ Cq [X] N M of the representation Vq,λ (V ) of the standard quantum group GLq (V ). It is of interest to know explicit transformation matrices connecting the ¯ with their standard monomial ¯ and ∧d (X) Gelfand-Tsetlin bases of Cdq [X] q bases in Proposition 4.1. In other words, we want to know the decompositions in Proposition 4.3 (1) and (2) explicitly. When W = V ∗ , this amounts to finding explicit formulae for the matrix coefficients of irreducible representations of GLq (V ). This problem has been studied intensively in the literature. When dim(V ) = 2, explicit formulae for matrix coefficients in terms little q-Jacobi polynomials are known. In general, the problem is not completely understood at present; see the survey [35] and the references therein. ¯ ∧q [X], ¯ The advantage of working with the Gelfand-Tsetlin bases of Cq [X], instead of the standard monomial bases in Proposition 4.1 is that multiplication is simpler in terms of the former, and have explicit formulae in terms of Clebsch-Gordon coefficients. We shall now state these formulae, assuming for the sake of simplicity, the multiplicity free case–which is enough for the purposes of this paper. That is, we shall assume that the multiplicity of each Vq,γ (V ) in the tensor product decomposition (25) for α’s and β’s under consideration is one. When W = V ∗ , we have the following multiplication formula for matrix coefficients; cf. [15, 35]: uαN R uβKS =
X
α,β,γ γ α,β,γ CN KM CRSL uM L .
(49)
γ,M,L
where α, β, γ are as in the decomposition (25), which is assumed to be multiplicity free. We also have the following identity: X α,β,γ α,β,γ α β uγM L = CN KM CRSL uN R uKS .
(50)
N,K,R,S
Multiplication formula for the Gelfnad-Tsetlin basis (47) is similar. Let α, β be Young diagrams of height less than dim(V ) and dim(W ), and γ be a Young diagram in the decomposition (25), which is assumed to be multiplicity free. Then X CNα Kβ Mγ ,r CRα Sβ Lγ |Mγ i ⊗ |Lγ i. (51) (|Nα i ⊗ |Rα i)(|Kβ i ⊗ |Sβ i) = γ,Mγ ,Lγ
22
Also X
|Mγ i ⊗ |Lγ i =
CNα Kβ Mγ CRα Sβ Lγ (|Nα i ⊗ |Rα i)(|Kβ i ⊗ |Sβ i).
Nα ,Kβ ,Rα ,Sβ
(52)
Multiplication formula for the Gelfand-Tsetlin basis (48) is also similar. Let α, β, γ be Young diagrams of height at most dim(V ) and width at most dim(W ), and α′ , β ′ , γ ′ their conjugates. Assume that the multiplicity of γ in the decomposition (25) is one –i.e. the Littlewood-Richardson coefficient cα,β,γ is one–and also that cα′ ,β ′ ,γ ′ is one. Then, (|Nα i⊗|Rα′ i)(|Kβ i⊗|Sβ ′ i) =
X
CNα Kβ Mγ CRα′ Sβ ′ Lγ′ |Mγ i⊗|Lγ ′ i. (53)
γ,Mγ ,Lγ
Also |Mγ i ⊗ |Lγ ′ i =
X
CNα Kβ Mγ CRα′ Sβ ′ Lγ′ (|Nα i ⊗ |Rα′ i)(|Kβ i ⊗ |Sβ ′ i),
Nα ,Kβ ,Rα′ ,Sβ ′
5
(54)
Quantum matrix space ¯
Let u = uX be a variable matrix, specifying a generic transformation from ¯ to X. ¯ Let Chui denote the free algebra over the variable entries of u. X ¯ of the virtual quantum We define the coordinate algebra O(Mq (X)) ¯ to be the quotient of Chui modulo the relations space Mq (X) ¯
¯
P+X (u ⊗ u) = (u ⊗ u)P+X . ¯
(55)
¯
Since I = P−X + P+X , these are equivalent to the relations ¯
¯
P−X (u ⊗ u) = (u ⊗ u)P−X .
(56) ¯
¯ is an FRT-algebra [32], with a singular P X playing Thus O(Mq (X)) − the role of an R-matrix in eq.(6). But, as we shall see below, it is not coquasitriangular, and hence the main theory of FRT-algbras [32] does not ¯ apply to O(Mq (X)).
23
Proposition 5.1 1. The relation (55) is equivalent to the relations on the entries of u which say that the defining equation (31) of the quan¯ sym is preserved by the left and right actions tum symmetric space X of u on x given by x → ux and xt → xt u. This means the quan¯ is a left and right comodule-algebra of tum symmetric algebra Cq [X] ¯ O(Mq (X). 2. The relation (56) is equivalent to the relations on the entries of u which say that the defining equation (32) of the quantum antisymmetric space ¯ ∧ is preserved by the left and right actions of u on x. This means X ¯ is a left and right comodule the quantum antisymmetric algebra ∧q [X] ¯ algebra of O(Mq (X)). Proof: Left to the reader. Q.E.D. ¯ can be In view of this proposition, the quantum matrix space Mq (X) ¯ sym , or equivalently, of viewed as the space of linear transformations of X ¯ ¯ ¯ sym or X ¯ ∧ preserve X∧ ; i.e., so that the left and right actions of Mq (X) on X their defining equations. This definition is akin to that of the standard quan¯ sym tum matrix space Mq (X), but with the quantum symmetric space X playing the role of the standard quantum symmetric space Xsym and the ¯ ∧ playing the role of the standard quantum quantum antisymmetric space X antisymmetric space X∧ . Remark 1:
The relations (55) and (56) are equivalent to the relations ˆ ¯ ¯ (a, b), SˆX, ¯ X ¯ (a, b)(u ⊗ u) = (u ⊗ u)S X,X where
(57)
¯ ¯ X X SˆX, ¯ X ¯ (a, b) = aP+ + bP− ,
¯ is an FRT-algebra for any constants a, b, not both zero. Thus O(Mq (X)) ˆ ¯ ¯ (a, b), where τ denotes the with the R-matrix being SX, ¯ (a, b) = τ ◦ S ¯ X X,X ¯ ⊗d , flip operator. Fix a, b, and let S = SX, ¯ (a, b). Given a tensor product X ¯ X let Si denote the transformation which acts like S on the i-th and (i + 1)st factors, the other factors remaining unaffected. Then it can be shown that Si ’s do not satisfy the Quantum Yang Baxter Equations (QYBE)– equivalently Sˆi ’s do not satisfy the braid identities; cf. Section 11. Remark 2: ˆ ¯ ¯ . One can also consider an FRT-algebra A(R ¯ ¯ ) Let RX, ¯ X ¯ =τ ◦R X,X X,X ¯ associated with the R-matrix RX, ¯ X ¯ . It is not isomorphic to O(Mq (X)), 24
¯ ˆ ¯ ¯ . Furthermore, it can be shown that the since, P+X is nonlinear in R X,X dimension of the quantum group associated with A(RX, ¯ X ¯ ) is much smaller than the classical dimension of GL(X). Hence, it cannot be considered as a deformation of GL(X). ˆ ˆ ¯ ∗ ¯ ∗ be an R ¯ X) ¯ =X ¯ ∗ ⊗ X. ¯ Let R ˆ U,U = R ˆ¯ ¯ ⊗R Let U = End(X, X ,X X,X matrix associated with U . It is diagonalizable. Let
I = P−U + P+U be the associated spectral decomposition of the identity, where P−U and P+U ˆ U,U for the eigenvalues with denote the projections onto the eigenspaces of R sign − and +, respectively. The following is an analogue of Proposition 4.2. Proposition 5.2 Eq.(55) or eq.(56) is equivalent to the relation P−U (u ⊗ u) = 0.
(58)
Proof: Left to the reader. Q.E.D. When q = 1 this relation says that uij ’s commute. Thus it expresses the q-analogue of commutativity in the present context. ¯ ⊗ X) ¯ and Aq (X ¯ ⊗ X), ¯ respectively; let Let AX¯ , BX¯ be bases of Sq (X ∗ ∗ ¯ ⊗X ¯ and d ∈ X ¯ ⊗ X, ¯ we AX¯ ∗ and BX¯ ∗ be defined similarly. Given c ∈ X ∗ ∗ ¯ ¯ ¯ ¯ define the restitution c ∗ d ∈ (X ⊗ X) ⊗ (X ⊗ X) = U ⊗ U very much as in (33). Then an eigenbasis of P−U in U ⊗ U is given by {a ∗ b | a ∈ AX¯ ∗ , b ∈ BX¯ } ∪ {b ∗ a | b ∈ BX¯ ∗ , a ∈ AX¯ }. Hence, the following is an explicit form of the defining relation in Proposi¯ tion 5.2 for O(Mq (X)): a∗b=0
a ∈ AX¯ ∗ , b ∈ BX¯ (59)
b∗a=0
b ∈ BX¯ ∗ , a ∈ AX¯ .
¯ is a bialgebra such that Proposition 5.3 O(Mq (X)) ¯
¯ ·
¯
¯
∆(uX ) = uX ⊗ uX , and ǫ(uX ) = I. Here ⊗ denotes tensor product and · denotes matrix multiplication. Proof: Follows from Proposition 9.1 in [15] applied to the FRT-algebra ¯ = A(S ¯ ¯ (a, b)). Q.E.D. O(Mq (X)) X,X 25
5.1
Example
Let dim(V ) = dim(W ) = 2. Let AX¯ = {a11, a22, a12, a33, a44, a34, a13, a24, a14, a23}, and BX¯ = {b12, b34, b13, b24, b14, b23}, where aij and bkl are as in eq.(38) and (39). We shall identify AX¯ ∗ and BX¯ ∗ with AX¯ and BX¯ , respectively. We use the notation zi1 i2 ,k1 k2 = xi1 i2 ∗ xk1 k2 , ¯ are as in eq.(38) and (39). We also drop ⊗ symbol. So that where xij ∈ X xi1 i2 xj1 j2 actually means xi1 i2 ⊗ xj1 j2 . For example, z12,11 z21,22 = (x12 x21 ) ∗ (x11 x22 ). ¯ are now 120 in number. We The defining relations (59) of O(Mq (X)) show one such typical relation below: 0 = a14 ∗ b14 = (x11 x22 + px12 x21 + px21 x12 + p2 x22 x11 )∗ (x11 x22 − qx12 x21 + px21 x12 − x22 x11 ) = z11,11 z22,22 + pz12,11 z21,22 + pz21,11 z12,22 + p2 z22,11 z11,22 −qz11,12 z22,21 − z12,12 z21,21 − z21,12 z12,21 − pz22,12 z11,21 +pz11,21 z22,12 + p2 z12,21 z21,12 + p2 z21,21 z12,12 + p3 z22,21 z11,12 −z11,22 z22,11 − pz12,22 z21,11 − pz21,22 z12,11 − p2 z22,22 z11,11 .
(60)
All these 120 relations together, after taking appropriate linear combinations, can be recast in the form of a reduction system–just as the relations (35) were recast in the form of a reduction system (40)–where each reduction rule is of the form X zA zA′ = α(A, A′ , B, B ′ )zB zB ′ , B,B ′
each zB zB ′ being standard; i.e., B > B ′ (say, lexicographically). The resulting reduction system is described in Appendix. It turns out that this system does not satisfy the diamond property. For example, the monomial mm = z1111 z1112 z1221 , when reduced in two different ways, yields the following two distinct standard expressions: 26
q3 − q q2 − 1 ·z z z + ·z1212 z1121 z1111 + 1211 1122 1111 1 + q2 1 + q2 1 − q2 q 2q 2 · z1221 z1112 z1111 + · z1222 z1111 z1111 , 1 + q2 1 + q2
l121 = (−1+q 2 )·z1211 z1121 z1112 +
and
l212 =
q 6 + q 4 − 3q 2 + 1 2q 3 − 2q · z1211 z1122 z1111 · z z z + 1211 1121 1112 q 2 (1 + q 2 ) (1 + q 2 )2
4q 2 + q 4 − 1 (1 + q 2 )2
·z1221 z1112 z1111 +
2q 2 − 2 (1 + q 2 )2
·[z1212 z1121 z1111 − q ·z1222 z1111 z1111 ].
This means we have the following nontrivial relation among standard monomials: l121 − l212 = 0. See Appendix for the details. The example above has the following consequence: ¯ does not, in general, Proposition 5.4 The Poincare series of O(Mq (X)) ¯ (at q = 1). coincide with the classical Poincare series of O(M (X)) ¯ we mean the series Here by Poincare series of O(Mq (X)) X ¯ d )td , dim(O(Mq (X)) d
¯ d denotes the degree d component of O(Mq (X)). ¯ where O(Mq (X)) As an example, when dim(V ) = dim(W ) = 2, using computer it was found that ¯ 3 ) = 688, whereas the classical dim(O(M (X))3 ) = 816. dim(O(Mq (X)) But we have: ¯ as a noncommutative algebraic Conjecture 5.5 The dimension of Mq (X), ¯ variety, as determined from the Poicare series of O(Mq (X)), is the same as 2 dim(X) , the dimension of the algebraic variety M (X).
27
¯ is the This would imply that dimension of the new quantum group GLq (X) same as the dimension of the classical GL(X). Proposition 5.4 has important consequences. In the standard case, the Poincare series of O(Mq (V )) coincides with the classical Poincare series of O(M (V ) (at q = 1). By the Peter-Weyl theorem, this implies that the dimensions of the irreducible representations of GLq (V ) are in one-to-one correspondence with the dimensions of the irreducible representations of GL(V ). Intuitively, this is why the irreducible representations of GLq (V ) turn out to be deformations of the irreducible representations of GL(V ). Proposition 5.4 implies that this would no longer be so for the new quantum group.
5.2
Homomorphism
Let O(Mq (V )) and O(Mq (W )) be coordinate algebras of the standard quantum matrix spaces Mq (V ) and Mq (W ), respectively (Section 2). ¯ to Proposition 5.6 There is a natural homomorphism ψ from O(Mq (X)) O(Mq (V )) ⊗ O(Mq (W )). ¯
Proof: In the matrix form the homomorphism ψ is u = uX → uV ⊗ uW . ¯ One has to check that the relations obtained by substituting uX = uV ⊗ uW ¯ are implied by the equations defining O(Mq (V )) in (55) defining O(Mq (X)) and O(Mq (W )). The defining equation (20) of O(Mq (V )) is P+V (uV ⊗ uV ) = (uV ⊗ uV )P+V ,
(61)
which is equivalent to (21); i.e., P−V (uV ⊗ uV ) = (uV ⊗ uV )P−V .
(62)
Similarly, the defining relation of O(Mq (W )) is P+W (uW ⊗ uW ) = (uW ⊗ uW )P+W ,
(63)
P−W (uW ⊗ uW ) = (uW ⊗ uW )P−W .
(64)
or equivalently, ¯
Since P−X = P−V ⊗ P+W + P+V ⊗ P−W , these relations imply (55) when ¯ uX = uV ⊗ uW . To show that ψ is a bialgebra homomorphism, one has to verify that 28
1. ψ(ab) = ψ(a)ψ(b). 2. ∆ ◦ ψ = (ψ ⊗ ψ) ◦ ∆. 3. ǫ = ǫ ◦ ψ. This is easy. Q.E.D.
6
Quantum determinant and minors
Let Ωr be the set of subsets of {1, . . . , N = nm} of size r, n = dim(V ), m = dim(W ). We write any I ∈ Ωk as I = {i1 , . . . , ir }, i1 < · · · < ik . We let xI be the monomial xi1 · · · xir . By Proposition 4.1, {xI } is the basis ¯ ¯ In particular, ∧nm of ∧q [X]. q [X] is a one-dimensional corepresentation of ¯ O(Mq (X)) with basis vector x1 · · · xN . ¯ is a left and right comodule algebra Proposition 5.1(2) says that ∧q [X] ¯ of O(Mq (X)). Let ¯ → O(Mq (X)) ¯ ⊗ ∧q [X] ¯ φL : ∧q [X] φL,r :
¯ ∧rq [X]
¯ ¯ ⊗ ∧r [X] → O(Mq (X)) q
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¯ and ∧rq [X]. ¯ The right be the maps defining the left corepresentations ∧q [X] corepresentation maps φR and φR,r are similar. We define the left determi¯ to be the matrix coefficient of the left comodule ∧nm ¯ nant DqL = DqL (X) q [X]. R R ¯ The right determinant Dq = Dq (X) is the matrix coefficient of the right conm ¯ nm [X] ¯ module ∧nm q [X]. Since ∧q [X] is one dimensional like the classical ∧ (Proposition 4.1), both DqL and DqR are nonzero. ¯ in the standard More generally, the left quantum r-minors of Mq (X) monomial basis are defined to be the matrix coefficients of the left corepresentation map φL,r in the standard monomial basis (cf. Proposition 4.1) ¯ Specifically, for I, J ∈ Ωr , we define the left quantum r-minors of ∧rq [X]. L,I ¯ DJ (X) such that X L,I ¯ ⊗ xJ . φL,r (xI ) = DJ (X) J
¯ are such that The right quantum r-minors DIR,J (X) X ¯ φR,r (xI ) = xJ ⊗ DIR,J (X). J
29
¯ in We can similarly define left and right quantum minors of Mq (X) the Gelfand-Tsetlin basis to be the matrix coefficients of the left and right corepresentation maps φL,r and φR,r in the Gelfand-Tsetlin basis (48) of ˜ ∧¯ , with α and β being ¯ Specifically, for |Mα i ⊗ |Nα′ i, |Mβ i ⊗ |Nβ ′ i ∈ B ∧rq [X]. X L,Mα ,Nα′ ¯ young diagrams of size r, we define the left quantum r-minors D (X) Mβ ,Nβ ′
so that φL,r (|Mα i ⊗ |Nα′ i) =
X
L,M ,Nα′
α DMβ ,N
β′
¯ ⊗ |Mβ i ⊗ |Nβ ′ i. (X)
Mβ ,Mβ ′ R,M ,Nα′
The right quantum r-minors DMβ ,Nα
β′
¯ are defined similarly. (X)
The quantum determinants DqL and DqR are the same whether we use the Gelfand-Tsetlin or the standard monomial basis.
6.1
Explicit formulae
We wish to give explicit formulae for quantum determinants DqL and DqR and, more generally, the left and right quantum minors in the Gelfand-Tsetlin basis. This is possible because of the explicit formulae for multiplication in the Gelfand-Tsetlin basis (Section 4.1). We do not have similar formulae in the standard monomial basis. 6.1.1
Example
Let us first give an explicit formula for DqL and DqR when dim(V ) = dim(W ) = ¯ has a basis {yI = yi · · · yir }, i1 < i2 · · · , where 2. Then ∧rq [X] 1 y1 = x11 , y2 = x12 , y3 = x21 , y4 = x22 , satisfying the relations a11, a22, a12, a33, a44, a34, a13, a24, a14, a23 = 0; P ¯ ⊗d ¯ = cf. (38) and Proposition 4.1. Let I∧ denote the ideal in T (X) dX generated by these relations. The last two relations imply that y4 y1 = −y1 y4 . Since y1 and y4 quasicommute with all yi ’s and yi2 = 0, for allQi, it is easy to show that yi yj yk yl is zero modulo I∧ , unless it is of the form yσ(i) , for some permutation σ, or is either y2 y3 y2 y3 or y3 y2 y3 y2 . Furthermore, we have Y yσ(i) = (−1)l(σ) q r(σ) y1 y2 y3 y4 , 30
where l(σ) is the number of inversions in σ, and r(σ) is the number of inversions in σ not involving (2, 3) or (1, 4). Also y2 y3 y2 y3 = (p − q)q 2 y1 y2 y3 y4 y3 y2 y3 y2 = (q − p)q 2 y1 y2 y3 y4 . ¯ considThe left determinant Dq,L is the the matrix coefficient of ∧4q (X), ered as a left comodule, and the right determinant Dq,R is the the matrix ¯ considered as a right comodule. From the preceding coefficient of ∧4q (X), remarks, it easily follows that X Dq,L = ( (−1)l(σ) q r(σ) uiσ(i) )+(p−q)q 2 u12 u23 u32 u43 +(q−p)q 2 u13 u22 u33 u42 . σ
The expression for Dq,R is similar. Compare this with the formula (23) for the standard quantum determinant. More generally, the following result gives an explicit formula in terms of the fundamental Clebch-Gordan (Wigner) coefficients (Section 4.1) for ¯ (in the Gelfandexpanding the left or the right quantum minor of Mq (X) Tsetlin basis) by row or column. Proposition 6.1 L,Mγ ,L ′ ¯ DEδ ,F ′ γ (X) Pδ = Nα ,Kβ ,R ′ ,S ′ CNα Kβ Mγ CRα′ Sβ ′ Lγ′ α
P
β
Aµ ,Bµ′ ,Cλ ,Dλ′
L,N ,Rα′
DAµ ,Bα
µ′
L,Kβ ,Sβ ′
¯ (X)D Cλ ,D
λ′
¯ Aµ C ,E CB ′ D ′ F ′ , (X)C λ δ µ λ δ
where CNα Kβ Mγ ’s etc. denote fundamental Clebsch-Gordon coefficients; cf. (27) and (28). A formula for the right minor is similar. ¯ Proof: Using (53) and (54) we calculate, dropping the symbol X:
31
φL,r (|Mγ i ⊗ |Lγ ′ i) P = φL,r ( Nα ,Kβ ,R
α′ ,Sβ ′
=
P
Nα ,Kβ ,Rα′ ,Sβ ′
CNα Kβ Mγ CRα′ Sβ ′ Lγ′ (|Nα i ⊗ |Rα′ i)(|Kβ i ⊗ |Sβ ′ i))
CNα Kβ Mγ CRα′ Sβ ′ Lγ′
(|Aµ i ⊗ |Bµ′ i)(|Cλ i ⊗ |Dλ′ i)
=
P
Nα ,Kβ ,Rα′ ,Sβ ′ L,N ,Rα′
DAµ ,Bα
µ′
=
P
P
Eδ ,Fδ′
µ′
λ′
L,N ,Rα′
DAµ ,Bα
µ′
L,Kβ ,Sβ ′
DCλ ,D
λ′
Aµ ,Bµ′ ,Cλ ,Dλ′
⊗
Nα ,Kβ ,Rα′ ,Sβ ′
L,N ,Rα′
DAµ ,Bα
L,Kβ ,Sβ ′
DCλ ,D
P
Aµ ,Bµ′ ,Cλ ,Dλ′
CAµ Cλ ,Eδ CBµ′ Dλ′ Fδ′ |Eδ i ⊗ |Fδ′ i
P
Eδ ,Fδ′ (
CNα Kβ Mγ CRα′ Sβ ′ Lγ′
P
L,Kβ ,Sβ ′
DCλ ,D
λ′
CNα Kβ Mγ CRα′ Sβ ′ Lγ′
P
Aµ ,Bµ′ ,Cλ ,Dλ′
CAµ Cλ ,Eδ CBµ′ Dλ′ Fδ′ ) ⊗ |Eδ i ⊗ |Fδ′ i.
Now the result follows from the definition of a left minor. Q.E.D. We can also give an analogue of a general Laplace expansion in this context. But that would require general Clebsch-Gordon coefficients for which no explicit expressions are known so far.
6.2
Symmetry of the determinants and minors
The following property of determinants is needed later for defining the real ¯ of the new quantum group GLq (X) ¯ (Cf. Proposi(unitary) form Uq (X) tion 8.1). ¯ More generally, ¯ = D R (X). Proposition 6.2 DqL (X) q ¯ = D R,J (X) ¯ DIL,J (X) I and
L,Mγ ,Lγ ′
DEδ ,F
δ′
R,Mγ ,Lγ ′
¯ =D (X) Eδ ,F
32
δ′
¯ (X).
⊗
¯ = DqL (X) ¯ = This allows us to define the quantum determinant Dq = Dq (X) R ¯ and the quantum minors Dq (X), ¯ = DL,J (X) ¯ = D R,J (X) ¯ DIJ (X) I I and
Mγ ,Lγ ′
DEδ ,F
δ′
L,Mγ ,Lγ ′
¯ =D (X) Eδ ,F
δ′
R,Mγ ,Lγ ′
¯ (X)D Eδ ,F
δ′
¯ (X).
By a standard property of matrix coefficients (cf. Proposition 1.13 in [15]), we also have that X I ¯ ¯ = ¯ ¯ = δIJ , I, J ∈ Ωr , ∆(DJI (X)) DK (X) ⊗ DJK (X), ǫ(DJI (X)) K
and a similar property for the minors in the Gelfand-Tsetlin basis. In particular, ¯ = Dq (X) ¯ ⊗ Dq (X), ¯ ¯ = 1. ∆(Dq (X)) ǫ(Dq (X)) (66) ¯ Thus the quantum determinant is a nonzero group like element of O(Mq (X)). The rest of this section is devoted to the proof of Proposition 6.2. The proof is technical and may be skipped on the first reading. The main problem that needs to be addressed is the following. In the standard case, the right determinant, when rewritten using the reduction system (24), coincides with the left determinant. Analogously, one may wish to prove that the explicit expressions for the left and right determinants (minors) in Proposition 6.1 are equal by reducing them with respect to the reduction system correspond¯ ing to the relations (59) defining O(Mq (X)). But this reduction system does not satisfy the diamond lemma, and the standard monomials do not form a ¯ cf. Section 5.1. Hence, the expressions for the left and basis of O(Mq (X); right minors in Proposition 6.1 need not reduce to the same expressions. So this strategy does not work. We start with a few general observations. Given a coalgebra A, and a right corepresentation φ : V → V ⊗ A given by X φ(ej ) = ek ⊗ vkj , (67) k
there is a left corepresentation φR : V → A ⊗ V given by X φL (ek ) = vkj ⊗ ej .
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j
We shall denote this dual left corepresentation to V by VL . The two corepresentations V and VL share the same matrix [vkj ] of coefficients, which we 33
shall denote by M (V ) = M (VL ). Furthermore, the span C(φ) =< vkj > of the matrix coefficients has a left and right A-coaction, and hence, is a A-bicomodule. Similarly, given a left corepresentation W → A ⊗ W : X τ (ek ) = wkj ⊗ ej ,
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j
there is a corresponding right corepresentation WR : X τR (ej ) = ek ⊗ wkj ,
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k
with the same matrix M (W ) = M (WR ) = [wkj ] of coefficients. Furthermore, C(τ ) = C(τR ) =< wkj > has the left and right coaction, and hence, is a Abicomodule. ¯ ¯ Now we return to the case A = O(Mq (X)). Let ∧d,L q (X) denote the left d,L ¯ ¯ We define ∧d,R ¯ ¯ on the space ∧d [X]. corepresentation of O(Mq (X)) q (X), Cq (X) q d,L ¯ and Cd,R and M∧d,R denote the coefficient matrices q (X) similarly. Let M∧ d,R ¯ d,L ¯ of ∧q (X) and ∧q (X) in the standard monomial bases (Proposition 4.1), ˜ ∧d,L and M ˜ ∧d,R the coefficient matrices in the Gelfand-Tsetlin basis and M d,L d,R d,L d,R ˜ sym ˜ sym (48). The coefficient matrices Msym and Msym , and M and M are defined similarly. Proposition 6.2 follows from the following: ˜ ∧d,L = M ˜ ∧d,R . Lemma 6.3 For every d, M∧d,L = M∧d,R , and M d,R ¯ d,R ¯ d,L ¯ ¯ ∧d,L q (X)R = ∧q (X), ∧q (X)L = ∧q (X).
Hence,
In particular, with d = dim(X) = dim(V ) dim(W ), this implies that the ¯ ¯ is equal to the right determinant D R (X). left determinant DqL (X) q We now turn to the proof of the lemma. 2,L 2,R Proposition 6.4 M∧2,L = M∧2,R and Msym = Msym . Hence,
¯ C2,R q (X)L 2,L ¯ Cq (X)R ¯ ∧2,R q (X)L ¯ ∧2,L q (X)R
∼ = ∼ = ∼ = ∼ =
34
¯ C2,L q (X), 2,R ¯ Cq (X), ¯ ∧2,L q (X), ¯ ∧2,R q (X).
Proof: Assume that dim(V ) = dim(W ) = 2; the general case is very similar. The result follows by explicit computation of the coefficient matrices and checking that they are equal. We omit the details. 2,L ¯ ¯ The matrix coefficients of C2,R q (X) and Cq (X) (ignoring constant factors) are a(23) ∗ a(23) a(ij) ∗ a(i′ j ′ ), (i, j), (i′ , j ′ ) 6= (2, 3) b(ij) ∗ b(i′ j ′ ), (i, j), (i′ , j ′ ) = (1, 2), (3, 4) or (1, 4) b(ij) ∗ b(i′ j ′ ), (i, j), (i′ , j ′ ) = (1, 3), (2, 3) or (2, 4),
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where, for clarity, we denote aij and bkl in (38) and (39) by a(ij) and b(kl) here and in what follows. 2,R ¯ ¯ Similarly, the matrix coefficients of ∧2,L q (X) and ∧q (X) (ignoring constant factors) are a(ij) ∗ a(23), a(23) ∗ a(ij) (i, j) 6= (2, 3) {b(12), b(34), b(14)} ∗ {b(13), b(23), b(24)} {b(13), b(23), b(24)} ∗ {b(12), b(34), b(14)}. Q.E.D. ∼ ⊕d (X ¯ ⊗X ¯ ∗ )⊗d be the ideal generated by (55) or (56) Let I ⊆ ChU i = ¯ = ChU i/I. Equivalently, it is generated by so that O(Mq (X)) a(ij) ∗ b(rs) and b(rs) ∗ a(ij); ¯ generated by M 2 = cf. (59). Let J be the two sided ideal in O(Mq (X)) sym 2,L 2,R ¯ Msym = Msym ; i.e., the matrix coefficients of C2,L ( X), or equivalently, q ¯ ¯ ¯ C2,R q (X). It is a O(Mq (X))-bicomodule. Let O(Mq (X))d and Jd be the ¯ and J respectively. We have chosen degree d components of O(Mq (X)) ¯ J so that it will turn out later (cf. (77) and (78)) that, as O(Mq (X))bicomodules, d,L ¯ d,R ¯ ∼ d,R ¯ ¯ d /Jd ∼ ¯ O(Mq (X)) = ∧d,L q (X) ⊗ ∧q (X)R = ∧q (X)L ⊗ ∧q (X).
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(This is obvious at q = 1.) ¯ We shall find an explicit basis of R = O(Mq (X))/J, and then deduce Lemma 6.3 from the form of this basis. We have ∼ ¯ ¯ ⊗X ¯ ∗ )⊗d /(I + J). R = O(Mq (X)/J = ⊕d (X 35
Hence, by the diamond lemma [15], to get an explicit standard monomial basis of R, it suffices to give a reduction system for the ideal I + J in which all ambiguities can be resolved. Let L(I + J) denote the set of explicit generators for I + J; cf. (59), (38), (39), and the proof of Proposition 6.4. For example, when dim(V ) = dim(W ) = 2, these are a(ij) ∗ b(i′ j ′ ) and b(i′ j ′ ) ∗ a(ij) a(23) ∗ a(23) a(ij) ∗ a(i′ j ′ ), (i, j), (i′ , j ′ ) 6= (2, 3) b(ij) ∗ b(i′ j ′ ), (i, j), (i′ , j ′ ) = (1, 2), (3, 4) or (1, 4). b(ij) ∗ b(i′ j ′ ), (i, j), (i′ , j ′ ) = (1, 3), (2, 3) or (2, 4).
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The basic lemma is: Lemma 6.5 The equations {l = 0|l ∈ L(I + J)} can be recast in the form of an equivalent reduction system–just as eq.(20) was recast in the form a reduction system (24)–whose ambiguities can be resolved. Hence by the diamond lemma [15], the standard monomials as per this reduction system ∼ ¯ ¯ ⊗X ¯ ∗ )⊗d /(I + J). form a basis of R = O(Mq (X)/J = ⊕d (X Proof: We begin with introducing some notation that will be convenient in the proof. By an abuse of notation, we shall denote the set of basis elements of V , ¯ ¯∗ ⊗ X ¯ by V, X ¯ and U again; what is intended will be clear X and U = X ¯ ∗ and X, ¯ V ∗ and V , so from the context. Furthermore, we shall identify X ¯ ⊗ X. ¯ that we write U = X In the proof we shall explicitly distinguish between the two tensoring operations that we use. The first one, which we call restitution, and denote by ⋆ creates new variables from old. The second, which we call tensor, and denote by ⊗, raises the degree of the variable, and implements the usual tensor algebra. ¯ = V ⋆ V creates the new Thus for example, when V = W = {v1 , v2 }, X variables (basis vectors) xij = vi ⋆ vj . On the other hand, V ⊗ V creates the set {vi ⊗ vj |1 ≤ i, j ≤ 2} The element vi ⊗ vj is also denoted by vi vj . ¯ is the set V ⋆ W , whence if V = W , In this notation X ¯ = {xij = vi ⋆ vj |1 ≤ i, j ≤ 2} X 36
¯ ⊗X ¯ is thus elements of the type The set X xij xkl = xij ⊗ xkl = (vi ⋆ vj ) ⊗ (vk ⋆ vl ) = (vi ⊗ vk ) ⋆ (vj ⊗ vl ) = vi vk ⋆ vj vl ¯ ⊗X ¯ = (V ⊗ V ) ⋆ (V ⊗ V ). In this sense X ¯ is given by X ¯ ⋆X ¯ = The set U of the matrix coefficients for the set X V ⋆ V ⋆ V ⋆ V . Thus uijkl = vi ⋆ vj ⋆ vk ⋆ vl We also see that: U ⊗ U = (V ⊗ V ) ⋆ (V ⊗ V ) ⋆ (V ⊗ V ) ⋆ (V ⊗ V ) We define the set of z-variables Z m as the set obtained by the m-way ¯ = restitution of the variables vi . In other words, if V = {v1 , v2 }, then X V ⋆ V and U = Z4 = V ⋆ V ⋆ V ⋆ V Note that dim(Z 4 ) = 16, when dim(V ) = dim(W ) = 2. We use the notation za1,a2,...,am = va1 ⋆ va2 ⋆ . . . ⋆ vam . Now let V be 2-dimensional as above, and let W = V . Recall (cf. (37)) the elements A11, A12, A22 and B12 of V ⊗ V , listed below: A11 A22 A12 B12
= = = =
v1 v1 v2 v2 v1 v2 + pv2 v1 v1 v2 − qv2 v1
In the space Z m ⊗ Z m , the subspace P − (corresponding to eigenvectors of ˆ Z m ,Z m with negative eigenvalues) is spanned by the m-way restitutions R T1 ⋆ T2 ⋆ . . . ⋆ Tm , where (i) each Ti ∈ {A11, A12, A22, B12}. (ii) the term B12 appears exactly odd number of times. We use the following notation: n1 p1 ni pi
= = = =
C · B12 C · A11 ⊕ C · A22 ⊕ C · A12 ni−1 ⋆ p1 ⊕ pi−1 ⋆ n1 ni−1 ⋆ n1 ⊕ pi−1 ⋆ p1 37
Note that pi , ni ⊆ Z i ⊗ Z i . ¯ is the tensor algebra T (U ) modulo the ideal The algebra O(Mq (X)) generated by n4 ⊆ U ⊗ U . Thus I in the statement of the lemma is n4 in this terminology, since U = Z 4 . First we shall assume that dim(V ) = dim(W ) = 2. Then Z 1 ⊗ Z 1 = ¯ = Z 2 is 4-dimensional and n1 ⊕ p1 , with dim(n1 ) = 1 and dim(p1 ) = 3, X ¯ = (Z 2 ⊗ Z 2 )/n2 is 10-dimensional. A representative decomposition C2q (X) ¯ is given by of C2q (X) ¯ = n 1 ⋆ n 1 ⊕ p1 ⋆ p1 . C2q (X) ¯ either as a left The space M spanned by the matrix coefficients of C2q (X), or right corepresentation (cf. Proposition 6.4), is thus given by the 100dimensional ¯ ⋆ C2 (X) ¯ = (n1 ⋆ n1 ⊕ p1 ⋆ p1 ) ⋆ (n1 ⋆ n1 ⊕ p1 ⋆ p1 ). C2q (X) q
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¯ We now turn to the algebra R = O(Mq (X))/J, where J is the ideal ¯ = T (U )/I, where generated by the subspace M ⊆ U ⊗ U . Thus O(Mq (X)) I = n4 is the ideal in T (U ) generated by P − ⊆ U ⊗ U . Thus R = T (U )/(I + J). We consider the degree 2-component of R above and construct an appropriate reduction system. Remembering that U = Z 4 , our task is to construct a reduction system for (Z 4 ⊗ Z 4 )/(M ⊕ n4 ). Note that this space is 36-dimensional, since dim(V ) = dim(W ) = 2. We use the fact that Z 4 = Z 2 ⋆ Z 2 and given A = (a1 , . . . , a4 ) and B = (b1 , . . . , b4 ), we write zA zB = zA1 zB1 ⋆ zA2 zB2 , where A1 = (a1 , a2 ) and A2 = (a3 , a4 ), with B1 and B2 similarly defined. We say that zA1 zA2 ⋆ zB1 zB2 is standard if A1 ≻ B1 and A2 ≻ B2; in other words, it must be that a1 > b1 or a1 = b1 and a2 > b2 , and a similar condition on A2 and B2 . We represent this as A ⊐ B. We say that zA1 zA2 ⋆ zB1 zB2 is nonstandard if A 6⊐ B. There are exactly 6 choices of tuples for A1 , B1 that satisfy A1 ≻ B1 . These are σ = {(22, 11), (22, 12), (22, 21), (21, 11), (21, 12), (12, 11)} If Σ = {(A, B)|A ⊐ B} 38
then |Σ| = |σ|2 = 36. Our reduction system will specify for each nonstandard zA zB an element βA,B ∈ M ⊕ n4 which contains zA zB and all other terms in it are either standard or lower than zA zB in a certain order. We first note that for zA zB ∈ Z 2 ⊗Z 2 with A 6≻ B, there is an αA,B ∈ p2 , such that X αA,B = zA zB + ai zAi zBi i
with Ai ≻ Bi . These expressions are readily computed as follows. Let Aij’s and Bij’s be as in (37). Let α11,11 α22,22 α12,12 α21,21 α11,12 α11,21 α21,22 α12,22
= = = = = = = =
A11 ⋆ A11 A22 ⋆ A22 A11 ⋆ A22 A22 ⋆ A11 A11 ⋆ A12 A12 ⋆ A11 A22 ⋆ A12 A12 ⋆ A22
= = = = = = = =
z11 z11 z22 z22 z12 z12 z21 z21 z11 z12 + pz12 z11 z11 z21 + pz21 z11 z21 z22 + pz22 z21 z12 z22 + pz22 z12
The more complicated ones are: α11,22 = (qA12 ⋆ A12 + pB12 ⋆ B12)/(p + q) = z11 z22 + z22 z11 α12,21 = (A12 ⋆ A12 − B12 ⋆ B12)/(p + q) = z12 z21 + z21 z12 + (p − q)z22 z11 Our next observation is that for any αA,B as above, and any i, j, k, l we have αA,B ⋆ zij zkl ∈ M ⊕ n4 . Indeed, we may wend through the following sequence of relations: p2 ⋆ Z 2 = (n1 ⋆ n1 ⊕ p1 ⋆ p1 ) ⋆ (n1 ⊕ p1 ) ⋆ (n1 ⊕ p1 ) = n 1 ⋆ n 1 ⋆ n 1 ⋆ n 1 ⊕ n 1 ⋆ n 1 ⋆ n 1 ⋆ p1 ⊕ n 1 ⋆ n 1 ⋆ p1 ⋆ n 1 ⊕ n1 ⋆ n1 ⋆ p1 ⋆ p1 ⊕ (p1 ⋆ p1 ) ⋆ Z 2 The first 4 terms are easily seen to belong either to M or to n4 . The other unexpanded terms behave similarly. Now, the promised βA,B are available forthwith. Given A, B with the nonstandard expression zA zB = (zA1 zB1 ) ⋆ (zA2 zB2 ), if A1 6⊐ B1, consider βA,B = αA1,B1 ⋆ zA2 zB2
(75)
In this, other than the term zA zB , we will have all terms zC zD with C1 ≻ D1 . Next, when A, B are such that A1 ⊐ B1 , we construct βA,B = zA1 zB1 ⋆ αA2,B2 39
(76)
This defines the reduction system completely. As an example, we have β(1212,2122) = = = β(2212,1122) = = =
α12,21 ⋆ z12 z22 (z12 z21 + z21 z12 + (p − q)z22 z11 ) ⋆ z12 z22 z1212 z2122 + z2112 z1222 + (p − q)z2212 z1122 z22 z11 ⋆ α12,22 z22 z11 ⋆ (z12 z22 + pz22 z12 ) z2212 z1122 + pz2222 z1112
Next, for arbitrary dim(V ) and dim(W ), modification of the βA,B as above is straightforward. Let A = (a1 , . . . , a4 ) and B = (b1 , . . . , b4 ). For each index i, depending on the three conditions ai < bi or ai = bi or ai > bi , we replace ai and bi by either of 1 or 2 which satisfy the same relation. We term this new seqeunce as A′ and B ′ . After obtaining βA′ ,B ′ we replace each occurrence of 1 or 2 in each subscript by its original entry. This is illustrated by the following example: Let A = (1313) and B = (3223). We put A′ = (1212) and B ′ = (2122). and note that a′i < b′i iff ai < bi and similarly for other comparisons. We obtain β(1212,2122) from the above example and make the substitutions to get: β(1212,2122) = z1212 z2122 + z2112 z1222 + (p − q)z2212 z1122 β(1313,3223) = z1313 z3223 + z3213 z1323 + (p − q)z3313 z1223 Explicit reduction rules which arise from β(2212,1122) and β(1313,3223) are shown below: z2212 z1122 → −pz2222 z1112 z1313 z3223 → −z3213 z1323 − (p − q)z3313 z1223 The reduction system arising from the above elements βA,B does satisfy the diamond lemma. This is an easy consequence of the same fact for Z 2 /p2 , ¯ the case that we have exhibited earlier as the antisymmetric algebra ∧q [X] ¯ (cf. the proof of Proposition 4.1 (2)). Q.E.D. for X Lemma 6.6
d,R ¯ 2 ¯ d /Jd ) = dim(∧d,L ¯ 2 1. dim(O(Mq (X)) q (X)) = dim(∧q (X)) .
2. The matrix elements of M∧d,L or M∧d,R (modulo Jd ) form a basis of ¯ d /Jd . O(Mq (X)) 40
Proof: From the diamond lemma [15] and the form of the reduction system above (cf. (75) and (76)), it follows that the standard monomials of the form zA1 zA2 · · · zAr , ¯ where A1 ⊐ A2 ⊐ · · · , form a basis of O(Mq (X))/J. This immediately implies the first statement. The second statement is easy to check at q = 1. Since the dimension ¯ d is the same at q = 1 and general transendental real q, the of O(Mq (X)) second statement follows. Q.E.D. ¯ and J Now let us turn to the proof of Lemma 6.3. Since O(Mq (X)) ¯ ¯ are O(Mq (X))-bicomodules, so is O(Mq (X))/J. The second statement of ¯ Lemma 6.6 implies that, as O(Mq (X))-bicomodules, d,L ¯ ¯ d /Jd ∼ ¯ O(Mq (X)) = ∧d,L q (X) ⊗ ∧q (X)R ,
(77)
d,R ¯ ¯ d /Jd ∼ ¯ O(Mq (X)) = ∧d,R q (X)L ⊗ ∧q (X).
(78)
and, similarly,
Hence,
¯ ∼ d,R ¯ ∧d,L q (X) = ∧q (X)L
and
¯ ∼ ¯ ∧d,R (X) = ∧d,L q q (X)R .
d,R ¯ ¯ It follows that coefficient matrices of ∧d,L q (X) and ∧q (X) are similar; i.e., there exists a nonsingular similarity matrix Q such that M∧d,L = Q−1 M∧d,R Q. We have to show that Q is the identity matrix. This follows from Proposition 5.6, Proposition 4.3, and the theory of the standard Drinfeld-Jimbo quantum group, the main point being uniqueness of the orthonormal Gelfand¯ we leave details to the reader. This proves Tsetlin basis (48) for ∧dq [X]; Lemma 6.3 and hence Proposition 6.2.
7
Hopf structure
To define a cofactor matrix of u we need the following. Proposition 7.1 There is a nondegenerate pairing ¯ ⊗ ∧nm−r [X] ¯ → ∧nm [X], ¯ ∧rq [X] q q given by (xI , xJ ) → xI xJ ; 41
Proof: This follows from Proposition 4.1 and nondegeneracy of the pairing in the classical q = 1 case. Q.E.D. ˜ so that Proposition 7.2 There exists a cofactor matrix u ¯ ˜ u = u˜ u u = Dq (X)I. Proof: The matrix form of the nondegenerate pairing in Proposition 7.1 ¯ in the present conyields a q-analogue of Laplace expansion for O(Mq (X)) text. In particular, we have nondegenerate pairings
and
¯ ⊗ ∧1q [X] ¯ → ∧nm ¯ ∧nm−1 [X] q q [X],
(79)
¯ ⊗ ∧nm−1 ¯ → ∧nm ¯ ∧1q [X] [X] q q [X],
(80)
¯ =X ¯ is the fundamental vector representation. where ∧1q [X] ¯ and B ′ the Gelfand-Tsetlin Let B = {xij } be the standard basis of X, nm−1 ¯ [X]; cf. (48). Then B and B ′ are dual bases as per the basis of ∧q pairings in (79) and (80). This follows from (1) the multiplication formula in the Gelfand-Tsetlin basis (cf. (53), and (2) the symmetry and other detailed properties of Clebsch-Gordon coefficients proved in [3]; e.g. the fundamental CBC’s CN KM and CKN M are obtained from each other by just replacing q by q −1 . We leave the details to the reader. Let {br } denote the elements of B and {b′s } the elements of B ′ so that br b′s = δrs ,
(81)
b′s br = δrs .
(82)
and Applying the homomorphism φR (cf. Section 6) to the first equation ˜ R so that here implies that there exists a cofactor matrix u ¯ u˜ uR = Dq (X)I. Applying the homomorphism φL to the second equation here implies ˜ L so that that there exists a cofactor matrix u ¯ ˜ L u = Dq (X)I. u
42
˜L = u ˜ R . So It follows from the symmetry result (Proposition 6.2) that u we let ˜=u ˜L = u ˜ R. u Q.E.D. ¯ belongs to This result implies just as in the standard case that Dq (X) ¯ ¯ the center of O(Mq (X)). The coordinate algebra O(GLq (X)) of the sought ¯ is obtained by adjoining the inverse Dq (X) ¯ −1 to quantum group GLq (X) −1 −1 ¯ ¯ ˜ . This allows us to define O(Mq (X)). We formally define u = Dq (X) u ¯ just as in the standard case (Section 2). a Hopf structure on O(GLq (X), ¯ Proposition 7.3 There is a unique Hopf algebra structure on O(GLq (X)) so that ·
1. ∆(u) = u ⊗ u. 2. ǫ(u) = I. ¯ −1 , S(Dq (X) ¯ −1 ) = Dq (X), ¯ where ui are the entries 3. S(uij ) = u ˜ij Dq (X) j ˜. of u and u ˜ij are the entries of u ¯ is an FRT-algebra, the proof is similar to that of Proof: Since O(Mq (X)) Proposition 9.10 in [15]. Q.E.D.
8
Compact real form
¯ is a Hopf ∗-algebra with the inProposition 8.1 The algebra O(GLq (X)) j volution ∗ determined by (uij )∗ = S(ui ). Proof: This follows from Proposition 3, Chapter 9 in [15]. The crucial fact here is that the left and the right determinants coincide (Proposition 6.2). Q.E.D. ¯ is a CMQG algebra (cf. Proposition 8.2 The Hopf ∗-algebra O(GLq (X)) Section 2.1). ¯ of O(GLq (X)) ¯ is unitary by Proof: The fundamental corepresentation X −1 ¯ ¯ Proposition 7.2, and Dq (X) is a unitary element of O(GLq (X). Fur¯ thermore, O(GLq (X)) is generated by the matrix elements of the unitary 43
¯ −1 ). Hence, the result follows from Theorem 2.1 corepresentation u⊕(Dq (X) (a). Q.E.D. Finally, Theorem 1.3 can be restated as: ¯ defined above has a compact Theorem 8.3 (a) The quantum group GLq (X) ¯ real form, which we shall denote by Uq (X). (b) There is a homomorphism ¯ GLq (V ) × GLq (W ) → GLq (X),
(83)
¯ is unitarizable, and (c) Every finite dimensional representation of GLq (X) hence, is a direct sum of irreducible representations. (d) The quantum analogue of the Peter-Weyl theorem holds. That is, ¯ = ⊕λ S ∗ ⊗ Sλ , O(GLq (X)) λ ¯ where Sλ ranges over all irreducible representations of GLq (X). This follows from Propositions 7.3,8.1,8.2 and Theorem 2.1.
9
Example
¯ = V ⊗ W , dim(V ) = dim(W ) = 2. Recall that Vq,α (V ) denotes Let X the q-Weyl module of the standard quantum group GLq (V ) labelled by the partition α. Let Vλ (X) denote the Weyl module of G = GL(X) labelled by the partition λ. There are three partitions of size 3, namely (3), (2, 1), (1, 1, 1). We have V(3) (X) = Sym3 (X), with dimension 20, V(1,1,1) (X) = ∧3 (X), with dimension 4, and V(2,1) (X) with dimension 20. Hence, by Peter-Weyl, dim(O(M (X))3 ) = 202 + 42 + 202 = 816. ¯ 3 ) = 688. In contrast, by computer it was verified that dim(O(Mq (X)) 2 2 2 2 ¯ Since, 688 = 20 + 4 + 16 + 4 , by the Peter-Weyl theorem for GLq (X) ¯ has four irreducible represen(Theorem 8.3), we can suspect that GLq (X) tations of dimensions 20, 4, 16 and 4, respectively. ¯ and ∧3q [X] ¯ of GLq (X) ¯ That is indeed so. The representations C3q [X] (cf. Proposition 5.1) turn out to be irreducible representations of dimensions 20 and 4 respectively. These are the q-deformations of the classical 44
representations C3 (X) and ∧3 (X) in this setting. We shall see below that the classical representation V(2,1) (X) also has a 20-dimensional q-analogue ¯ which, however, is reducible. It decomposes into two irreducible Vq,(2,1) (X), ¯ and Vq,(2,1),2 (X) ¯ of GLq (X) ¯ of dimensions 16 representations Vq,(2,1),1 (X) and 4, respectively. ¯ above decompose as GLq (V )× The irreducible representations of GLq (X) GLq (W )-modules, via the homomorphism in Theorem 8.3, as follows: ¯ C3q [X]
= Vq,(3) (V ) ⊗ Vq,(3) (W ) ⊕ Vq,(2,1) (V ) ⊗ Vq,(2,1) (W ),
∧3q (X)
= Vq,(2,1) (V ) ⊗ Vq,(2,1) (W ),
Vq,(2,1),1 (X) = Vq,(2,1) (V ) ⊗ Vq,(3) (W ) ⊕ Vq,(3) (V ) ⊗ Vq,(2,1) (W ) Vq,(2,1),2 (X) = Vq,(2,1) (V ) ⊗ Vq,(2,1) (W ). ¯ can be explicitly constructed as follows. The module Vq,(2,1) (X) ¯ ⊗ ∧2q [X]. ¯ It is a 24-dimensional representation of GLq (X), ¯ Let M = X and it can be shown to contain, just as in the classical case, the four di¯ has a compact real form Uq (X), ¯ mensional module ∧3q (X). Since GLq (X) ¯ we can define an inner product on M . Then Vq,(2,1) (X) is the orthogonal ¯ in M . Alternatively, it is the orthogonal complement complement of ∧3q [X] ¯ in N = X ¯ ⊗ C2 [X]. ¯ of C3 [X] q
q
¯ and Vq,(2,1),2 (X) ¯ can also be explicThe irreducible modules Vq,(2,1),1 (X) itly constructed. In fact, at the end of the the next section we shall see how ¯ can be constructed, in principle, all irreducible representations of GLq (X) ¯ for general GLq (X).
10
Deformation Br of C[Sr ]
By the Brauer-Schur-Weyl duality in the standard case [15], the left action of GLq (V ) on V ⊗r commutes with the right action of the Hecke algebra Hr on V ⊗r , and the two actions determine each other. We now wish to construct a semisimple sub-algebra Br ⊆ Hr ⊗ Hr which will play the role of the Hecke algebra in the present context. The embedding B r → Hr ⊗ Hr , 45
here can be considered as a deformation of the embedding C[Sr ] → C[Sr ] ⊗ C[Sr ], determined by the diagonal embedding Sr → Sr × Sr of the symmetric ¯ does not coincide with the group Sr . Just as the Poincare series of GLq (X) Poincare series of the classical GL(X), the dimension of Br will turn out to be different (larger) that that of C[Sn ]. In conjunction with the Wederburn structure theorem, this would imply that the irreducible representations of Br are no longer in one-to-one correspondence with those of C[Sn ]. Thus Br would turn out to be qualitatively similar, but at the same time, fundamentally different from Hr . ¯ We shall define Br by essentially translating the definition of GLq (X) acting on the left in the dual setting of right action.
10.1
Hecke algebra
Before we can state its definition, we need to review some notation and results for standard affine Hecke Algebras. We largely follow the notation of [9]. Let K = C(q) be the field of rational functions in the indeterminate q with complex coefficients. The algebra Hr = Hr (q) is the associative K-algebra, generated by the symbols T1 , . . . , Tr−1 with the following relations: Ti Tj = Tj Ti whenever |i − j| > 1 Ti Ti+1 Ti = Ti+1 Ti Ti+1 for i = 1, . . . , r − 2 Ti2 = (q − 1)Ti + q for i = 1, . . . , r − 1
(84)
If M is a right-Hr -module, then eigenvalues of the right-multiplication on M by each Ti are q and −1. The element Tj is invertible: Tj−1 = q −1 Tj − (1 − q −1 ) It is well known that Hr is finite-dimensional and semi-simple over K with dimension r!. Since the eigenvalues of Ti must be q and −1, the element Ti ∈ Hr (q 2 ) has eigenvalues q 2 and −1. Thus the element Tˆi = q −1 Ti ∈ Hr (q 2 ) satisfies • (Tˆi − qI)(Tˆi + q −1 I) = 0, and • Tˆi Tˆj Tˆi = Tˆj Tˆi Tˆj whenever |j − i| > 1. 46
• Tˆi Tˆj = Tˆj Tˆi , if |i − j| > 1. ˆ=R ˆ V,V operator. SpecifThese coincide with the relations satisfied by the R ically, the first relation corresponds to (10) and the second relation correˆ [15]. It follows that Hr (q 2 ) has sponds to the braid relation satisfied by R ⊗r i ˆ ˆ ˆi a right action on V given by Ti → RV,V , where R V,V acts on the i-th ⊗r ˆ and (i + 1)-st factors of V as RV,V . This action clearly commutes with the left action of GLq (V ), and the two actions determine each other by Brauer-Schur-Weyl duality [15]. Remark 10.1 We will henceforth ignore the power of 2 and assume that Hr (q) itself acts on V ⊗r . The algebra Hr has other natural set of generators that will be crucial to us. Specifically, let pi =
Ti + 1 q+1
and
q i = 1 − pi =
q − Ti . q+1
(85)
The pi ’s generate Hr . The relations defining them are: pi pj = pj pi
(86)
pi − (q + 2 + 1/q)pi pi+1 pi = pi+1 − (q + 2 + 1/q)pi+1 pi pi+1
(87)
if |i − j| > 1,
for i = 1, . . . , r − 2, and p2i = pi
(88)
for i = 1, . . . , r − 1. The second equation here is a reformulation of the braid relation–the second relation in (84)–in terms of pi ’s. If we consider the right action of Hr (q 2 ) on V ⊗r as above, then pi corresponds to the operator P+i , which acts as P+V , defined in (12), on the i-th and (i + 1)-st factors of V ⊗r , and qi corresponds to the the operator P−i , which acts as P−V , defined in (12), on the i-th and (i + 1)-st factors of V ⊗r . The following rescaled versions of pi and qi are also useful: p˜i = (q 1/2 + q −1/2 )pi = q −1/2 (Ti + 1) q˜i = (q 1/2 + q −1/2 )qi = q −1/2 (q − Ti ).
(89)
The advantage of this rescaling is that Ci = −˜ qi is then the basic KazhdanLusztig basis element of degree one [14]. 47
Another useful set of generators of Hr is given by fi =
2Ti + 1 − q . q+1
There is an important C-involution ι of Hr which is crucial in the Kazhdan-Lusztig theory, namely, ι(q) = 1/q ι(Tj ) = Tj−1 = q −1 Tj − (1 − q −1 ). It is extended to Hr naturally. We have ι(pi ) = pi and ι(fi ) = fi . Another involution is the K-involution θ given by: θ(Ti ) = −Ti + q − 1 It is easy to check that θ(pi ) = qi , and θ(fi ) = −fi .
10.2
The algebra Br
ˆ = V ⊗ W in the obvious manner. Of special The algebra Hr ⊗ Hr acts on X importance are its elements Pi = pi ⊗ pi + qi ⊗ qi Qi = p i ⊗ q i + q i ⊗ p i .
(90)
¯ ⊗r correspond to the operators P X¯ and P X¯ , which act Their actions on X i,+ i,− ¯ ¯ ¯ ⊗r . as P+X and P−X , defined in (30), on the i-th and (i + 1)-st factors of X
The algebra Br = Br (q) is defined to be the subalgebra of Hr ⊗ Hr generated by the elements P1 , . . . , Pr−1 , or equivalently, the elements Qi ’s. ¯ ⊗r . By the defining relation We have the natural right action Br (q) on X ¯ (55) of O(Mq (X)), the right action of Br (q) commutes with the left action ¯ of the quantum group GLq (X). Remark 10.2 The algebra Br can be defined by letting arbitrary Coxeter group play the role of the symmetric group. The proof of the semisimplicity result below can be extended to any finite Weyl group in place of the symmetric group.
48
10.3
Basic properties of Br
Let Ar = Hr ⊗ Hr . The involution θ lifts to Ar and will also be denoted as θ. Thus, for example, θ(a ⊗ b) = θ(a) ⊗ θ(b) There is also the involution τ with τ (a ⊗ b) = b ⊗ a Since θ(Pi ) = τ (Pi ) = Pi , Br ⊆ Aτ,θ r , the invariant subring of Ar . We also have the involution Θ : Ar → Ar where Θ(a ⊗ b) = θ(a) ⊗ b. We see that: Θ(Pi ) = θ(pi ) ⊗ pi + θ(1 − pi ) ⊗ (1 − pi ) = (1 − pi ) ⊗ pi + pi ⊗ (1 − pi ) = 1 − Pi Thus Θ is an involution on Br as well. Proposition 10.3 The algebra Br is a semi-simple sub-algebra of Ar . Proof: We know that there is a right action of Hr on V ⊗r , which is faithful when dim(V ) is large, and where the matrix of each pi is symmetric. Consequently there is a faithful representation of Ar on (V ⊗ V )⊗r , which is faithful, and where the matrix of Pi = pi ⊗ pi + (1 − pi ) ⊗ (1 − pi ) is symmetric. We may now use the following fact to complete the proof. Fact: Let A ⊆ Mm (R) be a sub-algebra of the real matrix algebra Mm (R). Furthermore, let A be such that if a ∈ A then the transpose aT is also in A. Then A is semi-simple. Proof: We produce a vector-space basis C = {c1 , . . . , ck } of A such that the k × k-matrix G = T r(ci cj ) is non-singular. The non-singularity of this matrix is equivalent to the semi-simplicity of A. Since transpose is an involution, and A is closed under transposition, we have A = {a1 , . . . , ar } and B = {b1 , . . . , bs } such that (i) C = A ∪ B. (ii) aTi = ai for all i, and bTj = −bj for all j. Now consider T r(ai bj ) = T r(bTj aTi ) = −T r(bj ai ), whence T r(ai bj ) = 0. Thus G has the following format: GA 0 G= 0 GB 49
where GA = (T r(ai aj )) and GB = (T r(bi bj )). But GrA = (T r(ai aTj )) is the Gram-matrix for A, and similarly GrB is that for B, whence both GrA and GrB are non-singular. Since GA = GrA and GB = −GrB , G is nonsingular.
10.4
¯ Relationship with GLq (X)
In analogy with the Brauer-Schur-Weyl duality in the standard case [15], it is a reasonable conjecture that: ¯ and Br (q) determine each other. 1. The commuting actions of GLq (X) 2. There is a one-to-one correspondence between the irreducible (poly¯ of degree r and the irreducible nomial) representations of GLq (X) representations of Br so that, as a Bimodule, ¯ ⊗r = ⊕α,α′ V (α) ⊗ W (α′ ), X
(91)
¯ where V (α) runs over irreducible polynomial representations of GLq (X) ′ of degree r, and W (α ) denotes the irreducible representation of Br (q) in correspondence with V (α). Proposition 10.4 The third statement above holds for r = 3. Proof: This is sketched at the end of Section 11. Q.E.D. ¯ Assuming the conjecture above, irreducible representations of GLq (X) of degree r can be constructed, in principle, by constructing the idempotents ¯ ⊗r , very much as in the of the algebra Br and taking their right actions on X ¯ ⊗r · a is standard case [7]. Specifically, if a ∈ Br is an idempotent, then X ¯ and all irreducible (conjecturally) an irreducible representation of GLq (X), ¯ polynomial representations of GLq (X) of degree r can be (conjecturally) obtained in this way.
11
The algebra B3
As an example, now we wish to analyze the structure of the algebra B = B3 , in particular, its decomposition into irreducible modules. To conform with the convention in the Kazhdan-Lusztig paper [14], instead of the generators Pi ’s of Br in (90), we shall consider the generators Pi = p˜i ⊗ p˜i + q˜i ⊗ q˜i = fPi , 50
where p˜i and q˜i are as in (89), and f = (q + 2 + 1/q) = (q 1/2 + q −1/2 )2 . We see that Pi2 = f · Pi . (92) Through experimentation, by letting c1 =
q 6 + 2q 5 + 3q 4 + 4q 3 + 3q 2 + 2q + 1 q3 c2 =
q 4 + q 3 + 4q 2 + q + 1 q2
we see that: c1 P1 − c2 P1 P2 P1 + P1 P2 P1 P2 P1 = c1 P2 − c2 P2 P1 P2 + P2 P1 P2 P1 P2 (93) This identity is quite different from the braid identity (87) for Hr . This ¯ is not coquasitriangular in the terminology means the algebra O(GLq (X)) of [32]. We name Σ as either of the two sides of (93). Thus Σ = c1 P1 − c2 P1 P2 P1 + P1 P2 P1 P2 P1 and at once see that ΣP1 = ΣP2 = f · Σ, and thus C · Σ is a trivial onedimensional B-module. At q = 1, this specializes to the trivial representation of the symmetric group S3 . Next, let α= β1 β2 β12 β21 β121 β212 β1212 β2121
= = = = = = = =
f4
1 − c2 · f2 + c1
P1 − αΣ P2 − αΣ P1 P2 − f · αΣ P2 P1 − f · αΣ P1 P2 P1 − f2 · αΣ P2 P1 P2 − f2 · αΣ P1 P2 P1 P2 − f3 · αΣ P2 P1 P2 P1 − f3 · αΣ
(94)
It can be verified that β’s span an 8-dimensional B-module, with the following multiplication table.
51
β1 β2 β12 β21 β121 β212 β1212 β2121
P1 P2 fβ1 β12 β21 fβ2 β121 fβ12 fβ21 β212 fβ121 β1212 β2121 fβ212 −c1 β1 + c2 β121 fβ1212 fβ2121 −c1 β2 + c2 β212
This 8-dimensional module splits as follows:
χ1 =
β1 β12 β121 β1212
P1 P2 fβ1 β12 β121 fβ12 fβ121 β1212 −c1 β1 + c2 β121 fβ1212
χ2 =
β2 β21 β212 β2121
P1 P2 β21 fβ2 fβ21 β212 β2121 fβ212 fβ2121 −c1 β2 + c2 β212
The representation χ1 (and similarly χ2 ) may be split further as follows. Let γ1 = β1 /a + ·β121 γ12 = β12 /a + ·β1212 , for an indeterminate a. We get the multiplication table:
γ1 γ12
P1 P2 fγ1 γ12 (1+ac2 ) −c1 β1 + a β121 fγ12
For γ12 · P1 to be a multiple of γ1 we need: 1 + ac2 =a −ac1 Or in other words: c1 a2 + c2 a + 1 = 0 This yields: a=
−c2 ±
p
c22 − 4c1 2c1
52
As expected c22 − 4c1 is a perfect square and we have: a1 = a2 =
−q 2 (q 2 +1)2 −q (q+1)2
1 and γ 2 , γ 2 . We thus By choosing a1 and a2 , we form the vectors γ11 , γ12 1 12 have: χ1 = χ11 ⊕ χ21 χ2 = χ12 ⊕ χ22
Further, χ1 = χ11 ∼ = χ22 . = χ12 and χ2 = χ21 ∼ At q = 1, the nonisomorphic two-dimensional B-modules χ1 and χ2 specialize to the Specht module of S3 corresponding to the partition (2, 1). Finally, we have a 1-dimensional (alternating) module Cµ, which specializes at q = 1 to the alternating representation of S3 . This can be computed as follows. Let µ = 1 + θ1 β1 + θ2 β2 + θ12 β12 + θ21 β21 + θ121 β121 + θ212 β212 +θ1212 β1212 + θ2121 β2121 + θΣ, with the various constants being as in the following table: θ1 f3 −fc2 − f4 −f2 c2 +c1 θ2 f3 −fc2 − f4 −f 2 c +c 2 1
θ12 f2 −c2 f4 −f2 c2 +c1 θ21 f2 −c2 f4 −f2 c2 +c1
θ121
θ1212
− f4 −f2fc2 +c1
1 f4 −f2 c2 +c1
θ212 − f4 −f2fc2 +c1
θ −α
θ2121
1 f4 −f2 c2 +c1
Then it can be verified that µ · P1 = µ · P2 = 0. To summarize, we have 4 non-isomorphic representations of B, namely
Σ χ1 χ2 µ
dim. mult. 1 1 2 2 2 2 1 1
where, at q = 1, Σ specializes to the trivial one-dimensional representation of S3 , µ to the one-dimensional alternating representation, and χ1 and χ2 to the 2-dimensional Specht module of S3 corresponding to the partition (2, 1). Note that 12 + 22 + 22 + 12 = 10 = dim(B). 53
The analysis above can be extended to obtain explicit expressions for the idempotents of B3 ; we skip the details. Using these idempotents, we can obtain all irreducible representations of degree 3 of the quantum group ¯ X ¯ = V ⊗ W , dim(V ) = dim(W ) = 2. Indeed, if a is an idempotent GLq (X), ¯ ⊗3 · a is a representation of GLq (X). ¯ of Br then X These turn out to be irreducible when r = 3 (as per the general conjecture in Section 10.2). ¯ within X ¯ ⊗3 can be In this way all irreducible representations of GLq (X) computed, and Proposition 10.4 can be verified for r = 3.
12
Canonical basis
We now ask if Br ⊆ Hr ⊗ Hr has a canonical basis B that is akin to the Kazhdan-Lusztig basis of Hr . We shall not specify here precisely what “akin to” means; cf. [24] for the precise meaning. But two properties (among others) that this basis should satisfy are as follows. Let C be the KazhdanLusztig basis of Hr ⊗ Hr ; i.e., each element in C is of the form c1 ⊗ c2 , where ci ’s are Kazhdan-Lusztig basis elements of Hr . Then 1. Each coefficient of any b ∈ B, when expressed in terms of the basis C, should be of the form q a/2 f (q), for some integer a and a polynomial f (q) with nonnegative integral coefficients, and furthermore, each coefficient should be ι-invariant. This is analogous to the fact the coefficients of the Kazhdan-Lusztig basis of Hr in terms of the standard basis of Hr are (Kazhdan-Lusztig) polynomials with nonnegative coefficients (up to a factor of the form + or − q a/2 ). Thus the role of the standard basis of Hr is played here by C. 2. B should have cellular decompositions into left, right, and two-sided cells, just like the Kazhdan-Lusztig basis.
12.1
Canonical basis of B3
We now construct such a basis B of B3 . We follow the notation as in Section 11. Let
54
βˆ1 βˆ2 βˆ12 βˆ21 ˆ β121 βˆ212 βˆ1212 βˆ2121
= = = = = = = =
P1 P2 P1 P2 P2 P1 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P2 P1 P2 P1
(95)
These are modified versions of the elements β’s in eq.(94). 1 , . . . of the elements γ 1 , γ 1 , . . . in We define the modified forms γˆ11 , γˆ12 1 12 ˆ in place of β’s in their definitions. Let µ Section 11 by substituting β’s ˆ=1 ˆ = Σ. Finally, let and Σ
B = Uσ ∪ Uµ ∪ V1 ∪ V2 ∪ W1 ∪ W2 , where Uσ V1 V2 W1 W2 Uµ =
= = = = = =
ˆ {Σ} 1 } {ˆ γ11 , γˆ12 2 } 2 {ˆ γ1 , γˆ12 1 } {ˆ γ21 , γˆ21 2 2 {ˆ γ2 , γˆ21 } {ˆ µ}.
The discussion in Section 11 shows that B has a cellular decomposition into left cells. Indeed, Uσ , Uµ , Vi ’s and Wi ’s are the left cells, with an obvious partial order among these cells, with Uσ at the bottom of the partial order, and Uµ at the top. Similarly, it can be verified that B also has cellular decomposition into right or two-sided cells. Furthermore, coefficients of each b ∈ B in the Kazhdan-Lusztig basis C of H3 ⊗H3 are indeed polynomials in q with nonnegative coefficients (up to a factor of the form q a/2 ). These coefficients are shown in the following tables; cf. Figures 1-3. The coefficients are shown in the symmetrized KazhdanLusztig basis C ′ of H3 ⊗ H3 defined as follows. Let cw , w ∈ S3 , denote the Kazhdan-Lusztig basis element corresponding to the permutation w ∈ S3 . We order these permutations as per the lexicographic order on the words denoting their reduced decomposition: id < s1 < s1 s2 < s1 s2 s1 < s2 < s2 s1 , 55
where si denotes a simple transposition. Let ci denote the i-th element of the Kazhdan-Lusztig basis of H3 in this order. Let C ′ = {ci ⊗ ci } ∪ {ci ⊗ cj ⊕ cj ⊗ ci |i < j}. When r = 3, C ′ contains 21 elements. Figure 1 shows the 21-dimensional ˆ and µ coefficient vectors of Σ ˆ, Figure 2 those of the elements of V1 and V2 , and Figure 3 those of the elements of W1 and W2 .
13
The algebra B4 and beyond
The algebra B4 turns out to be considerably more complicated and is of dimension 114. This was verified on a computer by a simple procedure of generating monomials systematically and of increasing degree while retaining only those which were not linear combinations of earlier monomials. The top degree obtained thus was 9. In other words, every monomial of degree 10 and above is a linear combination of some smaller monomials. However, this linear combination seems fairly complicated. The ideal of all relations among these monomials is not generated by relations of the type (93)–letting arbitrary i and i + 1 there in place of 1 and 2–and (92). We do not know an explicit presentation in terms of generators and relations for the algebra Br , in general, akin to the explicit presentation (84) for the Hecke algebra. This means the construction procedure for a canonical basis for B3 in Section 12.1 cannot be generalized to general r. What is needed is a procedure that does not need explicit presentation; this problem is studied in [24]. To give an idea of the difficulties involved, we give below the simplest relation among the generators of B4 which cannot be deduced from the relations of the type (93) or (92). This is done by expressing the degree 7 monomial P3232123 ∈ B4 as a linear combination of smaller monomials, in degree or in lexicographical order, where Pi1 ,i2 ,...,ir denotes the monomial Pi1 Pi2 · · · Pir . There are 74 terms in this linear combination which are reported in the table in Figures 4-6. Every coefficient c(q) is a rational function in q such that c(q) = c(1/q). The table contains 74 rows and each row lists the term, the coefficient and finally, the monomial. Thus, for example, the 33-rd row corresponds to the term: −9.q 0 − 6.q 1 − 55.q 2 + 12.q 3 − 55.q 4 − 6.q 5 − 9.q 6 P13232 . 2.q 1 + 12.q 2 + 4.q 3 + 12.q 4 + 2.q 5 56
(96)
The computation of these coefficients was done in MATLAB, but not directly. A large prime p was chosen and these rational functions where calculated on Zp . The form was verified on another large prime q. It may be observed that, unlike in the relation (93), coefficients of some monomials in the reported relation–e.g. the one above–involve polynomials whose coefficients are of mixed signs. This suggests that there is no natural monomial basis for Br , unlike the standard monomial basis for Hecke algebras. Fortunately, Br still seems to possess a canonical basis in general [24].
References [1] B. Adsul, M. Sohoni, K. Subrahmanyam, Geometric complexity theory IX: algebraic and combinatorial aspects of the Kronecker problem, under preparation. [2] M. Artin, W. Schelter, J. Tate: Quantum deformations of GLn , Commun. Pure. Appl. Math. 44 (1991), 879-895. [3] E. Date, M. Jimbo, T. Miwa, Representations of Uq (gl(n, C)) at q = 0 and the Robinson-Schensted correspondence, in Physics and Mathematics of strings, World Scientific, Singapore, 1990. [4] V. Drinfeld: Quantum groups, in proceedings of the International Congress of Mathematicians (A. M. Gleason, ed), Amer. Math. Soc., Providence, RI, 1986, pp. 254-258. [5] W. Fulton, J. Harris: Representation theory, Springer Verlag, 1991. [6] I. Grojnowski, G. Lusztig, On bases of irreducible representations of quantum GLn, in Kazhdan-Lusztig theory and related topics, Chicago, IL, 1989, Contemp. Math. 139, 167-174. [7] M. Jimbo: a q-analogue of U (G) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69. [8] T. Hayashi: Nonexistence of homomorphisms between quantum groups, Tokyo J. Math. 15 (1992), 431-435. [9] J. Humphreys, Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics, 29 (1990), Cambridge University Press. 57
[10] M. Kashiwara: Crystallizing the q-analogue of universal enveloping algebra, Commun. Math. Phys. 133 (1990) 249-260. [11] M. Kashiwara: On crystal bases of the q-analogue of the universal enveloping algebras, Duke Math. J. 63 (1991), pp. 465–516. [12] M. Kashiwara, Global crystal bases of quantum groups, Duke Mathematical Journal, vol. 69, no.2, 455-485. [13] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. [14] D. Kazhdan, G. Lusztig, Schubert varieties and Poincare duality, Proc. Symp. Pure Math., AMS, 36 (1980), 185-203. [15] A. Klimyk, K. Schm¨ udgen, Quantum groups and their representations, Springer, 1997. [16] P. Littelmann: A Littlewood-Richardson rule for symmetrizable Kac-Moody Lie algebras, Invent. math. 116 (1994), 329-346. [17] G. Lusztig: Introduction to quantum groups, Birkh¨auser, Boston, 1993. [18] G. Lusztig: Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), pp. 447–498. [19] G. Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. USA, vo. 89, pp 8177-8179, 1992. [20] I. Macdonald: Symmetric functions and Hall polynomials, Oxford Science Publications, 1995. [21] Y. Manin: Quantum groups and noncommutative geometry, CRM, Montreal, 1988. [22] K. Mulmuley, On P vs. N P , geometric complexity theory and the flip, under preparation. [23] K. Mulmuley, Geometric complexity theory VII: a quantum group for the plethysm problem, under preparation. [24] K. Mulmuley, Geometric complexity theory VIII: towards canonical bases for the Kronecker problem, under preparation.
58
[25] K. Mulmuley, M. Sohoni: Geometric complexity theory, P vs. NP and explicit obstructions, in “Advances in Algebra and Geometry”, Edited by C. Musili, the proceedings of the International Conference on Algebra and Geometry, Hyderabad 2001. Available at authors’ websites. [26] K. Mulmuley, M. Sohoni: Geometric complexity theory: An approach to the P vs. NP and related problems, SIAM J. comput. vol. 31, no. 2, pp 496-526, (2001) [27] K. Mulmuley, M. Sohoni: Geometric complexity theory II: towards explicit obstructions for embeddings among class varieties, arXiv cs.CC/0612134, December, 2006. [28] K. Mulmuley, M. Sohoni: Geometric complexity theory III: on deciding positivity of Littlewood-Richardson coefficients, arXiv cs.CC/0501076, January 2005. [29] T. Nakashima: Crystal base and a generalization of LittlewoodRichardson rule for the classical Lie algebras, Commun. Math. Phys. 154 (1993), 215-243. [30] M. Noumi, H. Yamada, K. Mimachi: Finite dimensional representations of the quantum group GLq (n, C) and the zonal spherical functions on Uq (n)/Uq (n − 1), Jap. J. Math. 19 (1993), 31-80. [31] N. Reshetikhin: Multiparameter quantum groups and twisted quasitriangularHopf algebras, Lett. Math. Phys. 20 (1990), 331-335. [32] N. Reshetikhin, L. Takhtajan, L. Faddeev: Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225. [33] R. Stanley: Positivity problems and conjectures in algebraic combinatorics, In Mathematics: frontiers and perspectives, 295-319, Amer. Math. Soc. Providence, RI (2000). [34] A. Sudbery: Consistent multiparametric quantization of GL(n), J. Phys. A 23 (1990), L697-L704. [35] N. Vilenkin, A. Klimyk, Representations of Lie groups and special functions, vol. 3, Kluwer Acad. Publ. 1992. [36] S. Woronowicz: Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), 613-665. 59
¯ Appendix: Reduction system for O(Mq (X)) ¯ in the form In this section we reformulate the relations (59) for O(Mq (X)) of a reduction system, which is used in Section 5.1. We follow the same terminology as in the proof of Lemma 6.5. ¯ is the tensor algebra T (U ) In that terminology, the algebra O(Mq (X)) modulo the ideal generated by n4 ⊆ U ⊗ U . We assume first that dim(V ) = dim(W ) = 2. Then the dimension of the degree 2 component of this algebra, viz., dim((Z 4 ⊗ Z 4 )/n4 ) equals 17 136 = 2 . For this, we construct a total order on the variables Z = {za1,a2,a3,a4 |1 ≤ a1 , a2 , a3 , a4 ≤ 2}
This order is specified easily enough: za1,a2,a3,a4 zb1,b2,b3,b4 iff there is an 1 ≤ i ≤ 4 so that aj = bj for all 1 ≤ j ≤ i and ai+1 ≥ bi+1 . Thus, for example z2112 z1222 . Let A and B be the tuples A = (a1 , a2 , a3 , a4 ) and B = (b1 , b2 , b3 , b4 ). We say that a monomial zA zB is standard only if zA zB , otherwise, zA zB is called non-standard. Note that if Γ = {(A, B)|A B} then |Γ| = 136. We shall now see that the degree 2 component above is spanned by the standard monomials. We say that the monomial zA zB is exceptional of order i if ai+1 < bi+1 and aj ≥ bj for all 1 ≤ j ≤ i. Thus z2212 z1221 is non-standard of order 2. Clearly, every non-standard monomial zA zB is exceptional of order m for some m < 4. For every exceptional monomial zA zB of order m we exhibit an element ψA,B of P − = n4 such that: X ψA,B = zA zB + αi zCi zDi i
where either (i) each monomial zCi zDi is either standard, or (ii) zCi zDi is exceptional of order exceeding m. We construct ψA,B via another simpler element ψA′ ,B ′ and “exploding” this simpler element by the term ηA,B . The tuples A′ and B ′ are defined as follows. Let I be the set {i|ai 6= bi } arranged as a tuple. Let J = (j1 , . . . , jr ) be the complement of I. The term ηA,B is defined as zJ zJ where zJ = xj1 ⋆ . . . ⋆ xjr 60
Note that ηA,B = T1 ⋆ . . . ⋆ Tr where Ti = A11 if aji = 1 and Ti = A22 otherwise. Thus ηA,B ∈ P + for all A, B. We define A′ = AI and B ′ = BI , i.e., the tuples A and B restricted to the coordinates I. Thus, for example, if A = (1221) and B = (2212) then I = (134) and A′ = (121) and B ′ = (212). The factor ηA,B is (x2 )2 and thus equals A22. It is now clear that ψA,B = ψA′ ,B ′ ⋆ ηA,B where every coordinate of ηA,B goes in the right place in every term of ψA′ ,B ′ . Furthermore, if ψA′ ,B ′ is an element of P − then so is ψA,B . This reduces the computation for those monomial zA zB which are non-standard and for which ai 6= bi . The non-standardness implies that a1 = 1. For m = 4 there are 15 such tuples, the A values are listed below. B is precisely the complement of A. 1111 1112 1121 1122 1211 1212 1221 1222 111 112 121 122 11 12 1 For each A above, we inductively construct ψA ∈ P − and φA ∈ P + such that either of them is of the form X zA zB + αi zAi zBi i
where the first entry of Ai is 2. Thus, zA zB is indeed the solitary leading term, and all other terms, if non-standard are exceptional of a higher order. We illustrate this process by a small example: ψ1 = z1 z2 − qz2 z1 φ1 = z1 z2 + pz2 z1 Thus, ψ1 = B12 and φ1 = A12. Next, we construct the terms: (φ1 ⋆ A12 − ψ1 ⋆ B12)/(p + q) = z12 z21 + z21 z12 + (p − q)z22 z11 (q · φ1 ⋆ A12 + p · ψ1 ⋆ B12)/(p + q) = z11 z22 + z22 z11 Since both these terms are in P + , these qualify to be called φ11 and φ12 . The terms ψ11 and ψ12 are constructed similarly from φ1 ⋆B12 and ψ1 ⋆A12. 61
The net results is expressed simply as: q p 0 φ11 p+q p+q −1 1 φ12 0 p+q p+q q ψ11 = 0 0 p+q 1 ψ12 0 0 p+q In general, we note that: φA1 φA2 ψA1 = ψA2
q p+q 1 p+q
p p+q −1 p+q
0 0
0 0
0 0 p p+q −1 p+q
0 0
0 0
q p+q 1 p+q
p p+q −1 p+q
φ1 ⋆ A12 ψ1 ⋆ B12 φ1 ⋆ B12 ψ1 ⋆ A12
φA ⋆ A12 ψA ⋆ B12 φA ⋆ B12 ψA ⋆ A12
This gives us the contsruction of all ψA,B . As an example, for A = [1122] and B = [1212], we have ψA,B = A11 ⋆ ψ12 ⋆ A22 = −z1122 z1212 − z1212 z1122 This yields a reduction system for (Z ⊗ Z)/P − wherein we have X zA zB = αi zAi zBi i
where all zAi zBi are standard. By this reduction rule, there is a method of expanding any monomial zA zB zC as X zA zB zC = zAi zBi zCi i
wherein Ai Bi Ci . Thus every non-standard monomial may be expanded into a linear combination of standard monomials. Unfortunately, this reduction system does not obey the diamond lemma. In other words, there exist monomials zA zB zC wherein teo different simplifications using the above reduction rules yield two different expansions into standard monomials. Consider the monomial mm = z1111 z1112 z1221 . For any monomial zA zB zC , if A 6 B, then we may apply the reduction system above for the first two terms and this is denoted as (zA zB )zC . We say that R1 applies and display the result. Similar, we say that R2 applies if zB 6 zC and denote this 62
application by zA (zB zC ). The monomial above has two exapnsions, viz., R1 R2 R1 R2 and R2 R1 R2 , reading both strings from left to right. The first expansion yields (m)R1 = l1 = (q) · z1112 z1111 z1221 (m)R1 R2 = l12 = (−1 + q 2 ) · z1112 z1211 z1121 + q · z1112 z1221 z1111 The expression l12 has two terms t1 and t2 . We have: (t1 )R1 = z1211 z1112 z1121 = z1211 z1121 z1112 ; the last equality follows since z1121 and z1112 commute, as is easy to show. (t2 )R1 = (
−1 + q 2 −1 + q 2 2q )·z z z + ·z1212 z1121 z1111 + ·z1221 z1112 z1111 + 1211 1122 1111 2 2 1+q q (1 + q ) 1 + q2 1 − q2 · z1222 z1111 z1111 1 + q2
Combining all this, we have l121 = (m)R1 R2 R1 : q3 − q q2 − 1 ·z z z + ·z1212 z1121 z1111 + 1211 1122 1111 1 + q2 1 + q2 1 − q2 q 2q 2 · z1221 z1112 z1111 + · z1222 z1111 z1111 1 + q2 1 + q2
l121 = (−1+q 2 )·z1211 z1121 z1112 +
l121 has 5 terms, viz., t1 , . . . , t5 . Applying R2 to each we get: T erm1 = (1) · z1211 z1121 z1112 T erm2 = (1) · z1211 z1122 z1111 T erm3 = (1) · z1212 z1121 z1111 T erm4 = (1) · z1221 z1112 z1111 T erm5 = (1) · z1222 z1111 z1111 Finally, (m)R1 R2 R1 R2 = l1212 equals: q2 − 1 q3 − q ·z z z + ·z1212 z1121 z1111 + 1211 1122 1111 1 + q2 1 + q2 1 − q2 q 2q 2 · z1221 z1112 z1111 + · z1222 z1111 z1111 1 + q2 1 + q2
l1212 = (q 2 −1)·z1211 z1121 z1112 +
63
The second expansion is (m)R2 R1 R2 . We have l2 = (m)R2 as below: l2 =
q2 − 1 2q q2 − 1 ·z1111 z1212 z1121 + ·z z z + ·z1111 z1221 z1112 + 1111 1211 1122 1 + q2 q (1 + q 2 ) 1 + q2 1 − q2 · z1111 z1222 z1111 1 + q2
This has 4 terms, and applying R1 to each term yields: T erm1 = q · z1211 z1111 z1122 T erm2 = T erm3 =
q2 − 1 · z1211 z1112 z1121 + z1212 z1111 z1121 q q2 − 1 ) · z1211 z1121 z1112 + ·z1221 z1111 z1112 q
q2 − 1 q q2 − 1 q2 − 1 ·z z z + ·z z z + ·z1221 z1112 z1111 + T erm4 = 1211 1122 1111 1212 1121 1111 1 + q2 1 + q2 1 + q2 q (2 ) · z1222 z1111 z1111 1 + q2 Whence, l21 = (m)R2 R1 equals: q2 − 1 q 1 − 2q 2 + q 4 2q 2 − 2 ·z z z + ·z z z + )·z1211 z1121 z1112 + l21 = 1211 1111 1122 1211 1112 1121 1 + q2 q 2 (1 + q 2 ) 1 + q2 2q 3 − q − q 5
1 − 2 q2 + q4 q2 − 1 ·z z z − ·z1212 z1121 z1111 + 1212 1111 1121 q (1 + q 2 ) (1 + q 2 )2 (1 + q 2 )2 2q 2 − 2 q 1 − 2 q2 + q4 2q ·z1221 z1111 z1112 − ·z1221 z1112 z1111 − ·z1222 z1111 z1111 1 + q2 (1 + q 2 )2 (1 + q 2 )2 ·z1211 z1122 z1111 +
This has 9 terms, which on applying R2 yield: T erm1 =
q2 − 1 · z1211 z1121 z1112 + ·z1211 z1122 z1111 q T erm2 = z1211 z1121 z1112 T erm3 = z1211 z1121 z1112 T erm4 = z1211 z1122 z1111 T erm5 = q · z1212 z1121 z1111 64
T erm6 = z1212 z1121 z1111 T erm7 = q · z1221 z1112 z1111 T erm8 = z1221 z1112 z1111 T erm9 = z1222 z1111 z1111 Finally, collating this, we get l212 = (m)R2 R1 R2 as follows: l212 =
q 6 + q 4 − 3q 2 + 1 2q 3 − 2q · z1211 z1122 z1111 · z z z + 1211 1121 1112 q 2 (1 + q 2 ) (1 + q 2 )2
4q 2 + q 4 − 1 (1 + q 2 )2
· z1221 z1112 z1111 +
2q 2 − 2 (1 + q 2 )2
· [z1212 z1121 z1111 − q · z1222 z1111 z1111 ]
It is easy to see that (m)R1 R2 R1 R2 6= (m)R2 R1 R2 .
65
4 4 4 4 4 4
(q2 +q+1)
2
(1+q)4
q4
(q2 +q+1)(1+q)5 q 7/2
(q2 +q+1)(1+q)4 q3
(
)(1+q)3
q 2 +q+1
q 5/2
(q2 +q+1)(1+q)5 q 7/2
(q2 +q+1)(1+q)4 q3
6 (1+q) q3
6
6 (1+q) q 5/2
5
6 (1+q) q2
4
4
2 (q+2)(2 q+1)(1+q) q3 6 (1+q) q 5/2 2 2
4
(q2 +4 q+1)(1+q)4 q3
(
)(1+q)3
q 2 +4 q+1
q 5/2
6 (1+q) q 5/2
5
6 (1+q) q2
4
(q2 +4 q+1)(1+q)2 q2
6 (1+q) q2 2
2
5
4
(q2 +4 q+1)(1+q)3 q 5/2
6 (1+q) q3
6
6 (1+q) q 5/2
5
(q2 +4 q+1)(1+q)4 q3
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
ˆ and µ Figure 1: Coefficient vectors of the elements Σ ˆ in the symmetrized 66 Kazhdan-Lusztig basis
4
(q2 +q+1)(1+q)2 q2
(1+q) 5 q2 +6 q+5 ( ) 3/2 q (1+q)4 q2 (1+q)3 q 3/2 (1+q)5 q 5/2 (1+q)4 q2 2 8 (1+q) q (1+q)(q2 +6 q+1) q 3/2 8 (q2 +4 q+1)(1+q)2 q2 (1+q)(q2 +6 q+1) q 3/2 2 4 (1+q) q √ 4 1+q q (1+q)3 2 q 3/2 2 2 (1+q) q 8 2 2 (1+q) q 1+q 4 √q 4 2 (1+q) q2 3 2 (1+q) 3/2 q 2 4 (1+q) q
4
(q2 +q+1)(1+q)4 q3
(5 q2 +6 q+5)(1+q)3 q 5/2
(5 q2 +6 q+5)(1+q)2
4
(q2 +q+1)(1+q)2 q2 5
(1+q) q 5/2
q2
(1+q)4 q2
(1+q)5 q 5/2
(1+q)3 q 3/2
(5 q2 +6 q+5)(1+q)3 q 5/2
(1+q)(5 q 2 +6 q+5) q 3/2
(1+q)6 q3
(1+q)4 q2
8 (1+q) q2
4
2 (1+q) q2
4
8 (1+q) q 3/2
3
2 (1+q) q 3/2
3
2 (1+q) q
2
(q2 +6 q+1)(1+q)2 q2
6 (1+q) q2
4
q2
(q2 +6 q+1)(1+q)3 q 5/2
2
(q2 +4 q+1)(1+q)2
(q2 +6 q+1)(1+q)2 q2
2 (1+q) q 3/2
3
4 (1+q) q
2
4 (1+q) q 3/2
3
√ 4 1+q q
8 (1+q) q 3/2
3
(1+q)(q 2 +6 q+1) q 3/2
(q2 +6 q+1)(1+q)2 q2
8 (1+q) q
2
(q2 +6 q+1)(1+q)2 q2
2 (1+q) q 8 8
4 (1+q) q 3/2
3
√ 4 1+q q
8 (1+q) q2
4
8 (1+q) q
(q2 +6 q+1)(1+q)3 q 5/2
4 (1+q) q2
4
2
2
(1+q)(q 2 +6 q+1) q 3/2
4 (1+q) q
2
4
(q2 +q+1)(1+q)4 q3
(5 q2 +6 q+5)(1+q)3 q 5/2 (1+q)6 3 q 5 (1+q) 5/2 q (5 q2 +6 q+5)(1+q)3 q 5/2 2 (5 q2 +6 q+5)(1+q) q2 (1+q)4 8 q2 3 2 (q +6 q+1)(1+q) q 5/2 2 +6 q+1 (1+q)2 q ( ) q2 4 (1+q) 6 q2 3 (1+q) 8 q3/2 (1+q)4 4 q2 (1+q)3 4 q3/2 3 2 (q +6 q+1)(1+q) q 5/2 2 +6 q+1 (1+q)2 q ( ) q2 2 8 (1+q) q (q2 +6 q+1)(1+q)2 2 q 3 (1+q) 4 q3/2 (1+q)4 8 q2 3 8 (1+q) 3/2 q (q2 +6 q+1)(1+q)2 2 q2
67 Figure 2: Coefficient vectors of the elements of V1 and V2 in the symmetrized Kazhdan-Lusztig basis
(q2 +q+1)(1+q)4
(q2 +q+1)(1+q)6
(q2 +q+1)(1+q)4
(q2 +q+1)(1+q)6
q3
q4
q3
q4
(1+q)7 q 7/2
(1+q)5 q 5/2
(1+q)6 q3
(1+q)4 q2
(1+q)5 q 5/2
(1+q)3 q 3/2
(1+q)7 q 7/2
(1+q)5 q 5/2
(1+q)6 q3
(1+q)4 q2
(1+q)5 q 5/2 (1+q)4 q2 (1+q)3 q 3/2 (1+q)5 q 5/2 (1+q)4 q2 2 3 q3 +1+3 q+8 q2 +q4 q2 (1+q)(q2 +6 q+1) q 3/2 8 (q2 +4 q+1)(1+q)2 q2 (1+q)(q2 +6 q+1) q 3/2 2 4 (1+q) q √ 4 1+q q 3 2 (1+q) q 3/2 (1+q)2 2 q 8 2 2 (1+q) q √ 4 1+q q 4 2 (1+q) q2 3 2 (1+q) q 3/2 2 4 (1+q) q
2 2
(3 q3 +1+3 q+8 q2 +q4 )(1+q)2
2 (1+q) q2
4
q3 (1+q)(3 q 3 +1+3 q+8 q 2 +q 4 ) q 5/2
2 (1+q) q 3/2
3
2 (1+q) q
2
(q2 +6 q+1)(1+q)2 q2
(q4 +5 q3 +12 q2 +5 q+1)(1+q)2
(q2 +4 q+1)(1+q)2
q3
q2 3
(q2 +6 q+1)(1+q) q 5/2
2 3q
4 +6 q 3 +14 q 2 +6 q+3
q2
4 (1+q) q 3/2 2
3
(1+q)(3 q 3 +1+3 q+8 q 2 +q 4 ) q 5/2
2
8
3
q3
(q2 +6 q+1)(1+q)3 q 5/2
4 (1+q) q2
4
2
8
q2
(3 q3 +1+3 q+8 q2 +q4 )(1+q)2
2
2 (1+q) q
(q2 +6 q+1)(1+q)2
2
4 (1+q) q
(1+q)(q 2 +6 q+1) q 3/2
q2
4 (1+q) q 3/2
3
√ 4 1+q q
(q2 +6 q+1)(1+q)2 8 (1+q) q
2 (1+q) q 3/2
√ 4 1+q q
2 3q
3 +1+3 q+8 q 2 +q 4
q2
(1+q)(q 2 +6 q+1) q 3/2
4 (1+q) q
2
(1+q)6 q3 5 (1+q) q 5/2 7 (1+q) q 7/2 (1+q)6 q3 2 3 2 4 (3 q +1+3 q+8 q +q )(1+q) 2 q3 (q2 +6 q+1)(1+q)3 5/2 q 2 (q2 +6 q+1)(1+q) 2 q (q4 +5 q3 +12 q2 +5 q+1)(1+q)2 q3 (1+q)(3 q 3 +1+3 q+8 q 2 +q 4 ) 2 q 5/2 4 (1+q) 4 q2 3 4 (1+q) q 3/2 (q2 +6 q+1)(1+q)3 q 5/2 2 2 (q +6 q+1)(1+q) 2 q (1+q)2 8 q 2 2 +6 q+1 (1+q) q ( ) q2 (1+q)3 4 q3/2 2 3 2 4 (3 q +1+3 q+8 q +q )(1+q) 2 q3 (1+q)(3 q 3 +1+3 q+8 q 2 +q 4 ) 2 q 5/2 3 q 4 +6 q 3 +14 q 2 +6 q+3 2 q2
Figure 3: Coefficient vectors of the 68 elements of W1 and W2 in the symmetrized Kazhdan-Lusztig basis
(1+q)7 q 7/2
N umber 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Coef f icient 20.q 0 +104.q 1 +256.q 2 −113.q 3 −49.q 4 −113.q 5 +256.q 6 +104.q 7 +20.q 8 2.q 3 +12.q 4 +2.q 5 −16.q 0 −64.q 1 −128.q 2 −192.q 3 −224.q 4 −192.q 5 −128.q 6 −64.q 7 −16.q 8 2.q 3 +12.q 4 +2.q 5 −4.q 0 −16.q 1 −28.q 2 −32.q 3 −28.q 4 −16.q 5 −4.q 6 2.q 3 1.q 0 −4.q 2 +6.q 4 −4.q 6 +1.q 8 2.q 3 +2.q 5 −1.q 0 −18.q 1 −65.q 2 −128.q 3 −190.q 4 −220.q 5 −190.q 6 −128.q 7 −65.q 8 −18.q 9 −1.q 10 2.q 4 +12.q 5 +2.q 6 1.q 0 +5.q 1 +17.q 2 +36.q 3 +46.q 4 +46.q 5 +46.q 6 +36.q 7 +17.q 8 +5.q 9 +1.q 10 2.q 4 +2.q 6 7.q 0 +26.q 1 +75.q 2 +152.q 3 +174.q 4 +156.q 5 +174.q 6 +152.q 7 +75.q 8 +26.q 9 +7.q 10 2.q 3 +12.q 4 +4.q 5 +12.q 6 +2.q 7 −1.q 0 −8.q 1 −20.q 2 −24.q 3 −22.q 4 −24.q 5 −20.q 6 −8.q 7 −1.q 8 2.q 3 +2.q 5 −22.q 0 −92.q 1 −170.q 2 −200.q 3 −170.q 4 −92.q 5 −22.q 6 2.q 2 +12.q 3 +2.q 4 2.q 0 +2.q 1 +12.q 2 +14.q 3 +4.q 4 +14.q 5 +12.q 6 +2.q 7 +2.q 8 2.q 3 +2.q 5 −2.q 0 −12.q 1 −40.q 2 −52.q 3 −44.q 4 −52.q 5 −40.q 6 −12.q 7 −2.q 8 2.q 3 +12.q 4 +2.q 5 −1.q 0 −2.q 1 −12.q 2 −14.q 3 −6.q 4 −14.q 5 −12.q 6 −2.q 7 −1.q 8 2.q 3 +2.q 5 1.q 0 +22.q 1 +88.q 2 +170.q 3 +206.q 4 +170.q 5 +88.q 6 +22.q 7 +1.q 8 2.q 3 +12.q 4 +2.q 5 6.q 0 +8.q 1 +4.q 2 +8.q 3 +6.q 4 2.q 2 3.q 0 +6.q 1 +5.q 2 +4.q 3 +5.q 4 +6.q 5 +3.q 6 2.q 2 +2.q 4 12.q 0 +32.q 1 +40.q 2 +32.q 3 +12.q 4 2.q 1 +12.q 2 +2.q 3 −3.q 0 −2.q 1 −5.q 2 −12.q 3 −5.q 4 −2.q 5 −3.q 6 2.q 2 +2.q 4 1.q 0 +4.q 1 +11.q 2 +16.q 3 +11.q 4 +4.q 5 +1.q 6 2.q 3 8.q 0 +12.q 1 +24.q 2 +40.q 3 +24.q 4 +12.q 5 +8.q 6 2.q 2 +12.q 3 +2.q 4 −6.q 0 −8.q 1 −4.q 2 −8.q 3 −6.q 4 2.q 1 +2.q 3 −5.q 0 −4.q 1 −44.q 2 −60.q 3 −30.q 4 −60.q 5 −44.q 6 −4.q 7 −5.q 8 2.q 2 +12.q 3 +4.q 4 +12.q 5 +2.q 6 −1.q 0 −5.q 1 −11.q 2 −14.q 3 −11.q 4 −5.q 5 −1.q 6 2.q 3 −3.q 0 −6.q 1 −5.q 2 −4.q 3 −5.q 4 −6.q 5 −3.q 6 2.q 2 +2.q 4 2.q 0 +4.q 1 +4.q 2 +4.q 3 +2.q 4 2.q 2 −1.q 0 −4.q 1 −6.q 2 −4.q 3 −1.q 4 2.q 2 6.q 0 +8.q 1 +4.q 2 +8.q 3 +6.q 4 2.q 1 +2.q 3 16.q 0 +32.q 1 +16.q 2 2.q 0 +12.q 1 +2.q 2 4.q 0 +8.q 1 +40.q 2 +8.q 3 +4.q 4 2.q 1 +12.q 2 +2.q 3 −3.q 0 −8.q 1 −4.q 2 −8.q 3 +46.q 4 −8.q 5 −4.q 6 −8.q 7 −3.q 8 2.q 2 +12.q 3 +4.q 4 +12.q 5 +2.q 6 −8.q 0 2.q 0
69 Figure 4: A relation in B4
M onomial 1 2 3 12 13 21 23 32 121 132 212 213 232 323 1212 1213 1232 1321 1323 2121 2123 2321 2323 3212 3213 3232 12121 12123 12132 12321
N umber 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Coef f icient −4.q 0 −8.q 1 −40.q 2 −8.q 3 −4.q 4 2.q 1 +12.q 2 +2.q 3 −3.q 0 −4.q 1 −2.q 2 −4.q 3 −3.q 4 2.q 1 +2.q 3 −9.q 0 −6.q 1 −55.q 2 +12.q 3 −55.q 4 −6.q 5 −9.q 6 2.q 1 +12.q 2 +4.q 3 +12.q 4 +2.q 5 9.q 0 +6.q 1 +55.q 2 −12.q 3 +55.q 4 +6.q 5 +9.q 6 2.q 1 +12.q 2 +4.q 3 +12.q 4 +2.q 5 1.q 0 −1.q 1 +3.q 2 −6.q 3 +3.q 4 −1.q 5 +1.q 6 2.q 2 +2.q 4 −1.q 0 +2.q 2 −1.q 4 2.q 2 2.q 0 +3.q 1 +6.q 2 −3.q 3 −16.q 4 −3.q 5 +6.q 6 +3.q 7 +2.q 8 2.q 2 +12.q 3 +4.q 4 +12.q 5 +2.q 6 3.q 0 +4.q 1 +2.q 2 +4.q 3 +3.q 4 2.q 1 +2.q 3 −16.q 0 −32.q 1 −16.q 2 2.q 0 +12.q 1 +2.q 2 3.q 0 +4.q 1 +2.q 2 +4.q 3 +3.q 4 2.q 1 +2.q 3 8.q 0 2.q 0 1.q 0 −2.q 2 +1.q 4 2.q 2 −3.q 0 −4.q 1 −2.q 2 −4.q 3 −3.q 4 2.q 1 +2.q 3 −8.q 0 −16.q 1 −8.q 2 2.q 0 +12.q 1 +2.q 2 −1.q 0 −14.q 1 −15.q 2 −4.q 3 −15.q 4 −14.q 5 −1.q 6 2.q 1 +12.q 2 +4.q 3 +12.q 4 +2.q 5 −2.q 0 −4.q 1 −2.q 2 2.q 1 −2.q 0 +4.q 2 −2.q 4 2.q 1 +12.q 2 +2.q 3 8.q 0 +16.q 1 +8.q 2 2.q 0 +12.q 1 +2.q 2 −1.q 0 −2.q 1 −1.q 2 2.q 1 2.q 0 −4.q 2 +2.q 4 2.q 1 +12.q 2 +2.q 3 2.q 0 +8.q 1 +12.q 2 +8.q 3 +2.q 4 2.q 1 +12.q 2 +2.q 3 2.q 0 +4.q 1 +2.q 2 2.q 1 1.q 0 +14.q 1 +15.q 2 +4.q 3 +15.q 4 +14.q 5 +1.q 6 2.q 1 +12.q 2 +4.q 3 +12.q 4 +2.q 5 3.q 0 +8.q 1 +10.q 2 +8.q 3 +3.q 4 2.q 1 +12.q 2 +2.q 3 −2.q 0 −8.q 1 −12.q 2 −8.q 3 −2.q 4 2.q 1 +12.q 2 +2.q 3 −1.q 0 −2.q 1 −1.q 2 2.q 1 −3.q 0 −8.q 1 −10.q 2 −8.q 3 −3.q 4 2.q 1 +12.q 2 +2.q 3 1.q 0 +2.q 1 +1.q 2 2.q 1 −2.q 0 −4.q 1 −2.q 2 2.q 1 2.q 0 +4.q 1 +2.q 2 2.q 1
Figure 5: A relation 70 in B4 continued.
M onomial 12323 13212 13232 21213 21232 21321 21323 23213 23232 32121 32123 32132 32321 121213 121232 121321 123213 123232 132121 132123 212132 212321 212323 213212 213232 232121 232132 232321 321232 321323
N umber 61 62 63 64 65 66 67 68 69 70 71 72 73 74
Coef f icient 1.q 0 +2.q 1 +1.q 2 2.q 1 1.q 0 −2.q 1 +1.q 2 2.q 0 +2.q 2 2.q 0 2.q 0 1.q 0 −2.q 1 +1.q 2 2.q 0 +2.q 2 2.q 0 2.q 0 2.q 0 −4.q 1 +2.q 2 2.q 0 +12.q 1 +2.q 2 16.q 1 2.q 0 +12.q 1 +2.q 2 −2.q 0 2.q 0 −1.q 0 +2.q 1 −1.q 2 2.q 0 +2.q 2 −4.q 0 −8.q 1 −4.q 2 2.q 0 +12.q 1 +2.q 2 −16.q 1 2.q 0 +12.q 1 +2.q 2 −1.q 0 +2.q 1 −1.q 2 2.q 0 +2.q 2 −2.q 0 +4.q 1 −2.q 2 2.q 0 +12.q 1 +2.q 2 4.q 0 +8.q 1 +4.q 2 2.q 0 +12.q 1 +2.q 2
M onomial 323212 1212132 1213213 1213232 1232121 1232132 1321232 1321323 2121323 2123212 2123213 2123232 2132123 2321232
Figure 6: A relation in B4 continued.
71
Geometric Complexity Theory IV: quantum group for the Kronecker problem Dedicated to Sri Ramakrishna Ketan D. Mulmuley The University of Chicago and I.I.T., Mumbai∗ Milind Sohoni I.I.T., Mumbai February 1, 2008
Abstract A fundamental problem in representation theory is to find an explicit positive rule, akin to the Littlewood-Richardson rule, for decomposing the tensor product of two irreducible representations of the symmetric group (Kronecker problem). In this paper a generalization of the Drinfeld-Jimbo quantum group, with a compact real form, is constructed, and also an associated semisimple algebra that has conjecturally the same relationship with the generalized quantum group that the Hecke algebra has with the Drinfeld-Jimbo quantum group. In the sequel [24] it is observed that an explicit positive decomposition rule for the Kronecker problem exists assuming that the coordinate ring of the generalized quantum group has a basis analogous to the canonical basis for the coordinate ring of the Drinfeld-Jimbo quantum group, as per Kashiwara and Lusztig [12, 17], or in the dual setting–the associated algebra has a basis analogous to the Kazhdan-Lusztig basis for the Hecke algebra [13], as suggested by the experimental and theoretical results therein. In the other sequel [23], similar quantum group and algebra are constructed for the generalized plethysm problem, of which the Kronecker problem studied here is a special case. These problems play a central role in geometric complexity theory– an approach to the P vs. N P and related problems. ∗
Visiting faculty member
1
1
Introduction
This is a continuation of the series of articles [26, 27, 28] on geometric complexity theory, an approach to the P vs. N P and related problems. A basic philosophy of this approach is called the flip, which was proposed in [25], with a detailed exposition to appear in [22]. The flip, in essence, reduces the negative lower bound problems in complexity theory to positive problems in mathematics. One central positive problem that arises in the flip turns out to be the following fundamental problem in representation theory of the symmetric group Sn ; cf. [5, 20, 33] for its history and significance. Problem 1.1 (Kronecker problem) Find an explicit positive decomposition rule for the tensor product of two irreducible representations (Specht modules) of Sn over C. Specifically, given partitions (Young diagrams) λ, µ, π, find an explicit positive formula π , which is the multiplicity of the irreducible for the Kronecker coefficient kλ,µ representation (Specht module) Sπ of the symmetric group Sn in the tensor product Sλ ⊗ Sµ . By an explicit positive formula, we mean a formula involving no alternating signs, as in the Littlewood-Richardson rule. This problem is a special case of the following slightly more general problem [5]. Given a partition λ of height at most n, let Vλ (G) denote the corresponding irreducible Weyl module of G. Let V = C n , W = Cm , X = V ⊗ W , and consider the natural homomorphism H = GL(V ) × GL(W ) → G = GL(V ⊗ W ) = GL(X).
(1)
Any irreducible representation Vλ (G) of G decomposes as an H-module: Vλ (G) = ⊕α,β mλα,β Vα (GL(V )) ⊗ Vβ (GL(W )),
(2)
where α and β range over Young diagrams of height at most n and m, respectively, and Vα (GL(V )) and Vβ (GL(W )) denote the corresponding irreducible representations of GL(V ) and GL(W ), respectively. Problem 1.2 (Kronecker problem: second version) Find an explicit positive rule for the decomposition (2). Specifically, given partitions α, β, λ, find an explicit positive formula for the (generalized) Kronecker coefficient mλα,β , which is the multiplicity of the H-module Vα (GL(V )) ⊗ Vβ (GL(W )) in the G-module Vλ (G). 2
1.1
The basic idea
Our approach to this problem is motivated by a transparent proof for the positive Littlewood-Richardson rule based on the theory of quantum groups, which works for arbitrary symmetrizable Kac-Moody algebras [10, 11, 16, 18, 17, 29]. To recall the basic idea of this proof, we need some notation. Given Z = Cn , G = GL(Z), let G be the Lie algebra of G, U (G) its enveloping algebra, and O(G) its coordinate algebra. There are two equivalent ways of defining a standard quantum group associated with G: either by defining (1) a quantization of the enveloping algebra U (G), namely, the Drinfeld-Jimbo quantized enveloping algebra Uq (G) [4, 7], or (2) a quantization Oq (G) of the coordinate algebra O(G), namely, the FRT-algebra [32] that is dual to Uq (G). Let GLq (Z) denote the standard quantum group deformation of G = GL(Z)–which is only a virtual object–whose coordinate ring is Oq (G). Now a proof of the generalized Littlewood-Richardson rule mentioned above roughly goes as follows. Let H = GLn (C). Given partitions α and β of height at most n, the tensor product of the Weyl modules Vα (H) and Vβ (H) decomposes as: Vα (H) ⊗ Vβ (H) = ⊕λ cλα,β Vλ (H), where λ ranges over Young diagrams of height at most n and cλα,β is the Littlewood-Richardson coefficient. The problem is to find an explicit positive formula for this coefficient. Equivalently, letting G = H × H, the problem is to find an explicit positive decomposition of a given G–module Vα (H) ⊗ Vβ (H), when considered as an H-module via the diagonal embedding: H → G = H × H. Let H be the Lie algebra of H, and U (H) its enveloping algebra. The infinitesimal version the diagonal map above is the map U (H) →∆ U (H) ⊗ U (H), where ∆(x) = 1 ⊗ x + x ⊗ 1 for x ∈ H. Drinfeld and Jimbo [4, 7] have given a quantum deformation of this map: Uq (H) →∆ Uq (H) ⊗ Uq (H), where Uq (H) is the Drinfeld-Jimbo algebra, and such a deformation works for arbitrary symmetrizable Kac-Moody algebras. The (generalized) LittlewoodRichardson rule then comes out of the properties of canonical (local/global 3
crystal) bases of Kashiwara and Lusztig [10, 11, 16, 17, 29] for representations of Uq (H). This suggests the following analogous strategy to address the Kronecker problem (Problem 1.2). (1) Find a quantization of the homomorphism (1) H = GL(V ) × GL(W ) → G = GL(V ⊗ W ) = GL(X). Instead of working in the setting of Drinfeld-Jimbo quantized algebra, we shall work in the dual setting FRT-coordinate algebras [32]. So what we seek is a quantization of the form: ¯ GLq (V ) × GLq (W ) → GLq (X),
(3)
¯ where the quantum groups GLq (V ) and GLq (W ) are standard, and GLq (X) is a newly sought quantum group deformation of GL(V ⊗ W ). Furthermore, ¯ should have a compact real form in the sense like GLq (X), the new GLq (X) of Woronowicz [36]. This is required in the present context for two reasons. First, we want representation theory of the new quantum group to be rich like that of the standard quantum group. In particular, we want quantum analogues of the classical complete reducibility theorem, and Peter-Weyl theorem to hold, for which compactness is crucial. Second, we want representations of the new quantum group to have bases akin to the canonical bases for representations of the standard quantum group. Compactness is important for existence of such bases. Indeed, Kashiwara’s theory of local crystal bases [10], which was later globalized in [12], came out of an analysis of q-orthonormal Gelfand-Tsetlin bases for representations of the standard quantum group, whose construction depends on compactness of its real form in the sense of Woronowicz. (2) Develop a theory of canonical (local/global crystal) bases for represen¯ akin to the canonical bases for representations of the tations of GLq (X) standard quantum group GLq (X), as per Kashiwara and Lusztig [10, 11, 18]. (3) The required explicit positive decomposition rule should then follow from the properties of canonical bases. This is the route that we shall adopt. 4
1.2
Quantization
As expected, the theory of Drinfeld-Jimbo quantum group does not work for the Kronecker problem, because it can be shown (as in [8]) that the homomorphism (1) can not be quantized in the category of Drinfeld-Jimbo quantum groups. What is needed an analogue of the Drinfeld-Jimbo quantum group in the present setting. This is provided by the following result, which addresses the first step. Theorem 1.3 (a) The homomorphism H = GL(V ) × GL(W ) → G = GL(V ⊗ W ) = GL(X).
(4)
can be quantized in the form ¯ GLq (V ) × GLq (W ) → GLq (X),
(5)
¯ where the quantum groups GLq (V ) and GLq (W ) are standard, and GLq (X) is a new quantum group deformation of GL(V ⊗ W ) defined in this paper. ¯ has a real form Uq (X), ¯ which is a compact (b) The quantum group GLq (X) quantum group in the sense of Woronowicz [36]. This implies that ¯ is completely re1. Every finite dimensional representation of GLq (X) ducible as a direct sum of irreducible representations. 2. Quantum analogue of the Peter-Weyl theorem holds. This is proved in Sections 3-8. When W is trivial, the new quantum group specializes to the Drinfeld-Jimbo quantum group GLq (V ). Furthermore, let C[Sr ] be the group algebra of the symmetric group Sr , and C[Sr ] → C[Sr ] × C[Sr ] the embedding corresponding to the diagonal homomorphism Sr → Sr × Sr . Then Proposition 1.4 The embedding C[Sr ] ֒→ C[Sr ] × C[Sr ] can be similarly quantized. That is, there is a quantized semisimple algebra Br = Br (q) with an embedding Br → Hr ⊗ Hr , where Hr = Hr (q) denotes the Hecke algebra associated with the symmetric group Sr –it is a quantization of the group algebra C[Sr ]. This is proved in Sections 10-10.3. The relationship between Br (q) and ¯ here is conjecturally similar to that between the Hecke algebra GLq (X) Hr (q) and the standard quantum group GLq (X) (Section 10.2). 5
1.3
Explicit positive decomposition
The following addresses the second step. ¯ has a basis that Conjecture 1.5 (cf.[24]) The coordinate ring of GLq (X) is akin to the canonical basis of the coordinate ring of GLq (X) as per Kashiwara and Lusztig [12, 19, 17]. The algebra Br (q) has a basis that is akin to the canonical (KazhdanLusztig) basis of Hr (q) [13, 14]. The precise meaning of “akin to” will be made clear in [24], along with theoretical and experimental evidence in support of the second statement. See also Section 12 for an example of a canonical basis for B3 (q). The first and the second statements here are closely related, in view of the close relationship between analogous two statements for the standard quantum group and Hecke algebra [6]. Assuming this conjecture, it follows [24] that there is an explicit positive rule for the Kronecker problem. This addresses the third step.
1.4
Comparison
¯ and the algebra Br (q) are qualitatively similar, The quantum group GLq (X) but at the same time fundamentally different, in comparison to the standard quantum group GLq (X) and the Hecke algebra Hr (q), respectively. We now compare their properties. ¯ shares (1) Compactness Compactness is a crucial property that GLq (X) with the standard quantum group GLq (X). This means Woronowicz’ theory ¯ [36] of compact quantum groups is applicable to GLq (X). In particular, compactness implies existence of orthonormal bases for representations of ¯ GLq (X). We can also expect nice orthonormal bases for representations ¯ that yield local crystal bases, akin to the orthonormal Gelfandof GLq (X) Tsetlin bases for representations of GLq (X), which, after renormalization, yield local crystal bases [10]. It may be remarked that among the several quantum deformations of GL(Cn ) known by now, besides the one due to Drinfeld and Jimbo–e.g., Reshetikhin, Takhtadzhyan, Faddeev [32, 31], Manin [21], Sudbery [34], Artin, Schelter and Tate [2]–the standard Drinfeld-Jimbo deformation GLq (Cn ) is the only one that has a compact real form in the sense of Woronowicz [36]. 6
Not surprisingly, it is the only deformation of GL(Cn ) that has been studied in depth in the literature. (2) Determinants and minors The quantum determinant and minors of the standard quantum group have simple formulae, very similar to the classical ones. In contrast, explicit formulae for the quantum determinant ¯ turn out to be nonelementary; cf. Propoand quantum minors of GLq (X) sition 6.1. These involve Clebsch-Gordon coefficients and certain q-special functions that arise in the theory of the standard quantum group, which have been intensively studied (cf. the surveys [15]), but not yet completely understood. This difference in the complexity of the determinants and minors is the source of main problems that arise in the new setting. (3) Representations The irreducible representations of the standard quantum group GLq (X) are q-deformations of the irreducible representations of ¯ Because the poincare series of GL(X). This is no longer so for GLq (X). ¯ GLq (X) is different from the Poincare series of GL(X) (cf. Proposition 5.4). Thus, unlike in the standard case, the representation theory of the new quantum group does not run parallel to the representation theory of the classical general linear group. But conjecturally each irreducible representation Vα of G = GL(X) has a unique q-analogue that is a possibly reducible repre¯ cf. Section 9 for a basic example which illustrates this sentation of GL(X); phenomenon, and [24] for a detailed treatment. Thus the representations of GL(X) are not lost in a virtual sense in the transition from GL(X) to ¯ GLq (X). (4) Standard bases The main reason for the one-to-one correspondence between representations of GL(X) and of the standard quantum group GLq (X) is that the formulae for the determinants and minors of GLq (X) are very similar to those for GL(X). Hence, the standard monomial basis in terms of minors of X can be quantized [30] to get standard monomial bases for irreducible representations of GLq (X) [15]. Since, the formulae for the ¯ are no longer elementary, construction determinants and minors of GLq (X) ¯ and their standard bases turns of the irreducible representations of GLq (X) out to be a challenge that we have not been able to meet. (5) Explicit presentation: The Hecke algebra Hr (q) has an explicit presentation in terms of generators and relations. These relations are quantizations of the usual relations among the generators of C[Sr ], the degree three relations being the braid relations. This explicit presentation is necessary for the construction of irreducible representations of Hr (q) and their special bases as in [13]. Analogous explicit presentation for Br (q) turns out to be
7
a challenging problem, as an example for r = 4 in Section 13 illustrates. Hence, explicit construction of irreducible representations of Br (q) and their bases turns out to be challenge that we have not been able to meet. The algebraic structure of Br (q) and its combinatorial properties, which may shed light on this, will be studied in more detail in [1].
1.5
Plethysm and subgroup restriction problem
The Kronecker problem is a special case of the following generalized plethysm problem with H = GL(V ) × GL(W ) and G = GL(V ⊗ W ) = GL(X). Problem 1.6 (The plethysm problem) Given partitions λ, µ, π, give an explicit positive formula for the the plethysm constant aπλ,µ . This is the multiplicity of the irreducible representation Vπ (H) of H = GLn (C) in the irreducible representation Vλ (G) of G = GL(Vµ ), where Vµ = Vµ (H) is an irreducible representation H. Here Vλ (G) is considered an H-module via the representation map ρ : H → G = GL(Vµ ). (The generalized plethysm problem) As above, allowing H to be any connected reductive group, and letting µ and π be dominant weights of H. This problem will be addressed in [23]. Specifically, a quantization of the map H → G in Problem 1.6 is constructed there, which specializes to the one in Theorem 1.3, when H = GL(V ) × GL(W ) and G = GL(X). Analogue of Br (q) in this context will also be constructed. Assuming analogue of Conjecture 1.5 for this quantization, it then follows that there exists an explicit positive formula as sought in Problem 1.6. Also addressed in [23] is a more general subgroup restriction problem, where one is explicitly given a polynomial homomorphism H → G, H and G being arbitrary connected reductive groups.
1.6
Organization
The rest of this paper is organized as follows. In Section 2, we recall basic results and notions concerning the standard (Drinfeld-Jimbo) quantum group in the setting of FRT-algebras [32]. In Sections 3-8 we prove Theorem 1.3. In Sections 10-10.3, we prove Proposition 1.4. Sections 9,11,13 describe concrete examples. Section 12 gives a canonical basis of B3 . 8
2
The standard quantum group
In this section, we briefly recall basic notions concerning the standard quantum group; cf. [15, 32, 21]. It can be defined by specifying either its enveloping algebra, as in Drinfeld and Jimbo [4, 7], or its coordinate FRT-algebra, as in [32]. In this paper, we shall adopt the latter view. We shall mostly follow the terminology in [15]. Let V be vector space of dimension n, R = RV,V a linear mapping of ˆ V,V = R ˆ = τ ◦ R, where τ is the flip of V ⊗ V . Let V ⊗ V to itself, and R u be an n × n variable matrix, and Chui the free algebra over the variable entries of u. Then the defining relations of the FRT bialgebra A(R) [32] in the matrix form is ˆ 1 u2 = u1 u2 R, ˆ Ru (6) i.e.,
ˆ ⊗ u) = (u ⊗ u)R, ˆ R(u
(7)
where u1 = u ⊗ I and u2 = I ⊗ u, and I denotes the identity matrix. This is equivalent to: Ru1 u2 = u2 u1 R. (8) Thus,
ˆ 1 u2 − u1 u2 Ri. ˆ A(R) = Chui/hRu
(9)
This is a bialgebra with comultiplication and counit given by X ∆(uij ) = uik ⊗ ukj , ǫ(uij ) = δij , k
or in the matrix form ˙ ∆(u) = u⊗u, and ǫ(u) = I, where · denotes matrix multiplication. The coordinate algebra O(Mq (V )) of the standard quantum matrix space Mq (V ) is the FRT-algebra A(R) for the specific R given by: ji Rmn = q δij δin δjm + (q − q −1 )δim δjn θ(j − i),
where θ(k) is 1 if k > 0 and 0 otherwise. It is known that the corresponding ˆ satisfies the quadratic equation R ˆ − qI)(R ˆ + q −1 I) = 0, (R 9
(10)
and has the spectral decomposition ˆ = qP+ − q −1 P, R
(11)
where the projections P+ = P+V and P− = P−V are P+ =
ˆ + q −1 I R , q + q −1
P− =
ˆ + qI −R , q + q −1
(12)
so that I = P+ P−
(13)
is the spectral decomposition of the identity. These projections are quantum analogues of the symmetrization and antisymmetrization operators on Cl ⊗ Cl , respectively. Specifically, let the symmetric subspace Sq (V ⊗ V ) be the image of P+ , and the antisymmetric space Aq (V ⊗V ) the image of P− . Thus Sq (V ⊗ V ) is defined by the equation P− v1 v2 = 0,
(14)
where v1 = v ⊗ I and v2 = I ⊗ v. In terms of the entries vi ’s of v, this becomes vi vj = qvj vi , i < j. (15) The antisymmetric space Aq (V ⊗ V ) is defined by the equation P+ v1 v2 = 0,
(16)
or equivalently, vi2 = 0, and vi vj = −q −1 vj vi ,
i < j.
(17)
Let Cq [V ] be the algebra over the entries vi ’s of v subject to the relation (15). It will called the quantum symmetric algebra of V . Let ∧q [V ] be the algebra over the entries vi ’s of v subject to the relation (17). It will called the quantum exterior algebra of V . Let Crq [V ] and ∧rq [V ] be the degree r-components of Cq [V ] and ∧q [V ], respectively. Both Cq [V ] and ∧q [V ] are left and right corepresentations of O(Mq (V )). By (12), the defining relation (6) of O(Mq (V )) is equivalent to P+ u1 u2 = u1 u2 P+ ,
(18)
P− u1 u2 = u1 u2 P− .
(19)
or equivalently,
10
These can also be rewritten as: P+ (u ⊗ u) = (u ⊗ u)P+ ,
(20)
P− (u ⊗ u) = (u ⊗ u)P− .
(21)
or equivalently, It can be shown [15] that (20) is equivalent to saying that the defining relation (15) of the quantum symmetric algebra Cq [V ] is preserved by the left and right actions of u on v given by v → uv and vt → vt u. Similarly, (21) is equivalent to saying that the defining relation (17) of the quantum antisymmetric algebra ∧q [V ] is also preserved by the left and right actions of u on v. We think of Cq [V ] as the coordinate algebra of a virtual symmetric quantum space Vsym , isomorphic to V as a vector space, with commuting coordinates (in the quantum sense), and ∧q [V ] as the coordinate algebra of a virtual antisymmetric quantum space V∧ , isomorphic to V as a vector space, with anti-commuting coordinates (in the quantum sense). Thus Mq (V ) is the set of linear transformations of the symmetric quantum space Vsym or the antisymmetric quantum space V∧ , on which each transformation acts from the left and as well as the right. This view of the standard quantum group, emphasized by Manin [21], will be a starting point for the definition ¯ of the new quantum group GLq (X). Let φR,r be the right coaction of O(Mq (V )) on ∧rq [V ]: φR,r : ∧rq [V ] → ∧rq [V ] ⊗ O(Mq (V )), and φL,r the left coaction. Let Ωr be the set of subsets of {1, . . . , n} of size r. For a subset I ∈ Ωr , with I = {i1 , . . . , ir }, i1 < i2 < · · · , let vI = vi1 · · · vir . The left quantum r-minors of O(Mq (V )) are defined to be the matrix coefficients of the left corepresentation map φL,r . Specifically, for I, J ∈ Ωr , the left quantum r-minors DJL,I (V ) are such that φL,r (vI ) =
X
DJL,I (V ) ⊗ vJ .
J
The right quantum r-minors DIR,J (V ) are such that φR,r (vI ) =
X
vJ ⊗ DIR,J (V ).
J
11
Then
DJI = DJL,I (V ) = DJR,I (V ) 6= 0.
(22)
The quantum determinant Dq = Dq (V ) of u is defined to be DJL,I (V ) = DJR,I (V ), with I = J = [1, n]. Explicitly: DJI =
X
i
i
, (−q)l(σ) ujσ(1) · · · ujσ(n) n 1
(23)
σ∈Sn
where l(σ) is the number of inversions in the permutation σ. The coordinate algebra O(GLq (V )) of the quantum group GLq (V ) is obtained by adjoining the inverse Dq (V )−1 to O(Mq (V )). We have a nondegenerate pairing ∧qn−1 [V ] × ∧1q [V ] → ∧nq [V ], where ∧1q [V ] = V is the fundamental vector representation. Its matrix form ˜ so that lets us define the cofactor matrix u ˜ u = u˜ u u = Dq (V )I. Then we can formally define u−1 = Dq (V )−1 u. This gives the following Hopf structure on Oq (GLq (V )): ·
1. ∆(u) = u ⊗ u. 2. ǫ(u) = I. 3. S(uij ) = u ˜ij Dq−1 , S(Dq−1 ) = Dq , where uij are the entries of u and u ˜ij ˜. are the entries of u The Poincare series of O(Mq (V )) coincides with the Poincare series of the commutative algebra C[U ] = C[uij ]. Because, just as in the classical case, O(Mq (V )) has a basis consisting of the standard monomials (u11 )k11 (u12 )k12 · · · (unn )knn , kij being nonnegative integers. To show this [30, 2], the monomials are ordered lexicographically, and the defining equations (20) of O(Mq (V )) are recast in the form of a reduction system: ujk uik ukj uki ujk uil ujl uik
→ → → →
q −1 uik ujk (i < j) q −1 uki ukj (i < j) uil ujk (i < j, k < l) uik ujl − (q − q −1 )uil ujk
12
(24) (i < j, k < l).
Then, by the diamond lemma [15], it suffices to show that all ambiguities in this reduction system are resolvable. This means any term of the form uij ukl urs , when reduced in any way, leads to the same result. This has to be checked for 24 different types of configurations of the three indices (i, j), (k, l), (r, s); see [2, 15, 30] for details.
2.1
Compactness, unitary transformations
What sets the standard quantum group apart from other known deformations [2, 21, 32, 31, 34] of GL(V ) is that it has a real form that is compact. To see what this means, we have to recall the notion of compactness due to Woronowicz in the quantum setting; cf. [36] or Chapter 11 in [15] for details. Let A be the coordinate Hopf algebra of a quantum group Gq . Suppose there is an involution ∗ on A so that it is a Hopf ∗-algebra [15]. We say that ∗ defines a real form of the quantum group Gq . A finite dimensional corepresentation of A on a vector space V with a Hermitian form is called unitary if the matrix v = (vij ) of this corepresentation with respect to an a orthonormal basis {ei } of V satisfies v ∗ v = vv ∗ = I. The algebra A is called a compact matrix group algebra (CMQG) if (1) it is the linear span of all matrix elements of finite-dimensional corepresentations of A, (2) it is generated as an algebra by finitely many elements. Then Theorem 2.1 (Woronowicz [36]; cf. chapter 11 in [15]) (a) A Hopf ∗algebra A is a CMQG algebra iff there is a finite dimensional unitary corepresentation of A whose matrix elements generate A as an algebra. (b) If A is a CMQG algebra then the quantum analogue of the Peter-Weyl theorem holds, and any finite dimensional corepresentation of A is unitarizable, and hence, a direct sum of irreducible corepresentations. There is a unique involution ∗ on the algebra O(GLq (V ) such that = S(uji ). Equipped with this involution, O(GLq (V )) becomes a Hopf ∗-algebra, called denoted by O(Uq (V )), and called the coordinate algebra of the quantum unitary group Uq (V )–which is, again, a virtual object. Furthermore, O(GLq (V )) is a CMQG algebra. (uij )∗
Woronowicz [36] has shown that the usual results for real compact groups, such as Harmonic analysis, existence of orthonormal bases, and so on, generalize to CMQG algebras. 13
2.2
Gelfand-Tsetlin bases and Clebsch-Gordon coefficients
In particular, standard results for the unitary group U (V ) have their analogues for Uq (V ). In this section, we describe results of this kind that we need; cf. [15, 35] for their detailed description. Given a partition λ, let Vλ (V ) denote the corresponding Weyl module of GL(V ), and Vq,λ (V ) denote the corresponding irreducible representation (qWeyl module) of the standard quantum group GLq (V ). Let {|M i} denote the orthonormal Gelfand-Tsetlin basis for Vq,λ (V ), where M ranges over Gelfand-Tsetlin tableau of shape λ. The tensor product of two irreducible representations of GLq (V ) decomposes as Vq,α (V ) ⊗ Vq,β (V ) = ⊕γ,r Vq,γ,r (V ), (25) where r labels different copies of Vq,γ (V )–the number of these copies is the Littlewood-Richardson coefficient cα,β,γ . The Clebsch-Gordon (Wigner) coefficients (CGCs) of this tensor product are defined by the formula X α,β,γ |M ir = CN KM,r |N i ⊗ |Ki, (26) N,K
where N and K range over Gelfand-Tsetlin tableau of shapes α and β, respectively, and |M ir denotes the Gelfand-Tsetlin basis element of Vq,γ,r (V ) in (25) labelled by the Gelfand-Tsetlin tableau M of shape γ. We denote α,β,γ CN KM,r by simply CN KM,r if the shapes are understood in a context. Furthermore, if the multiplicity of Vq,γ (V ) is one in (25), we denote CN KM,r by CN KM . These coefficients have been intensively studied in the literature; cf. [15, 35] and the references therein. An explicit formula for them is known when either Vq,α (V ) or Vq,β (V ) is a fundamental vector representation, or more generally, a symmetric representation. In the presence of multiplicities, the Clebsch-Gordon coefficients are not uniquely determined, and do not have explicit formulae in general. We now briefly recall explicit formulae for the fundamental ClebschGordon coeffficients–i.e., when Vq,α (V ) or Vq,β (V ) is a fundamental representation. Suppose the multiplicity of each irreducible representation in the tensor product decomposition (25) is one, as in the fundamental case. Then each CGC CN KM is a product of so called reduced CGC’s, which depend on the consecutive two rows of N, K, M . The s-reduced CGC is denoted by the 14
symbol
Ns Ks Ms Ns−1 Ks−1 Ms−1
,
where Ns denote the s-th row of the Gelfand-Tsetlin tableau N . Let us assume that Vq,β (V ) is a fundamental representation. Then each (valid) reduced CGC is either of the form:
Ns (1, 0) Ns + ei Ns−1 (0, 0) Ns−1
,
where 0 = (0, . . . , 0) and ei denotes a unit-row with i-th entry one, or of the form
Ns (1, 0) Ns + ei Ns−1 (1, 0) Ns−1 + ej
.
The first CGC is equal to
q
−1/2(i+1+
P
j Nj,s−1 −
P
j6=i Nj,s )
×
Qs−1
j=1 [Nj,s−1
Q
− Ni,s − j + i − 1]
j6=i [Nj,s
− Ni,s − j + i]
!1/2
,
(27)
and the second CGC is equal to ν(j − i)q −(Nj,s−1 −Ni,s −j+i)/2 × Q [Nk,s −Nj,s−1 −k+j] Q k6=i
[Nk,s −Ni,s −k+i]
[Nk,s−1 −Ni,s −k+i−1] 1/2 , k6=j [Nk,s−1 −Nj,s−1 −k+j−1
(28)
where [m] = [m]q = (q m − q −m )/(q − q −1 ) is the q-number, Ni,k are the components of Ni , and ν(j − i) := 1 if j − i ≥ 0 and −1 otherwise.
3
Quantization of GL(V ) × GL(W ) ֒→ GL(V ⊗ W )
Now we turn to Theorem 1.3. Let H = GL(V ) × GL(W ), where V and W are vector spaces of dimension n and m. Let X = V ⊗ W be the fundamental representation of H, and ρ : H ֒→ G = GL(X) = GL(V ⊗ W ) the corresponding homomorphism. The goal is to quantize this homomorphism in the form: ¯ GLq (V ) × GLq (W ) → GLq (X), 15
(29)
¯ where the quantum groups GLq (V ) and GLq (W ) are standard, and GLq (X) is a newly sought quantum group deformation of GL(V ⊗W ) such that (1) it has compact real form and (2) its dimension is the same as that of GL(X). This then can be considered to be a correct quantization of ρ. We shall prove (1), and conjecture (2) (cf. Conjecture 5.5). When we think of the vector space X as a fundamental representation of ¯ defined GLq (V ) × GLq (W )–or as a fundamental representation of GLq (X) ¯ as in eq.(29). later–we denote it by X, ¯ is motivated by the view of the standard The construction of GLq (X) quantum group described in Section 2: 1. First we construct (Section 4) symmetric and antisymmetric quantum ¯ Cq (X) ¯ and ∧q (X), ¯ which can be thought of as coordialgebras of X, ¯ sym and X ¯∧ , respectively. nate algebras of (virtual) quantum spaces X ∗ ¯ When W = V , Xsym is isomorphic to the standard quantum space ¯ sym are the same as (24). Mq (V ); i.e., the defining equations of X ¯ is defined (Section 5) as the space 2. The quantum matrix space Mq (X) ¯ ¯ ∧ ; i.e., so that of linear transformations of Xsym , or equivalently, of X ¯ on X ¯ sym or X ¯ ∧ preserve their the left and right actions of Mq (X) defining equations. 3. It is shown that there is a natural bialgebra homomorphism from ¯ to O(Mq (V )) ⊗ O(Mq (W )); cf. Section 5.2. O(Mq (X)) ¯ r be the degree r component of ∧q [X]. ¯ Then the left and 4. Let ∧q [X] right quantum r-minors are defined as the matrix coefficients of the left ¯ on ∧rq [X], ¯ and their basic properties and right coactions of O(Mq (X)) are proved; cf. Section 6. ¯ are de5. The cofactor and the inverse of a generic matrix u ∈ Mq (X) ¯ fined using determinants and quantum minors, and GLq (X) is defined ¯ as the subset of nonsingular transformations in Mq (X). Formally, ¯ GLq (X) is defined by putting a Hopf structure on the coordinate al¯ gebra O(Mq (X)); cf. Section 7. ¯ 6. A natural ∗-structure is put on O(GLq (X)), and using Woronowicz’ ¯ results [36], it is shown that GLq (X) has a compact real form, which ¯ we shall denote by Uq (X)–this is the analogue of the unitary group in this setting; cf. Section 8.
16
7. Finally, it is shown that Theorem 1.3 follows from these results in conjunction with Woronowicz’s results [36]. In the subsequent sections we address these steps one at a time.
4
Quantum symmetric and antisymmetric algebras
ˆ V,V ⊗ R ˆ W,W be the R-matrix ˆ ¯ = V ⊗ W as ˆ¯ ¯ =R associated with X Let R X,X ˆ a representation of GLq (V ) × GLq (W ). This is different from the R-matrix ˆ X,X obtained by thinking of X = V ⊗ W as a vector representation of the R standard quantum group GLq (X). ˆ V,V and R ˆ W,W are diagonalizable with eigenvalues q and −q −1 . Both R ˆ V,V ⊗ R ˆ W,W is diagonalizable with eigenvalues q 2 , −1, q −2 . ˆ¯ ¯ =R Hence, R X,X In what follows, we assume that q is positive and transcendental. The ¯ ⊗ X) ¯ ⊆X ¯ ⊗X ¯ is defined to be the span quantum symmetric subspace Sq (X 2 of the eigenspaces for the positive eigenvalues q , q −2 (cf. Chapter 8,[15]). ¯ ⊗ X) ¯ ⊆X ¯ ⊗X ¯ is defined to be The quantum antisymmetric subspace Aq (X the eigenspace for the negative eigenvalue −1. Let ˆ ¯ ¯) ˆ ¯ ¯ ) + P− (R I = P+ (R X,X X,X be the spectral decomposition of the identity corresponding to the spectral ˆ ¯ ¯ . In what follows, we let decomposition of the diagonalizable R X,X ¯
P−X ¯ P+X
ˆ ¯ ¯) = P− (R X,X ˆ ¯ ¯ ). = P+ (R X,X
(30)
¯ ¯ ˆ ¯ ¯ . The symmetric subspace Both P+X and P−X are polynomials in R X,X ¯ ¯ ⊗ X) ¯ is the image of P+X and the antisymmetric space Aq (X ¯ ⊗ X) ¯ is Sq (X ¯ ¯ ⊗ X) ¯ is defined by the equation the image of P−X . In other words, Sq (X ¯
P−X x1 x2 = 0,
(31)
where x1 = x ⊗ I and x2 = I ⊗ x, and Aq (X ⊗ X) is defined by the equation ¯
P+X x1 x2 = 0,
(32)
¯ of X ¯ be the algebra over the Let the quantum symmetric algebra Cq [X] entries xi ’s of x subject to relation (31). It will be called the coordinate ring 17
¯ sym , which is only a virtual object. Let the quantum of the quantum space X ¯ ¯ be the algebra over the entries xi ’s of x subject exterior algebra ∧q [X] of X to relation (32). It will called the coordinate ring of the virtual quantum ¯ ∧ . We shall sometimes refer to X ¯ sym and X ¯ ∧ as symmetric and space X antisymmetric quantum spaces (though the meaning of the term symmetric ¯ quantum space here is different from the one in the literature). Let Crq [X] r ¯ ¯ ¯ and ∧q [X] be the degree r components of Cq [X] and ∧q [X], respectively. ˆ V,V , and Let AV be an eigenbasis of Sq (V ⊗ V ) for the eigenvalue q of R −1 BV an eigenbasis for the eigenvalue −q ; AW , BW are defined similarly. ¯ ⊗X ¯ Given c ∈ V ⊗ V and W ⊗ W , we define their c∗d ∈ X Pd ∈ P restitution 1 2 k 1 2 k as follows. Let c = i ci ⊗ ci , ci ∈ V , and d = j dj ⊗ dj , dj ∈ W . Then X (33) c∗d= (c1i ⊗ d1j ) ⊗ (c2i ⊗ d2j ). i,j
¯ ⊗ X) ¯ and Aq (X ¯ ⊗ X): ¯ Then the following are eigenbases of Sq (X ¯ ⊗ X) ¯ : Sq (X
{a ∗ a′ | a ∈ AV , a′ ∈ AW } ∪ {b ∗ b′ | b ∈ BV , b′ ∈ BW }.
¯ ⊗ X) ¯ : {a ∗ b′ | a ∈ AV , b′ ∈ BW } ∪ {b ∗ a′ | b ∈ BV , a′ ∈ AW }. Aq (X (34) An explicit form of the defining relation (31) for the symmetric quantum ¯ is obtained by setting the eigenbasis of Aq (X ¯ ⊗ X) ¯ to zero: algebra Cq [X] a ∗ b′ = 0 a ∈ AV , b′ ∈ BW , b ∗ a′ = 0 b ∈ BV , a′ ∈ AW .
(35)
An explicit form of the defining relation (32) for the antisymmetric quantum ¯ is obtained by setting the eigenbasis of Sq (X ¯ ⊗ X) ¯ to zero: algebra ∧q [X] a ∗ a′ = 0 a ∈ AV , a′ ∈ aW , b ∗ b′ = 0 b ∈ BV , b′ ∈ BW .
(36)
Example Let dim(V ) = dim(W ) = 2, V = hv1 , v2 i, W = hw1 , w2 i. Then AV = {A11, A22, B12} and BV = {B12}, where, deleting the ⊗ sign and letting p = 1/q, A11(V ) = v1 v1 A22(V ) = v2 v2 (37) A12(V ) = v1 v2 + pv2 v1 B12(V ) = v1 v2 − qv2 v1 18
¯ The following is a basis AW and BW are similar. Let xij = vi ⊗ wj ∈ X. ¯ ¯ of the symmetric space Sq (X ⊗ X): a11 := A11(V ) ∗ A11(W ) a22 := A11(V ) ∗ A22(W ) a12 := A11(V ) ∗ A12(W ) a33 := A22(V ) ∗ A11(W ) a44 := A22(V ) ∗ A22(W ) a34 := A22(V ) ∗ A12(W ) a13 := A12(V ) ∗ A11(W ) a24 := A12(V ) ∗ A22(W ) a14 := A12(V ) ∗ A12(W ) a23 := B12(V ) ∗ B12(W )
= x11 x11 = x12 x12 = x11 x12 + px12 x11 = x21 x21 = x22 x22 = x21 x22 + px22 x21 = x11 x21 + px21 x11 = x12 x22 + px22 x12 = x11 x22 + px12 x21 + px21 x12 + p2 x22 x11 = x11 x22 − qx12 x21 − qx21 x12 + q 2 x22 x11 (38) ¯ ¯ The following is a basis of the antisymmetric space Aq (X ⊗ X):
b12 := A11(V ) ∗ B12(W ) b34 := A22(V ) ∗ B12(W ) b13 := B12(V ) ∗ A11(W ) b24 := B12(V ) ∗ A22(W ) b14 := A12(V ) ∗ B12(W ) b23 := B12(V ) ∗ A12(W ) Proposition 4.1
= x11 x12 − qx12 x11 = x21 x22 − qx22 x21 = x11 x21 − qx21 x11 = x12 x22 − qx22 x12 = x11 x22 − qx12 x21 + px21 x12 − x22 x11 = x11 x22 + px12 x21 − qx21 x12 − x22 x11
(39)
Q kij ¯ is { 1. A basis of Cq [X] i,j xij | ki,j ∈ Z≥0 }.
¯ is {Q xkij | ki,j ∈ {0, 1}}. 2. A basis of ∧q [X] i,j ij
¯ and ∧q [X]. ¯ These bases will be called standard monomial bases of Cq [X]
Proof: (1) The relations (35) (cf. (39)) can be reformulated in the form of the following reduction system: xjk xik xkj xki xjk xil xjl xik
→ → → →
q −1 xik xjk (i < j) q −1 xki xkj (i < j) xil xjk (i < j, k < l) xik xjl − (q − q −1 )xil xjk
(40) (i < j, k < l).
When dim(V ) = dim(W ), these coincide with the defining relations (24) for the standard quantum matrix space Mq (V ) after the change of variables 19
¯ sym ∼ xij → uij . In other words, X = Mq (V ). In this case, the ambiguities in this reduction system can be resolved just as in the case of the reduction system for O(Mq (V )) [30, 2]; cf. Section 2. This is also so when dim(V ) 6= dim(W ). Hence the result follows from the diamond lemma [15]. (2) The relations (36) can be reformulated in the form of the following reduction system: x2ij → 0 xjk xik xkj xki xjl xik xjk xil
→ → → →
−qxik xjk (i < j) −qxki xkj (i < j) −xik xjl (i < j, k < l) −xil xjk + (q −1 − q)xik xjl
(41) (i < j, k < l).
Ambiguities in this reduction system can also be resolved just as in (1); we omit the details. So the result again follows from the diamond lemma [15]. Q.E.D. For future reference, we note down a corollary of the proof above. Let ˆ Y,Y = R ˆ V ∗ ,V ∗ ⊗ R ˆ V,V be an R ˆ matrix Y = End(V, V ) = V ∗ ⊗ V . Let R associated with Y . It is diagonalizable. Let I = P−Y + P+Y be the associated spectral decomposition of the identity, where P−Y and P+Y ˆ Y,Y for the eigenvalues with denote the projections onto the eigenspaces of R sign − and +, respectively. Proposition 4.2 Eq.(18) or eq.(19) defining O(Mq (V )) is equivalent to the relation P−Y (u ⊗ u) = 0. (42) At q = 1, this relation simply says that the entries of u commute with each other. So this is a quantized version of commutativity. Proof: See the proof of Proposition 4.1 (1), and the remark therein. Q.E.D. Proposition 4.3 (1) As a GLq (V ) × GLq (W )-module X ¯ = Symd (V ⊗ W ) ∼ Vq,λ (V ) ⊗ Vq,λ (W ), Cdq [X] = q
(43)
λ
where λ ranges over all Young diagrams of size d, with at most dim(V ) or dim(W ) rows. 20
(2) Similarly, ¯ = ∧dq (V ⊗ W ) ∼ ∧dq [X] = ⊕λ Vq,λ (V ) ⊗ Vq,λ′ (W ),
(44)
where λ ranges over all Young diagrams of size d, with at most dim(V ) rows and at most dim(W ) columns, and λ′ denotes the conjugate of λ obtained by interchanging rows and columns. ¯ as a GLq (V ) × Proof: (1) By Proposition 4.1, the character of Cdq [X] GLq (W )-module coincides with the character of Cd [X] as a GL(V )×GL(W )module. Since, finite dimensional representations of the GLq (V ) × GLq (W ) are completely reducible, the irreducible representations of GLq (V )×GLq (W ) are in one-to-one correspondence with the those of GL(V ) × GL(W ), and the characters of the corresponding representations coincide [15], (1) follows from the the classical result Cd [X] = Symd (V ⊗ W ) ∼ = ⊕λ Vλ (V ) ⊗ Vλ (W ).
(45)
Similarly (2) is a q-analogue of the classical result ∧d (V ⊗ W ) ∼ = ⊕λ Vλ (V ) ⊗ Vλ′ (W ),
(46)
Q.E.D. ¯ is isomorphic to the coordinate algeWe have already noted that Cdq [X] bra O(Mq (V )) of the standard quantum matrix space, when W = V ∗ . In this case, Proposition 4.3 (1) is the q-analogue of the Peter-Weyl theorem for the standard quantum group GLq (V ), and Proposition 4.3 (2) is the q-analogue of the antisymmetric form of the Peter-Weyl theorem.
4.1
Explicit product formulae
We wish to give explicit formulae for products in the symmetric and anti¯ and ∧q [X]. ¯ symmetric algebras Cq [X] Let
˜ s¯ = ∪λ {|Mλ i ⊗ |Nλ i} B X
(47)
¯ as per the decomposition in Propobe the Gelfand-Tsetlin basis for Cdq [X] sition 4.3 (1) and ˜ ∧¯ = ∪λ {|Mλ i ⊗ |Nλ′ i} (48) B X ¯ as per Proposition 4.3 (2). that for ∧dq [X] 21
When W = V ∗ , the basis element |M i ⊗ |N i ∈ Vq,λ (V ) ⊗ Vq,λ (V ∗ ) ⊆ ¯ stands for the matrix coefficient uλ Cq [X] N M of the representation Vq,λ (V ) of the standard quantum group GLq (V ). It is of interest to know explicit transformation matrices connecting the ¯ with their standard monomial ¯ and ∧d (X) Gelfand-Tsetlin bases of Cdq [X] q bases in Proposition 4.1. In other words, we want to know the decompositions in Proposition 4.3 (1) and (2) explicitly. When W = V ∗ , this amounts to finding explicit formulae for the matrix coefficients of irreducible representations of GLq (V ). This problem has been studied intensively in the literature. When dim(V ) = 2, explicit formulae for matrix coefficients in terms little q-Jacobi polynomials are known. In general, the problem is not completely understood at present; see the survey [35] and the references therein. ¯ ∧q [X], ¯ The advantage of working with the Gelfand-Tsetlin bases of Cq [X], instead of the standard monomial bases in Proposition 4.1 is that multiplication is simpler in terms of the former, and have explicit formulae in terms of Clebsch-Gordon coefficients. We shall now state these formulae, assuming for the sake of simplicity, the multiplicity free case–which is enough for the purposes of this paper. That is, we shall assume that the multiplicity of each Vq,γ (V ) in the tensor product decomposition (25) for α’s and β’s under consideration is one. When W = V ∗ , we have the following multiplication formula for matrix coefficients; cf. [15, 35]: uαN R uβKS =
X
α,β,γ γ α,β,γ CN KM CRSL uM L .
(49)
γ,M,L
where α, β, γ are as in the decomposition (25), which is assumed to be multiplicity free. We also have the following identity: X α,β,γ α,β,γ α β uγM L = CN KM CRSL uN R uKS .
(50)
N,K,R,S
Multiplication formula for the Gelfnad-Tsetlin basis (47) is similar. Let α, β be Young diagrams of height less than dim(V ) and dim(W ), and γ be a Young diagram in the decomposition (25), which is assumed to be multiplicity free. Then X CNα Kβ Mγ ,r CRα Sβ Lγ |Mγ i ⊗ |Lγ i. (51) (|Nα i ⊗ |Rα i)(|Kβ i ⊗ |Sβ i) = γ,Mγ ,Lγ
22
Also X
|Mγ i ⊗ |Lγ i =
CNα Kβ Mγ CRα Sβ Lγ (|Nα i ⊗ |Rα i)(|Kβ i ⊗ |Sβ i).
Nα ,Kβ ,Rα ,Sβ
(52)
Multiplication formula for the Gelfand-Tsetlin basis (48) is also similar. Let α, β, γ be Young diagrams of height at most dim(V ) and width at most dim(W ), and α′ , β ′ , γ ′ their conjugates. Assume that the multiplicity of γ in the decomposition (25) is one –i.e. the Littlewood-Richardson coefficient cα,β,γ is one–and also that cα′ ,β ′ ,γ ′ is one. Then, (|Nα i⊗|Rα′ i)(|Kβ i⊗|Sβ ′ i) =
X
CNα Kβ Mγ CRα′ Sβ ′ Lγ′ |Mγ i⊗|Lγ ′ i. (53)
γ,Mγ ,Lγ
Also |Mγ i ⊗ |Lγ ′ i =
X
CNα Kβ Mγ CRα′ Sβ ′ Lγ′ (|Nα i ⊗ |Rα′ i)(|Kβ i ⊗ |Sβ ′ i),
Nα ,Kβ ,Rα′ ,Sβ ′
5
(54)
Quantum matrix space ¯
Let u = uX be a variable matrix, specifying a generic transformation from ¯ to X. ¯ Let Chui denote the free algebra over the variable entries of u. X ¯ of the virtual quantum We define the coordinate algebra O(Mq (X)) ¯ to be the quotient of Chui modulo the relations space Mq (X) ¯
¯
P+X (u ⊗ u) = (u ⊗ u)P+X . ¯
(55)
¯
Since I = P−X + P+X , these are equivalent to the relations ¯
¯
P−X (u ⊗ u) = (u ⊗ u)P−X .
(56) ¯
¯ is an FRT-algebra [32], with a singular P X playing Thus O(Mq (X)) − the role of an R-matrix in eq.(6). But, as we shall see below, it is not coquasitriangular, and hence the main theory of FRT-algbras [32] does not ¯ apply to O(Mq (X)).
23
Proposition 5.1 1. The relation (55) is equivalent to the relations on the entries of u which say that the defining equation (31) of the quan¯ sym is preserved by the left and right actions tum symmetric space X of u on x given by x → ux and xt → xt u. This means the quan¯ is a left and right comodule-algebra of tum symmetric algebra Cq [X] ¯ O(Mq (X). 2. The relation (56) is equivalent to the relations on the entries of u which say that the defining equation (32) of the quantum antisymmetric space ¯ ∧ is preserved by the left and right actions of u on x. This means X ¯ is a left and right comodule the quantum antisymmetric algebra ∧q [X] ¯ algebra of O(Mq (X)). Proof: Left to the reader. Q.E.D. ¯ can be In view of this proposition, the quantum matrix space Mq (X) ¯ sym , or equivalently, of viewed as the space of linear transformations of X ¯ ¯ ¯ sym or X ¯ ∧ preserve X∧ ; i.e., so that the left and right actions of Mq (X) on X their defining equations. This definition is akin to that of the standard quan¯ sym tum matrix space Mq (X), but with the quantum symmetric space X playing the role of the standard quantum symmetric space Xsym and the ¯ ∧ playing the role of the standard quantum quantum antisymmetric space X antisymmetric space X∧ . Remark 1:
The relations (55) and (56) are equivalent to the relations ˆ ¯ ¯ (a, b), SˆX, ¯ X ¯ (a, b)(u ⊗ u) = (u ⊗ u)S X,X where
(57)
¯ ¯ X X SˆX, ¯ X ¯ (a, b) = aP+ + bP− ,
¯ is an FRT-algebra for any constants a, b, not both zero. Thus O(Mq (X)) ˆ ¯ ¯ (a, b), where τ denotes the with the R-matrix being SX, ¯ (a, b) = τ ◦ S ¯ X X,X ¯ ⊗d , flip operator. Fix a, b, and let S = SX, ¯ (a, b). Given a tensor product X ¯ X let Si denote the transformation which acts like S on the i-th and (i + 1)st factors, the other factors remaining unaffected. Then it can be shown that Si ’s do not satisfy the Quantum Yang Baxter Equations (QYBE)– equivalently Sˆi ’s do not satisfy the braid identities; cf. Section 11. Remark 2: ˆ ¯ ¯ . One can also consider an FRT-algebra A(R ¯ ¯ ) Let RX, ¯ X ¯ =τ ◦R X,X X,X ¯ associated with the R-matrix RX, ¯ X ¯ . It is not isomorphic to O(Mq (X)), 24
¯ ˆ ¯ ¯ . Furthermore, it can be shown that the since, P+X is nonlinear in R X,X dimension of the quantum group associated with A(RX, ¯ X ¯ ) is much smaller than the classical dimension of GL(X). Hence, it cannot be considered as a deformation of GL(X). ˆ ˆ ¯ ∗ ¯ ∗ be an R ¯ X) ¯ =X ¯ ∗ ⊗ X. ¯ Let R ˆ U,U = R ˆ¯ ¯ ⊗R Let U = End(X, X ,X X,X matrix associated with U . It is diagonalizable. Let
I = P−U + P+U be the associated spectral decomposition of the identity, where P−U and P+U ˆ U,U for the eigenvalues with denote the projections onto the eigenspaces of R sign − and +, respectively. The following is an analogue of Proposition 4.2. Proposition 5.2 Eq.(55) or eq.(56) is equivalent to the relation P−U (u ⊗ u) = 0.
(58)
Proof: Left to the reader. Q.E.D. When q = 1 this relation says that uij ’s commute. Thus it expresses the q-analogue of commutativity in the present context. ¯ ⊗ X) ¯ and Aq (X ¯ ⊗ X), ¯ respectively; let Let AX¯ , BX¯ be bases of Sq (X ∗ ∗ ¯ ⊗X ¯ and d ∈ X ¯ ⊗ X, ¯ we AX¯ ∗ and BX¯ ∗ be defined similarly. Given c ∈ X ∗ ∗ ¯ ¯ ¯ ¯ define the restitution c ∗ d ∈ (X ⊗ X) ⊗ (X ⊗ X) = U ⊗ U very much as in (33). Then an eigenbasis of P−U in U ⊗ U is given by {a ∗ b | a ∈ AX¯ ∗ , b ∈ BX¯ } ∪ {b ∗ a | b ∈ BX¯ ∗ , a ∈ AX¯ }. Hence, the following is an explicit form of the defining relation in Proposi¯ tion 5.2 for O(Mq (X)): a∗b=0
a ∈ AX¯ ∗ , b ∈ BX¯ (59)
b∗a=0
b ∈ BX¯ ∗ , a ∈ AX¯ .
¯ is a bialgebra such that Proposition 5.3 O(Mq (X)) ¯
¯ ·
¯
¯
∆(uX ) = uX ⊗ uX , and ǫ(uX ) = I. Here ⊗ denotes tensor product and · denotes matrix multiplication. Proof: Follows from Proposition 9.1 in [15] applied to the FRT-algebra ¯ = A(S ¯ ¯ (a, b)). Q.E.D. O(Mq (X)) X,X 25
5.1
Example
Let dim(V ) = dim(W ) = 2. Let AX¯ = {a11, a22, a12, a33, a44, a34, a13, a24, a14, a23}, and BX¯ = {b12, b34, b13, b24, b14, b23}, where aij and bkl are as in eq.(38) and (39). We shall identify AX¯ ∗ and BX¯ ∗ with AX¯ and BX¯ , respectively. We use the notation zi1 i2 ,k1 k2 = xi1 i2 ∗ xk1 k2 , ¯ are as in eq.(38) and (39). We also drop ⊗ symbol. So that where xij ∈ X xi1 i2 xj1 j2 actually means xi1 i2 ⊗ xj1 j2 . For example, z12,11 z21,22 = (x12 x21 ) ∗ (x11 x22 ). ¯ are now 120 in number. We The defining relations (59) of O(Mq (X)) show one such typical relation below: 0 = a14 ∗ b14 = (x11 x22 + px12 x21 + px21 x12 + p2 x22 x11 )∗ (x11 x22 − qx12 x21 + px21 x12 − x22 x11 ) = z11,11 z22,22 + pz12,11 z21,22 + pz21,11 z12,22 + p2 z22,11 z11,22 −qz11,12 z22,21 − z12,12 z21,21 − z21,12 z12,21 − pz22,12 z11,21 +pz11,21 z22,12 + p2 z12,21 z21,12 + p2 z21,21 z12,12 + p3 z22,21 z11,12 −z11,22 z22,11 − pz12,22 z21,11 − pz21,22 z12,11 − p2 z22,22 z11,11 .
(60)
All these 120 relations together, after taking appropriate linear combinations, can be recast in the form of a reduction system–just as the relations (35) were recast in the form of a reduction system (40)–where each reduction rule is of the form X zA zA′ = α(A, A′ , B, B ′ )zB zB ′ , B,B ′
each zB zB ′ being standard; i.e., B > B ′ (say, lexicographically). The resulting reduction system is described in Appendix. It turns out that this system does not satisfy the diamond property. For example, the monomial mm = z1111 z1112 z1221 , when reduced in two different ways, yields the following two distinct standard expressions: 26
q3 − q q2 − 1 ·z z z + ·z1212 z1121 z1111 + 1211 1122 1111 1 + q2 1 + q2 1 − q2 q 2q 2 · z1221 z1112 z1111 + · z1222 z1111 z1111 , 1 + q2 1 + q2
l121 = (−1+q 2 )·z1211 z1121 z1112 +
and
l212 =
q 6 + q 4 − 3q 2 + 1 2q 3 − 2q · z1211 z1122 z1111 · z z z + 1211 1121 1112 q 2 (1 + q 2 ) (1 + q 2 )2
4q 2 + q 4 − 1 (1 + q 2 )2
·z1221 z1112 z1111 +
2q 2 − 2 (1 + q 2 )2
·[z1212 z1121 z1111 − q ·z1222 z1111 z1111 ].
This means we have the following nontrivial relation among standard monomials: l121 − l212 = 0. See Appendix for the details. The example above has the following consequence: ¯ does not, in general, Proposition 5.4 The Poincare series of O(Mq (X)) ¯ (at q = 1). coincide with the classical Poincare series of O(M (X)) ¯ we mean the series Here by Poincare series of O(Mq (X)) X ¯ d )td , dim(O(Mq (X)) d
¯ d denotes the degree d component of O(Mq (X)). ¯ where O(Mq (X)) As an example, when dim(V ) = dim(W ) = 2, using computer it was found that ¯ 3 ) = 688, whereas the classical dim(O(M (X))3 ) = 816. dim(O(Mq (X)) But we have: ¯ as a noncommutative algebraic Conjecture 5.5 The dimension of Mq (X), ¯ variety, as determined from the Poicare series of O(Mq (X)), is the same as 2 dim(X) , the dimension of the algebraic variety M (X).
27
¯ is the This would imply that dimension of the new quantum group GLq (X) same as the dimension of the classical GL(X). Proposition 5.4 has important consequences. In the standard case, the Poincare series of O(Mq (V )) coincides with the classical Poincare series of O(M (V ) (at q = 1). By the Peter-Weyl theorem, this implies that the dimensions of the irreducible representations of GLq (V ) are in one-to-one correspondence with the dimensions of the irreducible representations of GL(V ). Intuitively, this is why the irreducible representations of GLq (V ) turn out to be deformations of the irreducible representations of GL(V ). Proposition 5.4 implies that this would no longer be so for the new quantum group.
5.2
Homomorphism
Let O(Mq (V )) and O(Mq (W )) be coordinate algebras of the standard quantum matrix spaces Mq (V ) and Mq (W ), respectively (Section 2). ¯ to Proposition 5.6 There is a natural homomorphism ψ from O(Mq (X)) O(Mq (V )) ⊗ O(Mq (W )). ¯
Proof: In the matrix form the homomorphism ψ is u = uX → uV ⊗ uW . ¯ One has to check that the relations obtained by substituting uX = uV ⊗ uW ¯ are implied by the equations defining O(Mq (V )) in (55) defining O(Mq (X)) and O(Mq (W )). The defining equation (20) of O(Mq (V )) is P+V (uV ⊗ uV ) = (uV ⊗ uV )P+V ,
(61)
which is equivalent to (21); i.e., P−V (uV ⊗ uV ) = (uV ⊗ uV )P−V .
(62)
Similarly, the defining relation of O(Mq (W )) is P+W (uW ⊗ uW ) = (uW ⊗ uW )P+W ,
(63)
P−W (uW ⊗ uW ) = (uW ⊗ uW )P−W .
(64)
or equivalently, ¯
Since P−X = P−V ⊗ P+W + P+V ⊗ P−W , these relations imply (55) when ¯ uX = uV ⊗ uW . To show that ψ is a bialgebra homomorphism, one has to verify that 28
1. ψ(ab) = ψ(a)ψ(b). 2. ∆ ◦ ψ = (ψ ⊗ ψ) ◦ ∆. 3. ǫ = ǫ ◦ ψ. This is easy. Q.E.D.
6
Quantum determinant and minors
Let Ωr be the set of subsets of {1, . . . , N = nm} of size r, n = dim(V ), m = dim(W ). We write any I ∈ Ωk as I = {i1 , . . . , ir }, i1 < · · · < ik . We let xI be the monomial xi1 · · · xir . By Proposition 4.1, {xI } is the basis ¯ ¯ In particular, ∧nm of ∧q [X]. q [X] is a one-dimensional corepresentation of ¯ O(Mq (X)) with basis vector x1 · · · xN . ¯ is a left and right comodule algebra Proposition 5.1(2) says that ∧q [X] ¯ of O(Mq (X)). Let ¯ → O(Mq (X)) ¯ ⊗ ∧q [X] ¯ φL : ∧q [X] φL,r :
¯ ∧rq [X]
¯ ¯ ⊗ ∧r [X] → O(Mq (X)) q
(65)
¯ and ∧rq [X]. ¯ The right be the maps defining the left corepresentations ∧q [X] corepresentation maps φR and φR,r are similar. We define the left determi¯ to be the matrix coefficient of the left comodule ∧nm ¯ nant DqL = DqL (X) q [X]. R R ¯ The right determinant Dq = Dq (X) is the matrix coefficient of the right conm ¯ nm [X] ¯ module ∧nm q [X]. Since ∧q [X] is one dimensional like the classical ∧ (Proposition 4.1), both DqL and DqR are nonzero. ¯ in the standard More generally, the left quantum r-minors of Mq (X) monomial basis are defined to be the matrix coefficients of the left corepresentation map φL,r in the standard monomial basis (cf. Proposition 4.1) ¯ Specifically, for I, J ∈ Ωr , we define the left quantum r-minors of ∧rq [X]. L,I ¯ DJ (X) such that X L,I ¯ ⊗ xJ . φL,r (xI ) = DJ (X) J
¯ are such that The right quantum r-minors DIR,J (X) X ¯ φR,r (xI ) = xJ ⊗ DIR,J (X). J
29
¯ in We can similarly define left and right quantum minors of Mq (X) the Gelfand-Tsetlin basis to be the matrix coefficients of the left and right corepresentation maps φL,r and φR,r in the Gelfand-Tsetlin basis (48) of ˜ ∧¯ , with α and β being ¯ Specifically, for |Mα i ⊗ |Nα′ i, |Mβ i ⊗ |Nβ ′ i ∈ B ∧rq [X]. X L,Mα ,Nα′ ¯ young diagrams of size r, we define the left quantum r-minors D (X) Mβ ,Nβ ′
so that φL,r (|Mα i ⊗ |Nα′ i) =
X
L,M ,Nα′
α DMβ ,N
β′
¯ ⊗ |Mβ i ⊗ |Nβ ′ i. (X)
Mβ ,Mβ ′ R,M ,Nα′
The right quantum r-minors DMβ ,Nα
β′
¯ are defined similarly. (X)
The quantum determinants DqL and DqR are the same whether we use the Gelfand-Tsetlin or the standard monomial basis.
6.1
Explicit formulae
We wish to give explicit formulae for quantum determinants DqL and DqR and, more generally, the left and right quantum minors in the Gelfand-Tsetlin basis. This is possible because of the explicit formulae for multiplication in the Gelfand-Tsetlin basis (Section 4.1). We do not have similar formulae in the standard monomial basis. 6.1.1
Example
Let us first give an explicit formula for DqL and DqR when dim(V ) = dim(W ) = ¯ has a basis {yI = yi · · · yir }, i1 < i2 · · · , where 2. Then ∧rq [X] 1 y1 = x11 , y2 = x12 , y3 = x21 , y4 = x22 , satisfying the relations a11, a22, a12, a33, a44, a34, a13, a24, a14, a23 = 0; P ¯ ⊗d ¯ = cf. (38) and Proposition 4.1. Let I∧ denote the ideal in T (X) dX generated by these relations. The last two relations imply that y4 y1 = −y1 y4 . Since y1 and y4 quasicommute with all yi ’s and yi2 = 0, for allQi, it is easy to show that yi yj yk yl is zero modulo I∧ , unless it is of the form yσ(i) , for some permutation σ, or is either y2 y3 y2 y3 or y3 y2 y3 y2 . Furthermore, we have Y yσ(i) = (−1)l(σ) q r(σ) y1 y2 y3 y4 , 30
where l(σ) is the number of inversions in σ, and r(σ) is the number of inversions in σ not involving (2, 3) or (1, 4). Also y2 y3 y2 y3 = (p − q)q 2 y1 y2 y3 y4 y3 y2 y3 y2 = (q − p)q 2 y1 y2 y3 y4 . ¯ considThe left determinant Dq,L is the the matrix coefficient of ∧4q (X), ered as a left comodule, and the right determinant Dq,R is the the matrix ¯ considered as a right comodule. From the preceding coefficient of ∧4q (X), remarks, it easily follows that X Dq,L = ( (−1)l(σ) q r(σ) uiσ(i) )+(p−q)q 2 u12 u23 u32 u43 +(q−p)q 2 u13 u22 u33 u42 . σ
The expression for Dq,R is similar. Compare this with the formula (23) for the standard quantum determinant. More generally, the following result gives an explicit formula in terms of the fundamental Clebch-Gordan (Wigner) coefficients (Section 4.1) for ¯ (in the Gelfandexpanding the left or the right quantum minor of Mq (X) Tsetlin basis) by row or column. Proposition 6.1 L,Mγ ,L ′ ¯ DEδ ,F ′ γ (X) Pδ = Nα ,Kβ ,R ′ ,S ′ CNα Kβ Mγ CRα′ Sβ ′ Lγ′ α
P
β
Aµ ,Bµ′ ,Cλ ,Dλ′
L,N ,Rα′
DAµ ,Bα
µ′
L,Kβ ,Sβ ′
¯ (X)D Cλ ,D
λ′
¯ Aµ C ,E CB ′ D ′ F ′ , (X)C λ δ µ λ δ
where CNα Kβ Mγ ’s etc. denote fundamental Clebsch-Gordon coefficients; cf. (27) and (28). A formula for the right minor is similar. ¯ Proof: Using (53) and (54) we calculate, dropping the symbol X:
31
φL,r (|Mγ i ⊗ |Lγ ′ i) P = φL,r ( Nα ,Kβ ,R
α′ ,Sβ ′
=
P
Nα ,Kβ ,Rα′ ,Sβ ′
CNα Kβ Mγ CRα′ Sβ ′ Lγ′ (|Nα i ⊗ |Rα′ i)(|Kβ i ⊗ |Sβ ′ i))
CNα Kβ Mγ CRα′ Sβ ′ Lγ′
(|Aµ i ⊗ |Bµ′ i)(|Cλ i ⊗ |Dλ′ i)
=
P
Nα ,Kβ ,Rα′ ,Sβ ′ L,N ,Rα′
DAµ ,Bα
µ′
=
P
P
Eδ ,Fδ′
µ′
λ′
L,N ,Rα′
DAµ ,Bα
µ′
L,Kβ ,Sβ ′
DCλ ,D
λ′
Aµ ,Bµ′ ,Cλ ,Dλ′
⊗
Nα ,Kβ ,Rα′ ,Sβ ′
L,N ,Rα′
DAµ ,Bα
L,Kβ ,Sβ ′
DCλ ,D
P
Aµ ,Bµ′ ,Cλ ,Dλ′
CAµ Cλ ,Eδ CBµ′ Dλ′ Fδ′ |Eδ i ⊗ |Fδ′ i
P
Eδ ,Fδ′ (
CNα Kβ Mγ CRα′ Sβ ′ Lγ′
P
L,Kβ ,Sβ ′
DCλ ,D
λ′
CNα Kβ Mγ CRα′ Sβ ′ Lγ′
P
Aµ ,Bµ′ ,Cλ ,Dλ′
CAµ Cλ ,Eδ CBµ′ Dλ′ Fδ′ ) ⊗ |Eδ i ⊗ |Fδ′ i.
Now the result follows from the definition of a left minor. Q.E.D. We can also give an analogue of a general Laplace expansion in this context. But that would require general Clebsch-Gordon coefficients for which no explicit expressions are known so far.
6.2
Symmetry of the determinants and minors
The following property of determinants is needed later for defining the real ¯ of the new quantum group GLq (X) ¯ (Cf. Proposi(unitary) form Uq (X) tion 8.1). ¯ More generally, ¯ = D R (X). Proposition 6.2 DqL (X) q ¯ = D R,J (X) ¯ DIL,J (X) I and
L,Mγ ,Lγ ′
DEδ ,F
δ′
R,Mγ ,Lγ ′
¯ =D (X) Eδ ,F
32
δ′
¯ (X).
⊗
¯ = DqL (X) ¯ = This allows us to define the quantum determinant Dq = Dq (X) R ¯ and the quantum minors Dq (X), ¯ = DL,J (X) ¯ = D R,J (X) ¯ DIJ (X) I I and
Mγ ,Lγ ′
DEδ ,F
δ′
L,Mγ ,Lγ ′
¯ =D (X) Eδ ,F
δ′
R,Mγ ,Lγ ′
¯ (X)D Eδ ,F
δ′
¯ (X).
By a standard property of matrix coefficients (cf. Proposition 1.13 in [15]), we also have that X I ¯ ¯ = ¯ ¯ = δIJ , I, J ∈ Ωr , ∆(DJI (X)) DK (X) ⊗ DJK (X), ǫ(DJI (X)) K
and a similar property for the minors in the Gelfand-Tsetlin basis. In particular, ¯ = Dq (X) ¯ ⊗ Dq (X), ¯ ¯ = 1. ∆(Dq (X)) ǫ(Dq (X)) (66) ¯ Thus the quantum determinant is a nonzero group like element of O(Mq (X)). The rest of this section is devoted to the proof of Proposition 6.2. The proof is technical and may be skipped on the first reading. The main problem that needs to be addressed is the following. In the standard case, the right determinant, when rewritten using the reduction system (24), coincides with the left determinant. Analogously, one may wish to prove that the explicit expressions for the left and right determinants (minors) in Proposition 6.1 are equal by reducing them with respect to the reduction system correspond¯ ing to the relations (59) defining O(Mq (X)). But this reduction system does not satisfy the diamond lemma, and the standard monomials do not form a ¯ cf. Section 5.1. Hence, the expressions for the left and basis of O(Mq (X); right minors in Proposition 6.1 need not reduce to the same expressions. So this strategy does not work. We start with a few general observations. Given a coalgebra A, and a right corepresentation φ : V → V ⊗ A given by X φ(ej ) = ek ⊗ vkj , (67) k
there is a left corepresentation φR : V → A ⊗ V given by X φL (ek ) = vkj ⊗ ej .
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j
We shall denote this dual left corepresentation to V by VL . The two corepresentations V and VL share the same matrix [vkj ] of coefficients, which we 33
shall denote by M (V ) = M (VL ). Furthermore, the span C(φ) =< vkj > of the matrix coefficients has a left and right A-coaction, and hence, is a A-bicomodule. Similarly, given a left corepresentation W → A ⊗ W : X τ (ek ) = wkj ⊗ ej ,
(69)
j
there is a corresponding right corepresentation WR : X τR (ej ) = ek ⊗ wkj ,
(70)
k
with the same matrix M (W ) = M (WR ) = [wkj ] of coefficients. Furthermore, C(τ ) = C(τR ) =< wkj > has the left and right coaction, and hence, is a Abicomodule. ¯ ¯ Now we return to the case A = O(Mq (X)). Let ∧d,L q (X) denote the left d,L ¯ ¯ We define ∧d,R ¯ ¯ on the space ∧d [X]. corepresentation of O(Mq (X)) q (X), Cq (X) q d,L ¯ and Cd,R and M∧d,R denote the coefficient matrices q (X) similarly. Let M∧ d,R ¯ d,L ¯ of ∧q (X) and ∧q (X) in the standard monomial bases (Proposition 4.1), ˜ ∧d,L and M ˜ ∧d,R the coefficient matrices in the Gelfand-Tsetlin basis and M d,L d,R d,L d,R ˜ sym ˜ sym (48). The coefficient matrices Msym and Msym , and M and M are defined similarly. Proposition 6.2 follows from the following: ˜ ∧d,L = M ˜ ∧d,R . Lemma 6.3 For every d, M∧d,L = M∧d,R , and M d,R ¯ d,R ¯ d,L ¯ ¯ ∧d,L q (X)R = ∧q (X), ∧q (X)L = ∧q (X).
Hence,
In particular, with d = dim(X) = dim(V ) dim(W ), this implies that the ¯ ¯ is equal to the right determinant D R (X). left determinant DqL (X) q We now turn to the proof of the lemma. 2,L 2,R Proposition 6.4 M∧2,L = M∧2,R and Msym = Msym . Hence,
¯ C2,R q (X)L 2,L ¯ Cq (X)R ¯ ∧2,R q (X)L ¯ ∧2,L q (X)R
∼ = ∼ = ∼ = ∼ =
34
¯ C2,L q (X), 2,R ¯ Cq (X), ¯ ∧2,L q (X), ¯ ∧2,R q (X).
Proof: Assume that dim(V ) = dim(W ) = 2; the general case is very similar. The result follows by explicit computation of the coefficient matrices and checking that they are equal. We omit the details. 2,L ¯ ¯ The matrix coefficients of C2,R q (X) and Cq (X) (ignoring constant factors) are a(23) ∗ a(23) a(ij) ∗ a(i′ j ′ ), (i, j), (i′ , j ′ ) 6= (2, 3) b(ij) ∗ b(i′ j ′ ), (i, j), (i′ , j ′ ) = (1, 2), (3, 4) or (1, 4) b(ij) ∗ b(i′ j ′ ), (i, j), (i′ , j ′ ) = (1, 3), (2, 3) or (2, 4),
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where, for clarity, we denote aij and bkl in (38) and (39) by a(ij) and b(kl) here and in what follows. 2,R ¯ ¯ Similarly, the matrix coefficients of ∧2,L q (X) and ∧q (X) (ignoring constant factors) are a(ij) ∗ a(23), a(23) ∗ a(ij) (i, j) 6= (2, 3) {b(12), b(34), b(14)} ∗ {b(13), b(23), b(24)} {b(13), b(23), b(24)} ∗ {b(12), b(34), b(14)}. Q.E.D. ∼ ⊕d (X ¯ ⊗X ¯ ∗ )⊗d be the ideal generated by (55) or (56) Let I ⊆ ChU i = ¯ = ChU i/I. Equivalently, it is generated by so that O(Mq (X)) a(ij) ∗ b(rs) and b(rs) ∗ a(ij); ¯ generated by M 2 = cf. (59). Let J be the two sided ideal in O(Mq (X)) sym 2,L 2,R ¯ Msym = Msym ; i.e., the matrix coefficients of C2,L ( X), or equivalently, q ¯ ¯ ¯ C2,R q (X). It is a O(Mq (X))-bicomodule. Let O(Mq (X))d and Jd be the ¯ and J respectively. We have chosen degree d components of O(Mq (X)) ¯ J so that it will turn out later (cf. (77) and (78)) that, as O(Mq (X))bicomodules, d,L ¯ d,R ¯ ∼ d,R ¯ ¯ d /Jd ∼ ¯ O(Mq (X)) = ∧d,L q (X) ⊗ ∧q (X)R = ∧q (X)L ⊗ ∧q (X).
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(This is obvious at q = 1.) ¯ We shall find an explicit basis of R = O(Mq (X))/J, and then deduce Lemma 6.3 from the form of this basis. We have ∼ ¯ ¯ ⊗X ¯ ∗ )⊗d /(I + J). R = O(Mq (X)/J = ⊕d (X 35
Hence, by the diamond lemma [15], to get an explicit standard monomial basis of R, it suffices to give a reduction system for the ideal I + J in which all ambiguities can be resolved. Let L(I + J) denote the set of explicit generators for I + J; cf. (59), (38), (39), and the proof of Proposition 6.4. For example, when dim(V ) = dim(W ) = 2, these are a(ij) ∗ b(i′ j ′ ) and b(i′ j ′ ) ∗ a(ij) a(23) ∗ a(23) a(ij) ∗ a(i′ j ′ ), (i, j), (i′ , j ′ ) 6= (2, 3) b(ij) ∗ b(i′ j ′ ), (i, j), (i′ , j ′ ) = (1, 2), (3, 4) or (1, 4). b(ij) ∗ b(i′ j ′ ), (i, j), (i′ , j ′ ) = (1, 3), (2, 3) or (2, 4).
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The basic lemma is: Lemma 6.5 The equations {l = 0|l ∈ L(I + J)} can be recast in the form of an equivalent reduction system–just as eq.(20) was recast in the form a reduction system (24)–whose ambiguities can be resolved. Hence by the diamond lemma [15], the standard monomials as per this reduction system ∼ ¯ ¯ ⊗X ¯ ∗ )⊗d /(I + J). form a basis of R = O(Mq (X)/J = ⊕d (X Proof: We begin with introducing some notation that will be convenient in the proof. By an abuse of notation, we shall denote the set of basis elements of V , ¯ ¯∗ ⊗ X ¯ by V, X ¯ and U again; what is intended will be clear X and U = X ¯ ∗ and X, ¯ V ∗ and V , so from the context. Furthermore, we shall identify X ¯ ⊗ X. ¯ that we write U = X In the proof we shall explicitly distinguish between the two tensoring operations that we use. The first one, which we call restitution, and denote by ⋆ creates new variables from old. The second, which we call tensor, and denote by ⊗, raises the degree of the variable, and implements the usual tensor algebra. ¯ = V ⋆ V creates the new Thus for example, when V = W = {v1 , v2 }, X variables (basis vectors) xij = vi ⋆ vj . On the other hand, V ⊗ V creates the set {vi ⊗ vj |1 ≤ i, j ≤ 2} The element vi ⊗ vj is also denoted by vi vj . ¯ is the set V ⋆ W , whence if V = W , In this notation X ¯ = {xij = vi ⋆ vj |1 ≤ i, j ≤ 2} X 36
¯ ⊗X ¯ is thus elements of the type The set X xij xkl = xij ⊗ xkl = (vi ⋆ vj ) ⊗ (vk ⋆ vl ) = (vi ⊗ vk ) ⋆ (vj ⊗ vl ) = vi vk ⋆ vj vl ¯ ⊗X ¯ = (V ⊗ V ) ⋆ (V ⊗ V ). In this sense X ¯ is given by X ¯ ⋆X ¯ = The set U of the matrix coefficients for the set X V ⋆ V ⋆ V ⋆ V . Thus uijkl = vi ⋆ vj ⋆ vk ⋆ vl We also see that: U ⊗ U = (V ⊗ V ) ⋆ (V ⊗ V ) ⋆ (V ⊗ V ) ⋆ (V ⊗ V ) We define the set of z-variables Z m as the set obtained by the m-way ¯ = restitution of the variables vi . In other words, if V = {v1 , v2 }, then X V ⋆ V and U = Z4 = V ⋆ V ⋆ V ⋆ V Note that dim(Z 4 ) = 16, when dim(V ) = dim(W ) = 2. We use the notation za1,a2,...,am = va1 ⋆ va2 ⋆ . . . ⋆ vam . Now let V be 2-dimensional as above, and let W = V . Recall (cf. (37)) the elements A11, A12, A22 and B12 of V ⊗ V , listed below: A11 A22 A12 B12
= = = =
v1 v1 v2 v2 v1 v2 + pv2 v1 v1 v2 − qv2 v1
In the space Z m ⊗ Z m , the subspace P − (corresponding to eigenvectors of ˆ Z m ,Z m with negative eigenvalues) is spanned by the m-way restitutions R T1 ⋆ T2 ⋆ . . . ⋆ Tm , where (i) each Ti ∈ {A11, A12, A22, B12}. (ii) the term B12 appears exactly odd number of times. We use the following notation: n1 p1 ni pi
= = = =
C · B12 C · A11 ⊕ C · A22 ⊕ C · A12 ni−1 ⋆ p1 ⊕ pi−1 ⋆ n1 ni−1 ⋆ n1 ⊕ pi−1 ⋆ p1 37
Note that pi , ni ⊆ Z i ⊗ Z i . ¯ is the tensor algebra T (U ) modulo the ideal The algebra O(Mq (X)) generated by n4 ⊆ U ⊗ U . Thus I in the statement of the lemma is n4 in this terminology, since U = Z 4 . First we shall assume that dim(V ) = dim(W ) = 2. Then Z 1 ⊗ Z 1 = ¯ = Z 2 is 4-dimensional and n1 ⊕ p1 , with dim(n1 ) = 1 and dim(p1 ) = 3, X ¯ = (Z 2 ⊗ Z 2 )/n2 is 10-dimensional. A representative decomposition C2q (X) ¯ is given by of C2q (X) ¯ = n 1 ⋆ n 1 ⊕ p1 ⋆ p1 . C2q (X) ¯ either as a left The space M spanned by the matrix coefficients of C2q (X), or right corepresentation (cf. Proposition 6.4), is thus given by the 100dimensional ¯ ⋆ C2 (X) ¯ = (n1 ⋆ n1 ⊕ p1 ⋆ p1 ) ⋆ (n1 ⋆ n1 ⊕ p1 ⋆ p1 ). C2q (X) q
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¯ We now turn to the algebra R = O(Mq (X))/J, where J is the ideal ¯ = T (U )/I, where generated by the subspace M ⊆ U ⊗ U . Thus O(Mq (X)) I = n4 is the ideal in T (U ) generated by P − ⊆ U ⊗ U . Thus R = T (U )/(I + J). We consider the degree 2-component of R above and construct an appropriate reduction system. Remembering that U = Z 4 , our task is to construct a reduction system for (Z 4 ⊗ Z 4 )/(M ⊕ n4 ). Note that this space is 36-dimensional, since dim(V ) = dim(W ) = 2. We use the fact that Z 4 = Z 2 ⋆ Z 2 and given A = (a1 , . . . , a4 ) and B = (b1 , . . . , b4 ), we write zA zB = zA1 zB1 ⋆ zA2 zB2 , where A1 = (a1 , a2 ) and A2 = (a3 , a4 ), with B1 and B2 similarly defined. We say that zA1 zA2 ⋆ zB1 zB2 is standard if A1 ≻ B1 and A2 ≻ B2; in other words, it must be that a1 > b1 or a1 = b1 and a2 > b2 , and a similar condition on A2 and B2 . We represent this as A ⊐ B. We say that zA1 zA2 ⋆ zB1 zB2 is nonstandard if A 6⊐ B. There are exactly 6 choices of tuples for A1 , B1 that satisfy A1 ≻ B1 . These are σ = {(22, 11), (22, 12), (22, 21), (21, 11), (21, 12), (12, 11)} If Σ = {(A, B)|A ⊐ B} 38
then |Σ| = |σ|2 = 36. Our reduction system will specify for each nonstandard zA zB an element βA,B ∈ M ⊕ n4 which contains zA zB and all other terms in it are either standard or lower than zA zB in a certain order. We first note that for zA zB ∈ Z 2 ⊗Z 2 with A 6≻ B, there is an αA,B ∈ p2 , such that X αA,B = zA zB + ai zAi zBi i
with Ai ≻ Bi . These expressions are readily computed as follows. Let Aij’s and Bij’s be as in (37). Let α11,11 α22,22 α12,12 α21,21 α11,12 α11,21 α21,22 α12,22
= = = = = = = =
A11 ⋆ A11 A22 ⋆ A22 A11 ⋆ A22 A22 ⋆ A11 A11 ⋆ A12 A12 ⋆ A11 A22 ⋆ A12 A12 ⋆ A22
= = = = = = = =
z11 z11 z22 z22 z12 z12 z21 z21 z11 z12 + pz12 z11 z11 z21 + pz21 z11 z21 z22 + pz22 z21 z12 z22 + pz22 z12
The more complicated ones are: α11,22 = (qA12 ⋆ A12 + pB12 ⋆ B12)/(p + q) = z11 z22 + z22 z11 α12,21 = (A12 ⋆ A12 − B12 ⋆ B12)/(p + q) = z12 z21 + z21 z12 + (p − q)z22 z11 Our next observation is that for any αA,B as above, and any i, j, k, l we have αA,B ⋆ zij zkl ∈ M ⊕ n4 . Indeed, we may wend through the following sequence of relations: p2 ⋆ Z 2 = (n1 ⋆ n1 ⊕ p1 ⋆ p1 ) ⋆ (n1 ⊕ p1 ) ⋆ (n1 ⊕ p1 ) = n 1 ⋆ n 1 ⋆ n 1 ⋆ n 1 ⊕ n 1 ⋆ n 1 ⋆ n 1 ⋆ p1 ⊕ n 1 ⋆ n 1 ⋆ p1 ⋆ n 1 ⊕ n1 ⋆ n1 ⋆ p1 ⋆ p1 ⊕ (p1 ⋆ p1 ) ⋆ Z 2 The first 4 terms are easily seen to belong either to M or to n4 . The other unexpanded terms behave similarly. Now, the promised βA,B are available forthwith. Given A, B with the nonstandard expression zA zB = (zA1 zB1 ) ⋆ (zA2 zB2 ), if A1 6⊐ B1, consider βA,B = αA1,B1 ⋆ zA2 zB2
(75)
In this, other than the term zA zB , we will have all terms zC zD with C1 ≻ D1 . Next, when A, B are such that A1 ⊐ B1 , we construct βA,B = zA1 zB1 ⋆ αA2,B2 39
(76)
This defines the reduction system completely. As an example, we have β(1212,2122) = = = β(2212,1122) = = =
α12,21 ⋆ z12 z22 (z12 z21 + z21 z12 + (p − q)z22 z11 ) ⋆ z12 z22 z1212 z2122 + z2112 z1222 + (p − q)z2212 z1122 z22 z11 ⋆ α12,22 z22 z11 ⋆ (z12 z22 + pz22 z12 ) z2212 z1122 + pz2222 z1112
Next, for arbitrary dim(V ) and dim(W ), modification of the βA,B as above is straightforward. Let A = (a1 , . . . , a4 ) and B = (b1 , . . . , b4 ). For each index i, depending on the three conditions ai < bi or ai = bi or ai > bi , we replace ai and bi by either of 1 or 2 which satisfy the same relation. We term this new seqeunce as A′ and B ′ . After obtaining βA′ ,B ′ we replace each occurrence of 1 or 2 in each subscript by its original entry. This is illustrated by the following example: Let A = (1313) and B = (3223). We put A′ = (1212) and B ′ = (2122). and note that a′i < b′i iff ai < bi and similarly for other comparisons. We obtain β(1212,2122) from the above example and make the substitutions to get: β(1212,2122) = z1212 z2122 + z2112 z1222 + (p − q)z2212 z1122 β(1313,3223) = z1313 z3223 + z3213 z1323 + (p − q)z3313 z1223 Explicit reduction rules which arise from β(2212,1122) and β(1313,3223) are shown below: z2212 z1122 → −pz2222 z1112 z1313 z3223 → −z3213 z1323 − (p − q)z3313 z1223 The reduction system arising from the above elements βA,B does satisfy the diamond lemma. This is an easy consequence of the same fact for Z 2 /p2 , ¯ the case that we have exhibited earlier as the antisymmetric algebra ∧q [X] ¯ (cf. the proof of Proposition 4.1 (2)). Q.E.D. for X Lemma 6.6
d,R ¯ 2 ¯ d /Jd ) = dim(∧d,L ¯ 2 1. dim(O(Mq (X)) q (X)) = dim(∧q (X)) .
2. The matrix elements of M∧d,L or M∧d,R (modulo Jd ) form a basis of ¯ d /Jd . O(Mq (X)) 40
Proof: From the diamond lemma [15] and the form of the reduction system above (cf. (75) and (76)), it follows that the standard monomials of the form zA1 zA2 · · · zAr , ¯ where A1 ⊐ A2 ⊐ · · · , form a basis of O(Mq (X))/J. This immediately implies the first statement. The second statement is easy to check at q = 1. Since the dimension ¯ d is the same at q = 1 and general transendental real q, the of O(Mq (X)) second statement follows. Q.E.D. ¯ and J Now let us turn to the proof of Lemma 6.3. Since O(Mq (X)) ¯ ¯ are O(Mq (X))-bicomodules, so is O(Mq (X))/J. The second statement of ¯ Lemma 6.6 implies that, as O(Mq (X))-bicomodules, d,L ¯ ¯ d /Jd ∼ ¯ O(Mq (X)) = ∧d,L q (X) ⊗ ∧q (X)R ,
(77)
d,R ¯ ¯ d /Jd ∼ ¯ O(Mq (X)) = ∧d,R q (X)L ⊗ ∧q (X).
(78)
and, similarly,
Hence,
¯ ∼ d,R ¯ ∧d,L q (X) = ∧q (X)L
and
¯ ∼ ¯ ∧d,R (X) = ∧d,L q q (X)R .
d,R ¯ ¯ It follows that coefficient matrices of ∧d,L q (X) and ∧q (X) are similar; i.e., there exists a nonsingular similarity matrix Q such that M∧d,L = Q−1 M∧d,R Q. We have to show that Q is the identity matrix. This follows from Proposition 5.6, Proposition 4.3, and the theory of the standard Drinfeld-Jimbo quantum group, the main point being uniqueness of the orthonormal Gelfand¯ we leave details to the reader. This proves Tsetlin basis (48) for ∧dq [X]; Lemma 6.3 and hence Proposition 6.2.
7
Hopf structure
To define a cofactor matrix of u we need the following. Proposition 7.1 There is a nondegenerate pairing ¯ ⊗ ∧nm−r [X] ¯ → ∧nm [X], ¯ ∧rq [X] q q given by (xI , xJ ) → xI xJ ; 41
Proof: This follows from Proposition 4.1 and nondegeneracy of the pairing in the classical q = 1 case. Q.E.D. ˜ so that Proposition 7.2 There exists a cofactor matrix u ¯ ˜ u = u˜ u u = Dq (X)I. Proof: The matrix form of the nondegenerate pairing in Proposition 7.1 ¯ in the present conyields a q-analogue of Laplace expansion for O(Mq (X)) text. In particular, we have nondegenerate pairings
and
¯ ⊗ ∧1q [X] ¯ → ∧nm ¯ ∧nm−1 [X] q q [X],
(79)
¯ ⊗ ∧nm−1 ¯ → ∧nm ¯ ∧1q [X] [X] q q [X],
(80)
¯ =X ¯ is the fundamental vector representation. where ∧1q [X] ¯ and B ′ the Gelfand-Tsetlin Let B = {xij } be the standard basis of X, nm−1 ¯ [X]; cf. (48). Then B and B ′ are dual bases as per the basis of ∧q pairings in (79) and (80). This follows from (1) the multiplication formula in the Gelfand-Tsetlin basis (cf. (53), and (2) the symmetry and other detailed properties of Clebsch-Gordon coefficients proved in [3]; e.g. the fundamental CBC’s CN KM and CKN M are obtained from each other by just replacing q by q −1 . We leave the details to the reader. Let {br } denote the elements of B and {b′s } the elements of B ′ so that br b′s = δrs ,
(81)
b′s br = δrs .
(82)
and Applying the homomorphism φR (cf. Section 6) to the first equation ˜ R so that here implies that there exists a cofactor matrix u ¯ u˜ uR = Dq (X)I. Applying the homomorphism φL to the second equation here implies ˜ L so that that there exists a cofactor matrix u ¯ ˜ L u = Dq (X)I. u
42
˜L = u ˜ R . So It follows from the symmetry result (Proposition 6.2) that u we let ˜=u ˜L = u ˜ R. u Q.E.D. ¯ belongs to This result implies just as in the standard case that Dq (X) ¯ ¯ the center of O(Mq (X)). The coordinate algebra O(GLq (X)) of the sought ¯ is obtained by adjoining the inverse Dq (X) ¯ −1 to quantum group GLq (X) −1 −1 ¯ ¯ ˜ . This allows us to define O(Mq (X)). We formally define u = Dq (X) u ¯ just as in the standard case (Section 2). a Hopf structure on O(GLq (X), ¯ Proposition 7.3 There is a unique Hopf algebra structure on O(GLq (X)) so that ·
1. ∆(u) = u ⊗ u. 2. ǫ(u) = I. ¯ −1 , S(Dq (X) ¯ −1 ) = Dq (X), ¯ where ui are the entries 3. S(uij ) = u ˜ij Dq (X) j ˜. of u and u ˜ij are the entries of u ¯ is an FRT-algebra, the proof is similar to that of Proof: Since O(Mq (X)) Proposition 9.10 in [15]. Q.E.D.
8
Compact real form
¯ is a Hopf ∗-algebra with the inProposition 8.1 The algebra O(GLq (X)) j volution ∗ determined by (uij )∗ = S(ui ). Proof: This follows from Proposition 3, Chapter 9 in [15]. The crucial fact here is that the left and the right determinants coincide (Proposition 6.2). Q.E.D. ¯ is a CMQG algebra (cf. Proposition 8.2 The Hopf ∗-algebra O(GLq (X)) Section 2.1). ¯ of O(GLq (X)) ¯ is unitary by Proof: The fundamental corepresentation X −1 ¯ ¯ Proposition 7.2, and Dq (X) is a unitary element of O(GLq (X). Fur¯ thermore, O(GLq (X)) is generated by the matrix elements of the unitary 43
¯ −1 ). Hence, the result follows from Theorem 2.1 corepresentation u⊕(Dq (X) (a). Q.E.D. Finally, Theorem 1.3 can be restated as: ¯ defined above has a compact Theorem 8.3 (a) The quantum group GLq (X) ¯ real form, which we shall denote by Uq (X). (b) There is a homomorphism ¯ GLq (V ) × GLq (W ) → GLq (X),
(83)
¯ is unitarizable, and (c) Every finite dimensional representation of GLq (X) hence, is a direct sum of irreducible representations. (d) The quantum analogue of the Peter-Weyl theorem holds. That is, ¯ = ⊕λ S ∗ ⊗ Sλ , O(GLq (X)) λ ¯ where Sλ ranges over all irreducible representations of GLq (X). This follows from Propositions 7.3,8.1,8.2 and Theorem 2.1.
9
Example
¯ = V ⊗ W , dim(V ) = dim(W ) = 2. Recall that Vq,α (V ) denotes Let X the q-Weyl module of the standard quantum group GLq (V ) labelled by the partition α. Let Vλ (X) denote the Weyl module of G = GL(X) labelled by the partition λ. There are three partitions of size 3, namely (3), (2, 1), (1, 1, 1). We have V(3) (X) = Sym3 (X), with dimension 20, V(1,1,1) (X) = ∧3 (X), with dimension 4, and V(2,1) (X) with dimension 20. Hence, by Peter-Weyl, dim(O(M (X))3 ) = 202 + 42 + 202 = 816. ¯ 3 ) = 688. In contrast, by computer it was verified that dim(O(Mq (X)) 2 2 2 2 ¯ Since, 688 = 20 + 4 + 16 + 4 , by the Peter-Weyl theorem for GLq (X) ¯ has four irreducible represen(Theorem 8.3), we can suspect that GLq (X) tations of dimensions 20, 4, 16 and 4, respectively. ¯ and ∧3q [X] ¯ of GLq (X) ¯ That is indeed so. The representations C3q [X] (cf. Proposition 5.1) turn out to be irreducible representations of dimensions 20 and 4 respectively. These are the q-deformations of the classical 44
representations C3 (X) and ∧3 (X) in this setting. We shall see below that the classical representation V(2,1) (X) also has a 20-dimensional q-analogue ¯ which, however, is reducible. It decomposes into two irreducible Vq,(2,1) (X), ¯ and Vq,(2,1),2 (X) ¯ of GLq (X) ¯ of dimensions 16 representations Vq,(2,1),1 (X) and 4, respectively. ¯ above decompose as GLq (V )× The irreducible representations of GLq (X) GLq (W )-modules, via the homomorphism in Theorem 8.3, as follows: ¯ C3q [X]
= Vq,(3) (V ) ⊗ Vq,(3) (W ) ⊕ Vq,(2,1) (V ) ⊗ Vq,(2,1) (W ),
∧3q (X)
= Vq,(2,1) (V ) ⊗ Vq,(2,1) (W ),
Vq,(2,1),1 (X) = Vq,(2,1) (V ) ⊗ Vq,(3) (W ) ⊕ Vq,(3) (V ) ⊗ Vq,(2,1) (W ) Vq,(2,1),2 (X) = Vq,(2,1) (V ) ⊗ Vq,(2,1) (W ). ¯ can be explicitly constructed as follows. The module Vq,(2,1) (X) ¯ ⊗ ∧2q [X]. ¯ It is a 24-dimensional representation of GLq (X), ¯ Let M = X and it can be shown to contain, just as in the classical case, the four di¯ has a compact real form Uq (X), ¯ mensional module ∧3q (X). Since GLq (X) ¯ we can define an inner product on M . Then Vq,(2,1) (X) is the orthogonal ¯ in M . Alternatively, it is the orthogonal complement complement of ∧3q [X] ¯ in N = X ¯ ⊗ C2 [X]. ¯ of C3 [X] q
q
¯ and Vq,(2,1),2 (X) ¯ can also be explicThe irreducible modules Vq,(2,1),1 (X) itly constructed. In fact, at the end of the the next section we shall see how ¯ can be constructed, in principle, all irreducible representations of GLq (X) ¯ for general GLq (X).
10
Deformation Br of C[Sr ]
By the Brauer-Schur-Weyl duality in the standard case [15], the left action of GLq (V ) on V ⊗r commutes with the right action of the Hecke algebra Hr on V ⊗r , and the two actions determine each other. We now wish to construct a semisimple sub-algebra Br ⊆ Hr ⊗ Hr which will play the role of the Hecke algebra in the present context. The embedding B r → Hr ⊗ Hr , 45
here can be considered as a deformation of the embedding C[Sr ] → C[Sr ] ⊗ C[Sr ], determined by the diagonal embedding Sr → Sr × Sr of the symmetric ¯ does not coincide with the group Sr . Just as the Poincare series of GLq (X) Poincare series of the classical GL(X), the dimension of Br will turn out to be different (larger) that that of C[Sn ]. In conjunction with the Wederburn structure theorem, this would imply that the irreducible representations of Br are no longer in one-to-one correspondence with those of C[Sn ]. Thus Br would turn out to be qualitatively similar, but at the same time, fundamentally different from Hr . ¯ We shall define Br by essentially translating the definition of GLq (X) acting on the left in the dual setting of right action.
10.1
Hecke algebra
Before we can state its definition, we need to review some notation and results for standard affine Hecke Algebras. We largely follow the notation of [9]. Let K = C(q) be the field of rational functions in the indeterminate q with complex coefficients. The algebra Hr = Hr (q) is the associative K-algebra, generated by the symbols T1 , . . . , Tr−1 with the following relations: Ti Tj = Tj Ti whenever |i − j| > 1 Ti Ti+1 Ti = Ti+1 Ti Ti+1 for i = 1, . . . , r − 2 Ti2 = (q − 1)Ti + q for i = 1, . . . , r − 1
(84)
If M is a right-Hr -module, then eigenvalues of the right-multiplication on M by each Ti are q and −1. The element Tj is invertible: Tj−1 = q −1 Tj − (1 − q −1 ) It is well known that Hr is finite-dimensional and semi-simple over K with dimension r!. Since the eigenvalues of Ti must be q and −1, the element Ti ∈ Hr (q 2 ) has eigenvalues q 2 and −1. Thus the element Tˆi = q −1 Ti ∈ Hr (q 2 ) satisfies • (Tˆi − qI)(Tˆi + q −1 I) = 0, and • Tˆi Tˆj Tˆi = Tˆj Tˆi Tˆj whenever |j − i| > 1. 46
• Tˆi Tˆj = Tˆj Tˆi , if |i − j| > 1. ˆ=R ˆ V,V operator. SpecifThese coincide with the relations satisfied by the R ically, the first relation corresponds to (10) and the second relation correˆ [15]. It follows that Hr (q 2 ) has sponds to the braid relation satisfied by R ⊗r i ˆ ˆ ˆi a right action on V given by Ti → RV,V , where R V,V acts on the i-th ⊗r ˆ and (i + 1)-st factors of V as RV,V . This action clearly commutes with the left action of GLq (V ), and the two actions determine each other by Brauer-Schur-Weyl duality [15]. Remark 10.1 We will henceforth ignore the power of 2 and assume that Hr (q) itself acts on V ⊗r . The algebra Hr has other natural set of generators that will be crucial to us. Specifically, let pi =
Ti + 1 q+1
and
q i = 1 − pi =
q − Ti . q+1
(85)
The pi ’s generate Hr . The relations defining them are: pi pj = pj pi
(86)
pi − (q + 2 + 1/q)pi pi+1 pi = pi+1 − (q + 2 + 1/q)pi+1 pi pi+1
(87)
if |i − j| > 1,
for i = 1, . . . , r − 2, and p2i = pi
(88)
for i = 1, . . . , r − 1. The second equation here is a reformulation of the braid relation–the second relation in (84)–in terms of pi ’s. If we consider the right action of Hr (q 2 ) on V ⊗r as above, then pi corresponds to the operator P+i , which acts as P+V , defined in (12), on the i-th and (i + 1)-st factors of V ⊗r , and qi corresponds to the the operator P−i , which acts as P−V , defined in (12), on the i-th and (i + 1)-st factors of V ⊗r . The following rescaled versions of pi and qi are also useful: p˜i = (q 1/2 + q −1/2 )pi = q −1/2 (Ti + 1) q˜i = (q 1/2 + q −1/2 )qi = q −1/2 (q − Ti ).
(89)
The advantage of this rescaling is that Ci = −˜ qi is then the basic KazhdanLusztig basis element of degree one [14]. 47
Another useful set of generators of Hr is given by fi =
2Ti + 1 − q . q+1
There is an important C-involution ι of Hr which is crucial in the Kazhdan-Lusztig theory, namely, ι(q) = 1/q ι(Tj ) = Tj−1 = q −1 Tj − (1 − q −1 ). It is extended to Hr naturally. We have ι(pi ) = pi and ι(fi ) = fi . Another involution is the K-involution θ given by: θ(Ti ) = −Ti + q − 1 It is easy to check that θ(pi ) = qi , and θ(fi ) = −fi .
10.2
The algebra Br
ˆ = V ⊗ W in the obvious manner. Of special The algebra Hr ⊗ Hr acts on X importance are its elements Pi = pi ⊗ pi + qi ⊗ qi Qi = p i ⊗ q i + q i ⊗ p i .
(90)
¯ ⊗r correspond to the operators P X¯ and P X¯ , which act Their actions on X i,+ i,− ¯ ¯ ¯ ⊗r . as P+X and P−X , defined in (30), on the i-th and (i + 1)-st factors of X
The algebra Br = Br (q) is defined to be the subalgebra of Hr ⊗ Hr generated by the elements P1 , . . . , Pr−1 , or equivalently, the elements Qi ’s. ¯ ⊗r . By the defining relation We have the natural right action Br (q) on X ¯ (55) of O(Mq (X)), the right action of Br (q) commutes with the left action ¯ of the quantum group GLq (X). Remark 10.2 The algebra Br can be defined by letting arbitrary Coxeter group play the role of the symmetric group. The proof of the semisimplicity result below can be extended to any finite Weyl group in place of the symmetric group.
48
10.3
Basic properties of Br
Let Ar = Hr ⊗ Hr . The involution θ lifts to Ar and will also be denoted as θ. Thus, for example, θ(a ⊗ b) = θ(a) ⊗ θ(b) There is also the involution τ with τ (a ⊗ b) = b ⊗ a Since θ(Pi ) = τ (Pi ) = Pi , Br ⊆ Aτ,θ r , the invariant subring of Ar . We also have the involution Θ : Ar → Ar where Θ(a ⊗ b) = θ(a) ⊗ b. We see that: Θ(Pi ) = θ(pi ) ⊗ pi + θ(1 − pi ) ⊗ (1 − pi ) = (1 − pi ) ⊗ pi + pi ⊗ (1 − pi ) = 1 − Pi Thus Θ is an involution on Br as well. Proposition 10.3 The algebra Br is a semi-simple sub-algebra of Ar . Proof: We know that there is a right action of Hr on V ⊗r , which is faithful when dim(V ) is large, and where the matrix of each pi is symmetric. Consequently there is a faithful representation of Ar on (V ⊗ V )⊗r , which is faithful, and where the matrix of Pi = pi ⊗ pi + (1 − pi ) ⊗ (1 − pi ) is symmetric. We may now use the following fact to complete the proof. Fact: Let A ⊆ Mm (R) be a sub-algebra of the real matrix algebra Mm (R). Furthermore, let A be such that if a ∈ A then the transpose aT is also in A. Then A is semi-simple. Proof: We produce a vector-space basis C = {c1 , . . . , ck } of A such that the k × k-matrix G = T r(ci cj ) is non-singular. The non-singularity of this matrix is equivalent to the semi-simplicity of A. Since transpose is an involution, and A is closed under transposition, we have A = {a1 , . . . , ar } and B = {b1 , . . . , bs } such that (i) C = A ∪ B. (ii) aTi = ai for all i, and bTj = −bj for all j. Now consider T r(ai bj ) = T r(bTj aTi ) = −T r(bj ai ), whence T r(ai bj ) = 0. Thus G has the following format: GA 0 G= 0 GB 49
where GA = (T r(ai aj )) and GB = (T r(bi bj )). But GrA = (T r(ai aTj )) is the Gram-matrix for A, and similarly GrB is that for B, whence both GrA and GrB are non-singular. Since GA = GrA and GB = −GrB , G is nonsingular.
10.4
¯ Relationship with GLq (X)
In analogy with the Brauer-Schur-Weyl duality in the standard case [15], it is a reasonable conjecture that: ¯ and Br (q) determine each other. 1. The commuting actions of GLq (X) 2. There is a one-to-one correspondence between the irreducible (poly¯ of degree r and the irreducible nomial) representations of GLq (X) representations of Br so that, as a Bimodule, ¯ ⊗r = ⊕α,α′ V (α) ⊗ W (α′ ), X
(91)
¯ where V (α) runs over irreducible polynomial representations of GLq (X) ′ of degree r, and W (α ) denotes the irreducible representation of Br (q) in correspondence with V (α). Proposition 10.4 The third statement above holds for r = 3. Proof: This is sketched at the end of Section 11. Q.E.D. ¯ Assuming the conjecture above, irreducible representations of GLq (X) of degree r can be constructed, in principle, by constructing the idempotents ¯ ⊗r , very much as in the of the algebra Br and taking their right actions on X ¯ ⊗r · a is standard case [7]. Specifically, if a ∈ Br is an idempotent, then X ¯ and all irreducible (conjecturally) an irreducible representation of GLq (X), ¯ polynomial representations of GLq (X) of degree r can be (conjecturally) obtained in this way.
11
The algebra B3
As an example, now we wish to analyze the structure of the algebra B = B3 , in particular, its decomposition into irreducible modules. To conform with the convention in the Kazhdan-Lusztig paper [14], instead of the generators Pi ’s of Br in (90), we shall consider the generators Pi = p˜i ⊗ p˜i + q˜i ⊗ q˜i = fPi , 50
where p˜i and q˜i are as in (89), and f = (q + 2 + 1/q) = (q 1/2 + q −1/2 )2 . We see that Pi2 = f · Pi . (92) Through experimentation, by letting c1 =
q 6 + 2q 5 + 3q 4 + 4q 3 + 3q 2 + 2q + 1 q3 c2 =
q 4 + q 3 + 4q 2 + q + 1 q2
we see that: c1 P1 − c2 P1 P2 P1 + P1 P2 P1 P2 P1 = c1 P2 − c2 P2 P1 P2 + P2 P1 P2 P1 P2 (93) This identity is quite different from the braid identity (87) for Hr . This ¯ is not coquasitriangular in the terminology means the algebra O(GLq (X)) of [32]. We name Σ as either of the two sides of (93). Thus Σ = c1 P1 − c2 P1 P2 P1 + P1 P2 P1 P2 P1 and at once see that ΣP1 = ΣP2 = f · Σ, and thus C · Σ is a trivial onedimensional B-module. At q = 1, this specializes to the trivial representation of the symmetric group S3 . Next, let α= β1 β2 β12 β21 β121 β212 β1212 β2121
= = = = = = = =
f4
1 − c2 · f2 + c1
P1 − αΣ P2 − αΣ P1 P2 − f · αΣ P2 P1 − f · αΣ P1 P2 P1 − f2 · αΣ P2 P1 P2 − f2 · αΣ P1 P2 P1 P2 − f3 · αΣ P2 P1 P2 P1 − f3 · αΣ
(94)
It can be verified that β’s span an 8-dimensional B-module, with the following multiplication table.
51
β1 β2 β12 β21 β121 β212 β1212 β2121
P1 P2 fβ1 β12 β21 fβ2 β121 fβ12 fβ21 β212 fβ121 β1212 β2121 fβ212 −c1 β1 + c2 β121 fβ1212 fβ2121 −c1 β2 + c2 β212
This 8-dimensional module splits as follows:
χ1 =
β1 β12 β121 β1212
P1 P2 fβ1 β12 β121 fβ12 fβ121 β1212 −c1 β1 + c2 β121 fβ1212
χ2 =
β2 β21 β212 β2121
P1 P2 β21 fβ2 fβ21 β212 β2121 fβ212 fβ2121 −c1 β2 + c2 β212
The representation χ1 (and similarly χ2 ) may be split further as follows. Let γ1 = β1 /a + ·β121 γ12 = β12 /a + ·β1212 , for an indeterminate a. We get the multiplication table:
γ1 γ12
P1 P2 fγ1 γ12 (1+ac2 ) −c1 β1 + a β121 fγ12
For γ12 · P1 to be a multiple of γ1 we need: 1 + ac2 =a −ac1 Or in other words: c1 a2 + c2 a + 1 = 0 This yields: a=
−c2 ±
p
c22 − 4c1 2c1
52
As expected c22 − 4c1 is a perfect square and we have: a1 = a2 =
−q 2 (q 2 +1)2 −q (q+1)2
1 and γ 2 , γ 2 . We thus By choosing a1 and a2 , we form the vectors γ11 , γ12 1 12 have: χ1 = χ11 ⊕ χ21 χ2 = χ12 ⊕ χ22
Further, χ1 = χ11 ∼ = χ22 . = χ12 and χ2 = χ21 ∼ At q = 1, the nonisomorphic two-dimensional B-modules χ1 and χ2 specialize to the Specht module of S3 corresponding to the partition (2, 1). Finally, we have a 1-dimensional (alternating) module Cµ, which specializes at q = 1 to the alternating representation of S3 . This can be computed as follows. Let µ = 1 + θ1 β1 + θ2 β2 + θ12 β12 + θ21 β21 + θ121 β121 + θ212 β212 +θ1212 β1212 + θ2121 β2121 + θΣ, with the various constants being as in the following table: θ1 f3 −fc2 − f4 −f2 c2 +c1 θ2 f3 −fc2 − f4 −f 2 c +c 2 1
θ12 f2 −c2 f4 −f2 c2 +c1 θ21 f2 −c2 f4 −f2 c2 +c1
θ121
θ1212
− f4 −f2fc2 +c1
1 f4 −f2 c2 +c1
θ212 − f4 −f2fc2 +c1
θ −α
θ2121
1 f4 −f2 c2 +c1
Then it can be verified that µ · P1 = µ · P2 = 0. To summarize, we have 4 non-isomorphic representations of B, namely
Σ χ1 χ2 µ
dim. mult. 1 1 2 2 2 2 1 1
where, at q = 1, Σ specializes to the trivial one-dimensional representation of S3 , µ to the one-dimensional alternating representation, and χ1 and χ2 to the 2-dimensional Specht module of S3 corresponding to the partition (2, 1). Note that 12 + 22 + 22 + 12 = 10 = dim(B). 53
The analysis above can be extended to obtain explicit expressions for the idempotents of B3 ; we skip the details. Using these idempotents, we can obtain all irreducible representations of degree 3 of the quantum group ¯ X ¯ = V ⊗ W , dim(V ) = dim(W ) = 2. Indeed, if a is an idempotent GLq (X), ¯ ⊗3 · a is a representation of GLq (X). ¯ of Br then X These turn out to be irreducible when r = 3 (as per the general conjecture in Section 10.2). ¯ within X ¯ ⊗3 can be In this way all irreducible representations of GLq (X) computed, and Proposition 10.4 can be verified for r = 3.
12
Canonical basis
We now ask if Br ⊆ Hr ⊗ Hr has a canonical basis B that is akin to the Kazhdan-Lusztig basis of Hr . We shall not specify here precisely what “akin to” means; cf. [24] for the precise meaning. But two properties (among others) that this basis should satisfy are as follows. Let C be the KazhdanLusztig basis of Hr ⊗ Hr ; i.e., each element in C is of the form c1 ⊗ c2 , where ci ’s are Kazhdan-Lusztig basis elements of Hr . Then 1. Each coefficient of any b ∈ B, when expressed in terms of the basis C, should be of the form q a/2 f (q), for some integer a and a polynomial f (q) with nonnegative integral coefficients, and furthermore, each coefficient should be ι-invariant. This is analogous to the fact the coefficients of the Kazhdan-Lusztig basis of Hr in terms of the standard basis of Hr are (Kazhdan-Lusztig) polynomials with nonnegative coefficients (up to a factor of the form + or − q a/2 ). Thus the role of the standard basis of Hr is played here by C. 2. B should have cellular decompositions into left, right, and two-sided cells, just like the Kazhdan-Lusztig basis.
12.1
Canonical basis of B3
We now construct such a basis B of B3 . We follow the notation as in Section 11. Let
54
βˆ1 βˆ2 βˆ12 βˆ21 ˆ β121 βˆ212 βˆ1212 βˆ2121
= = = = = = = =
P1 P2 P1 P2 P2 P1 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P2 P1 P2 P1
(95)
These are modified versions of the elements β’s in eq.(94). 1 , . . . of the elements γ 1 , γ 1 , . . . in We define the modified forms γˆ11 , γˆ12 1 12 ˆ in place of β’s in their definitions. Let µ Section 11 by substituting β’s ˆ=1 ˆ = Σ. Finally, let and Σ
B = Uσ ∪ Uµ ∪ V1 ∪ V2 ∪ W1 ∪ W2 , where Uσ V1 V2 W1 W2 Uµ =
= = = = = =
ˆ {Σ} 1 } {ˆ γ11 , γˆ12 2 } 2 {ˆ γ1 , γˆ12 1 } {ˆ γ21 , γˆ21 2 2 {ˆ γ2 , γˆ21 } {ˆ µ}.
The discussion in Section 11 shows that B has a cellular decomposition into left cells. Indeed, Uσ , Uµ , Vi ’s and Wi ’s are the left cells, with an obvious partial order among these cells, with Uσ at the bottom of the partial order, and Uµ at the top. Similarly, it can be verified that B also has cellular decomposition into right or two-sided cells. Furthermore, coefficients of each b ∈ B in the Kazhdan-Lusztig basis C of H3 ⊗H3 are indeed polynomials in q with nonnegative coefficients (up to a factor of the form q a/2 ). These coefficients are shown in the following tables; cf. Figures 1-3. The coefficients are shown in the symmetrized KazhdanLusztig basis C ′ of H3 ⊗ H3 defined as follows. Let cw , w ∈ S3 , denote the Kazhdan-Lusztig basis element corresponding to the permutation w ∈ S3 . We order these permutations as per the lexicographic order on the words denoting their reduced decomposition: id < s1 < s1 s2 < s1 s2 s1 < s2 < s2 s1 , 55
where si denotes a simple transposition. Let ci denote the i-th element of the Kazhdan-Lusztig basis of H3 in this order. Let C ′ = {ci ⊗ ci } ∪ {ci ⊗ cj ⊕ cj ⊗ ci |i < j}. When r = 3, C ′ contains 21 elements. Figure 1 shows the 21-dimensional ˆ and µ coefficient vectors of Σ ˆ, Figure 2 those of the elements of V1 and V2 , and Figure 3 those of the elements of W1 and W2 .
13
The algebra B4 and beyond
The algebra B4 turns out to be considerably more complicated and is of dimension 114. This was verified on a computer by a simple procedure of generating monomials systematically and of increasing degree while retaining only those which were not linear combinations of earlier monomials. The top degree obtained thus was 9. In other words, every monomial of degree 10 and above is a linear combination of some smaller monomials. However, this linear combination seems fairly complicated. The ideal of all relations among these monomials is not generated by relations of the type (93)–letting arbitrary i and i + 1 there in place of 1 and 2–and (92). We do not know an explicit presentation in terms of generators and relations for the algebra Br , in general, akin to the explicit presentation (84) for the Hecke algebra. This means the construction procedure for a canonical basis for B3 in Section 12.1 cannot be generalized to general r. What is needed is a procedure that does not need explicit presentation; this problem is studied in [24]. To give an idea of the difficulties involved, we give below the simplest relation among the generators of B4 which cannot be deduced from the relations of the type (93) or (92). This is done by expressing the degree 7 monomial P3232123 ∈ B4 as a linear combination of smaller monomials, in degree or in lexicographical order, where Pi1 ,i2 ,...,ir denotes the monomial Pi1 Pi2 · · · Pir . There are 74 terms in this linear combination which are reported in the table in Figures 4-6. Every coefficient c(q) is a rational function in q such that c(q) = c(1/q). The table contains 74 rows and each row lists the term, the coefficient and finally, the monomial. Thus, for example, the 33-rd row corresponds to the term: −9.q 0 − 6.q 1 − 55.q 2 + 12.q 3 − 55.q 4 − 6.q 5 − 9.q 6 P13232 . 2.q 1 + 12.q 2 + 4.q 3 + 12.q 4 + 2.q 5 56
(96)
The computation of these coefficients was done in MATLAB, but not directly. A large prime p was chosen and these rational functions where calculated on Zp . The form was verified on another large prime q. It may be observed that, unlike in the relation (93), coefficients of some monomials in the reported relation–e.g. the one above–involve polynomials whose coefficients are of mixed signs. This suggests that there is no natural monomial basis for Br , unlike the standard monomial basis for Hecke algebras. Fortunately, Br still seems to possess a canonical basis in general [24].
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[10] M. Kashiwara: Crystallizing the q-analogue of universal enveloping algebra, Commun. Math. Phys. 133 (1990) 249-260. [11] M. Kashiwara: On crystal bases of the q-analogue of the universal enveloping algebras, Duke Math. J. 63 (1991), pp. 465–516. [12] M. Kashiwara, Global crystal bases of quantum groups, Duke Mathematical Journal, vol. 69, no.2, 455-485. [13] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. [14] D. Kazhdan, G. Lusztig, Schubert varieties and Poincare duality, Proc. Symp. Pure Math., AMS, 36 (1980), 185-203. [15] A. Klimyk, K. Schm¨ udgen, Quantum groups and their representations, Springer, 1997. [16] P. Littelmann: A Littlewood-Richardson rule for symmetrizable Kac-Moody Lie algebras, Invent. math. 116 (1994), 329-346. [17] G. Lusztig: Introduction to quantum groups, Birkh¨auser, Boston, 1993. [18] G. Lusztig: Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), pp. 447–498. [19] G. Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. USA, vo. 89, pp 8177-8179, 1992. [20] I. Macdonald: Symmetric functions and Hall polynomials, Oxford Science Publications, 1995. [21] Y. Manin: Quantum groups and noncommutative geometry, CRM, Montreal, 1988. [22] K. Mulmuley, On P vs. N P , geometric complexity theory and the flip, under preparation. [23] K. Mulmuley, Geometric complexity theory VII: a quantum group for the plethysm problem, under preparation. [24] K. Mulmuley, Geometric complexity theory VIII: towards canonical bases for the Kronecker problem, under preparation.
58
[25] K. Mulmuley, M. Sohoni: Geometric complexity theory, P vs. NP and explicit obstructions, in “Advances in Algebra and Geometry”, Edited by C. Musili, the proceedings of the International Conference on Algebra and Geometry, Hyderabad 2001. Available at authors’ websites. [26] K. Mulmuley, M. Sohoni: Geometric complexity theory: An approach to the P vs. NP and related problems, SIAM J. comput. vol. 31, no. 2, pp 496-526, (2001) [27] K. Mulmuley, M. Sohoni: Geometric complexity theory II: towards explicit obstructions for embeddings among class varieties, arXiv cs.CC/0612134, December, 2006. [28] K. Mulmuley, M. Sohoni: Geometric complexity theory III: on deciding positivity of Littlewood-Richardson coefficients, arXiv cs.CC/0501076, January 2005. [29] T. Nakashima: Crystal base and a generalization of LittlewoodRichardson rule for the classical Lie algebras, Commun. Math. Phys. 154 (1993), 215-243. [30] M. Noumi, H. Yamada, K. Mimachi: Finite dimensional representations of the quantum group GLq (n, C) and the zonal spherical functions on Uq (n)/Uq (n − 1), Jap. J. Math. 19 (1993), 31-80. [31] N. Reshetikhin: Multiparameter quantum groups and twisted quasitriangularHopf algebras, Lett. Math. Phys. 20 (1990), 331-335. [32] N. Reshetikhin, L. Takhtajan, L. Faddeev: Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225. [33] R. Stanley: Positivity problems and conjectures in algebraic combinatorics, In Mathematics: frontiers and perspectives, 295-319, Amer. Math. Soc. Providence, RI (2000). [34] A. Sudbery: Consistent multiparametric quantization of GL(n), J. Phys. A 23 (1990), L697-L704. [35] N. Vilenkin, A. Klimyk, Representations of Lie groups and special functions, vol. 3, Kluwer Acad. Publ. 1992. [36] S. Woronowicz: Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), 613-665. 59
¯ Appendix: Reduction system for O(Mq (X)) ¯ in the form In this section we reformulate the relations (59) for O(Mq (X)) of a reduction system, which is used in Section 5.1. We follow the same terminology as in the proof of Lemma 6.5. ¯ is the tensor algebra T (U ) In that terminology, the algebra O(Mq (X)) modulo the ideal generated by n4 ⊆ U ⊗ U . We assume first that dim(V ) = dim(W ) = 2. Then the dimension of the degree 2 component of this algebra, viz., dim((Z 4 ⊗ Z 4 )/n4 ) equals 17 136 = 2 . For this, we construct a total order on the variables Z = {za1,a2,a3,a4 |1 ≤ a1 , a2 , a3 , a4 ≤ 2}
This order is specified easily enough: za1,a2,a3,a4 zb1,b2,b3,b4 iff there is an 1 ≤ i ≤ 4 so that aj = bj for all 1 ≤ j ≤ i and ai+1 ≥ bi+1 . Thus, for example z2112 z1222 . Let A and B be the tuples A = (a1 , a2 , a3 , a4 ) and B = (b1 , b2 , b3 , b4 ). We say that a monomial zA zB is standard only if zA zB , otherwise, zA zB is called non-standard. Note that if Γ = {(A, B)|A B} then |Γ| = 136. We shall now see that the degree 2 component above is spanned by the standard monomials. We say that the monomial zA zB is exceptional of order i if ai+1 < bi+1 and aj ≥ bj for all 1 ≤ j ≤ i. Thus z2212 z1221 is non-standard of order 2. Clearly, every non-standard monomial zA zB is exceptional of order m for some m < 4. For every exceptional monomial zA zB of order m we exhibit an element ψA,B of P − = n4 such that: X ψA,B = zA zB + αi zCi zDi i
where either (i) each monomial zCi zDi is either standard, or (ii) zCi zDi is exceptional of order exceeding m. We construct ψA,B via another simpler element ψA′ ,B ′ and “exploding” this simpler element by the term ηA,B . The tuples A′ and B ′ are defined as follows. Let I be the set {i|ai 6= bi } arranged as a tuple. Let J = (j1 , . . . , jr ) be the complement of I. The term ηA,B is defined as zJ zJ where zJ = xj1 ⋆ . . . ⋆ xjr 60
Note that ηA,B = T1 ⋆ . . . ⋆ Tr where Ti = A11 if aji = 1 and Ti = A22 otherwise. Thus ηA,B ∈ P + for all A, B. We define A′ = AI and B ′ = BI , i.e., the tuples A and B restricted to the coordinates I. Thus, for example, if A = (1221) and B = (2212) then I = (134) and A′ = (121) and B ′ = (212). The factor ηA,B is (x2 )2 and thus equals A22. It is now clear that ψA,B = ψA′ ,B ′ ⋆ ηA,B where every coordinate of ηA,B goes in the right place in every term of ψA′ ,B ′ . Furthermore, if ψA′ ,B ′ is an element of P − then so is ψA,B . This reduces the computation for those monomial zA zB which are non-standard and for which ai 6= bi . The non-standardness implies that a1 = 1. For m = 4 there are 15 such tuples, the A values are listed below. B is precisely the complement of A. 1111 1112 1121 1122 1211 1212 1221 1222 111 112 121 122 11 12 1 For each A above, we inductively construct ψA ∈ P − and φA ∈ P + such that either of them is of the form X zA zB + αi zAi zBi i
where the first entry of Ai is 2. Thus, zA zB is indeed the solitary leading term, and all other terms, if non-standard are exceptional of a higher order. We illustrate this process by a small example: ψ1 = z1 z2 − qz2 z1 φ1 = z1 z2 + pz2 z1 Thus, ψ1 = B12 and φ1 = A12. Next, we construct the terms: (φ1 ⋆ A12 − ψ1 ⋆ B12)/(p + q) = z12 z21 + z21 z12 + (p − q)z22 z11 (q · φ1 ⋆ A12 + p · ψ1 ⋆ B12)/(p + q) = z11 z22 + z22 z11 Since both these terms are in P + , these qualify to be called φ11 and φ12 . The terms ψ11 and ψ12 are constructed similarly from φ1 ⋆B12 and ψ1 ⋆A12. 61
The net results is expressed simply as: q p 0 φ11 p+q p+q −1 1 φ12 0 p+q p+q q ψ11 = 0 0 p+q 1 ψ12 0 0 p+q In general, we note that: φA1 φA2 ψA1 = ψA2
q p+q 1 p+q
p p+q −1 p+q
0 0
0 0
0 0 p p+q −1 p+q
0 0
0 0
q p+q 1 p+q
p p+q −1 p+q
φ1 ⋆ A12 ψ1 ⋆ B12 φ1 ⋆ B12 ψ1 ⋆ A12
φA ⋆ A12 ψA ⋆ B12 φA ⋆ B12 ψA ⋆ A12
This gives us the contsruction of all ψA,B . As an example, for A = [1122] and B = [1212], we have ψA,B = A11 ⋆ ψ12 ⋆ A22 = −z1122 z1212 − z1212 z1122 This yields a reduction system for (Z ⊗ Z)/P − wherein we have X zA zB = αi zAi zBi i
where all zAi zBi are standard. By this reduction rule, there is a method of expanding any monomial zA zB zC as X zA zB zC = zAi zBi zCi i
wherein Ai Bi Ci . Thus every non-standard monomial may be expanded into a linear combination of standard monomials. Unfortunately, this reduction system does not obey the diamond lemma. In other words, there exist monomials zA zB zC wherein teo different simplifications using the above reduction rules yield two different expansions into standard monomials. Consider the monomial mm = z1111 z1112 z1221 . For any monomial zA zB zC , if A 6 B, then we may apply the reduction system above for the first two terms and this is denoted as (zA zB )zC . We say that R1 applies and display the result. Similar, we say that R2 applies if zB 6 zC and denote this 62
application by zA (zB zC ). The monomial above has two exapnsions, viz., R1 R2 R1 R2 and R2 R1 R2 , reading both strings from left to right. The first expansion yields (m)R1 = l1 = (q) · z1112 z1111 z1221 (m)R1 R2 = l12 = (−1 + q 2 ) · z1112 z1211 z1121 + q · z1112 z1221 z1111 The expression l12 has two terms t1 and t2 . We have: (t1 )R1 = z1211 z1112 z1121 = z1211 z1121 z1112 ; the last equality follows since z1121 and z1112 commute, as is easy to show. (t2 )R1 = (
−1 + q 2 −1 + q 2 2q )·z z z + ·z1212 z1121 z1111 + ·z1221 z1112 z1111 + 1211 1122 1111 2 2 1+q q (1 + q ) 1 + q2 1 − q2 · z1222 z1111 z1111 1 + q2
Combining all this, we have l121 = (m)R1 R2 R1 : q3 − q q2 − 1 ·z z z + ·z1212 z1121 z1111 + 1211 1122 1111 1 + q2 1 + q2 1 − q2 q 2q 2 · z1221 z1112 z1111 + · z1222 z1111 z1111 1 + q2 1 + q2
l121 = (−1+q 2 )·z1211 z1121 z1112 +
l121 has 5 terms, viz., t1 , . . . , t5 . Applying R2 to each we get: T erm1 = (1) · z1211 z1121 z1112 T erm2 = (1) · z1211 z1122 z1111 T erm3 = (1) · z1212 z1121 z1111 T erm4 = (1) · z1221 z1112 z1111 T erm5 = (1) · z1222 z1111 z1111 Finally, (m)R1 R2 R1 R2 = l1212 equals: q2 − 1 q3 − q ·z z z + ·z1212 z1121 z1111 + 1211 1122 1111 1 + q2 1 + q2 1 − q2 q 2q 2 · z1221 z1112 z1111 + · z1222 z1111 z1111 1 + q2 1 + q2
l1212 = (q 2 −1)·z1211 z1121 z1112 +
63
The second expansion is (m)R2 R1 R2 . We have l2 = (m)R2 as below: l2 =
q2 − 1 2q q2 − 1 ·z1111 z1212 z1121 + ·z z z + ·z1111 z1221 z1112 + 1111 1211 1122 1 + q2 q (1 + q 2 ) 1 + q2 1 − q2 · z1111 z1222 z1111 1 + q2
This has 4 terms, and applying R1 to each term yields: T erm1 = q · z1211 z1111 z1122 T erm2 = T erm3 =
q2 − 1 · z1211 z1112 z1121 + z1212 z1111 z1121 q q2 − 1 ) · z1211 z1121 z1112 + ·z1221 z1111 z1112 q
q2 − 1 q q2 − 1 q2 − 1 ·z z z + ·z z z + ·z1221 z1112 z1111 + T erm4 = 1211 1122 1111 1212 1121 1111 1 + q2 1 + q2 1 + q2 q (2 ) · z1222 z1111 z1111 1 + q2 Whence, l21 = (m)R2 R1 equals: q2 − 1 q 1 − 2q 2 + q 4 2q 2 − 2 ·z z z + ·z z z + )·z1211 z1121 z1112 + l21 = 1211 1111 1122 1211 1112 1121 1 + q2 q 2 (1 + q 2 ) 1 + q2 2q 3 − q − q 5
1 − 2 q2 + q4 q2 − 1 ·z z z − ·z1212 z1121 z1111 + 1212 1111 1121 q (1 + q 2 ) (1 + q 2 )2 (1 + q 2 )2 2q 2 − 2 q 1 − 2 q2 + q4 2q ·z1221 z1111 z1112 − ·z1221 z1112 z1111 − ·z1222 z1111 z1111 1 + q2 (1 + q 2 )2 (1 + q 2 )2 ·z1211 z1122 z1111 +
This has 9 terms, which on applying R2 yield: T erm1 =
q2 − 1 · z1211 z1121 z1112 + ·z1211 z1122 z1111 q T erm2 = z1211 z1121 z1112 T erm3 = z1211 z1121 z1112 T erm4 = z1211 z1122 z1111 T erm5 = q · z1212 z1121 z1111 64
T erm6 = z1212 z1121 z1111 T erm7 = q · z1221 z1112 z1111 T erm8 = z1221 z1112 z1111 T erm9 = z1222 z1111 z1111 Finally, collating this, we get l212 = (m)R2 R1 R2 as follows: l212 =
q 6 + q 4 − 3q 2 + 1 2q 3 − 2q · z1211 z1122 z1111 · z z z + 1211 1121 1112 q 2 (1 + q 2 ) (1 + q 2 )2
4q 2 + q 4 − 1 (1 + q 2 )2
· z1221 z1112 z1111 +
2q 2 − 2 (1 + q 2 )2
· [z1212 z1121 z1111 − q · z1222 z1111 z1111 ]
It is easy to see that (m)R1 R2 R1 R2 6= (m)R2 R1 R2 .
65
4 4 4 4 4 4
(q2 +q+1)
2
(1+q)4
q4
(q2 +q+1)(1+q)5 q 7/2
(q2 +q+1)(1+q)4 q3
(
)(1+q)3
q 2 +q+1
q 5/2
(q2 +q+1)(1+q)5 q 7/2
(q2 +q+1)(1+q)4 q3
6 (1+q) q3
6
6 (1+q) q 5/2
5
6 (1+q) q2
4
4
2 (q+2)(2 q+1)(1+q) q3 6 (1+q) q 5/2 2 2
4
(q2 +4 q+1)(1+q)4 q3
(
)(1+q)3
q 2 +4 q+1
q 5/2
6 (1+q) q 5/2
5
6 (1+q) q2
4
(q2 +4 q+1)(1+q)2 q2
6 (1+q) q2 2
2
5
4
(q2 +4 q+1)(1+q)3 q 5/2
6 (1+q) q3
6
6 (1+q) q 5/2
5
(q2 +4 q+1)(1+q)4 q3
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
ˆ and µ Figure 1: Coefficient vectors of the elements Σ ˆ in the symmetrized 66 Kazhdan-Lusztig basis
4
(q2 +q+1)(1+q)2 q2
(1+q) 5 q2 +6 q+5 ( ) 3/2 q (1+q)4 q2 (1+q)3 q 3/2 (1+q)5 q 5/2 (1+q)4 q2 2 8 (1+q) q (1+q)(q2 +6 q+1) q 3/2 8 (q2 +4 q+1)(1+q)2 q2 (1+q)(q2 +6 q+1) q 3/2 2 4 (1+q) q √ 4 1+q q (1+q)3 2 q 3/2 2 2 (1+q) q 8 2 2 (1+q) q 1+q 4 √q 4 2 (1+q) q2 3 2 (1+q) 3/2 q 2 4 (1+q) q
4
(q2 +q+1)(1+q)4 q3
(5 q2 +6 q+5)(1+q)3 q 5/2
(5 q2 +6 q+5)(1+q)2
4
(q2 +q+1)(1+q)2 q2 5
(1+q) q 5/2
q2
(1+q)4 q2
(1+q)5 q 5/2
(1+q)3 q 3/2
(5 q2 +6 q+5)(1+q)3 q 5/2
(1+q)(5 q 2 +6 q+5) q 3/2
(1+q)6 q3
(1+q)4 q2
8 (1+q) q2
4
2 (1+q) q2
4
8 (1+q) q 3/2
3
2 (1+q) q 3/2
3
2 (1+q) q
2
(q2 +6 q+1)(1+q)2 q2
6 (1+q) q2
4
q2
(q2 +6 q+1)(1+q)3 q 5/2
2
(q2 +4 q+1)(1+q)2
(q2 +6 q+1)(1+q)2 q2
2 (1+q) q 3/2
3
4 (1+q) q
2
4 (1+q) q 3/2
3
√ 4 1+q q
8 (1+q) q 3/2
3
(1+q)(q 2 +6 q+1) q 3/2
(q2 +6 q+1)(1+q)2 q2
8 (1+q) q
2
(q2 +6 q+1)(1+q)2 q2
2 (1+q) q 8 8
4 (1+q) q 3/2
3
√ 4 1+q q
8 (1+q) q2
4
8 (1+q) q
(q2 +6 q+1)(1+q)3 q 5/2
4 (1+q) q2
4
2
2
(1+q)(q 2 +6 q+1) q 3/2
4 (1+q) q
2
4
(q2 +q+1)(1+q)4 q3
(5 q2 +6 q+5)(1+q)3 q 5/2 (1+q)6 3 q 5 (1+q) 5/2 q (5 q2 +6 q+5)(1+q)3 q 5/2 2 (5 q2 +6 q+5)(1+q) q2 (1+q)4 8 q2 3 2 (q +6 q+1)(1+q) q 5/2 2 +6 q+1 (1+q)2 q ( ) q2 4 (1+q) 6 q2 3 (1+q) 8 q3/2 (1+q)4 4 q2 (1+q)3 4 q3/2 3 2 (q +6 q+1)(1+q) q 5/2 2 +6 q+1 (1+q)2 q ( ) q2 2 8 (1+q) q (q2 +6 q+1)(1+q)2 2 q 3 (1+q) 4 q3/2 (1+q)4 8 q2 3 8 (1+q) 3/2 q (q2 +6 q+1)(1+q)2 2 q2
67 Figure 2: Coefficient vectors of the elements of V1 and V2 in the symmetrized Kazhdan-Lusztig basis
(q2 +q+1)(1+q)4
(q2 +q+1)(1+q)6
(q2 +q+1)(1+q)4
(q2 +q+1)(1+q)6
q3
q4
q3
q4
(1+q)7 q 7/2
(1+q)5 q 5/2
(1+q)6 q3
(1+q)4 q2
(1+q)5 q 5/2
(1+q)3 q 3/2
(1+q)7 q 7/2
(1+q)5 q 5/2
(1+q)6 q3
(1+q)4 q2
(1+q)5 q 5/2 (1+q)4 q2 (1+q)3 q 3/2 (1+q)5 q 5/2 (1+q)4 q2 2 3 q3 +1+3 q+8 q2 +q4 q2 (1+q)(q2 +6 q+1) q 3/2 8 (q2 +4 q+1)(1+q)2 q2 (1+q)(q2 +6 q+1) q 3/2 2 4 (1+q) q √ 4 1+q q 3 2 (1+q) q 3/2 (1+q)2 2 q 8 2 2 (1+q) q √ 4 1+q q 4 2 (1+q) q2 3 2 (1+q) q 3/2 2 4 (1+q) q
2 2
(3 q3 +1+3 q+8 q2 +q4 )(1+q)2
2 (1+q) q2
4
q3 (1+q)(3 q 3 +1+3 q+8 q 2 +q 4 ) q 5/2
2 (1+q) q 3/2
3
2 (1+q) q
2
(q2 +6 q+1)(1+q)2 q2
(q4 +5 q3 +12 q2 +5 q+1)(1+q)2
(q2 +4 q+1)(1+q)2
q3
q2 3
(q2 +6 q+1)(1+q) q 5/2
2 3q
4 +6 q 3 +14 q 2 +6 q+3
q2
4 (1+q) q 3/2 2
3
(1+q)(3 q 3 +1+3 q+8 q 2 +q 4 ) q 5/2
2
8
3
q3
(q2 +6 q+1)(1+q)3 q 5/2
4 (1+q) q2
4
2
8
q2
(3 q3 +1+3 q+8 q2 +q4 )(1+q)2
2
2 (1+q) q
(q2 +6 q+1)(1+q)2
2
4 (1+q) q
(1+q)(q 2 +6 q+1) q 3/2
q2
4 (1+q) q 3/2
3
√ 4 1+q q
(q2 +6 q+1)(1+q)2 8 (1+q) q
2 (1+q) q 3/2
√ 4 1+q q
2 3q
3 +1+3 q+8 q 2 +q 4
q2
(1+q)(q 2 +6 q+1) q 3/2
4 (1+q) q
2
(1+q)6 q3 5 (1+q) q 5/2 7 (1+q) q 7/2 (1+q)6 q3 2 3 2 4 (3 q +1+3 q+8 q +q )(1+q) 2 q3 (q2 +6 q+1)(1+q)3 5/2 q 2 (q2 +6 q+1)(1+q) 2 q (q4 +5 q3 +12 q2 +5 q+1)(1+q)2 q3 (1+q)(3 q 3 +1+3 q+8 q 2 +q 4 ) 2 q 5/2 4 (1+q) 4 q2 3 4 (1+q) q 3/2 (q2 +6 q+1)(1+q)3 q 5/2 2 2 (q +6 q+1)(1+q) 2 q (1+q)2 8 q 2 2 +6 q+1 (1+q) q ( ) q2 (1+q)3 4 q3/2 2 3 2 4 (3 q +1+3 q+8 q +q )(1+q) 2 q3 (1+q)(3 q 3 +1+3 q+8 q 2 +q 4 ) 2 q 5/2 3 q 4 +6 q 3 +14 q 2 +6 q+3 2 q2
Figure 3: Coefficient vectors of the 68 elements of W1 and W2 in the symmetrized Kazhdan-Lusztig basis
(1+q)7 q 7/2
N umber 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Coef f icient 20.q 0 +104.q 1 +256.q 2 −113.q 3 −49.q 4 −113.q 5 +256.q 6 +104.q 7 +20.q 8 2.q 3 +12.q 4 +2.q 5 −16.q 0 −64.q 1 −128.q 2 −192.q 3 −224.q 4 −192.q 5 −128.q 6 −64.q 7 −16.q 8 2.q 3 +12.q 4 +2.q 5 −4.q 0 −16.q 1 −28.q 2 −32.q 3 −28.q 4 −16.q 5 −4.q 6 2.q 3 1.q 0 −4.q 2 +6.q 4 −4.q 6 +1.q 8 2.q 3 +2.q 5 −1.q 0 −18.q 1 −65.q 2 −128.q 3 −190.q 4 −220.q 5 −190.q 6 −128.q 7 −65.q 8 −18.q 9 −1.q 10 2.q 4 +12.q 5 +2.q 6 1.q 0 +5.q 1 +17.q 2 +36.q 3 +46.q 4 +46.q 5 +46.q 6 +36.q 7 +17.q 8 +5.q 9 +1.q 10 2.q 4 +2.q 6 7.q 0 +26.q 1 +75.q 2 +152.q 3 +174.q 4 +156.q 5 +174.q 6 +152.q 7 +75.q 8 +26.q 9 +7.q 10 2.q 3 +12.q 4 +4.q 5 +12.q 6 +2.q 7 −1.q 0 −8.q 1 −20.q 2 −24.q 3 −22.q 4 −24.q 5 −20.q 6 −8.q 7 −1.q 8 2.q 3 +2.q 5 −22.q 0 −92.q 1 −170.q 2 −200.q 3 −170.q 4 −92.q 5 −22.q 6 2.q 2 +12.q 3 +2.q 4 2.q 0 +2.q 1 +12.q 2 +14.q 3 +4.q 4 +14.q 5 +12.q 6 +2.q 7 +2.q 8 2.q 3 +2.q 5 −2.q 0 −12.q 1 −40.q 2 −52.q 3 −44.q 4 −52.q 5 −40.q 6 −12.q 7 −2.q 8 2.q 3 +12.q 4 +2.q 5 −1.q 0 −2.q 1 −12.q 2 −14.q 3 −6.q 4 −14.q 5 −12.q 6 −2.q 7 −1.q 8 2.q 3 +2.q 5 1.q 0 +22.q 1 +88.q 2 +170.q 3 +206.q 4 +170.q 5 +88.q 6 +22.q 7 +1.q 8 2.q 3 +12.q 4 +2.q 5 6.q 0 +8.q 1 +4.q 2 +8.q 3 +6.q 4 2.q 2 3.q 0 +6.q 1 +5.q 2 +4.q 3 +5.q 4 +6.q 5 +3.q 6 2.q 2 +2.q 4 12.q 0 +32.q 1 +40.q 2 +32.q 3 +12.q 4 2.q 1 +12.q 2 +2.q 3 −3.q 0 −2.q 1 −5.q 2 −12.q 3 −5.q 4 −2.q 5 −3.q 6 2.q 2 +2.q 4 1.q 0 +4.q 1 +11.q 2 +16.q 3 +11.q 4 +4.q 5 +1.q 6 2.q 3 8.q 0 +12.q 1 +24.q 2 +40.q 3 +24.q 4 +12.q 5 +8.q 6 2.q 2 +12.q 3 +2.q 4 −6.q 0 −8.q 1 −4.q 2 −8.q 3 −6.q 4 2.q 1 +2.q 3 −5.q 0 −4.q 1 −44.q 2 −60.q 3 −30.q 4 −60.q 5 −44.q 6 −4.q 7 −5.q 8 2.q 2 +12.q 3 +4.q 4 +12.q 5 +2.q 6 −1.q 0 −5.q 1 −11.q 2 −14.q 3 −11.q 4 −5.q 5 −1.q 6 2.q 3 −3.q 0 −6.q 1 −5.q 2 −4.q 3 −5.q 4 −6.q 5 −3.q 6 2.q 2 +2.q 4 2.q 0 +4.q 1 +4.q 2 +4.q 3 +2.q 4 2.q 2 −1.q 0 −4.q 1 −6.q 2 −4.q 3 −1.q 4 2.q 2 6.q 0 +8.q 1 +4.q 2 +8.q 3 +6.q 4 2.q 1 +2.q 3 16.q 0 +32.q 1 +16.q 2 2.q 0 +12.q 1 +2.q 2 4.q 0 +8.q 1 +40.q 2 +8.q 3 +4.q 4 2.q 1 +12.q 2 +2.q 3 −3.q 0 −8.q 1 −4.q 2 −8.q 3 +46.q 4 −8.q 5 −4.q 6 −8.q 7 −3.q 8 2.q 2 +12.q 3 +4.q 4 +12.q 5 +2.q 6 −8.q 0 2.q 0
69 Figure 4: A relation in B4
M onomial 1 2 3 12 13 21 23 32 121 132 212 213 232 323 1212 1213 1232 1321 1323 2121 2123 2321 2323 3212 3213 3232 12121 12123 12132 12321
N umber 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Coef f icient −4.q 0 −8.q 1 −40.q 2 −8.q 3 −4.q 4 2.q 1 +12.q 2 +2.q 3 −3.q 0 −4.q 1 −2.q 2 −4.q 3 −3.q 4 2.q 1 +2.q 3 −9.q 0 −6.q 1 −55.q 2 +12.q 3 −55.q 4 −6.q 5 −9.q 6 2.q 1 +12.q 2 +4.q 3 +12.q 4 +2.q 5 9.q 0 +6.q 1 +55.q 2 −12.q 3 +55.q 4 +6.q 5 +9.q 6 2.q 1 +12.q 2 +4.q 3 +12.q 4 +2.q 5 1.q 0 −1.q 1 +3.q 2 −6.q 3 +3.q 4 −1.q 5 +1.q 6 2.q 2 +2.q 4 −1.q 0 +2.q 2 −1.q 4 2.q 2 2.q 0 +3.q 1 +6.q 2 −3.q 3 −16.q 4 −3.q 5 +6.q 6 +3.q 7 +2.q 8 2.q 2 +12.q 3 +4.q 4 +12.q 5 +2.q 6 3.q 0 +4.q 1 +2.q 2 +4.q 3 +3.q 4 2.q 1 +2.q 3 −16.q 0 −32.q 1 −16.q 2 2.q 0 +12.q 1 +2.q 2 3.q 0 +4.q 1 +2.q 2 +4.q 3 +3.q 4 2.q 1 +2.q 3 8.q 0 2.q 0 1.q 0 −2.q 2 +1.q 4 2.q 2 −3.q 0 −4.q 1 −2.q 2 −4.q 3 −3.q 4 2.q 1 +2.q 3 −8.q 0 −16.q 1 −8.q 2 2.q 0 +12.q 1 +2.q 2 −1.q 0 −14.q 1 −15.q 2 −4.q 3 −15.q 4 −14.q 5 −1.q 6 2.q 1 +12.q 2 +4.q 3 +12.q 4 +2.q 5 −2.q 0 −4.q 1 −2.q 2 2.q 1 −2.q 0 +4.q 2 −2.q 4 2.q 1 +12.q 2 +2.q 3 8.q 0 +16.q 1 +8.q 2 2.q 0 +12.q 1 +2.q 2 −1.q 0 −2.q 1 −1.q 2 2.q 1 2.q 0 −4.q 2 +2.q 4 2.q 1 +12.q 2 +2.q 3 2.q 0 +8.q 1 +12.q 2 +8.q 3 +2.q 4 2.q 1 +12.q 2 +2.q 3 2.q 0 +4.q 1 +2.q 2 2.q 1 1.q 0 +14.q 1 +15.q 2 +4.q 3 +15.q 4 +14.q 5 +1.q 6 2.q 1 +12.q 2 +4.q 3 +12.q 4 +2.q 5 3.q 0 +8.q 1 +10.q 2 +8.q 3 +3.q 4 2.q 1 +12.q 2 +2.q 3 −2.q 0 −8.q 1 −12.q 2 −8.q 3 −2.q 4 2.q 1 +12.q 2 +2.q 3 −1.q 0 −2.q 1 −1.q 2 2.q 1 −3.q 0 −8.q 1 −10.q 2 −8.q 3 −3.q 4 2.q 1 +12.q 2 +2.q 3 1.q 0 +2.q 1 +1.q 2 2.q 1 −2.q 0 −4.q 1 −2.q 2 2.q 1 2.q 0 +4.q 1 +2.q 2 2.q 1
Figure 5: A relation 70 in B4 continued.
M onomial 12323 13212 13232 21213 21232 21321 21323 23213 23232 32121 32123 32132 32321 121213 121232 121321 123213 123232 132121 132123 212132 212321 212323 213212 213232 232121 232132 232321 321232 321323
N umber 61 62 63 64 65 66 67 68 69 70 71 72 73 74
Coef f icient 1.q 0 +2.q 1 +1.q 2 2.q 1 1.q 0 −2.q 1 +1.q 2 2.q 0 +2.q 2 2.q 0 2.q 0 1.q 0 −2.q 1 +1.q 2 2.q 0 +2.q 2 2.q 0 2.q 0 2.q 0 −4.q 1 +2.q 2 2.q 0 +12.q 1 +2.q 2 16.q 1 2.q 0 +12.q 1 +2.q 2 −2.q 0 2.q 0 −1.q 0 +2.q 1 −1.q 2 2.q 0 +2.q 2 −4.q 0 −8.q 1 −4.q 2 2.q 0 +12.q 1 +2.q 2 −16.q 1 2.q 0 +12.q 1 +2.q 2 −1.q 0 +2.q 1 −1.q 2 2.q 0 +2.q 2 −2.q 0 +4.q 1 −2.q 2 2.q 0 +12.q 1 +2.q 2 4.q 0 +8.q 1 +4.q 2 2.q 0 +12.q 1 +2.q 2
M onomial 323212 1212132 1213213 1213232 1232121 1232132 1321232 1321323 2121323 2123212 2123213 2123232 2132123 2321232
Figure 6: A relation in B4 continued.
71
