Quantum double Schubert polynomials, quantum Schubert ...
arXiv:q-alg/9610022v3 10 Nov 1996
Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa–Intriligator formula. Anatol N. Kirillov∗ and Toshiaki Maeno† Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
Abstract We study the algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quane w (x, y), which are the Lascoux– tum double Schubert polynomials S Sch¨ utzenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy e w (x) as the identity. We define also quantum Schubert polynomials S Gram–Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Use w (x) = S e w (x, y)|y=0 ing quantum Cauchy identity, we prove that S and as corollary obtain a simple formula for the quantum Schubert (y) e e w (x) = ∂ww polynomials S 0 Sw0 (x, y)|y=0 . We also prove the higher genus analog of Vafa–Intriligator’s formula for the flag manifolds and study the quantum residues generating function. We introduce the extended Ehresman–Bruhat order on the symmetric group and formulate the equivariant quantum Pieri rule.
∗ On leave from Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 191011, Russia † Supported by JSPS Research Fellowships for Young Scientists
1
1
Introduction.
The structure constants of the quantum cohomology ring are given by the third derivatives of the Gromov-Witten potential F. The Gromov-Witten potential F is a generating function of the Gromov-Witten invariants. The axioms of the tree level Gromov-Witten invariants V hI0,m,β i : H ∗ (V, Q)⊗m −→ Q,
β ∈ H2 (V, Z), for a target space V are given by Kontsevich and Manin [KM]. ∗ Let X1 , . . . , Xm be cycles on V and X1∗ , . . . , Xm their dual classes. Then the V ∗ ∗ invariant hI0,m,β i(X1 ⊗ · · · ⊗ Xm ) can be considered as the virtual number of the stable maps f from m-pointed rational curve (P1 ; p1 , . . . , pm ) to V, such that the image of f represents the homology class β and f (pi) ∈ Xi . In case of flag variety F ln := SLn /B of type An−1 the potential F is given as follows. Let Ωv be the dual class of the Schubert cycle Xv corresponding to a permutation v ∈ Sn . Then the potential Fω ((tv )v∈Sn ) is defined by
Fω ((tv )v∈Sn ) =
X
X
exp(−
β m=Σmv ≥3
Z
β
V hI0,m,β i(
ω)
O
v Ω⊗m ) v
v∈Sn
Y
mv !
v∈Sn
Y
tmv ,
v∈Sn
where ω is a K¨ahler form. For each point t ∈ H ∗ (F ln ), the quantum multiplication law is given by Ωu ∗ Ωv =
∂ 3 Fω (t)Ωww0 , w∈Sn ∂tu ∂tv ∂tw X
where w0 is the permutation of maximal length. The algebra with this multiplication law is called a quantum cohomology ring, which is denoted by QHt∗ (F ln ). The associativity of the quantum multiplication is equivalent to the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation X
v∈Sn
X ∂ 3 Fω ∂ 3 Fω ∂ 3 Fω ∂ 3 Fω = , ∂tu1 ∂tu2 ∂tv ∂tvw0 ∂tu3 ∂tu4 v∈Sn ∂tu2 ∂tu3 ∂tv ∂tvw0 ∂tu1 ∂tu4
for any u1 , u2, u3 , u4 ∈ Sn . From [KM, Proposition 4.4], the potential Fω satisfies Fω (t) = Fω−t(2) (t − t(2) ), 2
where t(2) =
X
tv Ωv . Hence, we may assume ω = 0. The potential F is
l(v)=1
decomposed as a sum of the classical part fcl and the quantum correction f, where ! X 1 δu,vw0 tu tv , fcl = tid 6 u,v and f=
X
X
β m=Σmv ≥3
Fl hI0,m,β i(
O
v Ω⊗m ) v
l(v)≥1
Y
mv !
Y
tmv .
l(v)≥1
l(v)≥1
From the axioms of the Gromov-Witten invariants ([KM,(2.2.4)]), we have Fl i( hI0,m,β
O
v ) Ω⊗m v
=
Fl hI0,m−Σ i( l(u)=1 mu ,β
l(v)≥1
O
v ) Ω⊗m v
l(v)>1
Y Z
l(u)=1
β
Ωu .
Hence, the quantum correction f is expressed as f=
X
l(u)=1
v tm v , mv ! l(v)>1
N((mv )l(v)>1 | (bu )l(u)=1 ) exp(
mv ,bu
where
X
Fl N((mv ) | (bu )) = hI0,Σm i v ,Σbu Xu
(bu tu ))
O
l(v)>1
Y
v Ω⊗m . v
If n = 3, the WDVV equations with the initial condition N(0, 0, 1 | 1, 0) = N(0, 0, 1 | 0, 1) = 1 or N(2, 0, 0 | 1, 0) = N(0, 2, 0 | 0, 1) = 1
determine all the coefficients N(λ, µ, ν | a, b) uniquely. In fact, Di Francesco and Itzykson [FI] gave the coefficients N(λ, µ, ν | a, b) for a + b ≤ 10. Let x1 = Ω(1,2) and xi = Ω(i,i+1) − Ω(i−1,i) for 2 ≤ i ≤ n − 1, where (i, j) is the transposition that interchanges i and j. Since the classical cohomology ring is generated by x1 , . . . , xn−1 , the quantum cohomology ring QHt∗ (F ln ) is also generated by x1 , . . . , xn−1 in the neighborhood of the origin t = 0. Moreover, we can choose a neighborhood of the origin on which QHt∗ (F ln ) 3
is a complete intersection ring. Then, let I ⊂ C[x1 , . . . , xn−1 ] be the defining ideal of the quantum cohomology ring QHt∗ (F ln ). The Schubert class Ωv is expressed by the Schubert polynomial Sv (x1 , . . . , xn−1 ) in the classical cohomology ring. However, in the quantum cohomology ring, the class corresponding to the Schubert polynomial Sv is no longer the Schubert class Ωv . e t (x1 , . . . , xn−1 ) expressing Ωv gives a deformation of Hence, the polynomial S v the Schubert polynomial. We call it a big quantum Schubert polynomial. We identify the residue pairing defined by I with the intersection form on the cohomology ring, so the big quantum Schubert polynomials are obtained by orthogonalization the basis consisting of the classical Schubert polynomials. It is difficult to describe the defining ideal I of the big quantum cohomology ring for generic t, so the big quantum Schubert polynomials are complicated in general. However, in the case where parameters tv = 0 for all permutations v ∈ Sn such that l(v) > 1, the defining relations of the quantum cohomology ring are known by the results of A. Givental and B. Kim [GK] and I. Ciocan-Fontanine [C]. We call it the small quantum cohomology ring. The structure constants of the small quantum cohomology ring are given by ∂3F (t(2) ) = ∂tv1 ∂tv2 ∂tv3
=
X X
Fl hI0,Σm i(Ωv1 ⊗ Ωv2 ⊗ Ωv3 ⊗ ( u +3,β
Y
β mu ≥0
=
X
O
l(u)=1
mu !
l(u)=1
u Ω⊗m )) u
Y
u tm u
l(u)=1
Fl hI0,3,β i(Ωv1 ⊗ Ωv2 ⊗ Ωv2 )eΣbi t(i,i+1) ,
β
where the sum runs over β = b1 X(1,2) + · · · + bn−1 X(n−1,n) with bi ∈ Z≥0 . Hence the small quantum cohomology ring is determined by Fl the invariants hI0,3,β i. For a monomial xi1 ∗ · · · ∗ xim , the m-point correlation function determined by the small quantum cohomology ring (the so-called small quantum cohomology ring correlation function) is defined to be hxi1 · · · xim i =
Z
F ln
4
xi1 ∗ · · · ∗ xim .
If m ≥ 4, the small quantum cohomology ring correlation function can be expressed as follows: hxi1 · · · xim i = X
X
−
e
β=β1 +···+βm−3 v1 ,...,vm−3
R
β
ω
hI0,3,β1 i(xi1 ⊗ xi2 ⊗ Ωv1 )hI0,3,β2 i(Ωv1 w0 ⊗ xi3 ⊗ Ωv2 )
· · · hI0,3,βm−2 i(Ωvm−3 w0 ⊗ xim−1 ⊗ xim ). Hence, if m ≥ 4, the Gromov-Witten invariants hI0,m,β i(xi1 ⊗ · · · ⊗ xim ) do not appear as coefficients of small quantum cohomology ring correlation functions. Let us explain briefly the main results obtained in our paper. Follow to A. Givental and B. Kim [GK], and I. Ciocan–Fontanine [C], we define the quantum elementary symmetric polynomials ee1 , . . . , een by the formula
det
x1 + t q1 0 −1 x2 + t q2 0 −1 x3 + t .. .. .. . . . 0 ... 0 0 ... ... 0 ... ...
... 0 q3 .. .
... ... 0 .. .
... ... ... .. .
0 0 0 .. .
−1 xn−2 + t qn−2 0 0 −1 xn−1 + t qn−1 ... 0 −1 xn + t
= tn + ee1 tn−1 + ee2 tn−2 + · · · + een ,
where qi = et(i,i+1) . The defining ideal Ie of the small quantum cohomology ring is generated by the quantum elementary symmetric polynomials, namely QH ∗ (F ln , Z) := QH ∗ (F ln ) = Z[x1 , . . . , xn ; q1 , . . . , qn−1 ]/(ee1 , . . . , een ).
(1)
In the classical case q1 = · · · = qn−1 = 0, on the quotient ring A
:= Z[x1 , . . . , xn ]/(e1 (x), . . . , en (x)) ≃ H ∗ (F ln , Z)
there exists a natural pairing hf, gi = η(∂w0 (f g)) which comes from the intersection pairing in the homology group H∗ (F ln , Z) of the flag variety. We can interpret the pairing h, i as the Grothendieck residue pairing with respect to the ideal I (see Subsection 2.5): hf, gi = ResI (f g), 5
e where I = I| q=0 . Our first observation is that a natural residue pairing (we call it the quantum residue pairing)
hf.giQ = ResIe(f g)
on the quotient ring A := Z[x1 , . . . , xn ]/Ie corresponds to the intersection pairing in quantum cohomology QH ∗ (F ln , Z) under a natural isomorphism (1). It is well-known (e.g. [LS2], [M]) that the classical Schubert polynomials form an orthonormal basis (with respect to the pairing h, i) in the cohomology ring of flag manifold and also give a linear basis in the quantum cohomology ring QH ∗ (F ln , Z), [GK], [MS]. However, the classical Schubert polynomials do not orthogonal with respect to the quantum pairing any more. Thus, it is natural to ask: what kind of polynomials one can obtain applying the Gram–Schmidt orthogonalization to the classical Schubert polynomials with respect to the quantum pairing h, iQ ? Omiting some details with ordering (see Definition 5), the answer is: quantum Schubert polynomials. Our second observation is: to work with the equivariant quantum cohomology algebra ([GK], [K2]) is more convenient then with quantum cohomology ring itself. The main reason is that one can find Lascoux–Sch¨ utzenberger’s type representative for any equivariant quantum cohomology class. In other e w (x, y) can be obtained words, each quantum double Schubert polynomial S from the top one by using the divided difference operators acting on the y variables. Theorem-Definition A Let x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) be two sets of variables, and e (q) (x, y) S w0
where ∆k (t | x1 , . . . , xk ) :=
:=
k X
n−1 Y
∆i (yn−i | x1 , . . . , xi ),
i=1
tk−j ej (x1 , . . . , xk | q1 , . . . , qk−1 ) is the generat-
j=0
ing function for the quantum elementary symmetric functions in x1 , . . . , xk . e (q) (x, y) = ∂ (y) S e (q) Then S w ww0 w0 (x, y). e (q) (x) as Gram–Schmidt’s We define the quantum Schubert polynomials S w orthogonalization of the set of lexicographically ordered monomials 6
{xI | I ⊂ (n − 1, n − 2, . . . , 1, 0)} with respect to the quantum pairing h, iQ , see Definition 5. One of our main results is the quantum analog of Cauchy’s identity for (classical) Schubert polynomials, [M], (5.10). Theorem B (Quantum Cauchy’s identity) X
w∈Sn
e (q) (x)Sww (y) S 0 w
(q)
e (x, y). =S w0
(2)
We give a geometric proof of Theorem B in Section 7 using the arguments due to I. Ciocan-Fontanine [C]; more particularly, we reduce directly a proof of Theorem B to that of the following geometric statement: Lemma Let I ⊂ δ = (n − 1, n − 2, . . . , 1, 0) and w ∈ Sn be a permutation, then e w (x)iQ = heI (x), Sw (x)i, heeI (x), S (3)
where eI (x) :=
(resp. eeI (x) :=
n−1 Y
eik (x1 , . . . , xn−k )
k=1
eeik (x1 , . . . , xn−k | q1 , . . . , qn−k−1))
k=1 n−1 Y
is the elementary polynomial (resp. quantum elementary polynomial), see Section 5.2. It is the formula (3) that we prove in Section 7 using the geometrical arguments from [C] and [K2]. By product, it follows from our proof that ˆ w (x) defined geometrically (see Section 6) quantum Schubert polynomials S coincide with those defined algebraically (see Definition 5): ˆ w (x) S
e e w−1 (x) (mod I). ≡S
It is interesting to note, that the intersection numbers heI (x), Sw (x)i (which are nonnegative!) are precisely the coefficients of corresponding Schubert polynomial: X Sw (x) = heI (x), Sw (x)ixδ−I . I⊂δ
The quantum Cauchy formula (2) plays the important role in our approach to the quantum Schubert polynomials. As a direct consequence of (2), we obtain the Lascoux–Sch¨ utzenberger type formula for quantum Schubert polynomials (cf. Theorem-Definition A). 7
e w (x, y) be as in Theorem-Definition A, then Theorem C Let S 0 e w (x) S
(y) e = ∂ww Sw0 (x, y)|y=0 . 0
In Section 5 we introduce a quantization map Pn → P n , f 7→ fe.
The quantization is a linear map which preserves the pairings, i.e., hfe, geiQ = hf, gi, f, g ∈ Pn .
Using the quantum Cauchy formula (2), we prove that quantum double Schubert polynomials are the quantization of classical ones. Another class of polynomials having a nice quantization is the set of elementary polynomials eI (x) :=
n−1 Y
eik (x1 , . . . xn−k ), I = (i1 , . . . , in−1 ) ⊂ δ.
k=1
It follows from Theorem B that quantization eeI (x) of elementary polynomial eI (x) is given by eeI (x) =
n−1 Y
eik (x1 , . . . , xn−k | q1 , . . . , qn−k−1).
k=1
More generally, we make a conjecture (”quantum Schur functions ”) that quantization of the flagged Schur function (see [M], (3.1), (4.9) and (6.16))
sλ/µ (X1 , . . . , Xn ) = det hλi −µj −i+j (Xi )
1≤i,j≤n
is given by e seλ/µ (X1 , . . . , Xn ) = det h λi −µj −i+j (Xi )
1≤i,j≤n
,
e (X) is the quantum complete homogeneous symmetric function of where h k degree k, and X1 ⊂ · · · ⊂ Xn are the flagged sets of variables (see Section 5). In Section 5.2 we consider a problem how to quantize monomials. It seems to be difficult to find an explicit determinantal formula for a quantum monomial xeI , i.e., to find a quantum analog of the Billey-Jockusch-Stanley
8
formula for Schubert polynomials in terms of compatible sequences [BJS]. We prove the following formulae for quantum monomials xeI =
X
w∈Sn
e w (x), I ⊂ δ, η(∂w xI )S
e w (x, y) S 0
=
X
I⊂δ
xeI eδ−I (y).
In section 8.1 we give a proof of the higher genus analog of the Vafa– Intriligator type formula for the flag manifold. In Section 8.3 we study a problem how to compute the quantum residues. This is important for computation of small quantum cohomology ring correlation functions (or correlation functions, for short) and the Gromov–Witten invariants, see Introduction and Theorem 11. We introduce the generating function n−1 Y ti Ψ(t) = h i i=1 ti − xi for quantum residues and give a characterization of this function as the unique solution to some system of differential equations, see Proposition 14. In Appendix B we calculate the generating function Ψ(t) for the case n = 3 explicitly. In Section 9 we introduce the extended Ehresman–Bruhat order and give a sketch of a proof of equivariant quantum Pieri rule. Details will appear elsewhere. In Appendix A one can find a list of explicit expressions for the quantum double Schubert polynomials for the symmetric group S4 . We would like to mention, that in the recent preprint “Quantum Schubert polynomials” by S. Fomin, S. Gelfand and A. Postnikov, [FGP], developed a different approach to the theory of quantum Schubert polynomials, based on the remarkable family of commuting operators Xi ([FGP], (3.2)). Among main results, obtained by S. Fomin, S. Gelfand and A. Postnikov, are definitions, orthogonality, quantum Monk’s formula and other properties of quantum Schubert polynomials; definition of quantization map and quantum multiplication. Besides some overlap with the preprint of S. Fomin, S. Gelfand and A. Postnikov, our works were done independently and based on the different approaches, which allow to obtain the mutually complementary results. 9
2
Classical Schubert polynomials.
In this section we give a brief review of the theory of Schubert polynomials created by A. Lascoux and M.-P. Sch¨ utzenberger. In exposition we follow to the I. Macdonald book [M1] where proofs and more details can be found.
2.1
Divided differences.
Let x1 , . . . , xn , . . . be independent variables, and let Pn := Z[x1 , . . . , xn ] for each n ≥ 1, and P∞ := Z[x1 , x2 , . . .] =
∞ [
Pn .
(4)
n=1 Sn Let us denote by Λn := Z[x1 , . . . , xn ]X ⊂ Pn the ring of symmetric polynomials in x1 , . . . , xn , and by Hn := { aI xI | aI ∈ Z, 0 ≤ ik ≤ n − k, ∀k} I=(i1 ,...,in )
the additive subgroup of Pn spanned by all monomials xI := xi11 xi22 . . . xinn with I ⊂ δ := δn = (n − 1, n − 2, . . . , 1, 0). For 1 ≤ i ≤ n − 1 let us define a linear operator ∂i acting on Pn (∂i f )(x) =
f (x1 , . . . , xi , xi+1 , . . . , xn ) − f (x1 , . . . , xi+1 , xi , . . . , xn ) . xi − xi+1
(5)
Divided difference operators ∂i satisfy the following relations ∂i2 = 0, ∂i ∂j = ∂j ∂i , if | i − j |> 1, ∂i ∂i+1 ∂i = ∂i+1 ∂i ∂i+1 ,
(6)
and the Leibnitz rule ∂i (f g) = ∂i (f )g + si (f )∂i (g).
(7)
It follows from (7) that ∂i is a Λn -linear operator. For any permutation w ∈ Sn , let us denote by R(w) the set of reduced words for w, i.e. sequences (a1 , . . . , ap ) such that w = sa1 · · · sap , where 10
p = l(w) is the length of permutation w ∈ Sn , and si = (i, i + 1) is the simple transposition that interchanges i and i + 1. For any sequence a = (a1 , . . . , ap ) of positive integers, we define ∂a = ∂a1 · · · ∂ap .
Proposition 1 ([M1], (2.5),(2.6)) • If a, b ∈ R(w), then ∂a = ∂b . • If a is not reduced, then ∂a = 0. From Proposition 1 it follows that an operator ∂w = ∂a is well-defined, where a is any reduced word for w. By (7), the operators ∂w , w ∈ Sn , are Λn linear, i.e. if f ∈ Λn , then ∂w (f g) = f ∂w (g).
2.2
Schubert polynomials.
Let δ = δn = (n − 1, n − 2, . . . , 1, 0), so that xδ = x1n−1 x2n−2 . . . xn−1 . Definition 1 (Lascoux–Sch¨ utzenberger [LS1]). For each permutation w ∈ Sn the Schubert polynomial Sw is defined to be Sw (x)
= ∂w−1 w0 (xδ ),
where w0 is the longest element of Sn . Proposition 2 ([M1], (4.2),(4.5),(4.11),(4.15)). • Let v, w ∈ Sn . Then ∂v Sw =
(
Swv−1 ,
0,
if l(wv −1 ) = l(w) − l(v), otherwise.
11
• (Stability). Let m > n and let i : Sn ֒→ Sm to be the natural embedding. Then Sw = Si(w) . • The Schubert polynomials Sw , w ∈ Sn form a Z–basis of Hn . • (Monk’s formula). Let f =
n X
αi xi , w ∈ Sn . Then
i=1
f Sw =
X
(αi − αj )Swtij ,
∂w (f g) = w(f )∂w g +
X
(αi − αj )∂wtij g,
where tij is the transposition that interchanges i and j, and both sums are over all pairs i < j such that l(wtij ) = l(w) + 1.
2.3
Scalar product.
Let us define a scalar product on Pn with values in Λn , by the rule hf, gi = ∂w0 (f g), f, g ∈ Pn ,
(8)
where w0 is the longest element of Sn . The scalar product h, i defines a non-degenerate pairing h, i0 on the quotient ring Pn /In ∼ = H ∗ (F ln , Z), where In is the ideal in Pn generated by the elementary symmetric polynomials e1 (x), . . . , en (x). Proposition 3 ([M1], (5.3),(5.4),(5.6),(4.13),(5.10)). • If f ∈ Λn , then hf h, gi = f hh, gi; • If f, g ∈ Pn , w ∈ Sn , then h∂w f, gi = hf, ∂w−1 gi; (
1, if u = w0 v, 0, otherwise. • The Schubert polynomials Sw , w ∈ Sn , form a Λn –basis of Pn ;
• (Orthogonality) If l(u) + l(v) = l(w0 ), then hSu , Sv i =
• The Schubert polynomials Sw , w ∈ S (n) , form a Z–basis of Pn , where for each n ≥ 1, S (n) is the set of all permutations w such that the code w has length ≤ n; • (Cauchy’s formula) X
Sw (x)Sww0 (y)
=
Y
(xi + yj ).
i+j≤n
w∈Sn
12
Proposition 4 Schubert polynomials are uniquely characterized by the following properties ( 1, if u = w0 v, 1. (Orthogonality) hSu , Sv i0 = 0, otherwise. 2. Let w be a permutation in Sn and c(w) = (c1 , c2 , . . . , cn ) its code, then Sw (x)
= xc(w) +
X
αI xI ,
where I ⊂ δ, αI > 0 and I lexicographically smaller then c(w). Remark 1 1) (Definition of the code, [M1], p.9). For a permutation w ∈ Sn , we define ci = ♯{j | i < j, w(i) > w(j)}. The sequence c(w) = (c1 , c2 , . . . , cn ) is called the code of w. 2) Schubert polynomials are obtained as Gram-Schmidt’s orthogonalization of the set of monomials {xI }I⊂δ ordered lexicographically.
2.4
Double Schubert polynomials.
Let x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) be two sets of independent variables, and Y Sw0 (x, y) := (xi + yj ). i+j≤n
Definition 2 (Lascoux–Sch¨ utzenberger [LS2]). For each permutation w ∈ Sn , the double Schubert polynomial Sw (x, y) is defined to be Sw (x, y)
(x)
= ∂w−1 w0 Sw0 (x, y), (x)
where divided difference operator ∂w−1 w0 acts on the x variables. Proposition 5 X ([M1], (6.3),(6.8)). • Sw (x, y) = Su (x)Suw−1 (y), summed over all u ∈ Sn , such that u
l(u) + l(uw −1) = l(w); • (Interpolation formula). For all f ∈ Z[x1 , . . . , xn ] we have f (x) =
X
(y) Sw (x, −y)∂w f (y)
w
summed over all permutations w ∈ S (n) . 13
Double Schubert polynomials appear in algebra and geometry as cohomology classes related to degeneracy loci of flagged vector bundles. If h : E → F is a map of rank n vector bundles on a smooth variety X, E1 ⊂ E2 ⊂ · · · ⊂ En = E, F := Fn → Fn−1 → · · · → F1 are flags of subbundles and quotient bundles, then there is a degeneracy locus Ωw (h) for each permutation w in the symmetric group Sn , described by the conditions Ωw (h) = {x ∈ X | rank(Ep (x) → Fq (x)) ≤ #{i ≤ q, wi ≤ p}, ∀p, q}. For generic h, Ωw (h) is irreducible, codim Ωw (h) = l(w), and the class [Ωw (h)] of this locus in the Chow ring of X is equal to the double Schubert polynomial Sw0 w (x, −y), where xi = c1 (ker(Fi → Fi−1 )), yi = c1 (Ei /Ei−1 ), 1 ≤ i ≤ n. It is well-known [F] that the Chow ring of flag variety F ln admits the following description CH ∗(F ln ) ∼ = Z[x1 , . . . , xn , y1 , . . . , yn ]/J, where J is the ideal generated by ei (x1 , . . . , xn ) − ei (y1 , . . . , yn ), 1 ≤ i ≤ n, and ei (x) is the i-th elementary symmetric function in the variables x1 , . . . , xn . • ([LS2], [KV]) The ring Z[x1 , . . . , xn , y1 , . . . , yn ]/J is a free module of dimension n! over the ring R, with basis either Sw (x), or Sw (x, y), w ∈ Sn , where Z[x1 , . . . , xn ] ⊗ Sym[y1 , . . . , yn ] . R := J
2.5
Residue pairing.
Let I be an ideal in P¯n = R[x1 , . . . , xn ], R ⊂ C, generated by a regular system of parameters ϕ1 , . . . , ϕn , and A := P¯n /I.
14
Proposition 6 ([GH], [EL]). • dimR A < ∞. ! ∂ϕi • H := det 6∈ I. ∂xj Let d0 := deg H, where we assume that deg xi = 1 for all 1 ≤ i ≤ n. Proposition 7 ([EL]) • If f ∈ P¯n and deg f = d0 , then there exists a non zero α ∈ R such that f≡
α H (mod I). n!
• If f ∈ P¯n , f 6= 0, and deg f > d0 , then there exists g ∈ P¯n such that deg g ≤ d0 and g ≡ f (mod I). Definition 3 (Grothendieck residue with respect to the ideal I). Let f ∈ P¯n and deg f < d0 , then we define ResI (f ) = 0. α H (mod I) and we define ResI (f ) := α. n! Finally, if deg f > d0 , then choose g ∈ P¯n such that g ≡ f (mod I) and deg g ≤ d0 , and define ResI (f ) := ResI (g). If deg f = d0 , then f ≡
We will use also notation hf iI instead of ResI (f ). Finally, let us define a residue pairing h, iI on P n using the Grothendieck residue hf, giI = ResI (f, g), f, g ∈ P n .
Proposition 8 ([GH]). • If f ∈ I, then ResI (f ) = 0. • The residue pairing h, iI induces a non-degenerate pairing on A = P /I.
15
We will use this general construction of residue pairing in the following two cases: i) R = Z, In ⊂ Pn is an ideal generated by elementary symmetric polynomials e1 (x), . . . , en (x). It is well-known that if F ln := SL(n)/B is the flag variety of type An−1 , then H ∗ (F ln , Z) ≃ Pn /In , and residue pairing h, i on Pn /In coincides with the scalar product on Pn /In induced by (8). ii) R = Z[q1 , . . . , qn−1 ], Ien ⊂ P n is an ideal generated by the quantum elementary symmetric functions ee1 (x), . . . , een (x). It is a result of A. Grivental and B. Kim, and I. Ciocan–Fontanine, that QH ∗ (F ln ) ≃ P n /Ien ,
and the residue pairing defined by Ien may be naturally identified with the intersection form on the quantum cohomology ring. We will call this residue pairing as quantum pairing on P n /Ien and denote it by h, iQ .
3
Quantum double Schubert polynomials.
Quantum double Schubert polynomials are closely related with the equivariant quantum cohomology. Let us remind the result of A. Givental and B. Kim [GK] (see also [K2]) on the structure of the equivariant quantum cohomology algebra of the flag variety F ln : e QHU∗ n (F ln ) ∼ = Z[x1 , . . . , xn , y1 , . . . , yn , q1 , . . . , qn−1 ]/J,
where the ideal Je generated by
ei (x1 , . . . , xn | q1 , . . . , qn−1 ) − ei (y1, . . . , yn ), 1 ≤ i ≤ n.
In classical case q = 0, the double Schubert polynomials Sw (x, y) represent the equivariant cohomology classes [F]. Quantum double Schubert polynomials have to play the similar role for the quantum equivariant cohomology ring. Let us define at first the “top” quantum double Schubert polynomial e w (x, y). S 0 16
Let x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) be two sets of variables, put e w (x, y) S 0
:=
e (q) (x, y) S w0
where ∆k (t | x1 , . . . , xk ) :=
k X
=
n−1 Y
∆i (yn−i | x1 , . . . , xi ),
i=1
tk−j ej (x1 , . . . , xk | q1 , . . . , qk−1 ) is the generat-
j=0
ing function for the quantum elementary symmetric polynomials in x1 , . . . , xk , i.e.
∆k (t|x) :=
k X
ei (x|q)ti =
i=1
det
x1 + t q1 0 −1 x2 + t q2 0 −1 x3 + t .. .. .. . . . 0 ... 0 0 ... ... 0 ... ...
... 0 q3 .. .
... ... 0 .. .
... ... ... .. .
0 0 0 .. .
−1 xk−2 + t qk−2 0 0 −1 xk−1 + t qk−1 ... 0 −1 xk + t
.
(9)
Definition 4 For each permutation w ∈ Sn , the quantum double Schubert e w (x, y) is defined to be polynomial S e w (x, y) S
(y) e = ∂ww Sw0 (x, y), 0
(y) where divided difference operator ∂ww acts on the y variables. 0
Remark 2 i) In the ”classical limit” q1 = · · · = qn−1 = 0, e w (x, y)|q=0 S
(y) = ∂ww Sw0 (x, y) = Sw−1 (y, x) = Sw (x, y), 0
e w (x, y)|q=0 = Sw (x, y). i.e. S ii) (Stability) Let m > n and let i : Sn ֒→ Sm be the embedding. Then e w (x, y) S
iii) One can check that the ring
e i(w) (x, y). =S
Z[x1 , . . . , xn , y1, . . . , yn , q1 , . . . , qn−1 ]/Je 17
e with basis either is a free module of dimension n! over the quotient ring R e w (x) or S e w (x, y), w ∈ Sn , where S e := R
Z[x1 , . . . , xn , q1 , . . . , qn−1 ] ⊗ Sym[y1 , . . . , yn ] . Je
Example. Quantum double Schubert polynomials for S3 : e s s s (x, y) S 1 2 1 e s s (x, y) S 2 1
e s s (x, y) S 1 2 e Ss (x, y) 1
e s (x, y) S 2 e Sid (x, y)
= = = = = =
(x1 + y2 )(x1 + y1 )(x2 + y1 ) + q1 (x1 + y2 ), (x1 + y1 )(x1 + y2 ) − q1 , (x1 + y1 )(x2 + y1 ) + q1 , x1 + y1 , x1 + x2 + y1 + y2 , 1.
For the list of the quantum double Schubert polynomials corresponding to the symmetric group S4 , see appendix A. Theorem 1 Let z = (z1 , . . . , zn ) be a third set of variables. Then (x)
e w (x, y), S e w (x, z)i hS 0 0 Q = C(y, z),
(10)
where the upper index x means that the quantum pairing is taken in the x variables, and X Sw (x)Sw0 w (y) C(x, y) = w∈Sn
is the ”canonical” element in the tensor product H ∗ (F ln ) ⊗ H ∗ (F ln ). Theorem 1 plays the important role in our approach to the quantum Schubert polynomials. We will give a proof later, and now let us consider some applications of the formula (10).
4
Quantum Schubert polynomials.
18
4.1
Definition.
Let us remind the result of A. Givental and B. Kim, and I. Ciocan-Fontanine on the structure of the small quantum cohomology ring of flag variety F ln e QH ∗ (F ln ) ∼ = Z[x1 , . . . , xn , q1 , . . . , qn−1 ]/I,
where the ideal Ie is generated by the quantum elementary symmetric polynomials eei (x) := ei (x1 , . . . , xn |q1 , . . . , qn−1 ), 1 ≤ i ≤ n with generating function ∆n (t|x), see (9). We define a pairing on the ring of polynomials Z[x; q] and the quantum cohomology ring QH ∗ (F ln ) ≃ Z[x; q]/Ie using the Grothendieck residue hf, giQ = ResIe(f g), f, g ∈ Z[x1 , . . . , xn , q1 , . . . , qn−1 ].
Then e 1) hf, giQ = 0 if f ∈ I;
2) hf, giQ defines a nondegenerate pairing in QH ∗ (F ln ).
e w := S e w (x) as Definition 5 Define the quantum Schubert polynomials S Gram–Schmidt’s orthogonalization of the set of lexicographically ordered monomials {xI | I ⊂ δ} with respect(to the quantum residue pairing hf, giQ: 1, if v = w0 u e u, S e v iQ = hSu , Sv i = 1) hS 0, otherwise X c(w) e w (x) = x 2) S + aI (q)xI , where aI (q) ∈ Z[q1 , . . . , qn−1 ] and I n then h n k e (x) ∈ Ie . I 6⊂ δ, then hI (x) ∈ In and h I n It is not difficult to see that Pn is a free Λn -module of rank n! with basis {hI (x) | I ∈ T}. e (x) is the quantization of complete polynomial Theorem 5’ If I ⊂ δ, then h I hI (x). Proof. See Remark 10 in Section 7.
5.3
Canonical involution ω. ←
←
There exists an involution ω of the ring P n [y] given by ω(x) = x, ω(y) = y , ← ← ω(q) = q , where for any sequences z = (z1 , . . . , zm ) we define z to be equal to 28
←
(zm , zm−1 , . . . , z1 ) := z . It is clear from the definition of quantum elementary symmetric functions ei (x|q), see (9), Section 3, that ω(ei (x|q)) = ei (x|q) and thus the involution ω preserves the ideal Ien (as well as the ideals In , Jn and Jen ). Proposition 14
e e (q) (x, y)) ≡ ǫ(u)S e (q) ω(S u w0 uw0 (x, y) mod Jn ,
where ǫ(u) = (−1)l(u) .
Proof. First of all, if u = w0 , then ←
← e e (q) (x, y)) ≡ ǫ(w0 )S e ( q ) (← ω(S w0 w0 x, y ) mod Jn .
(24)
But ω∂u = ǫ(u)∂w0 uw0 ω (see [M1], (2.12)). Thus, applying the divided difference operator ǫ(u)∂w(y)0 uw0 to the both sides of (24), we obtain ←
← e e (q) (x, y)) ≡ ǫ(w0 )S e ( q ) (← ω(S uw0 w0 u x, y ) mod Jn .
Finally, let us describe the action of involution ω on the elementary polynomials. Proposition 15 e ← (x) mod Ie . ω(eeI (x)) ≡ (−1)|I| h n I
Remark 9 To our knowledge, originaly, construction of the quantization map, using a remarkable family of commuting operators Xi , appeared in [FGP]. We use a different definition of quantization map, but it can be shown that two forms of quantizations are equivalent. For original proofs of Theorem 5 and 5’, and Proposition 15, see Corollary 4.6, Corollary 7.16 and Proposition 7.13 in [FGP].
29
6
Quantum cohomology ring of flag variety.
Quantum cohomology ring of the flag variety F ln is a deformed ring of the ordinary cohomology ring H ∗ (F ln , Z). The structure constants of the quantum cohomology ring are given by the Gromov–Witten invariants. Let Ωw1 , . . . , Ωwm (wi ∈ Sn ) be Schubert cycles. We denote by Md¯(P1 , F ln ) the moduli space of morphisms from P1 to F ln of multidegree d¯ = (d1 , . . . , dn−1 ). We consider the restriction of the universal map for t ∈ P1 : ev
evt : Md¯(P1 , F ln ) × {t} ֒→ Md¯(P1 , F ln ) × P1 →F ln , (f, p) 7→ f (p). Let Ωw (t) = evt−1 (Ωw ). m X
X n(n − 1) +2 di and 2 i=1 t1 , . . . , tm ∈ P1 are distinct, then for general translates of Ωwi , the number
Theorem 7 (I. Ciocan-Fontanine). If
of points in
m \
l(wi ) =
Ωwi (ti ) is finite and independent of ti and the translates of
i=1
Ωwi .
Definition 8 The Gromov–Witten invariant is defined as an intersection number hΩw1 . . . Ωwm id¯ =
#
\
Ωwi (ti ), if
i
0,
P
l(wi ) =
n(n−1) 2
+2
otherwise
P
di
.
Now we can define the quantum multiplication as a linear map mq : Sym (H ∗ (F ln , Z)[q1 , . . . , qn−1 ]) → H ∗ (F ln , Z)[q1 , . . . , qn−1 ] given by mq (
m Y
i=1 d¯
Ωwi ) =
X d¯
¯X
qd
hΩw Ωw1 · · · Ωwm id¯Ω∗w ,
w
dn−1 q1d1 · · · qn−1
where q = and (Ω∗w ) is the dual basis of (Ωw ). Then the quantum cohomology ring QH ∗ (F ln ) is a commutative and associative Z[q1 , . . . , qn−1 ] – algebra. Let 0 = E0 ⊂ E1 ⊂ · · · ⊂ En = Cn ⊗ OF be the universal flag of subbundles on F ln . 30
Theorem 8 (A. Givental and B. Kim, I. Ciocan–Fontanine). The small quantum cohomology ring is generated by xi = c1 (En−i+1 /En−i ), i = 1, . . . , n, as a Z[q1 , . . . , qn−1 ]-algebra and QH ∗ (F ln ) ∼ = Z[x1 , . . . , xn , q1 , . . . , qn−1 ]/(e1 (x|q), . . . , en (x|q)), where ei (x|q) is given by the expansion of the following determinant
det
x1 + t q1 0 −1 x2 + t q2 0 −1 x3 + t .. .. .. . . . 0 ... 0 0 ... ... 0 ... ...
... 0 q3 .. .
... ... 0 .. .
... ... ... .. .
0 0 0 .. .
−1 xn−2 + t qn−2 0 0 −1 xn−1 + t qn−1 ... 0 −1 xn + t
= tn + e1 (x|q)tn−1 + · · · + en (x|q). It follows from Theorem 8 that any Schubert cycle Ωw may be expressed b w (x, q) in QH ∗ (F ln ). The polynomial S b w (x, q) is a deforas a polynomial S b w (x, 0) = Sw (x). Consider mation of the Schubert polynomial Sw (x) and S the correlation function hΩw1 . . . Ωwm i =
X
¯
q d hΩw1 . . . Ωwm id¯.
d¯
b w (x; q) is characterized by the condition Then S
b w (x, q)Ωw . . . Ωw i hΩw Ωw1 . . . Ωwm i = hS m 1
for any w1 , . . . , wm ∈ Sn . b w (x; q) is called a geometric quantum Schubert polynomial. By definiS b w (x; q) ∈ QH 2l(w) (F ln ). tion S
7
Proofs of Theorem 3 and quantum Cauchy formula. 31
Theorem 9 Let I ∈ T. Then e w (x)iQ = heI (x), Sw (x)i, heeI (x), S
for any permutation w ∈ Sn .
Proof. The proof is based on the arguments due to I. Ciocan-Fontanine [C]. To begin with, let us remind his results. We consider the hyper-quot scheme HQd¯(P1 , F ln ) associated to P1 with multidegree d¯ = (d1 , . . . , dn−1 ). Let Cn ⊗ O → Tn−1 → · · · → T2 → T1 → 0 be the universal sequence of quotients on P1 × HQd¯(P1 , F ln ) and Si = Ker{Cn ⊗ O → Tn−i }. We also consider the dual sequence ∗ Cn ⊗ O → Sn−1 → · · · → S1∗ .
We fix a flag 0 = V 0 ⊂ V 1 ⊂ · · · ⊂ V n = Cn and define the subschema Dwp,q of P1 × HQd¯(P1 , F ln ) as the locus where rank(Vp ⊗ O → Sq∗ ) ≤ rw (q, p), and rw (q, p) := ♯{i | i ≤ q, wi ≤ p}. Let Dwp,q (t) = Dwp,q and
\
{{t} × HQd¯(P1 , F ln )}
Ωw (t) =
n−1 \
Dwp,q (t).
p,q=1
¯ w (t) in the Chow ring CH l(w)(HQd¯(P1 , F ln )) is indepenThen the class of Ω dent of t ∈ P1 and the flag V0 ⊂ · · · ⊂ Vn . The boundary HQd¯(P1 , F ln ) \ Md (P1 , F ln ) consists of n − 1 divisors D1 , . . . , Dn−1 , which are birational to P1 × HQd¯1 (P1 , F ln ), . . . , P1 × HQd¯n−1 (P1, F ln )
32
¯ s (t), respectively, where d¯i = (d1 , . . . , di −1, . . . , dn−1 ). Let xi (t) = Ωsi (t)−Ω i−1 then for any permutation w ∈ Sn there exists an element Gw (t) ∈ CH∗ (
n−2 [
Di )
i=1
such that
¯ w−1 (t) − Sw (x1 (t), . . . , xn−1 (t)) = j∗ (Gw−1 (t)), Ω
where j:
n−2 [
Di → HQd¯(P1 , F ln )
i=1
is the inclusion. Let [m, k] ∈ Sn be the permutation 1 2 ... m− k − 1 m− k m− k + 1 ... m m + 1 ... n 1 2 ... m− k − 1 m m−k ... m −1 m + 1 ... n
!
.
ˆ [m,k]−1 (x) is the elementary symThen the geometric Schubert polynomial S metric function in x1 , . . . , xm−1 of degree k. Let I = (i1 , . . . , in ). We have to calculate in CH ∗ (HQd¯(P1 , F ln ))
n−1 \
Ω[n−ν+1,iν
ν=1
]−1 (t) ·
N Y
Ωwj (tj ) −
n−1 Y
S[n−ν+1,iν ] (x(t)) ·
ν=1
j=1
N Y
Ωwj (tj ), (25)
j=1
for distinct t, t1 , . . . , tN and w1 , . . . , wN ∈ Sn such that n−1 X
iν +
ν=1
N X
l(wj ) = n(n − 1)/2 + 2
j=1
n−1 X
dk ,
k=1
where ∩ is the classical intersection product and
n−1 \ ν=1
Ω[n−ν+1,iν ]−1 (t)
is the corresponding degeneracy locus on HQd¯(P1 , F ln ). In order to calculate the expression (25), we are going to prove at first that
n−1 \ ν=1
Ω[n−ν+1,iν
]−1 (t) ·
N Y
j=1
¯ w (tj ) − Ω j
n−1 Y ν=1
33
¯ [n−ν+1,i ]−1 (t) · Ω ν
N Y
j=1
¯ w (tj ) = 0. Ω j
The LHS of the last expression can be computed as the number of points in n−1 Y
¯ [n−ν+1,i ]−1 (t) · Ω ν
ν=1
N Y
¯ w (tj ) Ω j
j=1
n−2 suppoted on ∪i=1 Di . Let jd¯k be the natural rational map
P1 × HQd¯k (P1 , F ln )− → HQd¯(P1 , F ln ). From Remark 3 in [C], ¯ jd−1 ¯k (Ω[n−ν+1,iν ]−1 (t)) = (
¯ [n−ν+1,i ]−1 (t) ∪ Tn−ν−i +1≤p≤n−ν−1 D p,n−ν−1 −1 (t), if k = n − ν, P1 × Ω ν ν [n−ν+1,iν ] 1 ¯ otherwise, P × Ω[n−ν+1,iν ]−1 (t)
where \
p,n−ν−1 ¯ D[n−ν+1,i −1 (t) = {t} × Ω[n−ν,iν −1]−1 (t). ν]
n−ν−iν +1≤p≤n−ν−1
Because, by assumption, n−1 X
iν +
n−1 X
l(wj ) =
ν=1
ν=1
n−1 X n(n − 1) +2 dk , 2 k=1
we have ¯ [k,i −1]−1 (t) · Ω n−k
Y
¯ [n−ν+1,i ]−1 (t) · Ω ν
N Y
¯ w (tj ) = 0 Ω j
j=1
ν6=n−k
on the hyper-quot scheme HQd¯k (P1 , F ln ). Hence, we have equality
n−1 \ ν=1
Ω[n−ν+1,iν
]−1
(t) ·
N Y
¯ w (tj ) = Ω j
n−1 Y
¯ [n−ν+1,i ]−1 (t) · Ω ν
ν=1
j=1
N Y
¯ w (tj ). Ω j
j=1
Our next observation is that the intersection number in the RHS of the last equality is equal to n−1 Y
Ω[n−ν+1,iν ]−1 (sν ) ·
ν=1
N Y
j=1
34
Ωwj (tj ),
where we can chose s1 , . . . , sn−1 , t1 , . . . , tN ∈ P 1 to be the pairwise distinct points, since the class [Ωw (t′ )] in the Chow ring CH ∗ (HQd (P 1 , F ln )) does not depends on the chose of t′ ∈ P 1 . Now we are going to use the following identity m Y
ak −
k=1
m Y
bk =
m k−1 X Y
bj (ak − bk )
k=1 j=1
k=1
m Y
aj .
j=k+1
Let us take in the last equality m = n − 1 ak := Ω[n−k+1,ik ]−1 (sk ), bk := S[n−k+1,ik ] (x(sk )). Then we obtain the following equality n−1 Y
Ω[n−ν+1,iν ]−1 (sν )
ν=1
n−1 Y X k−1
=
N Y
Ωwj (tj ) −
n−1 Y
S[n−ν+1,iν ] (x(sν ))
ν=1
j=1
N Y
Ωwj (tj )
j=1
Ω[n−j+1,ij ]−1 (sj ) · j∗ (G[n,ik ]−1 (t))
k=1 j=1
n−1 Y
S[n−j+1,ij ] (x(sj ))
j=k+1
Q
· B,
where B := N j=1 Ωwj (tj ). The contributions from j∗ ((G[n,ik ]−1 (t)), 1 ≤ k ≤ n − 1, can be computed by using the arguments in [C]. Inded, as in the proof of Theorem 4 in [C], the intersection number (1 ≤ k ≤ n − 1) k−1 Y
Ω[n−l+1,il]−1 (sl ) · j∗ (G[n,ik ]−1 (t))
n−1 Y
S[n−l+1,il ] (x(sl )) · B
l=k+1
l=1
is the number of points in k−1 Y
Ω[n−l+1,il]−1 (sj )
S[n−l+1,il ] (x(sl )) · B
l=k
l=1
S
n−1 Y
n−2 supported on j=1 Di . Hence, by induction, we have the following identity for correlation functions
h
n−1 \ ν=1
!
Ω[n−ν+1,iν ]−1 ·
N Y
Ωwj i = h
n−1 Y ν=1
j=1
35
e [n−ν+1,i ] S ν
·
N Y
j=1
Ωwj i,
where
T
is the classical intersection product and · is the product in Sym H ∗ (F ln , Z)[q1 , . . . , qn−1 ].
The last equality for correlation functions is equivalent to the following one mq (Ω[n,i1 ]−1 ∩ Ω[n−1,i2 ]−1 , ∩ . . . ∩ Ω[l,in−1 ]−1 , ∗) = mq (eei1 · eei2 · · · eein−1 , ∗).
This completes the proof.
Remark 10 Using the similar geometrical arguments we can prove an analog of Theorem 9 for the quantum complete polynomials: Theorem 9’ Let I ⊂ δn . Then e (x), S e w (x)iQ = hhI (x), Sw (x)i, hh I
for any permutation ω ∈ Sn . It is easy to see that Theorem 5’ is a corollary of Theorem 9’.
8 8.1
Correlation functions. Higher genus correlation function and the Vafa– Intriligator type formula.
Fix a Riemann surface C of genus g. We denote by Md (C, F ) the moduli space of morphism from C to F ln . One can define the higher genus Gromov– Witten invariants by method which is similar to that in the case of genus zero, [RT]. We have the following recursion relation for higher genus correlation function corresponding to the generating function for higher genus Gromov– Witten invariants hΩw1 . . . ΩwN ig =
X
hΩw1 . . . ΩwN Ωv Ω∗v ig−1
v∈Sn
(cf. Ruan–Tian [RT]). From Corollary 2 and Theorem 11 we can deduce the Vafa–Intriligator type formula for higher genus correlation functions, namely, let hP (x1 , . . . , xn )ig be the genus g correlation function corresponding to a polynomial P , then 36
hP (x1 , . . . , xn )ig = ResIe(P Φg ) = X
∂ eei = P (x1 , . . . , xn ) det ∂xj e e1 =···=e en =0
e w (x, y), S e w (x, y)i(y) = where Φ(x) = hS 0 0
X
w∈Sn
!−1
(Φ(x1 , . . . , xn ))g ,
e w (x)S e w w (x) S 0
= C (q,q) (x, x).
To simplify!the formula above, we use the following observations ∂ eei e e w (x) (mod I); • det ≡ n! S 0 ∂xj e e w (x) (mod I). • Φ(x) := C (q,q) (x, x) ≡ n! S 0 Hence, we obtain
Theorem 10 (Higher genus Vafa-Intriligator formula) hP (x1 , . . . , xn )ig = ResIe(P Φg ) =
X
e e1 =···=e en =0
e w (x) P (x1 , . . . , xn ) S 0
g−1
,
e w (x) = eeδ (x) := ee1 (x1 )ee2 (x1 , x2 ) . . . een−1 (x1 , . . . , xn−1 ), and eei (z) is where S 0 the quantum elementary polynomial of degree i in the variable z = (z1 , . . . , zm ), see Section 5.2.
Remark 11 The polynomial ′
C (q,q ) (x, y) :=
X
w∈Sn
′
e (q) (x)S e (q ) (y) S w w0 w
corresponds to the dual class of the diagonal in the quantum cohomology ring QH ∗ (F ln × F ln , (q, q ′)) = QH ∗ (F ln , q) ⊗ QH ∗ (F ln , q ′ ).
8.2
Witten–Dijkgraaf–Verlinde–Verlinde equations for symmetric group.
The Witten–Dijkgraaf–Verlinde–Verlinde equations (WDVV-equations) are e uS e vS e w i ∈ Z[q1 , . . . , qn−1 ], where equations on the correlation functions hS u, v, w ∈ Sn . The correlation functions satisfy the following conditions 1) Normalization: e vS e w i = hSv , Sw i. h1S 37
2) Initial data: e w i = qk . es S es S hS 0 k k
3) Degree conditions:
e uS e vS e wi = 0 hS
if either l(u) + l(v) + l(w) < l(w0 ), or difference l(u) + l(v) + l(w) − l(w0 ) is an odd positive integer. 4) WDVV-equations: X v
ew S ew S e v ihS e w vS ew S ew i = hS 1 2 0 3 4
X v
ew S e v ihS e w vS e w i. ew S ew S hS 2 3 0 1 4
for any w1 , w2 , w3 , w4 ∈ Sn . Conjecture 1 Conditions 1)–4) uniquely determine the correlation funce uS e vS e w i. tions hS ew S ew S e w i is a generating function Remark 12 1) Correlation function hS 1 2 3 for the Gromov–Witten invariants (see Definition 4): ew S ew S e w i := hS 1 2 3
X d
ew S e wS ew i . q d hS 1 3 d
2) More generally, es S ew i es S hS 0 ij k
8.3
=
(
qi . . . qj−1, if 1 ≤ i ≤ k < j ≤ n, 0, otherwise.
Residue formula.
Theorem 11 Correlation function hP (x1 , . . . , xn )i is given by the formula hP (x1 , . . . , xn )i =
X
Resp
e e1 (p)=···=e en (p)=0
!
P (x1 , . . . , xn )dx1 ∧ · · · ∧ dxn . ee1 · · · een
Proof. If the polynomial P (x1 , . . . , xn ) is in the ideal generated by ee1 , . . . , een , then the left and right hand sides of the formula are zero. Hence, it is enough to prove that ! νn−1 X xν11 · · · xn−1 dx1 ∧ · · · ∧ dxn Resp ee1 · · · een 38
=
(
1, if (ν1 , . . . , νn−1 ) = (n − 1, n − 2, . . . , 1) 0, if 0 ≤ ν1 + · · · + νn−1 < n(n − 1)/2, 0 ≤ νi ≤ n − i.
We can extend the meromorphic form ν
n−1 xν11 · · · xn−1 dx1 ∧ · · · ∧ dxn ω= e e1 · · · een
on the affine space An(x1 ,...,xn) to (P1 )n . For each subset I ⊂ {1, . . . , n}, we consider the coordinate chart UJ = An(z J ,...,z J ) , where n
1
ziJ
=
(
xi , if i 6∈ J, 1/xi , otherwise.
Then (P1 )n =
[
UJ .
J
Let e¯Jj (z1J , . . . , znJ ) = (
Y
i∈J
Then
ν
n−1 xν11 · · · xn−1 (
ω = (−1)♯J
ziJ )eej (x1 , . . . , xn ).
Y
ziJ )n−2 dz1J ∧ · · · ∧ dznJ
i∈J
e¯J1 · · · e¯Jn
on Uφ ∩ UJ . If ♯J = j, then there exists a polynomial Qi (z1J , . . . , znJ ) for i ∈ J such that X e¯Ij (z1J , . . . , znJ ) = 1 + ziJ Qi . i∈J
This follows from eej (x1 , . . . , xn ) = ej (x1 , . . . , xn ) + (terms of lower degree).
Therefore e¯J1 , . . . e¯Jn do not have common zero on
BJ = {(z1J , . . . , znJ ) ∈ UJ | ziJ = 0, i ∈ J}. From the residue theorem, if 0 ≤ ν1 + · · · + νn < n(n − 1)/2, 0 ≤ νi ≤ n − i, then ! νn−1 X xν11 · · · xn−1 dx1 ∧ · · · ∧ dxn = 0. Resp ee1 · · · een 39
On the other hand, X
x1n−1 · · · xn−1 dx1 ∧ · · · ∧ dxn ee1 · · · een
Resp
e ei (p)=0
!
=−
where p runs over the common zeros of e¯J1 , . . . , e¯Jn in
X
Resp ω,
p
[
BJ . Let y1 = 1/x1 ,
1∈J
z = (y1 , x2 , . . . , xn ) and
e¯∗1 (z) = y1 (1 + y1 (x2 + · · · + xn )). Then we have −
{1} e¯2
e¯∗1
P
p
Resp ω =
X
Resp
x2n−2 · · · xn−1 dy1 ∧ · · · ∧ dxn {1}
= = ··· = =0 in the locus {y1 = 0}
=
X p
by induction.
Resp
{1}
e¯∗1 (z) · e¯2 (z) · · · e¯N (z)
{1} e¯n
x2n−2 · · · xn−1 dx2 ∧ · · · ∧ dxn ee1 (x2 , . . . , xn ) · · · een−1 (x2 , . . . , xn )
!
!
= 1,
From the residue formula, the correlation function is given by the quantum residue ResIe, namely, hP (x1 , . . . , xn−1 )i = ResIeP (t1 , . . . , tn−1 ).
In order to relate the quantum residue with the classical one, we consider the quantum residue generating function Ψ(t) = h
n−1 Y i=1
X ti i= hxν it−ν . ti − xi n−1 ν∈(Z ) ≥0
Then, we have ResIeP (t1 , . . . , tn−1 ) = ResI (P (x1 , . . . , xn−1 )Ψ(x)) .
Hence, it is important to determine the generating function Ψ(t). Let fi (t) = tn +
n−1 X j=0
40
(i)
γn−j tj
be the characteristic polynomial of the quantum multiplication by xi with respect to the basis consisting of the quantum Schubert polynomials. Let us consider the (n! + 1) × (n! + 1)-matrix Cn (t) such that (Cn (t))1,j =
(Cn (t))i,j =
(−1)j−1 tn−j+2 ; (j − 1)!
!
(−1)n i−2 tj−2 , if i ≥ 2, i + j ≥ n + 2, (j − 1)! n − j + 1 0,
otherwise.
We define the differential operator Di by (i)
(i)
Di = (γn! , γn!−1, . . . , 1) · Cn (ti ) ·t (1, ∂/∂ti , . . . , (∂/∂ti )n! ).
Proposition 16 The generating function Ψ(t) satisfies the system of differential equations Di Ψ(t) = 0, 1 ≤ i ≤ n − 1. ν
n−1 Conversely, these differential equations and the initial values hxν11 · · · xn−1 i for 0 ≤ νi ≤ n! − 1 determine the generating function uniquely.
Proof. Let x be a variable. Since
we have
1 x x2 x3 .. . xn!
= Cn
Di Ψ(t) = h
·
t(t − x)n! −x(t − x)n!−1 2!x(t − x)n!−2 −3!x(t − x)n!−3 .. . (−1)n! (n!)!x
,
Y fi (xi ) tj i = 0. n!+1 (ti − xi ) j6=i tj − xj
On the other hand, the recursive relations hfi (xi )P (x1 , . . . , xn−1 )i = 0 41
ν
n−1 with the initial values hxν11 · · · xn−1 i for 0 ≤ νi ≤ n! − 1 determine the correlation function uniquely. Remark 13 We can also consider another generating function
hexp(x1 t1 + · · · + xn tn )i. This is the generating volume function in [GK]. This generating function satisfies ! ∂ ∂ hexp(x1 t1 + · · · + xn tn )i = 0, ,···, eei ∂t1 ∂tn for 1 ≤ i ≤ n.
In the case of n = 3, we can calculate the generating function Ψ(t) explicitly. The results of calculation one can find in the Appendix B.
9
Extended Ehresman–Bruhat order and quantum Pieri rule.
Let us remind that the Ehresman–Bruhat order denoted by ≤, is the partial order on Sn that is the transitive closure of the relation →. Relation v → w means that 1) l(w) = l(v) + 1, 2) w = v · t where t is a transposition. In other words, if v and w are permutations, v ≤ w means that there exists r ≥ 0 and v0 , v1 , . . . , vr in Sn such that v = v0 → v1 → · · · → vr = w. Now let us define the extended Ehresman–Bruhat order v ⇐ w on Sn . First of all, we define a relation v ← w (see, also, [FGP]). Relation v ← w means that 1) w = v · t, where t is a transposition, 2) l(w) ≥ l(v) + l(v −1 w). Remark 14 i) It follows from [M], (1.10), that condition 2) is equivalent to the following one 2′ ) w(i) < w(j) and for all k such that i < k < j we have w(i) < w(k) < w(j). 42
ii) If w = vti,i+1 and l(w) = l(v) + 1 (i.e. v → w in the Bruhat order), then we have also an arrow v ← w. This is clear because in our case we have l(w) = l(v) + l(ti,i+1 ). Example. Symmetric group S3 (see Figure 1). 121 q 3 k Q q q q qq q q Q qq Q q qq q q q q qQ q qqQ q q q q qq Q q q q qq Q q s + q 12 * YH H q 21 q qq 6 H 6qq q H q q q H HH q q q q q q H q q q H q q q H q q q H q q q H q ? H ? q H q 1 q q 2 qq q k qqqq Q 3 q q q Q q qq q q Q qq q q qq Q qq Q q q q qq q q q q Q s ?+ Q id
Figure 1: Extended Ehresman-Bruhat order for S3 . We define a weight of an arrow v ← w, denoted by wt(v ← w), to be equal to the product qi . . . qi+s−1 , if t = tij and 2s := l(w) + 1 − l(v). We assume that weight of any arrow v → w is equal to 1 (see, also, [FGP]). Let us say that an arrow v ← w (resp. v → w) has a color k if w = vtij and 1 ≤ i ≤ k < j ≤ n. Extended Ehresman–Bruhat order on Sn (notation v ⇐ w) is the transitive closure of the relations ←, and →. In other words, there exists r ≥ 0 and v0 , v1 , . . . , vr in Sn such that v = v0 ⇀ ↽ vr = w, ↽ ··· ⇀ ↽ v2 ⇀ ↽ v1 ⇀
(26)
where symbol vi ⇀ ↽ vi+1 means either vi → vi+1 or vi ← vi+1 . For given pair v ⇐ w, we consider a sequence of arrows (26) as a path between v and w (notation v 7→ w) in the extended Ehresman–Bruhat order and call it as a BE–path (Bruhat–Ehresman path). We denote the number r in a representation (26) by l(v 7→ w). 43
Let us define a weight of a BE–path v 7→ w as follows wt(v 7→ w) =
r−1 Y
wt(vi ⇀ ↽ vi+1 ).
i=0
k
We will say that BE–path v 7→ w has a color k, notation v 7→ w, if in the representation (26) all arrows vi ⇀ ↽ vi+1 (i = 0, . . . , r −1) have the same color k. Theorem 12 (Quantum Pieri’s rule). Let us consider the Grassmanian permutation [b, d] = (1, 2, . . . , b − d − 1, b, b − d, b − d + 1 . . . , b − 1, b + 1, . . . , n), for 2 ≤ b ≤ n, 1 ≤ d ≤ b. Then e [b,d] · S ev S
≡
X
b e w (mod Ien ), wt(v 7→ w)S b
where the sum runs over all BE–paths v 7→ w, s.t. 1) l(v → 7 w) = d;
2) if vl = vl+1 (il jl ) (l = 0, . . . , d − 1), then all il are different. (Note that S[b,d] = ed (x1 , . . . , xb−1 )). Sketch of the proof. It is enough to consider the case d = 1 (induction!). In the case d = 1, we use a quantum analog of Kohnert–Veigneau’s method [KV]. Namely, at first we prove the quantum Pieri rule (for d = 1) for double quantum Schubert polynomials and then take y = 0 (see Theorem 4). e w (x, y)). Proposition 17 (Quantum Pieri’s rule for S 0 e w (x, y) ≡ (xj + yn+1−j )S 0
X i<j
e w t (x, y) − qij S 0 ij
X
j
Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa–Intriligator formula. Anatol N. Kirillov∗ and Toshiaki Maeno† Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
Abstract We study the algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quane w (x, y), which are the Lascoux– tum double Schubert polynomials S Sch¨ utzenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy e w (x) as the identity. We define also quantum Schubert polynomials S Gram–Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Use w (x) = S e w (x, y)|y=0 ing quantum Cauchy identity, we prove that S and as corollary obtain a simple formula for the quantum Schubert (y) e e w (x) = ∂ww polynomials S 0 Sw0 (x, y)|y=0 . We also prove the higher genus analog of Vafa–Intriligator’s formula for the flag manifolds and study the quantum residues generating function. We introduce the extended Ehresman–Bruhat order on the symmetric group and formulate the equivariant quantum Pieri rule.
∗ On leave from Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 191011, Russia † Supported by JSPS Research Fellowships for Young Scientists
1
1
Introduction.
The structure constants of the quantum cohomology ring are given by the third derivatives of the Gromov-Witten potential F. The Gromov-Witten potential F is a generating function of the Gromov-Witten invariants. The axioms of the tree level Gromov-Witten invariants V hI0,m,β i : H ∗ (V, Q)⊗m −→ Q,
β ∈ H2 (V, Z), for a target space V are given by Kontsevich and Manin [KM]. ∗ Let X1 , . . . , Xm be cycles on V and X1∗ , . . . , Xm their dual classes. Then the V ∗ ∗ invariant hI0,m,β i(X1 ⊗ · · · ⊗ Xm ) can be considered as the virtual number of the stable maps f from m-pointed rational curve (P1 ; p1 , . . . , pm ) to V, such that the image of f represents the homology class β and f (pi) ∈ Xi . In case of flag variety F ln := SLn /B of type An−1 the potential F is given as follows. Let Ωv be the dual class of the Schubert cycle Xv corresponding to a permutation v ∈ Sn . Then the potential Fω ((tv )v∈Sn ) is defined by
Fω ((tv )v∈Sn ) =
X
X
exp(−
β m=Σmv ≥3
Z
β
V hI0,m,β i(
ω)
O
v Ω⊗m ) v
v∈Sn
Y
mv !
v∈Sn
Y
tmv ,
v∈Sn
where ω is a K¨ahler form. For each point t ∈ H ∗ (F ln ), the quantum multiplication law is given by Ωu ∗ Ωv =
∂ 3 Fω (t)Ωww0 , w∈Sn ∂tu ∂tv ∂tw X
where w0 is the permutation of maximal length. The algebra with this multiplication law is called a quantum cohomology ring, which is denoted by QHt∗ (F ln ). The associativity of the quantum multiplication is equivalent to the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation X
v∈Sn
X ∂ 3 Fω ∂ 3 Fω ∂ 3 Fω ∂ 3 Fω = , ∂tu1 ∂tu2 ∂tv ∂tvw0 ∂tu3 ∂tu4 v∈Sn ∂tu2 ∂tu3 ∂tv ∂tvw0 ∂tu1 ∂tu4
for any u1 , u2, u3 , u4 ∈ Sn . From [KM, Proposition 4.4], the potential Fω satisfies Fω (t) = Fω−t(2) (t − t(2) ), 2
where t(2) =
X
tv Ωv . Hence, we may assume ω = 0. The potential F is
l(v)=1
decomposed as a sum of the classical part fcl and the quantum correction f, where ! X 1 δu,vw0 tu tv , fcl = tid 6 u,v and f=
X
X
β m=Σmv ≥3
Fl hI0,m,β i(
O
v Ω⊗m ) v
l(v)≥1
Y
mv !
Y
tmv .
l(v)≥1
l(v)≥1
From the axioms of the Gromov-Witten invariants ([KM,(2.2.4)]), we have Fl i( hI0,m,β
O
v ) Ω⊗m v
=
Fl hI0,m−Σ i( l(u)=1 mu ,β
l(v)≥1
O
v ) Ω⊗m v
l(v)>1
Y Z
l(u)=1
β
Ωu .
Hence, the quantum correction f is expressed as f=
X
l(u)=1
v tm v , mv ! l(v)>1
N((mv )l(v)>1 | (bu )l(u)=1 ) exp(
mv ,bu
where
X
Fl N((mv ) | (bu )) = hI0,Σm i v ,Σbu Xu
(bu tu ))
O
l(v)>1
Y
v Ω⊗m . v
If n = 3, the WDVV equations with the initial condition N(0, 0, 1 | 1, 0) = N(0, 0, 1 | 0, 1) = 1 or N(2, 0, 0 | 1, 0) = N(0, 2, 0 | 0, 1) = 1
determine all the coefficients N(λ, µ, ν | a, b) uniquely. In fact, Di Francesco and Itzykson [FI] gave the coefficients N(λ, µ, ν | a, b) for a + b ≤ 10. Let x1 = Ω(1,2) and xi = Ω(i,i+1) − Ω(i−1,i) for 2 ≤ i ≤ n − 1, where (i, j) is the transposition that interchanges i and j. Since the classical cohomology ring is generated by x1 , . . . , xn−1 , the quantum cohomology ring QHt∗ (F ln ) is also generated by x1 , . . . , xn−1 in the neighborhood of the origin t = 0. Moreover, we can choose a neighborhood of the origin on which QHt∗ (F ln ) 3
is a complete intersection ring. Then, let I ⊂ C[x1 , . . . , xn−1 ] be the defining ideal of the quantum cohomology ring QHt∗ (F ln ). The Schubert class Ωv is expressed by the Schubert polynomial Sv (x1 , . . . , xn−1 ) in the classical cohomology ring. However, in the quantum cohomology ring, the class corresponding to the Schubert polynomial Sv is no longer the Schubert class Ωv . e t (x1 , . . . , xn−1 ) expressing Ωv gives a deformation of Hence, the polynomial S v the Schubert polynomial. We call it a big quantum Schubert polynomial. We identify the residue pairing defined by I with the intersection form on the cohomology ring, so the big quantum Schubert polynomials are obtained by orthogonalization the basis consisting of the classical Schubert polynomials. It is difficult to describe the defining ideal I of the big quantum cohomology ring for generic t, so the big quantum Schubert polynomials are complicated in general. However, in the case where parameters tv = 0 for all permutations v ∈ Sn such that l(v) > 1, the defining relations of the quantum cohomology ring are known by the results of A. Givental and B. Kim [GK] and I. Ciocan-Fontanine [C]. We call it the small quantum cohomology ring. The structure constants of the small quantum cohomology ring are given by ∂3F (t(2) ) = ∂tv1 ∂tv2 ∂tv3
=
X X
Fl hI0,Σm i(Ωv1 ⊗ Ωv2 ⊗ Ωv3 ⊗ ( u +3,β
Y
β mu ≥0
=
X
O
l(u)=1
mu !
l(u)=1
u Ω⊗m )) u
Y
u tm u
l(u)=1
Fl hI0,3,β i(Ωv1 ⊗ Ωv2 ⊗ Ωv2 )eΣbi t(i,i+1) ,
β
where the sum runs over β = b1 X(1,2) + · · · + bn−1 X(n−1,n) with bi ∈ Z≥0 . Hence the small quantum cohomology ring is determined by Fl the invariants hI0,3,β i. For a monomial xi1 ∗ · · · ∗ xim , the m-point correlation function determined by the small quantum cohomology ring (the so-called small quantum cohomology ring correlation function) is defined to be hxi1 · · · xim i =
Z
F ln
4
xi1 ∗ · · · ∗ xim .
If m ≥ 4, the small quantum cohomology ring correlation function can be expressed as follows: hxi1 · · · xim i = X
X
−
e
β=β1 +···+βm−3 v1 ,...,vm−3
R
β
ω
hI0,3,β1 i(xi1 ⊗ xi2 ⊗ Ωv1 )hI0,3,β2 i(Ωv1 w0 ⊗ xi3 ⊗ Ωv2 )
· · · hI0,3,βm−2 i(Ωvm−3 w0 ⊗ xim−1 ⊗ xim ). Hence, if m ≥ 4, the Gromov-Witten invariants hI0,m,β i(xi1 ⊗ · · · ⊗ xim ) do not appear as coefficients of small quantum cohomology ring correlation functions. Let us explain briefly the main results obtained in our paper. Follow to A. Givental and B. Kim [GK], and I. Ciocan–Fontanine [C], we define the quantum elementary symmetric polynomials ee1 , . . . , een by the formula
det
x1 + t q1 0 −1 x2 + t q2 0 −1 x3 + t .. .. .. . . . 0 ... 0 0 ... ... 0 ... ...
... 0 q3 .. .
... ... 0 .. .
... ... ... .. .
0 0 0 .. .
−1 xn−2 + t qn−2 0 0 −1 xn−1 + t qn−1 ... 0 −1 xn + t
= tn + ee1 tn−1 + ee2 tn−2 + · · · + een ,
where qi = et(i,i+1) . The defining ideal Ie of the small quantum cohomology ring is generated by the quantum elementary symmetric polynomials, namely QH ∗ (F ln , Z) := QH ∗ (F ln ) = Z[x1 , . . . , xn ; q1 , . . . , qn−1 ]/(ee1 , . . . , een ).
(1)
In the classical case q1 = · · · = qn−1 = 0, on the quotient ring A
:= Z[x1 , . . . , xn ]/(e1 (x), . . . , en (x)) ≃ H ∗ (F ln , Z)
there exists a natural pairing hf, gi = η(∂w0 (f g)) which comes from the intersection pairing in the homology group H∗ (F ln , Z) of the flag variety. We can interpret the pairing h, i as the Grothendieck residue pairing with respect to the ideal I (see Subsection 2.5): hf, gi = ResI (f g), 5
e where I = I| q=0 . Our first observation is that a natural residue pairing (we call it the quantum residue pairing)
hf.giQ = ResIe(f g)
on the quotient ring A := Z[x1 , . . . , xn ]/Ie corresponds to the intersection pairing in quantum cohomology QH ∗ (F ln , Z) under a natural isomorphism (1). It is well-known (e.g. [LS2], [M]) that the classical Schubert polynomials form an orthonormal basis (with respect to the pairing h, i) in the cohomology ring of flag manifold and also give a linear basis in the quantum cohomology ring QH ∗ (F ln , Z), [GK], [MS]. However, the classical Schubert polynomials do not orthogonal with respect to the quantum pairing any more. Thus, it is natural to ask: what kind of polynomials one can obtain applying the Gram–Schmidt orthogonalization to the classical Schubert polynomials with respect to the quantum pairing h, iQ ? Omiting some details with ordering (see Definition 5), the answer is: quantum Schubert polynomials. Our second observation is: to work with the equivariant quantum cohomology algebra ([GK], [K2]) is more convenient then with quantum cohomology ring itself. The main reason is that one can find Lascoux–Sch¨ utzenberger’s type representative for any equivariant quantum cohomology class. In other e w (x, y) can be obtained words, each quantum double Schubert polynomial S from the top one by using the divided difference operators acting on the y variables. Theorem-Definition A Let x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) be two sets of variables, and e (q) (x, y) S w0
where ∆k (t | x1 , . . . , xk ) :=
:=
k X
n−1 Y
∆i (yn−i | x1 , . . . , xi ),
i=1
tk−j ej (x1 , . . . , xk | q1 , . . . , qk−1 ) is the generat-
j=0
ing function for the quantum elementary symmetric functions in x1 , . . . , xk . e (q) (x, y) = ∂ (y) S e (q) Then S w ww0 w0 (x, y). e (q) (x) as Gram–Schmidt’s We define the quantum Schubert polynomials S w orthogonalization of the set of lexicographically ordered monomials 6
{xI | I ⊂ (n − 1, n − 2, . . . , 1, 0)} with respect to the quantum pairing h, iQ , see Definition 5. One of our main results is the quantum analog of Cauchy’s identity for (classical) Schubert polynomials, [M], (5.10). Theorem B (Quantum Cauchy’s identity) X
w∈Sn
e (q) (x)Sww (y) S 0 w
(q)
e (x, y). =S w0
(2)
We give a geometric proof of Theorem B in Section 7 using the arguments due to I. Ciocan-Fontanine [C]; more particularly, we reduce directly a proof of Theorem B to that of the following geometric statement: Lemma Let I ⊂ δ = (n − 1, n − 2, . . . , 1, 0) and w ∈ Sn be a permutation, then e w (x)iQ = heI (x), Sw (x)i, heeI (x), S (3)
where eI (x) :=
(resp. eeI (x) :=
n−1 Y
eik (x1 , . . . , xn−k )
k=1
eeik (x1 , . . . , xn−k | q1 , . . . , qn−k−1))
k=1 n−1 Y
is the elementary polynomial (resp. quantum elementary polynomial), see Section 5.2. It is the formula (3) that we prove in Section 7 using the geometrical arguments from [C] and [K2]. By product, it follows from our proof that ˆ w (x) defined geometrically (see Section 6) quantum Schubert polynomials S coincide with those defined algebraically (see Definition 5): ˆ w (x) S
e e w−1 (x) (mod I). ≡S
It is interesting to note, that the intersection numbers heI (x), Sw (x)i (which are nonnegative!) are precisely the coefficients of corresponding Schubert polynomial: X Sw (x) = heI (x), Sw (x)ixδ−I . I⊂δ
The quantum Cauchy formula (2) plays the important role in our approach to the quantum Schubert polynomials. As a direct consequence of (2), we obtain the Lascoux–Sch¨ utzenberger type formula for quantum Schubert polynomials (cf. Theorem-Definition A). 7
e w (x, y) be as in Theorem-Definition A, then Theorem C Let S 0 e w (x) S
(y) e = ∂ww Sw0 (x, y)|y=0 . 0
In Section 5 we introduce a quantization map Pn → P n , f 7→ fe.
The quantization is a linear map which preserves the pairings, i.e., hfe, geiQ = hf, gi, f, g ∈ Pn .
Using the quantum Cauchy formula (2), we prove that quantum double Schubert polynomials are the quantization of classical ones. Another class of polynomials having a nice quantization is the set of elementary polynomials eI (x) :=
n−1 Y
eik (x1 , . . . xn−k ), I = (i1 , . . . , in−1 ) ⊂ δ.
k=1
It follows from Theorem B that quantization eeI (x) of elementary polynomial eI (x) is given by eeI (x) =
n−1 Y
eik (x1 , . . . , xn−k | q1 , . . . , qn−k−1).
k=1
More generally, we make a conjecture (”quantum Schur functions ”) that quantization of the flagged Schur function (see [M], (3.1), (4.9) and (6.16))
sλ/µ (X1 , . . . , Xn ) = det hλi −µj −i+j (Xi )
1≤i,j≤n
is given by e seλ/µ (X1 , . . . , Xn ) = det h λi −µj −i+j (Xi )
1≤i,j≤n
,
e (X) is the quantum complete homogeneous symmetric function of where h k degree k, and X1 ⊂ · · · ⊂ Xn are the flagged sets of variables (see Section 5). In Section 5.2 we consider a problem how to quantize monomials. It seems to be difficult to find an explicit determinantal formula for a quantum monomial xeI , i.e., to find a quantum analog of the Billey-Jockusch-Stanley
8
formula for Schubert polynomials in terms of compatible sequences [BJS]. We prove the following formulae for quantum monomials xeI =
X
w∈Sn
e w (x), I ⊂ δ, η(∂w xI )S
e w (x, y) S 0
=
X
I⊂δ
xeI eδ−I (y).
In section 8.1 we give a proof of the higher genus analog of the Vafa– Intriligator type formula for the flag manifold. In Section 8.3 we study a problem how to compute the quantum residues. This is important for computation of small quantum cohomology ring correlation functions (or correlation functions, for short) and the Gromov–Witten invariants, see Introduction and Theorem 11. We introduce the generating function n−1 Y ti Ψ(t) = h i i=1 ti − xi for quantum residues and give a characterization of this function as the unique solution to some system of differential equations, see Proposition 14. In Appendix B we calculate the generating function Ψ(t) for the case n = 3 explicitly. In Section 9 we introduce the extended Ehresman–Bruhat order and give a sketch of a proof of equivariant quantum Pieri rule. Details will appear elsewhere. In Appendix A one can find a list of explicit expressions for the quantum double Schubert polynomials for the symmetric group S4 . We would like to mention, that in the recent preprint “Quantum Schubert polynomials” by S. Fomin, S. Gelfand and A. Postnikov, [FGP], developed a different approach to the theory of quantum Schubert polynomials, based on the remarkable family of commuting operators Xi ([FGP], (3.2)). Among main results, obtained by S. Fomin, S. Gelfand and A. Postnikov, are definitions, orthogonality, quantum Monk’s formula and other properties of quantum Schubert polynomials; definition of quantization map and quantum multiplication. Besides some overlap with the preprint of S. Fomin, S. Gelfand and A. Postnikov, our works were done independently and based on the different approaches, which allow to obtain the mutually complementary results. 9
2
Classical Schubert polynomials.
In this section we give a brief review of the theory of Schubert polynomials created by A. Lascoux and M.-P. Sch¨ utzenberger. In exposition we follow to the I. Macdonald book [M1] where proofs and more details can be found.
2.1
Divided differences.
Let x1 , . . . , xn , . . . be independent variables, and let Pn := Z[x1 , . . . , xn ] for each n ≥ 1, and P∞ := Z[x1 , x2 , . . .] =
∞ [
Pn .
(4)
n=1 Sn Let us denote by Λn := Z[x1 , . . . , xn ]X ⊂ Pn the ring of symmetric polynomials in x1 , . . . , xn , and by Hn := { aI xI | aI ∈ Z, 0 ≤ ik ≤ n − k, ∀k} I=(i1 ,...,in )
the additive subgroup of Pn spanned by all monomials xI := xi11 xi22 . . . xinn with I ⊂ δ := δn = (n − 1, n − 2, . . . , 1, 0). For 1 ≤ i ≤ n − 1 let us define a linear operator ∂i acting on Pn (∂i f )(x) =
f (x1 , . . . , xi , xi+1 , . . . , xn ) − f (x1 , . . . , xi+1 , xi , . . . , xn ) . xi − xi+1
(5)
Divided difference operators ∂i satisfy the following relations ∂i2 = 0, ∂i ∂j = ∂j ∂i , if | i − j |> 1, ∂i ∂i+1 ∂i = ∂i+1 ∂i ∂i+1 ,
(6)
and the Leibnitz rule ∂i (f g) = ∂i (f )g + si (f )∂i (g).
(7)
It follows from (7) that ∂i is a Λn -linear operator. For any permutation w ∈ Sn , let us denote by R(w) the set of reduced words for w, i.e. sequences (a1 , . . . , ap ) such that w = sa1 · · · sap , where 10
p = l(w) is the length of permutation w ∈ Sn , and si = (i, i + 1) is the simple transposition that interchanges i and i + 1. For any sequence a = (a1 , . . . , ap ) of positive integers, we define ∂a = ∂a1 · · · ∂ap .
Proposition 1 ([M1], (2.5),(2.6)) • If a, b ∈ R(w), then ∂a = ∂b . • If a is not reduced, then ∂a = 0. From Proposition 1 it follows that an operator ∂w = ∂a is well-defined, where a is any reduced word for w. By (7), the operators ∂w , w ∈ Sn , are Λn linear, i.e. if f ∈ Λn , then ∂w (f g) = f ∂w (g).
2.2
Schubert polynomials.
Let δ = δn = (n − 1, n − 2, . . . , 1, 0), so that xδ = x1n−1 x2n−2 . . . xn−1 . Definition 1 (Lascoux–Sch¨ utzenberger [LS1]). For each permutation w ∈ Sn the Schubert polynomial Sw is defined to be Sw (x)
= ∂w−1 w0 (xδ ),
where w0 is the longest element of Sn . Proposition 2 ([M1], (4.2),(4.5),(4.11),(4.15)). • Let v, w ∈ Sn . Then ∂v Sw =
(
Swv−1 ,
0,
if l(wv −1 ) = l(w) − l(v), otherwise.
11
• (Stability). Let m > n and let i : Sn ֒→ Sm to be the natural embedding. Then Sw = Si(w) . • The Schubert polynomials Sw , w ∈ Sn form a Z–basis of Hn . • (Monk’s formula). Let f =
n X
αi xi , w ∈ Sn . Then
i=1
f Sw =
X
(αi − αj )Swtij ,
∂w (f g) = w(f )∂w g +
X
(αi − αj )∂wtij g,
where tij is the transposition that interchanges i and j, and both sums are over all pairs i < j such that l(wtij ) = l(w) + 1.
2.3
Scalar product.
Let us define a scalar product on Pn with values in Λn , by the rule hf, gi = ∂w0 (f g), f, g ∈ Pn ,
(8)
where w0 is the longest element of Sn . The scalar product h, i defines a non-degenerate pairing h, i0 on the quotient ring Pn /In ∼ = H ∗ (F ln , Z), where In is the ideal in Pn generated by the elementary symmetric polynomials e1 (x), . . . , en (x). Proposition 3 ([M1], (5.3),(5.4),(5.6),(4.13),(5.10)). • If f ∈ Λn , then hf h, gi = f hh, gi; • If f, g ∈ Pn , w ∈ Sn , then h∂w f, gi = hf, ∂w−1 gi; (
1, if u = w0 v, 0, otherwise. • The Schubert polynomials Sw , w ∈ Sn , form a Λn –basis of Pn ;
• (Orthogonality) If l(u) + l(v) = l(w0 ), then hSu , Sv i =
• The Schubert polynomials Sw , w ∈ S (n) , form a Z–basis of Pn , where for each n ≥ 1, S (n) is the set of all permutations w such that the code w has length ≤ n; • (Cauchy’s formula) X
Sw (x)Sww0 (y)
=
Y
(xi + yj ).
i+j≤n
w∈Sn
12
Proposition 4 Schubert polynomials are uniquely characterized by the following properties ( 1, if u = w0 v, 1. (Orthogonality) hSu , Sv i0 = 0, otherwise. 2. Let w be a permutation in Sn and c(w) = (c1 , c2 , . . . , cn ) its code, then Sw (x)
= xc(w) +
X
αI xI ,
where I ⊂ δ, αI > 0 and I lexicographically smaller then c(w). Remark 1 1) (Definition of the code, [M1], p.9). For a permutation w ∈ Sn , we define ci = ♯{j | i < j, w(i) > w(j)}. The sequence c(w) = (c1 , c2 , . . . , cn ) is called the code of w. 2) Schubert polynomials are obtained as Gram-Schmidt’s orthogonalization of the set of monomials {xI }I⊂δ ordered lexicographically.
2.4
Double Schubert polynomials.
Let x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) be two sets of independent variables, and Y Sw0 (x, y) := (xi + yj ). i+j≤n
Definition 2 (Lascoux–Sch¨ utzenberger [LS2]). For each permutation w ∈ Sn , the double Schubert polynomial Sw (x, y) is defined to be Sw (x, y)
(x)
= ∂w−1 w0 Sw0 (x, y), (x)
where divided difference operator ∂w−1 w0 acts on the x variables. Proposition 5 X ([M1], (6.3),(6.8)). • Sw (x, y) = Su (x)Suw−1 (y), summed over all u ∈ Sn , such that u
l(u) + l(uw −1) = l(w); • (Interpolation formula). For all f ∈ Z[x1 , . . . , xn ] we have f (x) =
X
(y) Sw (x, −y)∂w f (y)
w
summed over all permutations w ∈ S (n) . 13
Double Schubert polynomials appear in algebra and geometry as cohomology classes related to degeneracy loci of flagged vector bundles. If h : E → F is a map of rank n vector bundles on a smooth variety X, E1 ⊂ E2 ⊂ · · · ⊂ En = E, F := Fn → Fn−1 → · · · → F1 are flags of subbundles and quotient bundles, then there is a degeneracy locus Ωw (h) for each permutation w in the symmetric group Sn , described by the conditions Ωw (h) = {x ∈ X | rank(Ep (x) → Fq (x)) ≤ #{i ≤ q, wi ≤ p}, ∀p, q}. For generic h, Ωw (h) is irreducible, codim Ωw (h) = l(w), and the class [Ωw (h)] of this locus in the Chow ring of X is equal to the double Schubert polynomial Sw0 w (x, −y), where xi = c1 (ker(Fi → Fi−1 )), yi = c1 (Ei /Ei−1 ), 1 ≤ i ≤ n. It is well-known [F] that the Chow ring of flag variety F ln admits the following description CH ∗(F ln ) ∼ = Z[x1 , . . . , xn , y1 , . . . , yn ]/J, where J is the ideal generated by ei (x1 , . . . , xn ) − ei (y1 , . . . , yn ), 1 ≤ i ≤ n, and ei (x) is the i-th elementary symmetric function in the variables x1 , . . . , xn . • ([LS2], [KV]) The ring Z[x1 , . . . , xn , y1 , . . . , yn ]/J is a free module of dimension n! over the ring R, with basis either Sw (x), or Sw (x, y), w ∈ Sn , where Z[x1 , . . . , xn ] ⊗ Sym[y1 , . . . , yn ] . R := J
2.5
Residue pairing.
Let I be an ideal in P¯n = R[x1 , . . . , xn ], R ⊂ C, generated by a regular system of parameters ϕ1 , . . . , ϕn , and A := P¯n /I.
14
Proposition 6 ([GH], [EL]). • dimR A < ∞. ! ∂ϕi • H := det 6∈ I. ∂xj Let d0 := deg H, where we assume that deg xi = 1 for all 1 ≤ i ≤ n. Proposition 7 ([EL]) • If f ∈ P¯n and deg f = d0 , then there exists a non zero α ∈ R such that f≡
α H (mod I). n!
• If f ∈ P¯n , f 6= 0, and deg f > d0 , then there exists g ∈ P¯n such that deg g ≤ d0 and g ≡ f (mod I). Definition 3 (Grothendieck residue with respect to the ideal I). Let f ∈ P¯n and deg f < d0 , then we define ResI (f ) = 0. α H (mod I) and we define ResI (f ) := α. n! Finally, if deg f > d0 , then choose g ∈ P¯n such that g ≡ f (mod I) and deg g ≤ d0 , and define ResI (f ) := ResI (g). If deg f = d0 , then f ≡
We will use also notation hf iI instead of ResI (f ). Finally, let us define a residue pairing h, iI on P n using the Grothendieck residue hf, giI = ResI (f, g), f, g ∈ P n .
Proposition 8 ([GH]). • If f ∈ I, then ResI (f ) = 0. • The residue pairing h, iI induces a non-degenerate pairing on A = P /I.
15
We will use this general construction of residue pairing in the following two cases: i) R = Z, In ⊂ Pn is an ideal generated by elementary symmetric polynomials e1 (x), . . . , en (x). It is well-known that if F ln := SL(n)/B is the flag variety of type An−1 , then H ∗ (F ln , Z) ≃ Pn /In , and residue pairing h, i on Pn /In coincides with the scalar product on Pn /In induced by (8). ii) R = Z[q1 , . . . , qn−1 ], Ien ⊂ P n is an ideal generated by the quantum elementary symmetric functions ee1 (x), . . . , een (x). It is a result of A. Grivental and B. Kim, and I. Ciocan–Fontanine, that QH ∗ (F ln ) ≃ P n /Ien ,
and the residue pairing defined by Ien may be naturally identified with the intersection form on the quantum cohomology ring. We will call this residue pairing as quantum pairing on P n /Ien and denote it by h, iQ .
3
Quantum double Schubert polynomials.
Quantum double Schubert polynomials are closely related with the equivariant quantum cohomology. Let us remind the result of A. Givental and B. Kim [GK] (see also [K2]) on the structure of the equivariant quantum cohomology algebra of the flag variety F ln : e QHU∗ n (F ln ) ∼ = Z[x1 , . . . , xn , y1 , . . . , yn , q1 , . . . , qn−1 ]/J,
where the ideal Je generated by
ei (x1 , . . . , xn | q1 , . . . , qn−1 ) − ei (y1, . . . , yn ), 1 ≤ i ≤ n.
In classical case q = 0, the double Schubert polynomials Sw (x, y) represent the equivariant cohomology classes [F]. Quantum double Schubert polynomials have to play the similar role for the quantum equivariant cohomology ring. Let us define at first the “top” quantum double Schubert polynomial e w (x, y). S 0 16
Let x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) be two sets of variables, put e w (x, y) S 0
:=
e (q) (x, y) S w0
where ∆k (t | x1 , . . . , xk ) :=
k X
=
n−1 Y
∆i (yn−i | x1 , . . . , xi ),
i=1
tk−j ej (x1 , . . . , xk | q1 , . . . , qk−1 ) is the generat-
j=0
ing function for the quantum elementary symmetric polynomials in x1 , . . . , xk , i.e.
∆k (t|x) :=
k X
ei (x|q)ti =
i=1
det
x1 + t q1 0 −1 x2 + t q2 0 −1 x3 + t .. .. .. . . . 0 ... 0 0 ... ... 0 ... ...
... 0 q3 .. .
... ... 0 .. .
... ... ... .. .
0 0 0 .. .
−1 xk−2 + t qk−2 0 0 −1 xk−1 + t qk−1 ... 0 −1 xk + t
.
(9)
Definition 4 For each permutation w ∈ Sn , the quantum double Schubert e w (x, y) is defined to be polynomial S e w (x, y) S
(y) e = ∂ww Sw0 (x, y), 0
(y) where divided difference operator ∂ww acts on the y variables. 0
Remark 2 i) In the ”classical limit” q1 = · · · = qn−1 = 0, e w (x, y)|q=0 S
(y) = ∂ww Sw0 (x, y) = Sw−1 (y, x) = Sw (x, y), 0
e w (x, y)|q=0 = Sw (x, y). i.e. S ii) (Stability) Let m > n and let i : Sn ֒→ Sm be the embedding. Then e w (x, y) S
iii) One can check that the ring
e i(w) (x, y). =S
Z[x1 , . . . , xn , y1, . . . , yn , q1 , . . . , qn−1 ]/Je 17
e with basis either is a free module of dimension n! over the quotient ring R e w (x) or S e w (x, y), w ∈ Sn , where S e := R
Z[x1 , . . . , xn , q1 , . . . , qn−1 ] ⊗ Sym[y1 , . . . , yn ] . Je
Example. Quantum double Schubert polynomials for S3 : e s s s (x, y) S 1 2 1 e s s (x, y) S 2 1
e s s (x, y) S 1 2 e Ss (x, y) 1
e s (x, y) S 2 e Sid (x, y)
= = = = = =
(x1 + y2 )(x1 + y1 )(x2 + y1 ) + q1 (x1 + y2 ), (x1 + y1 )(x1 + y2 ) − q1 , (x1 + y1 )(x2 + y1 ) + q1 , x1 + y1 , x1 + x2 + y1 + y2 , 1.
For the list of the quantum double Schubert polynomials corresponding to the symmetric group S4 , see appendix A. Theorem 1 Let z = (z1 , . . . , zn ) be a third set of variables. Then (x)
e w (x, y), S e w (x, z)i hS 0 0 Q = C(y, z),
(10)
where the upper index x means that the quantum pairing is taken in the x variables, and X Sw (x)Sw0 w (y) C(x, y) = w∈Sn
is the ”canonical” element in the tensor product H ∗ (F ln ) ⊗ H ∗ (F ln ). Theorem 1 plays the important role in our approach to the quantum Schubert polynomials. We will give a proof later, and now let us consider some applications of the formula (10).
4
Quantum Schubert polynomials.
18
4.1
Definition.
Let us remind the result of A. Givental and B. Kim, and I. Ciocan-Fontanine on the structure of the small quantum cohomology ring of flag variety F ln e QH ∗ (F ln ) ∼ = Z[x1 , . . . , xn , q1 , . . . , qn−1 ]/I,
where the ideal Ie is generated by the quantum elementary symmetric polynomials eei (x) := ei (x1 , . . . , xn |q1 , . . . , qn−1 ), 1 ≤ i ≤ n with generating function ∆n (t|x), see (9). We define a pairing on the ring of polynomials Z[x; q] and the quantum cohomology ring QH ∗ (F ln ) ≃ Z[x; q]/Ie using the Grothendieck residue hf, giQ = ResIe(f g), f, g ∈ Z[x1 , . . . , xn , q1 , . . . , qn−1 ].
Then e 1) hf, giQ = 0 if f ∈ I;
2) hf, giQ defines a nondegenerate pairing in QH ∗ (F ln ).
e w := S e w (x) as Definition 5 Define the quantum Schubert polynomials S Gram–Schmidt’s orthogonalization of the set of lexicographically ordered monomials {xI | I ⊂ δ} with respect(to the quantum residue pairing hf, giQ: 1, if v = w0 u e u, S e v iQ = hSu , Sv i = 1) hS 0, otherwise X c(w) e w (x) = x 2) S + aI (q)xI , where aI (q) ∈ Z[q1 , . . . , qn−1 ] and I n then h n k e (x) ∈ Ie . I 6⊂ δ, then hI (x) ∈ In and h I n It is not difficult to see that Pn is a free Λn -module of rank n! with basis {hI (x) | I ∈ T}. e (x) is the quantization of complete polynomial Theorem 5’ If I ⊂ δ, then h I hI (x). Proof. See Remark 10 in Section 7.
5.3
Canonical involution ω. ←
←
There exists an involution ω of the ring P n [y] given by ω(x) = x, ω(y) = y , ← ← ω(q) = q , where for any sequences z = (z1 , . . . , zm ) we define z to be equal to 28
←
(zm , zm−1 , . . . , z1 ) := z . It is clear from the definition of quantum elementary symmetric functions ei (x|q), see (9), Section 3, that ω(ei (x|q)) = ei (x|q) and thus the involution ω preserves the ideal Ien (as well as the ideals In , Jn and Jen ). Proposition 14
e e (q) (x, y)) ≡ ǫ(u)S e (q) ω(S u w0 uw0 (x, y) mod Jn ,
where ǫ(u) = (−1)l(u) .
Proof. First of all, if u = w0 , then ←
← e e (q) (x, y)) ≡ ǫ(w0 )S e ( q ) (← ω(S w0 w0 x, y ) mod Jn .
(24)
But ω∂u = ǫ(u)∂w0 uw0 ω (see [M1], (2.12)). Thus, applying the divided difference operator ǫ(u)∂w(y)0 uw0 to the both sides of (24), we obtain ←
← e e (q) (x, y)) ≡ ǫ(w0 )S e ( q ) (← ω(S uw0 w0 u x, y ) mod Jn .
Finally, let us describe the action of involution ω on the elementary polynomials. Proposition 15 e ← (x) mod Ie . ω(eeI (x)) ≡ (−1)|I| h n I
Remark 9 To our knowledge, originaly, construction of the quantization map, using a remarkable family of commuting operators Xi , appeared in [FGP]. We use a different definition of quantization map, but it can be shown that two forms of quantizations are equivalent. For original proofs of Theorem 5 and 5’, and Proposition 15, see Corollary 4.6, Corollary 7.16 and Proposition 7.13 in [FGP].
29
6
Quantum cohomology ring of flag variety.
Quantum cohomology ring of the flag variety F ln is a deformed ring of the ordinary cohomology ring H ∗ (F ln , Z). The structure constants of the quantum cohomology ring are given by the Gromov–Witten invariants. Let Ωw1 , . . . , Ωwm (wi ∈ Sn ) be Schubert cycles. We denote by Md¯(P1 , F ln ) the moduli space of morphisms from P1 to F ln of multidegree d¯ = (d1 , . . . , dn−1 ). We consider the restriction of the universal map for t ∈ P1 : ev
evt : Md¯(P1 , F ln ) × {t} ֒→ Md¯(P1 , F ln ) × P1 →F ln , (f, p) 7→ f (p). Let Ωw (t) = evt−1 (Ωw ). m X
X n(n − 1) +2 di and 2 i=1 t1 , . . . , tm ∈ P1 are distinct, then for general translates of Ωwi , the number
Theorem 7 (I. Ciocan-Fontanine). If
of points in
m \
l(wi ) =
Ωwi (ti ) is finite and independent of ti and the translates of
i=1
Ωwi .
Definition 8 The Gromov–Witten invariant is defined as an intersection number hΩw1 . . . Ωwm id¯ =
#
\
Ωwi (ti ), if
i
0,
P
l(wi ) =
n(n−1) 2
+2
otherwise
P
di
.
Now we can define the quantum multiplication as a linear map mq : Sym (H ∗ (F ln , Z)[q1 , . . . , qn−1 ]) → H ∗ (F ln , Z)[q1 , . . . , qn−1 ] given by mq (
m Y
i=1 d¯
Ωwi ) =
X d¯
¯X
qd
hΩw Ωw1 · · · Ωwm id¯Ω∗w ,
w
dn−1 q1d1 · · · qn−1
where q = and (Ω∗w ) is the dual basis of (Ωw ). Then the quantum cohomology ring QH ∗ (F ln ) is a commutative and associative Z[q1 , . . . , qn−1 ] – algebra. Let 0 = E0 ⊂ E1 ⊂ · · · ⊂ En = Cn ⊗ OF be the universal flag of subbundles on F ln . 30
Theorem 8 (A. Givental and B. Kim, I. Ciocan–Fontanine). The small quantum cohomology ring is generated by xi = c1 (En−i+1 /En−i ), i = 1, . . . , n, as a Z[q1 , . . . , qn−1 ]-algebra and QH ∗ (F ln ) ∼ = Z[x1 , . . . , xn , q1 , . . . , qn−1 ]/(e1 (x|q), . . . , en (x|q)), where ei (x|q) is given by the expansion of the following determinant
det
x1 + t q1 0 −1 x2 + t q2 0 −1 x3 + t .. .. .. . . . 0 ... 0 0 ... ... 0 ... ...
... 0 q3 .. .
... ... 0 .. .
... ... ... .. .
0 0 0 .. .
−1 xn−2 + t qn−2 0 0 −1 xn−1 + t qn−1 ... 0 −1 xn + t
= tn + e1 (x|q)tn−1 + · · · + en (x|q). It follows from Theorem 8 that any Schubert cycle Ωw may be expressed b w (x, q) in QH ∗ (F ln ). The polynomial S b w (x, q) is a deforas a polynomial S b w (x, 0) = Sw (x). Consider mation of the Schubert polynomial Sw (x) and S the correlation function hΩw1 . . . Ωwm i =
X
¯
q d hΩw1 . . . Ωwm id¯.
d¯
b w (x; q) is characterized by the condition Then S
b w (x, q)Ωw . . . Ωw i hΩw Ωw1 . . . Ωwm i = hS m 1
for any w1 , . . . , wm ∈ Sn . b w (x; q) is called a geometric quantum Schubert polynomial. By definiS b w (x; q) ∈ QH 2l(w) (F ln ). tion S
7
Proofs of Theorem 3 and quantum Cauchy formula. 31
Theorem 9 Let I ∈ T. Then e w (x)iQ = heI (x), Sw (x)i, heeI (x), S
for any permutation w ∈ Sn .
Proof. The proof is based on the arguments due to I. Ciocan-Fontanine [C]. To begin with, let us remind his results. We consider the hyper-quot scheme HQd¯(P1 , F ln ) associated to P1 with multidegree d¯ = (d1 , . . . , dn−1 ). Let Cn ⊗ O → Tn−1 → · · · → T2 → T1 → 0 be the universal sequence of quotients on P1 × HQd¯(P1 , F ln ) and Si = Ker{Cn ⊗ O → Tn−i }. We also consider the dual sequence ∗ Cn ⊗ O → Sn−1 → · · · → S1∗ .
We fix a flag 0 = V 0 ⊂ V 1 ⊂ · · · ⊂ V n = Cn and define the subschema Dwp,q of P1 × HQd¯(P1 , F ln ) as the locus where rank(Vp ⊗ O → Sq∗ ) ≤ rw (q, p), and rw (q, p) := ♯{i | i ≤ q, wi ≤ p}. Let Dwp,q (t) = Dwp,q and
\
{{t} × HQd¯(P1 , F ln )}
Ωw (t) =
n−1 \
Dwp,q (t).
p,q=1
¯ w (t) in the Chow ring CH l(w)(HQd¯(P1 , F ln )) is indepenThen the class of Ω dent of t ∈ P1 and the flag V0 ⊂ · · · ⊂ Vn . The boundary HQd¯(P1 , F ln ) \ Md (P1 , F ln ) consists of n − 1 divisors D1 , . . . , Dn−1 , which are birational to P1 × HQd¯1 (P1 , F ln ), . . . , P1 × HQd¯n−1 (P1, F ln )
32
¯ s (t), respectively, where d¯i = (d1 , . . . , di −1, . . . , dn−1 ). Let xi (t) = Ωsi (t)−Ω i−1 then for any permutation w ∈ Sn there exists an element Gw (t) ∈ CH∗ (
n−2 [
Di )
i=1
such that
¯ w−1 (t) − Sw (x1 (t), . . . , xn−1 (t)) = j∗ (Gw−1 (t)), Ω
where j:
n−2 [
Di → HQd¯(P1 , F ln )
i=1
is the inclusion. Let [m, k] ∈ Sn be the permutation 1 2 ... m− k − 1 m− k m− k + 1 ... m m + 1 ... n 1 2 ... m− k − 1 m m−k ... m −1 m + 1 ... n
!
.
ˆ [m,k]−1 (x) is the elementary symThen the geometric Schubert polynomial S metric function in x1 , . . . , xm−1 of degree k. Let I = (i1 , . . . , in ). We have to calculate in CH ∗ (HQd¯(P1 , F ln ))
n−1 \
Ω[n−ν+1,iν
ν=1
]−1 (t) ·
N Y
Ωwj (tj ) −
n−1 Y
S[n−ν+1,iν ] (x(t)) ·
ν=1
j=1
N Y
Ωwj (tj ), (25)
j=1
for distinct t, t1 , . . . , tN and w1 , . . . , wN ∈ Sn such that n−1 X
iν +
ν=1
N X
l(wj ) = n(n − 1)/2 + 2
j=1
n−1 X
dk ,
k=1
where ∩ is the classical intersection product and
n−1 \ ν=1
Ω[n−ν+1,iν ]−1 (t)
is the corresponding degeneracy locus on HQd¯(P1 , F ln ). In order to calculate the expression (25), we are going to prove at first that
n−1 \ ν=1
Ω[n−ν+1,iν
]−1 (t) ·
N Y
j=1
¯ w (tj ) − Ω j
n−1 Y ν=1
33
¯ [n−ν+1,i ]−1 (t) · Ω ν
N Y
j=1
¯ w (tj ) = 0. Ω j
The LHS of the last expression can be computed as the number of points in n−1 Y
¯ [n−ν+1,i ]−1 (t) · Ω ν
ν=1
N Y
¯ w (tj ) Ω j
j=1
n−2 suppoted on ∪i=1 Di . Let jd¯k be the natural rational map
P1 × HQd¯k (P1 , F ln )− → HQd¯(P1 , F ln ). From Remark 3 in [C], ¯ jd−1 ¯k (Ω[n−ν+1,iν ]−1 (t)) = (
¯ [n−ν+1,i ]−1 (t) ∪ Tn−ν−i +1≤p≤n−ν−1 D p,n−ν−1 −1 (t), if k = n − ν, P1 × Ω ν ν [n−ν+1,iν ] 1 ¯ otherwise, P × Ω[n−ν+1,iν ]−1 (t)
where \
p,n−ν−1 ¯ D[n−ν+1,i −1 (t) = {t} × Ω[n−ν,iν −1]−1 (t). ν]
n−ν−iν +1≤p≤n−ν−1
Because, by assumption, n−1 X
iν +
n−1 X
l(wj ) =
ν=1
ν=1
n−1 X n(n − 1) +2 dk , 2 k=1
we have ¯ [k,i −1]−1 (t) · Ω n−k
Y
¯ [n−ν+1,i ]−1 (t) · Ω ν
N Y
¯ w (tj ) = 0 Ω j
j=1
ν6=n−k
on the hyper-quot scheme HQd¯k (P1 , F ln ). Hence, we have equality
n−1 \ ν=1
Ω[n−ν+1,iν
]−1
(t) ·
N Y
¯ w (tj ) = Ω j
n−1 Y
¯ [n−ν+1,i ]−1 (t) · Ω ν
ν=1
j=1
N Y
¯ w (tj ). Ω j
j=1
Our next observation is that the intersection number in the RHS of the last equality is equal to n−1 Y
Ω[n−ν+1,iν ]−1 (sν ) ·
ν=1
N Y
j=1
34
Ωwj (tj ),
where we can chose s1 , . . . , sn−1 , t1 , . . . , tN ∈ P 1 to be the pairwise distinct points, since the class [Ωw (t′ )] in the Chow ring CH ∗ (HQd (P 1 , F ln )) does not depends on the chose of t′ ∈ P 1 . Now we are going to use the following identity m Y
ak −
k=1
m Y
bk =
m k−1 X Y
bj (ak − bk )
k=1 j=1
k=1
m Y
aj .
j=k+1
Let us take in the last equality m = n − 1 ak := Ω[n−k+1,ik ]−1 (sk ), bk := S[n−k+1,ik ] (x(sk )). Then we obtain the following equality n−1 Y
Ω[n−ν+1,iν ]−1 (sν )
ν=1
n−1 Y X k−1
=
N Y
Ωwj (tj ) −
n−1 Y
S[n−ν+1,iν ] (x(sν ))
ν=1
j=1
N Y
Ωwj (tj )
j=1
Ω[n−j+1,ij ]−1 (sj ) · j∗ (G[n,ik ]−1 (t))
k=1 j=1
n−1 Y
S[n−j+1,ij ] (x(sj ))
j=k+1
Q
· B,
where B := N j=1 Ωwj (tj ). The contributions from j∗ ((G[n,ik ]−1 (t)), 1 ≤ k ≤ n − 1, can be computed by using the arguments in [C]. Inded, as in the proof of Theorem 4 in [C], the intersection number (1 ≤ k ≤ n − 1) k−1 Y
Ω[n−l+1,il]−1 (sl ) · j∗ (G[n,ik ]−1 (t))
n−1 Y
S[n−l+1,il ] (x(sl )) · B
l=k+1
l=1
is the number of points in k−1 Y
Ω[n−l+1,il]−1 (sj )
S[n−l+1,il ] (x(sl )) · B
l=k
l=1
S
n−1 Y
n−2 supported on j=1 Di . Hence, by induction, we have the following identity for correlation functions
h
n−1 \ ν=1
!
Ω[n−ν+1,iν ]−1 ·
N Y
Ωwj i = h
n−1 Y ν=1
j=1
35
e [n−ν+1,i ] S ν
·
N Y
j=1
Ωwj i,
where
T
is the classical intersection product and · is the product in Sym H ∗ (F ln , Z)[q1 , . . . , qn−1 ].
The last equality for correlation functions is equivalent to the following one mq (Ω[n,i1 ]−1 ∩ Ω[n−1,i2 ]−1 , ∩ . . . ∩ Ω[l,in−1 ]−1 , ∗) = mq (eei1 · eei2 · · · eein−1 , ∗).
This completes the proof.
Remark 10 Using the similar geometrical arguments we can prove an analog of Theorem 9 for the quantum complete polynomials: Theorem 9’ Let I ⊂ δn . Then e (x), S e w (x)iQ = hhI (x), Sw (x)i, hh I
for any permutation ω ∈ Sn . It is easy to see that Theorem 5’ is a corollary of Theorem 9’.
8 8.1
Correlation functions. Higher genus correlation function and the Vafa– Intriligator type formula.
Fix a Riemann surface C of genus g. We denote by Md (C, F ) the moduli space of morphism from C to F ln . One can define the higher genus Gromov– Witten invariants by method which is similar to that in the case of genus zero, [RT]. We have the following recursion relation for higher genus correlation function corresponding to the generating function for higher genus Gromov– Witten invariants hΩw1 . . . ΩwN ig =
X
hΩw1 . . . ΩwN Ωv Ω∗v ig−1
v∈Sn
(cf. Ruan–Tian [RT]). From Corollary 2 and Theorem 11 we can deduce the Vafa–Intriligator type formula for higher genus correlation functions, namely, let hP (x1 , . . . , xn )ig be the genus g correlation function corresponding to a polynomial P , then 36
hP (x1 , . . . , xn )ig = ResIe(P Φg ) = X
∂ eei = P (x1 , . . . , xn ) det ∂xj e e1 =···=e en =0
e w (x, y), S e w (x, y)i(y) = where Φ(x) = hS 0 0
X
w∈Sn
!−1
(Φ(x1 , . . . , xn ))g ,
e w (x)S e w w (x) S 0
= C (q,q) (x, x).
To simplify!the formula above, we use the following observations ∂ eei e e w (x) (mod I); • det ≡ n! S 0 ∂xj e e w (x) (mod I). • Φ(x) := C (q,q) (x, x) ≡ n! S 0 Hence, we obtain
Theorem 10 (Higher genus Vafa-Intriligator formula) hP (x1 , . . . , xn )ig = ResIe(P Φg ) =
X
e e1 =···=e en =0
e w (x) P (x1 , . . . , xn ) S 0
g−1
,
e w (x) = eeδ (x) := ee1 (x1 )ee2 (x1 , x2 ) . . . een−1 (x1 , . . . , xn−1 ), and eei (z) is where S 0 the quantum elementary polynomial of degree i in the variable z = (z1 , . . . , zm ), see Section 5.2.
Remark 11 The polynomial ′
C (q,q ) (x, y) :=
X
w∈Sn
′
e (q) (x)S e (q ) (y) S w w0 w
corresponds to the dual class of the diagonal in the quantum cohomology ring QH ∗ (F ln × F ln , (q, q ′)) = QH ∗ (F ln , q) ⊗ QH ∗ (F ln , q ′ ).
8.2
Witten–Dijkgraaf–Verlinde–Verlinde equations for symmetric group.
The Witten–Dijkgraaf–Verlinde–Verlinde equations (WDVV-equations) are e uS e vS e w i ∈ Z[q1 , . . . , qn−1 ], where equations on the correlation functions hS u, v, w ∈ Sn . The correlation functions satisfy the following conditions 1) Normalization: e vS e w i = hSv , Sw i. h1S 37
2) Initial data: e w i = qk . es S es S hS 0 k k
3) Degree conditions:
e uS e vS e wi = 0 hS
if either l(u) + l(v) + l(w) < l(w0 ), or difference l(u) + l(v) + l(w) − l(w0 ) is an odd positive integer. 4) WDVV-equations: X v
ew S ew S e v ihS e w vS ew S ew i = hS 1 2 0 3 4
X v
ew S e v ihS e w vS e w i. ew S ew S hS 2 3 0 1 4
for any w1 , w2 , w3 , w4 ∈ Sn . Conjecture 1 Conditions 1)–4) uniquely determine the correlation funce uS e vS e w i. tions hS ew S ew S e w i is a generating function Remark 12 1) Correlation function hS 1 2 3 for the Gromov–Witten invariants (see Definition 4): ew S ew S e w i := hS 1 2 3
X d
ew S e wS ew i . q d hS 1 3 d
2) More generally, es S ew i es S hS 0 ij k
8.3
=
(
qi . . . qj−1, if 1 ≤ i ≤ k < j ≤ n, 0, otherwise.
Residue formula.
Theorem 11 Correlation function hP (x1 , . . . , xn )i is given by the formula hP (x1 , . . . , xn )i =
X
Resp
e e1 (p)=···=e en (p)=0
!
P (x1 , . . . , xn )dx1 ∧ · · · ∧ dxn . ee1 · · · een
Proof. If the polynomial P (x1 , . . . , xn ) is in the ideal generated by ee1 , . . . , een , then the left and right hand sides of the formula are zero. Hence, it is enough to prove that ! νn−1 X xν11 · · · xn−1 dx1 ∧ · · · ∧ dxn Resp ee1 · · · een 38
=
(
1, if (ν1 , . . . , νn−1 ) = (n − 1, n − 2, . . . , 1) 0, if 0 ≤ ν1 + · · · + νn−1 < n(n − 1)/2, 0 ≤ νi ≤ n − i.
We can extend the meromorphic form ν
n−1 xν11 · · · xn−1 dx1 ∧ · · · ∧ dxn ω= e e1 · · · een
on the affine space An(x1 ,...,xn) to (P1 )n . For each subset I ⊂ {1, . . . , n}, we consider the coordinate chart UJ = An(z J ,...,z J ) , where n
1
ziJ
=
(
xi , if i 6∈ J, 1/xi , otherwise.
Then (P1 )n =
[
UJ .
J
Let e¯Jj (z1J , . . . , znJ ) = (
Y
i∈J
Then
ν
n−1 xν11 · · · xn−1 (
ω = (−1)♯J
ziJ )eej (x1 , . . . , xn ).
Y
ziJ )n−2 dz1J ∧ · · · ∧ dznJ
i∈J
e¯J1 · · · e¯Jn
on Uφ ∩ UJ . If ♯J = j, then there exists a polynomial Qi (z1J , . . . , znJ ) for i ∈ J such that X e¯Ij (z1J , . . . , znJ ) = 1 + ziJ Qi . i∈J
This follows from eej (x1 , . . . , xn ) = ej (x1 , . . . , xn ) + (terms of lower degree).
Therefore e¯J1 , . . . e¯Jn do not have common zero on
BJ = {(z1J , . . . , znJ ) ∈ UJ | ziJ = 0, i ∈ J}. From the residue theorem, if 0 ≤ ν1 + · · · + νn < n(n − 1)/2, 0 ≤ νi ≤ n − i, then ! νn−1 X xν11 · · · xn−1 dx1 ∧ · · · ∧ dxn = 0. Resp ee1 · · · een 39
On the other hand, X
x1n−1 · · · xn−1 dx1 ∧ · · · ∧ dxn ee1 · · · een
Resp
e ei (p)=0
!
=−
where p runs over the common zeros of e¯J1 , . . . , e¯Jn in
X
Resp ω,
p
[
BJ . Let y1 = 1/x1 ,
1∈J
z = (y1 , x2 , . . . , xn ) and
e¯∗1 (z) = y1 (1 + y1 (x2 + · · · + xn )). Then we have −
{1} e¯2
e¯∗1
P
p
Resp ω =
X
Resp
x2n−2 · · · xn−1 dy1 ∧ · · · ∧ dxn {1}
= = ··· = =0 in the locus {y1 = 0}
=
X p
by induction.
Resp
{1}
e¯∗1 (z) · e¯2 (z) · · · e¯N (z)
{1} e¯n
x2n−2 · · · xn−1 dx2 ∧ · · · ∧ dxn ee1 (x2 , . . . , xn ) · · · een−1 (x2 , . . . , xn )
!
!
= 1,
From the residue formula, the correlation function is given by the quantum residue ResIe, namely, hP (x1 , . . . , xn−1 )i = ResIeP (t1 , . . . , tn−1 ).
In order to relate the quantum residue with the classical one, we consider the quantum residue generating function Ψ(t) = h
n−1 Y i=1
X ti i= hxν it−ν . ti − xi n−1 ν∈(Z ) ≥0
Then, we have ResIeP (t1 , . . . , tn−1 ) = ResI (P (x1 , . . . , xn−1 )Ψ(x)) .
Hence, it is important to determine the generating function Ψ(t). Let fi (t) = tn +
n−1 X j=0
40
(i)
γn−j tj
be the characteristic polynomial of the quantum multiplication by xi with respect to the basis consisting of the quantum Schubert polynomials. Let us consider the (n! + 1) × (n! + 1)-matrix Cn (t) such that (Cn (t))1,j =
(Cn (t))i,j =
(−1)j−1 tn−j+2 ; (j − 1)!
!
(−1)n i−2 tj−2 , if i ≥ 2, i + j ≥ n + 2, (j − 1)! n − j + 1 0,
otherwise.
We define the differential operator Di by (i)
(i)
Di = (γn! , γn!−1, . . . , 1) · Cn (ti ) ·t (1, ∂/∂ti , . . . , (∂/∂ti )n! ).
Proposition 16 The generating function Ψ(t) satisfies the system of differential equations Di Ψ(t) = 0, 1 ≤ i ≤ n − 1. ν
n−1 Conversely, these differential equations and the initial values hxν11 · · · xn−1 i for 0 ≤ νi ≤ n! − 1 determine the generating function uniquely.
Proof. Let x be a variable. Since
we have
1 x x2 x3 .. . xn!
= Cn
Di Ψ(t) = h
·
t(t − x)n! −x(t − x)n!−1 2!x(t − x)n!−2 −3!x(t − x)n!−3 .. . (−1)n! (n!)!x
,
Y fi (xi ) tj i = 0. n!+1 (ti − xi ) j6=i tj − xj
On the other hand, the recursive relations hfi (xi )P (x1 , . . . , xn−1 )i = 0 41
ν
n−1 with the initial values hxν11 · · · xn−1 i for 0 ≤ νi ≤ n! − 1 determine the correlation function uniquely. Remark 13 We can also consider another generating function
hexp(x1 t1 + · · · + xn tn )i. This is the generating volume function in [GK]. This generating function satisfies ! ∂ ∂ hexp(x1 t1 + · · · + xn tn )i = 0, ,···, eei ∂t1 ∂tn for 1 ≤ i ≤ n.
In the case of n = 3, we can calculate the generating function Ψ(t) explicitly. The results of calculation one can find in the Appendix B.
9
Extended Ehresman–Bruhat order and quantum Pieri rule.
Let us remind that the Ehresman–Bruhat order denoted by ≤, is the partial order on Sn that is the transitive closure of the relation →. Relation v → w means that 1) l(w) = l(v) + 1, 2) w = v · t where t is a transposition. In other words, if v and w are permutations, v ≤ w means that there exists r ≥ 0 and v0 , v1 , . . . , vr in Sn such that v = v0 → v1 → · · · → vr = w. Now let us define the extended Ehresman–Bruhat order v ⇐ w on Sn . First of all, we define a relation v ← w (see, also, [FGP]). Relation v ← w means that 1) w = v · t, where t is a transposition, 2) l(w) ≥ l(v) + l(v −1 w). Remark 14 i) It follows from [M], (1.10), that condition 2) is equivalent to the following one 2′ ) w(i) < w(j) and for all k such that i < k < j we have w(i) < w(k) < w(j). 42
ii) If w = vti,i+1 and l(w) = l(v) + 1 (i.e. v → w in the Bruhat order), then we have also an arrow v ← w. This is clear because in our case we have l(w) = l(v) + l(ti,i+1 ). Example. Symmetric group S3 (see Figure 1). 121 q 3 k Q q q q qq q q Q qq Q q qq q q q q qQ q qqQ q q q q qq Q q q q qq Q q s + q 12 * YH H q 21 q qq 6 H 6qq q H q q q H HH q q q q q q H q q q H q q q H q q q H q q q H q ? H ? q H q 1 q q 2 qq q k qqqq Q 3 q q q Q q qq q q Q qq q q qq Q qq Q q q q qq q q q q Q s ?+ Q id
Figure 1: Extended Ehresman-Bruhat order for S3 . We define a weight of an arrow v ← w, denoted by wt(v ← w), to be equal to the product qi . . . qi+s−1 , if t = tij and 2s := l(w) + 1 − l(v). We assume that weight of any arrow v → w is equal to 1 (see, also, [FGP]). Let us say that an arrow v ← w (resp. v → w) has a color k if w = vtij and 1 ≤ i ≤ k < j ≤ n. Extended Ehresman–Bruhat order on Sn (notation v ⇐ w) is the transitive closure of the relations ←, and →. In other words, there exists r ≥ 0 and v0 , v1 , . . . , vr in Sn such that v = v0 ⇀ ↽ vr = w, ↽ ··· ⇀ ↽ v2 ⇀ ↽ v1 ⇀
(26)
where symbol vi ⇀ ↽ vi+1 means either vi → vi+1 or vi ← vi+1 . For given pair v ⇐ w, we consider a sequence of arrows (26) as a path between v and w (notation v 7→ w) in the extended Ehresman–Bruhat order and call it as a BE–path (Bruhat–Ehresman path). We denote the number r in a representation (26) by l(v 7→ w). 43
Let us define a weight of a BE–path v 7→ w as follows wt(v 7→ w) =
r−1 Y
wt(vi ⇀ ↽ vi+1 ).
i=0
k
We will say that BE–path v 7→ w has a color k, notation v 7→ w, if in the representation (26) all arrows vi ⇀ ↽ vi+1 (i = 0, . . . , r −1) have the same color k. Theorem 12 (Quantum Pieri’s rule). Let us consider the Grassmanian permutation [b, d] = (1, 2, . . . , b − d − 1, b, b − d, b − d + 1 . . . , b − 1, b + 1, . . . , n), for 2 ≤ b ≤ n, 1 ≤ d ≤ b. Then e [b,d] · S ev S
≡
X
b e w (mod Ien ), wt(v 7→ w)S b
where the sum runs over all BE–paths v 7→ w, s.t. 1) l(v → 7 w) = d;
2) if vl = vl+1 (il jl ) (l = 0, . . . , d − 1), then all il are different. (Note that S[b,d] = ed (x1 , . . . , xb−1 )). Sketch of the proof. It is enough to consider the case d = 1 (induction!). In the case d = 1, we use a quantum analog of Kohnert–Veigneau’s method [KV]. Namely, at first we prove the quantum Pieri rule (for d = 1) for double quantum Schubert polynomials and then take y = 0 (see Theorem 4). e w (x, y)). Proposition 17 (Quantum Pieri’s rule for S 0 e w (x, y) ≡ (xj + yn+1−j )S 0
X i<j
e w t (x, y) − qij S 0 ij
X
j
