Calculating Invariant Rings of Finite Groups over Arbitrary ... - MIMUW
J. Symbolic Computation (1996) 21, 351–366
Calculating Invariant Rings of Finite Groups over Arbitrary Fields GREGOR KEMPER IWR, Universit¨ at Heidelberg, Im Neuenheimer Feld 368, 69 120 Heidelberg, Germany† (Received 3 April 1995)
An algorithm is presented which calculates rings of polynomial invariants of finite linear groups over an arbitrary field K. Up to now, such algorithms have been available only for the case that the characteristic of K does not divide the group order. Some applications to the question whether a modular invariant ring is Cohen–Macaulay or isomorphic to a polynomial ring are discussed. c 1996 Academic Press Limited °
1. Introduction If G is a finite linear group over a field K such that char(K) 6 | |G|, there are various effective methods to calculate the invariant ring I of G, i.e., to find a finite system of generators of I as an algebra over K (see Sturmfels, 1993, Section 2; McShane, 1992; Kemper, 1993, 1994). These methods make use of the Reynolds operator, Molien’s formula, the Cohen–Macaulay property of invariant rings, and of Noether’s degree bound in the case char(K) > |G| (Noether, 1916). If, on the other hand, |G| is a multiple of char(K) (which we shall call the modular case), none of these techniques are available since they all involve divisions by |G|. In fact, G is not a linearly reductive group in this case. Nevertheless, the invariant ring is finitely generated as a K-algebra (Noether, 1926). If G is a permutation group there is a very nice algorithm by G¨ obel (1995) which calculates generators over any commutative ground ring and which implies a degree bound for the generators. But for the case of finite linear groups there is no algorithm available at the moment to construct a system of generators, and modular invariant rings are calculated by ad hoc methods (see Benson, 1993, Chapter 8; Wilkerson, 1983; Adem and Milgram, 1994, Chapter III). The purpose of this paper is to fill this gap. With the knowledge of generators for the invariant ring, it is quite easy to calculate its depth and its Poincar´e series (see Section 4.1). It is even easier to check the Cohen–Macaulay property. So the algorithm presented here could be useful to gain some experience and to test hypotheses. The first section of this paper is concerned with the calculation of primary invariants, which serve as a kind of first approximation to the invariant ring. A new algorithm to †
E-mail: [email protected]
0747–7171/96/030351 + 16 $18.00/0
c 1996 Academic Press Limited °
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construct primary invariants is proposed. Section 3 contains the main algorithm which does the step from the primary invariants to the full invariant ring for arbitrary ground fields. The last section is devoted to some applications. Here we probe into the question of the Cohen–Macaulay property mentioned above, and give a simple algorithm to decide whether the invariant ring of a given group is isomorphic to a polynomial ring. The last question has its background in another “defect” of modular linear groups, that their invariant rings are not always polynomial rings if the group is a reflection group. The author has incorporated the algorithms presented in this paper into a new version of the Invar -package† for Maple (Kemper, 1993). Let us fix some notation. Throughout this paper, K is an arbitrary field and G ≤ GLn (K) is a finite matrix group of degree n over K. G operates on the polynomial ring K[x1 , . . . , xn ] by linear transformations of the indeterminates xi . We write I for the invariant ring: I = K[x1 , . . . , xn ]G = {f ∈ K[x1 , . . . , xn ] | σ(f ) = f ∀σ ∈ G}. This is a graded subalgebra of K[x1 , . . . , xn ]. 2. Calculating Primary Invariants Let R be any graded K-algebra of Krull dimension m with R0 = K, then by Noether’s normalization lemma (see, for example, Benson, 1993, Theorem 2.2.7), there exist homogeneous elements f1 , . . . , fm ∈ R such that R is finitely generated as a module over A := K[f1 , . . . , fm ]. If R is the invariant ring I, then m = n and f1 , . . . , fn are called primary invariants. In this case, f1 , . . . , fn are also primary invariants for any subgroup H ≤ G since K[x1 , . . . , xn ] is integral over I. The following proposition is the key to the calculation of primary invariants: Proposition 1. A set {f1 , . . . , fi } ⊂ I of homogeneous invariants can be extended to a system of primary invariants if and only if dim(f1 , . . . , fi ) = n − i. In particular, homogeneous f1 , . . . , fk ∈ I form a system of primary invariants if and only if k = n and (2.1) VK¯ (f1 , . . . , fk ) = {0}, where VK¯ denotes the set of zeros over the algebraic closure of K. Proof. By Smith (1995), Proposition 5.3.7, homogeneous invariants f1 , . . . , fn are primary invariants if and only if dim(f1 , . . . , fn ) = 0. By Krull’s Principal Ideal Theorem (see, for example, Eisenbud, 1995, Theorem 10.1) each fi can diminish the dimension of the ideal by at most one, hence dim(f1 , . . . , fi ) = n−i if f1 , . . . , fn are primary invariants. Conversely, if dim(f1 , . . . , fi ) = n − i there exist homogeneous fi+1 , . . . , fn ∈ R := I/(f1 , . . . , fi ) such that R is finitely generated as a module over K[fi+1 , . . . , fn ]. Hence ¢ ¢ ¡ ¡ dimK K[x1 , . . . , xn ]/(f1 , . . . , fn ) = dimK R/(fi+1 , . . . , fn ) < ∞, † This version can be obtained by anonymous ftp from the site ftp.iwr.uni-heidelberg.de under /pub/kemper/INVAR2.
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so dim(f1 , . . . , fn ) = 0 by Becker and Weispfenning (1993), Theorem 6.5.4. This completes the proof. 2 Primary invariants are far from being uniquely determined by G. But the choice of primary invariants f1 , . . . , fn determines the minimum number mf1 ,...,fn of generators for I as a module over A = K[f1 , . . . , fn ], which is mf1 ,...,fn = dimK (K ⊗A I) by Smith (1995), Corollary 5.2.5. It is of crucial importance for the effectiveness of subsequent calculations that mf1 ,...,fn be as low as possible, as this is most likely to be achieved if the degrees of the primary invariants are chosen as small as possible, as is illustrated by Proposition 12 and Example 5(b). Hence I will call an algorithm that constructs primary invariants good if it is likely to produce primary invariants of small degrees. So the best algorithm in my sense need not be the fastest. Algorithms for the construction of primary invariants go back as far as Hilbert (1893) and Weber (1899). A good modern treatment can be found in Sturmfels (1993), Algorithm 2.5.8. He proposed that as a first step one should collect invariants of increasing degrees until arriving at a set {f1 , . . . , fk } having the property (2.1). This set is obtained by successively calculating K-bases for the vector spaces of homogeneous invariants of degree 1, 2, . . . and including a basis element into the set if it does not lie in the radical of the ideal spanned be the fi gotten so far. In the next step, one tries to delete elements fi from the set while retaining (2.1). Experience shows that in most cases this will lead to a set of n elements, i.e., a system of primary invariants. If it does not, we are left with k polynomials fi , k > n, which satisfy (2.1), and must apply a third step which consists of first converting all fi to invariants of the same degree by taking suitable powers Pk and then choosing random n × k-matrices (ai,j ) and checking condition (2.1) for fei = j=1 ai,j · fj (i = 1, . . . , n), until the fei form a system of primary invariants. It is especially the third step of this algorithm that is unsatisfactory since it involves a random search and since the conversion to equal degrees makes the degrees explode. But even in the majority of cases when this step is unnecessary, the algorithm often yields primary invariants whose degrees are not the lowest possible. Let us look at a typical example for this. Example 2. Let
µ G=h
i 0
0 −i
¶ i ≤ GL2 (C).
Then the above algorithm would in the first step produce the invariants f1 = x1 x2 , f2 = x41 and f3 = x42 and omit f1 in the second step, yielding primary invariants of degrees 4 and 4. But there are primaries of degrees 2 and 4, namely f1 = x1 x2 and f2 = x41 +x42 , which the algorithm would miss. / Another algorithm due to E. Dade (see Stanley, 1979) constructs primary invariants by taking products over G-orbits of suitable linear forms. This algorithm is very quick but also has the disadvantage that it tends to produce primary invariants of too high degrees (as it would in Example 2). While the algorithm of Sturmfels always tries to find systems of primary invariants as a whole, Proposition 1 suggests a strategy of successively adding primary invariants to
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the system, the condition for a new primary invariant fi+1 being that it decreases the dimension of the ideal (f1 , . . . , fi ), or, equivalently, that it lies in none of the associated prime ideals of (f1 , . . . , fi ). This brings us to propose the following algorithm: Algorithm 3. (Constructing primary invariants) Input: A set of generators of G. Output: Primary invariants f1 , . . . , fn . Begin Set i := 1, d := 1, r := 1 and P1 := {}. While i ≤ n Do { Calculate a K-basis {b1 , . . . , bmd } of the space of invariants of degree d. (This can be done by writing down a general polynomial of degree d with unknown coefficients, applying the generators of G and solving the resulting system of linear equations for the unknown coefficients.) Pm d tj · bj with indeterminates tj . Form Id (t1 , . . . , tmd ) := j=1 For k = 1, . . . , r calculate the complete residues rk (t1 , . . . , tmd ) of Id w.r.t. the Gr¨ obner basis Pk . If there exist α1 , . . . , αmd ∈ K with rk (α1 , . . . , αmd ) 6= 0 ∀k = 1, . . . , r Then { (See below for how to decide this.) Set fi := Id (α1 , . . . , αmd ). Calculate the associated prime ideals of (f1 , . . . , fi ). (See below for algorithms which perform this.) Set r :=(number of associated primes). obner bases of the associated primes w.r.t. any monoLet P1 , . . . , Pr be Gr¨ mial order. Set i := i + 1. } (End Then) Else (There is no other primary invariant of degree d.) { Set d := d + 1. } (End Else) } (End While) End. Since the rk are the complete residues of Id w.r.t. the Pk , the condition rk (α1 , . . . , αmd ) 6= 0 ∀k is equivalent to the requirement that Id (α1 , . . . , αmd ) lies in none of the associated prime ideals of (f1 , . . . , fi ). If any of the rk is zero, then this condition is certainly false. If, on the other hand, all rk are non-zero, then the equations rk (α1 , . . . , αmd ) = 0 describe proper subspaces of K md , so in the case of an infinite K there are always αi satisfying the condition. It is easy to write an algorithm that specifies the αi successively to produce a non-solution of all these equations in this case. If on the other hand K is finite (and all rk non-zero), we can simply search the elements of K md for a non-solution. Before turning to the computation of associated prime ideals involved in Algorithm 3, we shall look at how it would handle the group considered in Example 2: Example 4. Let G be as in Example 2. Then the first primary invariant taken by Algorithm 3 is f1 = x1 x2 , yielding associated primes (x1 ) and (x2 ). Passing to degree 4, we
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get I4 = t1 · x41 + t2 · x21 x22 + t3 · x42 with complete residues r1 = t3 · x42 and r2 = t1 · x41 . So (α1 , α2 , α3 ) = (1, 0, 1) is a non-solution, leading to f2 = x41 + x42 as the second primary invariant. / According to my experience Algorithm 3 is very likely to produce a system of primary invariants of smallest possible degrees, but it is not guaranteed to do so, as there are counter examples: Example 5. (a) Consider the permutation group G = h(123)(456), (12)(45)i ∼ = S3 acting on the indeterminates of C[x1 , . . . , x6 ]. There are primary invariants of degrees 1, 1, 2, 2, 3, 3 given by the elementary symmetric functions in the first and last three variables. But by an unlucky choice Algorithm 3 might pick f1 = x1 + x2 + x3 , f2 = x4 + x5 + x6 , f3 = x1 x2 + x1 x3 + x2 x3 , f4 = x1 x4 + x2 x5 + x3 x6 , f5 = x4 x5 x6 as the first five primary invariants. Then x1 = x2 = x3 = x4 = 0, x5 = −x6 is a zero of these fi and also of all invariants of degree 3, hence there is no further primary invariant of this degree. So Algorithm 3 would proceed to the next degree and finally obtain a last primary invariant of degree 6. (b) Take the “first A5 in SL4 (F2 )” of Adem and Milgram (1994) p. 116, given by 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 G = h 0 0 1 1 , 1 0 1 0 0 0 1 0 i ≤ GL4 (F2 ). 0 0 1 0 0 1 0 1 0 0 0 1 We only sketch the results here. Adem and Milgram’s primary invariants are of degrees 5, 5, 12, 12, and Algorithm 3 would find primaries of degrees 3, 3, 12, 20. But the “best” primary invariants (i.e., those leading to the smallest mf1 ,...,fn , see p. 353) are of degrees 3, 5, 8 and 12. They were also obtained by Algorithm 3 together with some degree of human intervention. With these primary invariants there are 24 module-generators for I of degree at most 24 needed, while Adem and Milgram’s primary invariants require 60 module-generators of degree at most 30. We see that it is still a subtle business to find good primary invariants! / The most expensive part of the algorithm is the computation of the associated prime ideals of (f1 , . . . , fi−1 ). Algorithms which perform this over any field K that is finitely generated over its prime field are given in Gianni et al. (1988). Here we can assume K to be finitely generated over its prime field by the matrix entries of the elements of G. These algorithms automatically yield Gr¨ obner bases for the associated primes. For the characteristic zero case see also Becker and Weispfenning (1993), Table 8.10. The complete computation of the associated primes can in most cases be circumvented by using the “Gr¨ obner factorization algorithm”, which is likely to produce the associated primes since in our case (f1 , . . . , fi−1 ) has no embedded primes. This algorithm is implemented as the function gsolve in Maple (Char et al., 1990). Since this algorithm always
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yields ideals which lie between (f1 , . . . , fi−1 ) and an associated prime, the condition that rk (α1 , . . . , αmd ) 6= 0 ∀k in Algorithm 3 will be necessary for the existence of another obner factorization alprimary invariant of degree d, if the Pk are the output of the Gr¨ gorithm. After having found a candidate fi = Id (α1 , . . . , αmd ), we have to check if this really decreases the dimension of the ideal by a Gr¨ obner basis method which is much simpler than the calculation of associated primes (see Becker and Weispfenning, 1993, Table 9.6). Only if fi does not qualify, we will have to calculate the associated primes rigorously. In the non-modular case, i.e., if char(K) 6 | |G|, we have Molien’s formula to calculate the Poincar´e series of the invariant ring. From this, we can make good guesses at the degrees of the primary invariants (see, for example, Sloane, 1977). So another variant of Algorithm 3 would be to incorporate guesses for complete systems of primary invariants by taking any homogeneous invariants of the “right” degrees and checking condition (2.1) before entering the full Algorithm 3. With both these modifications, Algorithm 3 performs reasonably well and is, in the author’s opinion, the best algorithm available at the moment for calculating primary invariants, although it is certainly slower than Dade’s algorithm (see above).
3. Calculating Secondary Invariants The next task is to find a system of generators of I as a module over A = K[f1 , . . . , fn ], where f1 , . . . , fn are primary invariants. Such generators are called secondary invariants. In the non-modular case, Molien’s formula provides complete information about the number and degrees of the secondary invariants. This reduces their calculation to a simple exercise of filling up homogeneous subspaces of invariants (see McShane, 1992, or Kemper, 1994). The idea for the general case now is to calculate secondary invariants g1 , . . . , gr for a subgroup H ≤ G with char(K) 6 | |H| (the trivial group will always do) first. Imposing G-invariance conditions on a general element of K[x1 , . . . , xn ]H will then lead to a system of linear equations with coefficients in K[x1 , . . . , xn ], for which we have to calculate the solutions whose components lie in A. So we need an algorithm which intersects a submodule of K[x1 , . . . , xn ]r with Ar , where Rr denotes the free module of rank r over a ring R. Ps r Lemma 6. Let R = K[x1 , . . . , xn ], M = i=1 R · bi ≤ R a submodule and A = K[f1 , . . . , fk ] ≤ R the subalgebra generated by elements f1 , . . . , fk ∈ R. Take the polynomial rings S = K[x1 , . . . , xn , t1 , . . . , tk ] and T = K[t1 , . . . , tk ] with further indeterminates t1 , . . . , tk and form f= M
s X
S · bi +
i=1
k X
(tj − fj ) · S r ≤ S r
j=1
and f ∩ T r. fT = M M Then with Φ: T r → Ar , tj 7→ fj we have fT ) = M ∩ Ar . Φ(M
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Proof. We consider the homomorphism Ψ: S r → Rr / M obtained by substituting f. tj 7→ fj , and claim ker(Ψ) = M f clearly lies in the kernel. Let P (t1 , . . . , tk ) be a product of tj ’s By construction, M and ei ∈ S r a vector of the standard basis. We show by induction on deg(P ) that ¢ ¡ f. There is nothing to show for deg(P ) = 0. If P (t1 , . . . , tk ) − P (f1 , . . . , fk ) · ei ∈ M 0 deg(P ) > 0, then P (t1 , . . . , tk ) = tj ·P (t1 , . . . , tk ) with deg(P 0 ) < deg(P ), so by induction ¢ ¡ 0 f, and we conclude P (t1 , . . . , tk ) − P 0 (f1 , . . . , fk ) · ei ∈ M ¢ ¡ P (t1 , . . . , tk ) − P (f1 , . . . , fk ) · ei = ³ ¡ ¢´ f. (tj − fj ) · P 0 (t1 , . . . , tk ) + fj · P 0 (t1 , . . . , tk ) − P 0 (f1 , . . . , fk ) · ei ∈ M It follows that for (g1 , . . . , gr ) ∈ S r we have ¯ ¯ f. (g1 , . . . , gr ) − (g1 ¯tj =fj , . . . , gr ¯tj =fj ) ∈ M ¯ ¯ f and hence (g1 , . . . , gr ) Now if Ψ(g1 , . . . , gr ) = 0, then (g1 ¯tj =fj , . . . , gr ¯tj =fj ) ∈ M ⊂ M f, which proves the claim. ∈M fT , and due to It follows that ker(Ψ |T r ) = M Ψ(T r ) = (Ar + M )/ M ∼ = Ar /(M ∩ Ar ) we have Tr
.
fT ∼ M = Ar /(M ∩ Ar )
with an isomorphism induced by Φ. This completes the proof.
2
We obtain the following algorithm: Algorithm 7. (Intersecting a module with a subalgebra) Input: Generators b1 , . . . , bs of a submodule M ≤ Rr , where R = K[x1 , . . . , xn ], and generators f1 , . . . , fk of a subalgebra A = K[f1 , . . . , fk ] ≤ R. Output: Generators c1 , . . . , cm ∈ Ar of M ∩ Ar as a module over A. Begin Take additional indeterminates t1 , . . . , tk and set S = K[x1 , . . . , xn , t1 , . . . , tk ]. f ≤ S r generated by the bi (i = 1, . . . , s) and by the Form the submodule M (tj − fj ) · ei (j = 1, . . . , k, i = 1, . . . , r). f w.r.t. a term order ≺ with the property that Calculate a Gr¨ obner basis B of M every xi is greater than any monomial in the tj . (See below for Gr¨ obner bases of modules.) Form B ∩ (K[t1 , . . . , tk ])r and substitute tj 7→ fj in its elements. Let {c1 , . . . , cm } be the resulting set. End. In the above algorithm we are using Gr¨ obner bases of modules. These were introduced by M¨oller and Mora (1986), who also extended Buchberger’s algorithm to this case. This algorithm is implemented in Macaulay (see Stillman et al., 1989). The elimination property for these Gr¨ obner bases, which was used in the algorithm, follows easily and can be found in (loc. cit.).
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Also implemented in Macaulay and described by M¨ oller and Mora (1986) are algorithms to find syzygy modules, i.e., solution modules for systems of linear equations over a polynomial ring. These constitute the second computational ingredient of the general algorithm to find secondary invariants, which follows now. Algorithm 8. (Calculating secondary invariants ) Input: A set S of generators of G and homogeneous f1 , . . . , fk ∈ I such that I is finitely generated as a module over A = K[f1 , . . . , fk ] (optimally, a system of primary invariants of small degrees). Output: Secondary invariants g1 , . . . , gm . Begin Choose a subgroup H ≤ G (for example, H = {ι}) and calculate homogeneous generators h1 , . . . , hr of K[x1 , . . . , xn ]H as a module over A. Calculate the module M ≤ K[x1 , . . . , xn ]r of all (p1 , . . . , pr ) ∈ K[x1 , . . . , xn ]r with r X
(σ(hi ) − hi ) · pi = 0 for all σ ∈ S.
(3.1)
i=1
(This is the calculation Pmof a syzygy module.) CalculateP M ∩ Ar = i=1 A · ci using Algorithm 7. r Set gi := j=1 ci,j · hj (i = 1, . . . , m), where ci = (ci,1 , . . . , ci,r ) ∈ Ar . End. The gi lie in I by construction. Conversely, for f = have by Equation (3.1) (a1 , . . . , ar ) ∈ M ∩ Ar ⇒ (a1 , . . . , ar ) = hence I =
Pm i=1
m X
Pr i=1
ai · hi ∈ I with ai ∈ A we
e ai · ci with e ai ∈ A ⇒ f =
i=1
m X
e ai · g i ,
i=1
A · gi , which proves the correctness of Algorithm 8.
Remark 9. (a) If χ: G → K × is a linear character, then the set Mχ of relative invariants of weight χ (i.e., of f ∈ K[x1 , . . . , xn ] such that σ(f ) = χ(σ) · f ∀σ ∈ G) is a module over I and also over A = K[f1 , . . . , fk ] for any invariants fi . If the fi satisfy the assumption made in Algorithm 8, then a generating set for Mχ as an A-module can be constructed by Algorithm 8 with χ introduced into Equation (3.1). (b) Let χ be as above and H ≤ G a p-Sylow subgroup, where p = char(K) 6= 0. Then χ|H = 1. Taking left coset representatives σ1 , . . . , σr of G/H, we have a “relative Reynolds operator” given by X 1 χ(σi−1 ) · σi (G : H) i=1 r
πχG/H =
(see Campbell et al., 1991 or Smith, 1995, p. 28). This is a projection K[x1 , . . . , xn ]H → →Mχ of I-modules. Hence we only have to calculate generators for K[x1 , . . . , xn ]H G/H as a module over A and then apply πχ to obtain Mχ . K[x1 , . . . , xn ]H itself is most conveniently calculated by applying Algorithm 8 recursively to a chain of subgroups H1 ≤ H2 ≤ · · · ≤ Hs = H with |Hi | = pi .
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(c) The set {g1 , . . . , gm } from Algorithm 8 can be made into a minimal set of secondary invariants (w.r.t. the primary invariants f1 , . . . , fn chosen once and for all) by successively deleting those gi which are contained in the module generated by the others. This test for membership amounts to the solution of a system of linear equations over K. / Let us look at an example now. Example 10. Consider the cyclic group G = Z4 with its regular representation over K = F2 . We find primary invariants f1 = x1 + x2 + x3 + x4 , f2 = x1 x3 + x2 x4 , f 3 = x 1 x 2 + x 2 x 3 + x 3 x 4 + x 4 x 1 , f 4 = x1 x 2 x 3 x 4 . With H = {ι} ≤ G, there are 16 generators h1 , . . . , h16 of K[x1 , . . . , xn ] as a module over A = K[f1 , . . . , f4 ], and the solution module M of Equation (3.1) is generated by 16 f in Algorithm 7 then has 80 generators and a Gr¨ vectors. The module M obner basis of 113 elements, which was calculated by Macaulay in a few seconds. From these, five lie in K[t1 , . . . , t4 ], and we get the secondary invariants g1 = 1, g2 = x31 + x32 + x33 + x34 , g4 = x31 x2 + x32 x3 + x33 x4 + x34 x1 ,
g3 = x21 x2 + x22 x3 + x23 x4 + x24 x1 , g5 = x31 x22 + x32 x23 + x33 x24 + x34 x21 .
The calculation can be essentially accelerated by taking H = Z2 ≤ G and applying Algorithm 8 recursively. We shall continue this example on p. 360 and show that the invariant ring is not Cohen–Macaulay. From the above secondary invariants we already see that the invariants of degree ≤ 4 do not generate I as a K-algebra, i.e., Noether’s degree bound (Noether, 1916) does not hold in the modular case. See Richman (1990) for examples of linear groups where generators of arbitrarily high degree are necessary for the invariant ring although the group order remains the same. Bertin (1967) and Benson (1993), p. 104, mention this example, but do not calculate the complete invariant ring. / Clearly, the Gr¨ obner basis calculation involved in Algorithm 7 is the most time consuming part of Algorithm 8. It will set the limit to the practical scope of the algorithm. 4. Applications This section is concerned with calculating some data associated with invariant rings, in particular free resolutions, Poincar´e series and depth. Easy algorithms are presented to decide the Cohen–Macaulay property and the property of being isomorphic to a polynomial algebra. 4.1. depth and the Cohen–Macaulay property Suppose that we have calculated primary invariants f1 , . . . , fn and secondary invariants g1 , . . . , gm by Algorithm 3 and 8. Then we can compute the module M ≤ K[x1 , . . . , xn ]m of syzygies between the gi and intersect M with Am (where as above A = K[f1 , . . . , fn ]) using Algorithm 7. We can further calculate a free resolution · · · → F2 → F1 → M ∩ Am → 0
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of M ∩ Am as an A-module, which yields a free resolution · · · → F2 → F1 → F0 → I → 0
(4.1)
of the invariant ring I. From this the Poincar´e series of I can easily be obtained (see Stanley, 1979, p. 496). If {g1 , . . . , gm } is a minimal set of secondary invariants and if (4.1) is a minimal free resolution then the homological dimension hdimA (I) is (by definition) the largest i such that Fi 6= 0. By the Auslander–Buchsbaum formula (see Auslander and Buchsbaum, 1957 or Eisenbud, 1995, Theorem 19.9) we have depthA (I) = n − hdimA (I), where depthA (I) is the maximal length of a regular sequence for I whose elements lie in A and have no constant term. But depth(I) := depthI (I) is the same as depthA (I) by the following proposition, whose proof is a word by word adaption of the proof of Corollary 19.15 in Eisenbud (1995) to the graded case: Proposition 11. Let A ≤ B be Noetherian graded algebras over a field K such that A0 = B0 = K and assume that B is finitely generated as an A-module. Then depthA (B) = depthB (B). Thus we can calculate the depth of I. Recall that I is called Cohen–Macaulay if depth(I) = n. The following proposition provides a much easier criterion to decide the Cohen–Macaulay property. This proposition seems to be part of the folklore, but since I could not find a proof in the literature I shall present one here. Proposition 12. Let f1 , . . . , fn ∈ I be a system of primary invariants of degrees d1 , . . . , dQ n . Then the invariant ring I is Cohen–Macaulay if and only if it can be generated by n ( i=1 di )/ |G| secondary invariants as a module over A = K[f1 , . . . , fn ]. Otherwise, more secondary invariants are necessary. Proof. The invariant ring K[x1 , . . . , xn ] of the trivial group is Cohen–Macaulay and Qn generated by i=1 di polynomials over A (see, for Q example, Sturmfels, 1993, Proposin tion 2.3.6). Hence [K(x1 , . . . , xn ) : K(f1 , . . . , fn )] = i=1 di , and by Galois theory n ´. ³Y [K(x1 , . . . , xn )G : K(f1 , . . . , fn )] = |G| =: d. di i=1
Thus there are at least d invariants necessary to generate I over A, and more than d will be linearly dependent. 2 There is not very much known about the question of Cohen–Macaulayness of modular invariant rings. Good references are Smith (1995) and Landweber and Strong (1987). A remarkable result of Ellingsrud and Skjelbred (1980) states that if G is a cyclic p-group with p = char(K) 6= 0 then depth(I) = min(n, n − m + 2), where m is the dimension of the K-vector space generated by all σ(xi ) − xi (σ ∈ G, i = 1, . . . , n). For general G there is the inequality depth(I) ≥ min(n, n − m + 2). It is time now to take another look at Example 10 Example 13. Let G = Z4 with its regular representation over K = F2 . In Example 10
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we calculated primary and secondary invariants for I and saw that 5 secondary ¢ ¡Q4 invariants were needed. Hence by Proposition 12, I is not Cohen–Macaulay, since i=1 deg(fi ) / |G| = 4. Calculating syzygies between the secondary invariants and applying Algorithm 7, we obtain the following relation: ¡ ¡ ¡ ¢ ¢ ¢ f1 f14 + f22 + f2 f3 · g1 + f12 + f2 + f3 · g2 + f12 + f2 · g3 + f1 · g4 = 0, which generates the module of syzygies. Thus the Poincar´e series is P (I, t) =
t5 t4 + 2t3 + 1 t5 + t4 + 2t3 + 1 − = , (1 − t)(1 − t2 )2 (1 − t4 ) (1 − t)(1 − t2 )2 (1 − t4 ) (1 − t)(1 − t2 )2 (1 − t4 )
and hdim(I) = 1, so depth(I) = 3 in accordance with the result of Ellingsrud and Skjelbred quoted above. Since G is a permutation group the Poincar´e series does not depend on the ground field (see, for example, Smith, 1995, Proposition 4.3.4) and can thus be calculated by Molien’s formula, which leads to a confirmation of the above formula. / It is natural to ask for reduction principles for the question of Cohen–Macaulayness. We obtain the following result. Proposition 14. Let 0 6= p be the characteristic of K and let H ≤ G be a p-Sylow subgroup of G. Then we have depth(K[x1 , . . . , xn ]G ) ≥ depth(K[x1 , . . . , xn ]H ). Proof. By Proposition 11, depth(K[x1 , . . . , xn ]H ) = depthK[x1 ,...,xn ]G (K[x1 , . . . , xn ]H ). Hence there exists a maximal regular sequence f1 , . . . , fm for K[x1 , . . . , xn ]H such that all fi ∈ I = K[x1 , . . . , xn ]G . We claim that f1 , . . . , fm is regular for I, too, so we have to prove that fi is not a zero divisor in I/(f1 , . . . , fi−1 ) for i = 1, . . . , m. Let gi fi = g1 f1 + · · · + gi−1 fi−1 with g1 , . . . , gi ∈ I. Since f1 , . . . , fm is regular for K[x1 , . . . , xn ]H , there are h1 , . . . , hi−1 ∈ K[x1 , . . . , xn ]H such that gi = h1 f1 + · · · + hi−1 fi−1 . Applying the relative Reynolds operator π G/H : K[x1 , . . . , xn ]H → I (see Remark 9(b)) to this equation yields gi = π G/H (h1 ) · f1 + · · · + π G/H (hi−1 ) · fi−1 , hence gi is zero in I/(f1 , . . . , fi−1 ).
2
This proposition implies that if K[x1 , . . . , xn ]H is Cohen–Macaulay (with the notation from the proposition), then so is K[x1 , . . . , xn ]G . This result already appeared in Campbell et al. (1991) and Smith (1995), Proposition 8.3.1. Together with Ellingsrud and Skjelbred’s result this implies that I is Cohen–Macaulay if n ≤ 3, since a linear representation of a p-group H always has a fixed vector. The above proof carries over to the case of relative invariants (see Remark 9(a)). The converse of Proposition 14 does not hold. Consider the example given by Campbell et al. (1991) of the regular representation of the cyclic group H = Zp , p a prime, over a field K of characteristic p. H occurs as p-Sylow subgroup of the symmetric group G = Sp whose invariant ring is polynomial and in particular Cohen–Macaulay. But if p > 3, then K[x1 , . . . , xp ]H is not Cohen–Macaulay by Ellingsrud and Skjelbred (1980). The next proposition concerns the relation between Cohen–Macaulayness for I and for the invariant ring of subrepresentations.
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Pk Proposition 15. Let G act on the first k and last n − k variables, i.e., let i=1 Kxi Pn and i=k+1 Kxi be G-stable vector spaces. Then if K[x1 , . . . , xn ]G is Cohen–Macaulay, so are K[x1 , . . . , xk ]G and K[xk+1 , . . . , xn ]G . Proof. Let f1 , . . . , fk and fk+1 , . . . , fn be primary invariants for K[x1 , . . . , xk ]G and K[xk+1 , . . . , xn ]G , respectively. Then f1 , . . . , fn are primary invariants for I = K[x1 , . . . , xn ]G . By assumption, all minimal systems of homogeneous generators for I as a module over A := K[f1 , . . . , fn ] are linearly independent over A since they all have the same cardinality dimK (K ⊗A I) (see p. 353). Let g1 , . . . , gr be a minimal system of homogeneous generators for K[x1 , . . . , xk ]G over K[f1 , . . . , fk ] and choose homogeneous gr+1 , . . . , gm ∈ I which together with g1 , . . . , gr generate I over A, with m minimal. The proposition is proved if we can show that this is a minimal generating set. So suppose that this is not true. Then one of the first r generators, say g1 satisfies a relation g1 =
m X
hi g i
(4.2)
i=2
with hi ∈ A, where we can assume all hi to be homogeneous and deg(gi ) < deg(g1 ) for those i > r with hi 6= 0 by the minimal choice of m. Setting xk+1 , . . . , xn = 0 in (4.2) yields r m X X g1 = hi g i + hi g i i=2
i=r+1
with hi ∈ K[f1 , . . . , fk ] and gi ∈ K[x1 , . . . , xk ]G , so g1 is a K[f1 , . . . , fk ]-linear combination of g2 , . . . , gr and other invariants in K[x1 , . . . , xk ]G of strictly lower degree than g1 , which is a contradiction to the minimal choice of g1 , . . . , gr . 2 Once again, the converse of Proposition 15 does not hold: By the result of Ellingsrud and Skjelbred (1980), a counter example is given for each prime number p by the cyclic group G generated by the block matrix 1 1 1 ∈ GL5 (Fp ). 1 1 1 1 1 Calculating concretely for the case p = 2 by means of the algorithms 3 and 8 yields primary invariants of degrees 1, 1, 2, 2, 4 and secondary invariants of degrees 0, 2, 3, 3, 4, hence indeed the invariant ring is not Cohen–Macaulay by Proposition 12 since |G| = 2. In fact, there is one relation of degree 4 between the secondary invariants. 4.2. polynomial rings In the modular case, it still holds that if I is a polynomial ring, then G must be a reflection group, but the converse is no longer true in general. It is thus an interesting question to assess the exact scope of validity of this converse. Examples of groups whose invariant rings are polynomial rings are, to name a few, the
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general and special linear groups GLn (Fq ) and SLn (Fq ) (see Dickson, 1911, or Wilkerson, 1983), the group U ≤ GLn (q) of unipotent upper triangular matrices (Wilkerson, 1983), the orthogonal and unitary groups On (q) and Un (q 2 ) for n ≤ 3 and n ≤ 2, respectively (Nakajima, 1979; Kemper, 1994), and the complex reflection groups G29 and G31 of Shephard and Todd (1954) reduced modulo 5 (Xu, 1994). Counter examples are the Weyl group W (F4 ) of the root system F4 (Toda, 1972), the orthogonal and unitary groups for n ≥ 4 and n ≥ 3, respectively, and a reflection subgroup of the unipotent upper triangular matrices which we shall inspect in Example 20 below (Nakajima, 1979). The following proposition leads to an easy algorithm to check if the invariant ring is a polynomial ring. The case char(K) 6 | |G| is Proposition 7.4.2 of Smith (1995). Proposition 16. Let f1 , . . . , fn ∈ I be homogeneous invariants of degrees d1 , . . . , dn . Then the following statements are equivalent: (a) I = K[f1 , . . . , fn ], Qn (b) The fi are algebraically independent over K and i=1 di = |G|. Qn (c) The Jacobian determinant J = det(∂fi /∂xj ) is non-zero and i=1 di = |G|. Remark 17. This proposition appeared in Xu (1994), Proposition 2.1. In the proof, Xu concludes directly from the equality of the transcendence degrees that K[x1 , . . . , xn ] is integral over K[f1 , . . . , fn ]. This conclusion is by no means clear and in fact false in general (take n = 2 and f1 = x1 , f2 = x1 x2 ). Hence it appears appropriate to provide a proof here. / Proof. The implication “(a) ⇒ (b)” is well known and follows from Proposition 12. For the reverse implication, it suffices by Smith (1995), Proposition 5.5.5, to show that the ¯ where K ¯ is an algebraic system f1 = · · · = fn = 0 has no projective zero in Pn−1 (K), closure of K. e of We take additional indeterminates t1 , . . . , tn and x0 and an algebraic closure K ¯ K(t1 , . . . , tn ) which contains K. By B´ezout’s theorem (see Fulton, 1984, Example 12.3.7, where there is no assumption made on the dimension of the zero manifold), the projective e given by algebraic set V ⊂ Pn (K) (4.3) f1 (x1 , . . . , xn ) − t1 xd01 = · · · = fn (x1 , . . . , xn ) − tn xd0n = 0 Qn has at most i=1 di = |G| irreducible components. So we have to show that there are at least |G| components of V with x0 6= 0. K(x1 , . . . , xn ) is a finite extension of K(f1 , . . . , fn ), so each xi has a minimal polye n is a solution of nomial over K(f1 , . . . , fn ): gi (xi , f1 , . . . , fn ) = 0. If (ξ1 , . . . , ξn ) ∈ K f1 (x1 , . . . , xn ) − t1 = · · · = fn (x1 , . . . , xn ) − tn = 0, then gi (ξi , t1 , . . . , tn ) = 0, hence there are at most finitely many such solutions, and each constitutes a component of V with x0 6= 0. We shall complete the proof by giving |G| such solutions. Via the homomorphism K(f1 , . . . , fn ) → K(t1 , . . . , tn ), fi 7→ ti , form L = K(t1 , . . . , tn ) ⊗K(f1 ,...,fn ) K(x1 , . . . , xn ), e Take a which is a finite field extension of K(t1 , . . . , tn ) and can be assumed to lie in K. e then σ ∈ G and ξi := 1 ⊗ σ(xi ) ∈ K, fi (ξ1 , . . . , ξn ) = 1 ⊗ fi (σ(x1 ), . . . , σ(xn )) = 1 ⊗ σ(fi ) = 1 ⊗ fi = ti ⊗ 1.
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Hence the elements of G give rise to |G| distinct solutions of Equation (4.3) with x0 6= 0, and “(b) ⇒ (a)” is proved. Finally, J 6= 0 if and only if K(x1 , . . . , xn ) is a finite separable field extension of K(f1 , . . . , fn ) (see, for example, Benson, 1993, Proposition 5.4.2). Hence (a) and (b) imply (c) since K(x1 , . . . , xn ) is Galois over the field of fractions Quot(I) with group G, and (c) implies (b). 2 We obtain the following algorithm, which reduces the question whether the invariant ring of a given group is a polynomial ring to pure linear algebra. Algorithm 18. (Check if I is a polynomial ring) Input: A set of generators of G. The ground field K is assumed to be perfect. Output: False if I is not a polynomial ring, otherwise generators f1 , . . . , fn of I as an algebra over K. Begin Calculate the group order |G|. Set i := 0, m := 1, R := K and d := 1. While i < n Do { Calculate a K-basis of the space of invariants of degree d. Select a maximal subset {b1 , . . . , bmd } of this basis which is linearly independent modulo the homogeneous degree-d part Rd of R. If md > 0 And d 6 | |G| Then Return(False). For j = 1, . . . , md set fi+j := bj . Set m := m · dmd , i := i + md and R := K[f1 , . . . , fi ]. If i > n Or m > |G| Then Return(False). Set d := d + 1. } (End While) If m < |G| Then Return(False). Calculate the Jacobian determinant J = det(∂fi /∂xj ). If J = 0 Then Return(False) Else Return(f1 , . . . , fn ). End. We finish with two examples. Example 19. The complex reflection group G23 from Shephard and Todd (1954) is isomorphic to {±1} × A5 . It is generated by the matrices −1 0 0 0 0 −1 1 −1 α 0 −1 0 , −1 0 0 and 0 α −α 0 0 −1 0 1 0 0 α −1 √ with α = (1 + 5)/2. Algorithm 18 yields invariants f2 , f6 and f10 of degrees 2, 6 and 10, respectively, whose Jacobian determinant does not vanish. The degrees of the fi are of course classically known from Shephard and Todd (1954). Reducing the above matrices modulo a maximal ideal in Z[α] containing p for p = 3 or p = 5, we obtain a linear group G defined over K = F9 or F5 , respectively, which remains isomorphic to {±1} × A5 . Reducing the Jacobian determinant yields non-zero polynomials, hence I is a polynomial ring.
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Reducing modulo the maximal ideal containing p = 2 (which sends α to a root of X 2 + X + 1) yields a group G ≤ GL3 (F4 ) which is isomorphic to A5 . Here we find an additional invariant f5 of degree 5. The Jacobian determinant of f2 , f5 and f6 does not vanish modulo 2, hence these three invariants generate I. f6 is a non-split group extension of f6 , where A Similarly, we can take G27 ∼ = {±1} × A A6 with kernel Z3 , and reduce this modulo a maximal ideal containing 2 to obtain a f6 . Here Algorithm 18 produces invariants group G ≤ GL3 (F4 ) which is isomorphic to A of degrees 6, 12 and 15 with non-vanishing Jacobian determinant, hence I again is a polynomial ring. The calculations for this example where done with the computer programs mentioned at the end of the introduction. The resulting invariants are too long to be reprinted here. / Finally, we present a simple example of a modular reflection group whose invariant ring is not a polynomial ring. Example 20. The group G of all 1 0 0 0
matrices 0 1 0 0
a b b c ∈ GL4 (Fq ) 1 0 0 1
with a, b, c ∈ K := Fq is a reflection group of order q 3 . We assume that I is a polynomial ring. There are two invariants of degree 1, namely f1 = x1 and f2 = x2 . If f3 and f4 are the two remaining generators with deg(f3 ) ≤ deg(f4 ), then deg(f4 ) > q, and both degrees are powers of p, where p is the prime number dividing q. Hence any invariant of degree q + 1 must have the form g1 · f3 + g2 with g1 , g2 ∈ K[x1 , x2 ]. But it is easily verified that xq1 x3 − x1 xq3 + xq2 x4 − x2 xq4 is an invariant which is not of the above form. This contradiction shows that I is not a polynomial ring. Nakajima (1979) stated this result without proof. / Acknowledgements I would like to express my thanks to Prof. Matzat, who raised my interest in invariant theory, and to the referees for their valuable comments on the first version of this paper. References Adem, A., Milgram, R. J. (1994). Cohomology of Finite Groups. Berlin, Heidelberg, New York: SpringerVerlag. Auslander, M., Buchsbaum, D. A. (1957). Homological Dimension in Local Rings. Trans. of the Amer. Math. Soc. 85, 390–405. Becker, T., Weispfenning, V. (1993). Gr¨ obner Bases. Berlin, Heidelberg, New York: Springer-Verlag. Benson, D. J. (1993). Polynomial Invariants of Finite Groups. Lond. Math. Soc. Lecture Note Ser. 190. Cambridge: Cambridge Univ. Press. Bertin, M.-J. (1967). Anneaux d’invariants d’anneaux de polynomes, en caract´eristique p. Comptes Rendus Acad. Sci. Paris (S´ erie A) 264, 653–656. Campbell, H. E. A., Hughes, I., Pollack, R. D. (1991). Rings of Invariants and p-Sylow Subgroups. Canad. Math. Bull. 34(1), 42–47.
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Char, B., Geddes, K., Gonnet, G., Monagan, M., Watt, S. (1990). Maple Reference Manual. Waterloo, Ontario: Waterloo Maple Publishing. Dickson, L. E. (1911). A Fundamental System of Invariants of the General Modular Linear Group with a Solution of the Form Problem. Trans. Amer. Math. Soc. 12, 75–98. Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. New York: Springer-Verlag. Ellingsrud, G., Skjelbred, T. (1980). Profondeur d’anneaux d’invariants en caract´eristique p. Compos. Math. 41, 233–244. Fulton, W. (1984). Intersection Theory. Berlin, Heidelberg, New York: Springer-Verlag. Gianni, P., Trager, B., Zacharias, G. (1988). Gr¨ obner Bases and Primary Decomposition of Polynomial Ideals. J. of Symbolic Computation 6, 149–267. G¨ obel, M. (1995). Computing Bases for Rings of Permutation-invariant Polynomials. J. Symbolic Computation 19, 285–291. ¨ Hilbert, D. (1893). Uber die vollen Invariantensysteme. Math. Ann. 42, 313–370. Kemper, G. (1993). The Invar Package for Calculating Rings of Invariants. IWR Preprint 93-34. Heidelberg. Kemper, G. (1994). Das Noethersche Problem und generische Polynome. PhD Thesis. University of Heidelberg. Also available as: IWR Preprint 94-49, Heidelberg 1994. Landweber, P. S., Stong, R. E. (1987). The Depth of Rings of Invariants over Finite Fields. In: Proc. New York Number Theory Seminar, 1984. Lecture Notes in Math. 1240. New York: Springer-Verlag. McShane, J. M. (1992). Computation of Polynomial Invariants of Finite Groups. PhD Thesis. University of Arizona. M¨ oller, H. M., Mora, F. (1986). New Constructive Methods in Classical Ideal Theory. J. of Algebra 100, 138–178. Nakajima, H. (1979). Invariants of Finite Groups Generated by Pseudo-Reflections in Positive Characteristic. Tsukuba J. Math. 3, 109–122. Noether, E. (1916). Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann. 77, 89–92. Noether, E. (1926). Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charkteristik p. Nachr. Ges. Wiss. G¨ ottingen pp. 28–35. Richman, D. R. (1990). On Vector Invariants over Finite Fields. Adv. in Math. 81, 30–65. Shephard, G. C., Todd, J. A. (1954). Finite Unitary Reflection Groups. Canad. J. Math. 6, 274–304. Sloane, N. J. A. (1977). Error-Correcting Codes and Invariant Theory: New Applications of a NineteenthCentury Technique. Amer. Math. Monthly 84, 82–107. Smith, L. (1995). Polynomial Invariants of Finite Groups. Wellesley, Mass.: A. K. Peters. Stanley, R. P. (1979). Invariants of Finite Groups and their Applications to Combinatorics. Bull. Amer. Math. Soc. 1(3), 475–511. Stillman, M., Stillman, M., Bayer, D. (1989). Macaulay User Manual. Available by anonymous ftp from various sites. Sturmfels, B. (1993). Algorithms in Invariant Theory. Wien, New York: Springer-Verlag. Toda, H. (1972). Cohomology mod 3 of the Classifying Space BF4 of the Exceptional Group F4 . J. Math. Kyoto Univ. 13, 97–115. Weber, H. (1899). Lehrbuch der Algebra. Braunschweig: Viehweg Verlag. Wilkerson, C. (1983). A Primer on the Dickson Invariants. Amer. Math. Soc. Contemp. Math. Series 19, 421–434. Xu, C. (1994). Computing Invariant Polynomials of p-adic Reflection Groups. In: Proc. of Symposia in Appl. Math. 48. Providence, RI: Amer. Math. Soc.
Calculating Invariant Rings of Finite Groups over Arbitrary Fields GREGOR KEMPER IWR, Universit¨ at Heidelberg, Im Neuenheimer Feld 368, 69 120 Heidelberg, Germany† (Received 3 April 1995)
An algorithm is presented which calculates rings of polynomial invariants of finite linear groups over an arbitrary field K. Up to now, such algorithms have been available only for the case that the characteristic of K does not divide the group order. Some applications to the question whether a modular invariant ring is Cohen–Macaulay or isomorphic to a polynomial ring are discussed. c 1996 Academic Press Limited °
1. Introduction If G is a finite linear group over a field K such that char(K) 6 | |G|, there are various effective methods to calculate the invariant ring I of G, i.e., to find a finite system of generators of I as an algebra over K (see Sturmfels, 1993, Section 2; McShane, 1992; Kemper, 1993, 1994). These methods make use of the Reynolds operator, Molien’s formula, the Cohen–Macaulay property of invariant rings, and of Noether’s degree bound in the case char(K) > |G| (Noether, 1916). If, on the other hand, |G| is a multiple of char(K) (which we shall call the modular case), none of these techniques are available since they all involve divisions by |G|. In fact, G is not a linearly reductive group in this case. Nevertheless, the invariant ring is finitely generated as a K-algebra (Noether, 1926). If G is a permutation group there is a very nice algorithm by G¨ obel (1995) which calculates generators over any commutative ground ring and which implies a degree bound for the generators. But for the case of finite linear groups there is no algorithm available at the moment to construct a system of generators, and modular invariant rings are calculated by ad hoc methods (see Benson, 1993, Chapter 8; Wilkerson, 1983; Adem and Milgram, 1994, Chapter III). The purpose of this paper is to fill this gap. With the knowledge of generators for the invariant ring, it is quite easy to calculate its depth and its Poincar´e series (see Section 4.1). It is even easier to check the Cohen–Macaulay property. So the algorithm presented here could be useful to gain some experience and to test hypotheses. The first section of this paper is concerned with the calculation of primary invariants, which serve as a kind of first approximation to the invariant ring. A new algorithm to †
E-mail: [email protected]
0747–7171/96/030351 + 16 $18.00/0
c 1996 Academic Press Limited °
352
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construct primary invariants is proposed. Section 3 contains the main algorithm which does the step from the primary invariants to the full invariant ring for arbitrary ground fields. The last section is devoted to some applications. Here we probe into the question of the Cohen–Macaulay property mentioned above, and give a simple algorithm to decide whether the invariant ring of a given group is isomorphic to a polynomial ring. The last question has its background in another “defect” of modular linear groups, that their invariant rings are not always polynomial rings if the group is a reflection group. The author has incorporated the algorithms presented in this paper into a new version of the Invar -package† for Maple (Kemper, 1993). Let us fix some notation. Throughout this paper, K is an arbitrary field and G ≤ GLn (K) is a finite matrix group of degree n over K. G operates on the polynomial ring K[x1 , . . . , xn ] by linear transformations of the indeterminates xi . We write I for the invariant ring: I = K[x1 , . . . , xn ]G = {f ∈ K[x1 , . . . , xn ] | σ(f ) = f ∀σ ∈ G}. This is a graded subalgebra of K[x1 , . . . , xn ]. 2. Calculating Primary Invariants Let R be any graded K-algebra of Krull dimension m with R0 = K, then by Noether’s normalization lemma (see, for example, Benson, 1993, Theorem 2.2.7), there exist homogeneous elements f1 , . . . , fm ∈ R such that R is finitely generated as a module over A := K[f1 , . . . , fm ]. If R is the invariant ring I, then m = n and f1 , . . . , fn are called primary invariants. In this case, f1 , . . . , fn are also primary invariants for any subgroup H ≤ G since K[x1 , . . . , xn ] is integral over I. The following proposition is the key to the calculation of primary invariants: Proposition 1. A set {f1 , . . . , fi } ⊂ I of homogeneous invariants can be extended to a system of primary invariants if and only if dim(f1 , . . . , fi ) = n − i. In particular, homogeneous f1 , . . . , fk ∈ I form a system of primary invariants if and only if k = n and (2.1) VK¯ (f1 , . . . , fk ) = {0}, where VK¯ denotes the set of zeros over the algebraic closure of K. Proof. By Smith (1995), Proposition 5.3.7, homogeneous invariants f1 , . . . , fn are primary invariants if and only if dim(f1 , . . . , fn ) = 0. By Krull’s Principal Ideal Theorem (see, for example, Eisenbud, 1995, Theorem 10.1) each fi can diminish the dimension of the ideal by at most one, hence dim(f1 , . . . , fi ) = n−i if f1 , . . . , fn are primary invariants. Conversely, if dim(f1 , . . . , fi ) = n − i there exist homogeneous fi+1 , . . . , fn ∈ R := I/(f1 , . . . , fi ) such that R is finitely generated as a module over K[fi+1 , . . . , fn ]. Hence ¢ ¢ ¡ ¡ dimK K[x1 , . . . , xn ]/(f1 , . . . , fn ) = dimK R/(fi+1 , . . . , fn ) < ∞, † This version can be obtained by anonymous ftp from the site ftp.iwr.uni-heidelberg.de under /pub/kemper/INVAR2.
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so dim(f1 , . . . , fn ) = 0 by Becker and Weispfenning (1993), Theorem 6.5.4. This completes the proof. 2 Primary invariants are far from being uniquely determined by G. But the choice of primary invariants f1 , . . . , fn determines the minimum number mf1 ,...,fn of generators for I as a module over A = K[f1 , . . . , fn ], which is mf1 ,...,fn = dimK (K ⊗A I) by Smith (1995), Corollary 5.2.5. It is of crucial importance for the effectiveness of subsequent calculations that mf1 ,...,fn be as low as possible, as this is most likely to be achieved if the degrees of the primary invariants are chosen as small as possible, as is illustrated by Proposition 12 and Example 5(b). Hence I will call an algorithm that constructs primary invariants good if it is likely to produce primary invariants of small degrees. So the best algorithm in my sense need not be the fastest. Algorithms for the construction of primary invariants go back as far as Hilbert (1893) and Weber (1899). A good modern treatment can be found in Sturmfels (1993), Algorithm 2.5.8. He proposed that as a first step one should collect invariants of increasing degrees until arriving at a set {f1 , . . . , fk } having the property (2.1). This set is obtained by successively calculating K-bases for the vector spaces of homogeneous invariants of degree 1, 2, . . . and including a basis element into the set if it does not lie in the radical of the ideal spanned be the fi gotten so far. In the next step, one tries to delete elements fi from the set while retaining (2.1). Experience shows that in most cases this will lead to a set of n elements, i.e., a system of primary invariants. If it does not, we are left with k polynomials fi , k > n, which satisfy (2.1), and must apply a third step which consists of first converting all fi to invariants of the same degree by taking suitable powers Pk and then choosing random n × k-matrices (ai,j ) and checking condition (2.1) for fei = j=1 ai,j · fj (i = 1, . . . , n), until the fei form a system of primary invariants. It is especially the third step of this algorithm that is unsatisfactory since it involves a random search and since the conversion to equal degrees makes the degrees explode. But even in the majority of cases when this step is unnecessary, the algorithm often yields primary invariants whose degrees are not the lowest possible. Let us look at a typical example for this. Example 2. Let
µ G=h
i 0
0 −i
¶ i ≤ GL2 (C).
Then the above algorithm would in the first step produce the invariants f1 = x1 x2 , f2 = x41 and f3 = x42 and omit f1 in the second step, yielding primary invariants of degrees 4 and 4. But there are primaries of degrees 2 and 4, namely f1 = x1 x2 and f2 = x41 +x42 , which the algorithm would miss. / Another algorithm due to E. Dade (see Stanley, 1979) constructs primary invariants by taking products over G-orbits of suitable linear forms. This algorithm is very quick but also has the disadvantage that it tends to produce primary invariants of too high degrees (as it would in Example 2). While the algorithm of Sturmfels always tries to find systems of primary invariants as a whole, Proposition 1 suggests a strategy of successively adding primary invariants to
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the system, the condition for a new primary invariant fi+1 being that it decreases the dimension of the ideal (f1 , . . . , fi ), or, equivalently, that it lies in none of the associated prime ideals of (f1 , . . . , fi ). This brings us to propose the following algorithm: Algorithm 3. (Constructing primary invariants) Input: A set of generators of G. Output: Primary invariants f1 , . . . , fn . Begin Set i := 1, d := 1, r := 1 and P1 := {}. While i ≤ n Do { Calculate a K-basis {b1 , . . . , bmd } of the space of invariants of degree d. (This can be done by writing down a general polynomial of degree d with unknown coefficients, applying the generators of G and solving the resulting system of linear equations for the unknown coefficients.) Pm d tj · bj with indeterminates tj . Form Id (t1 , . . . , tmd ) := j=1 For k = 1, . . . , r calculate the complete residues rk (t1 , . . . , tmd ) of Id w.r.t. the Gr¨ obner basis Pk . If there exist α1 , . . . , αmd ∈ K with rk (α1 , . . . , αmd ) 6= 0 ∀k = 1, . . . , r Then { (See below for how to decide this.) Set fi := Id (α1 , . . . , αmd ). Calculate the associated prime ideals of (f1 , . . . , fi ). (See below for algorithms which perform this.) Set r :=(number of associated primes). obner bases of the associated primes w.r.t. any monoLet P1 , . . . , Pr be Gr¨ mial order. Set i := i + 1. } (End Then) Else (There is no other primary invariant of degree d.) { Set d := d + 1. } (End Else) } (End While) End. Since the rk are the complete residues of Id w.r.t. the Pk , the condition rk (α1 , . . . , αmd ) 6= 0 ∀k is equivalent to the requirement that Id (α1 , . . . , αmd ) lies in none of the associated prime ideals of (f1 , . . . , fi ). If any of the rk is zero, then this condition is certainly false. If, on the other hand, all rk are non-zero, then the equations rk (α1 , . . . , αmd ) = 0 describe proper subspaces of K md , so in the case of an infinite K there are always αi satisfying the condition. It is easy to write an algorithm that specifies the αi successively to produce a non-solution of all these equations in this case. If on the other hand K is finite (and all rk non-zero), we can simply search the elements of K md for a non-solution. Before turning to the computation of associated prime ideals involved in Algorithm 3, we shall look at how it would handle the group considered in Example 2: Example 4. Let G be as in Example 2. Then the first primary invariant taken by Algorithm 3 is f1 = x1 x2 , yielding associated primes (x1 ) and (x2 ). Passing to degree 4, we
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get I4 = t1 · x41 + t2 · x21 x22 + t3 · x42 with complete residues r1 = t3 · x42 and r2 = t1 · x41 . So (α1 , α2 , α3 ) = (1, 0, 1) is a non-solution, leading to f2 = x41 + x42 as the second primary invariant. / According to my experience Algorithm 3 is very likely to produce a system of primary invariants of smallest possible degrees, but it is not guaranteed to do so, as there are counter examples: Example 5. (a) Consider the permutation group G = h(123)(456), (12)(45)i ∼ = S3 acting on the indeterminates of C[x1 , . . . , x6 ]. There are primary invariants of degrees 1, 1, 2, 2, 3, 3 given by the elementary symmetric functions in the first and last three variables. But by an unlucky choice Algorithm 3 might pick f1 = x1 + x2 + x3 , f2 = x4 + x5 + x6 , f3 = x1 x2 + x1 x3 + x2 x3 , f4 = x1 x4 + x2 x5 + x3 x6 , f5 = x4 x5 x6 as the first five primary invariants. Then x1 = x2 = x3 = x4 = 0, x5 = −x6 is a zero of these fi and also of all invariants of degree 3, hence there is no further primary invariant of this degree. So Algorithm 3 would proceed to the next degree and finally obtain a last primary invariant of degree 6. (b) Take the “first A5 in SL4 (F2 )” of Adem and Milgram (1994) p. 116, given by 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 G = h 0 0 1 1 , 1 0 1 0 0 0 1 0 i ≤ GL4 (F2 ). 0 0 1 0 0 1 0 1 0 0 0 1 We only sketch the results here. Adem and Milgram’s primary invariants are of degrees 5, 5, 12, 12, and Algorithm 3 would find primaries of degrees 3, 3, 12, 20. But the “best” primary invariants (i.e., those leading to the smallest mf1 ,...,fn , see p. 353) are of degrees 3, 5, 8 and 12. They were also obtained by Algorithm 3 together with some degree of human intervention. With these primary invariants there are 24 module-generators for I of degree at most 24 needed, while Adem and Milgram’s primary invariants require 60 module-generators of degree at most 30. We see that it is still a subtle business to find good primary invariants! / The most expensive part of the algorithm is the computation of the associated prime ideals of (f1 , . . . , fi−1 ). Algorithms which perform this over any field K that is finitely generated over its prime field are given in Gianni et al. (1988). Here we can assume K to be finitely generated over its prime field by the matrix entries of the elements of G. These algorithms automatically yield Gr¨ obner bases for the associated primes. For the characteristic zero case see also Becker and Weispfenning (1993), Table 8.10. The complete computation of the associated primes can in most cases be circumvented by using the “Gr¨ obner factorization algorithm”, which is likely to produce the associated primes since in our case (f1 , . . . , fi−1 ) has no embedded primes. This algorithm is implemented as the function gsolve in Maple (Char et al., 1990). Since this algorithm always
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yields ideals which lie between (f1 , . . . , fi−1 ) and an associated prime, the condition that rk (α1 , . . . , αmd ) 6= 0 ∀k in Algorithm 3 will be necessary for the existence of another obner factorization alprimary invariant of degree d, if the Pk are the output of the Gr¨ gorithm. After having found a candidate fi = Id (α1 , . . . , αmd ), we have to check if this really decreases the dimension of the ideal by a Gr¨ obner basis method which is much simpler than the calculation of associated primes (see Becker and Weispfenning, 1993, Table 9.6). Only if fi does not qualify, we will have to calculate the associated primes rigorously. In the non-modular case, i.e., if char(K) 6 | |G|, we have Molien’s formula to calculate the Poincar´e series of the invariant ring. From this, we can make good guesses at the degrees of the primary invariants (see, for example, Sloane, 1977). So another variant of Algorithm 3 would be to incorporate guesses for complete systems of primary invariants by taking any homogeneous invariants of the “right” degrees and checking condition (2.1) before entering the full Algorithm 3. With both these modifications, Algorithm 3 performs reasonably well and is, in the author’s opinion, the best algorithm available at the moment for calculating primary invariants, although it is certainly slower than Dade’s algorithm (see above).
3. Calculating Secondary Invariants The next task is to find a system of generators of I as a module over A = K[f1 , . . . , fn ], where f1 , . . . , fn are primary invariants. Such generators are called secondary invariants. In the non-modular case, Molien’s formula provides complete information about the number and degrees of the secondary invariants. This reduces their calculation to a simple exercise of filling up homogeneous subspaces of invariants (see McShane, 1992, or Kemper, 1994). The idea for the general case now is to calculate secondary invariants g1 , . . . , gr for a subgroup H ≤ G with char(K) 6 | |H| (the trivial group will always do) first. Imposing G-invariance conditions on a general element of K[x1 , . . . , xn ]H will then lead to a system of linear equations with coefficients in K[x1 , . . . , xn ], for which we have to calculate the solutions whose components lie in A. So we need an algorithm which intersects a submodule of K[x1 , . . . , xn ]r with Ar , where Rr denotes the free module of rank r over a ring R. Ps r Lemma 6. Let R = K[x1 , . . . , xn ], M = i=1 R · bi ≤ R a submodule and A = K[f1 , . . . , fk ] ≤ R the subalgebra generated by elements f1 , . . . , fk ∈ R. Take the polynomial rings S = K[x1 , . . . , xn , t1 , . . . , tk ] and T = K[t1 , . . . , tk ] with further indeterminates t1 , . . . , tk and form f= M
s X
S · bi +
i=1
k X
(tj − fj ) · S r ≤ S r
j=1
and f ∩ T r. fT = M M Then with Φ: T r → Ar , tj 7→ fj we have fT ) = M ∩ Ar . Φ(M
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Proof. We consider the homomorphism Ψ: S r → Rr / M obtained by substituting f. tj 7→ fj , and claim ker(Ψ) = M f clearly lies in the kernel. Let P (t1 , . . . , tk ) be a product of tj ’s By construction, M and ei ∈ S r a vector of the standard basis. We show by induction on deg(P ) that ¢ ¡ f. There is nothing to show for deg(P ) = 0. If P (t1 , . . . , tk ) − P (f1 , . . . , fk ) · ei ∈ M 0 deg(P ) > 0, then P (t1 , . . . , tk ) = tj ·P (t1 , . . . , tk ) with deg(P 0 ) < deg(P ), so by induction ¢ ¡ 0 f, and we conclude P (t1 , . . . , tk ) − P 0 (f1 , . . . , fk ) · ei ∈ M ¢ ¡ P (t1 , . . . , tk ) − P (f1 , . . . , fk ) · ei = ³ ¡ ¢´ f. (tj − fj ) · P 0 (t1 , . . . , tk ) + fj · P 0 (t1 , . . . , tk ) − P 0 (f1 , . . . , fk ) · ei ∈ M It follows that for (g1 , . . . , gr ) ∈ S r we have ¯ ¯ f. (g1 , . . . , gr ) − (g1 ¯tj =fj , . . . , gr ¯tj =fj ) ∈ M ¯ ¯ f and hence (g1 , . . . , gr ) Now if Ψ(g1 , . . . , gr ) = 0, then (g1 ¯tj =fj , . . . , gr ¯tj =fj ) ∈ M ⊂ M f, which proves the claim. ∈M fT , and due to It follows that ker(Ψ |T r ) = M Ψ(T r ) = (Ar + M )/ M ∼ = Ar /(M ∩ Ar ) we have Tr
.
fT ∼ M = Ar /(M ∩ Ar )
with an isomorphism induced by Φ. This completes the proof.
2
We obtain the following algorithm: Algorithm 7. (Intersecting a module with a subalgebra) Input: Generators b1 , . . . , bs of a submodule M ≤ Rr , where R = K[x1 , . . . , xn ], and generators f1 , . . . , fk of a subalgebra A = K[f1 , . . . , fk ] ≤ R. Output: Generators c1 , . . . , cm ∈ Ar of M ∩ Ar as a module over A. Begin Take additional indeterminates t1 , . . . , tk and set S = K[x1 , . . . , xn , t1 , . . . , tk ]. f ≤ S r generated by the bi (i = 1, . . . , s) and by the Form the submodule M (tj − fj ) · ei (j = 1, . . . , k, i = 1, . . . , r). f w.r.t. a term order ≺ with the property that Calculate a Gr¨ obner basis B of M every xi is greater than any monomial in the tj . (See below for Gr¨ obner bases of modules.) Form B ∩ (K[t1 , . . . , tk ])r and substitute tj 7→ fj in its elements. Let {c1 , . . . , cm } be the resulting set. End. In the above algorithm we are using Gr¨ obner bases of modules. These were introduced by M¨oller and Mora (1986), who also extended Buchberger’s algorithm to this case. This algorithm is implemented in Macaulay (see Stillman et al., 1989). The elimination property for these Gr¨ obner bases, which was used in the algorithm, follows easily and can be found in (loc. cit.).
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Also implemented in Macaulay and described by M¨ oller and Mora (1986) are algorithms to find syzygy modules, i.e., solution modules for systems of linear equations over a polynomial ring. These constitute the second computational ingredient of the general algorithm to find secondary invariants, which follows now. Algorithm 8. (Calculating secondary invariants ) Input: A set S of generators of G and homogeneous f1 , . . . , fk ∈ I such that I is finitely generated as a module over A = K[f1 , . . . , fk ] (optimally, a system of primary invariants of small degrees). Output: Secondary invariants g1 , . . . , gm . Begin Choose a subgroup H ≤ G (for example, H = {ι}) and calculate homogeneous generators h1 , . . . , hr of K[x1 , . . . , xn ]H as a module over A. Calculate the module M ≤ K[x1 , . . . , xn ]r of all (p1 , . . . , pr ) ∈ K[x1 , . . . , xn ]r with r X
(σ(hi ) − hi ) · pi = 0 for all σ ∈ S.
(3.1)
i=1
(This is the calculation Pmof a syzygy module.) CalculateP M ∩ Ar = i=1 A · ci using Algorithm 7. r Set gi := j=1 ci,j · hj (i = 1, . . . , m), where ci = (ci,1 , . . . , ci,r ) ∈ Ar . End. The gi lie in I by construction. Conversely, for f = have by Equation (3.1) (a1 , . . . , ar ) ∈ M ∩ Ar ⇒ (a1 , . . . , ar ) = hence I =
Pm i=1
m X
Pr i=1
ai · hi ∈ I with ai ∈ A we
e ai · ci with e ai ∈ A ⇒ f =
i=1
m X
e ai · g i ,
i=1
A · gi , which proves the correctness of Algorithm 8.
Remark 9. (a) If χ: G → K × is a linear character, then the set Mχ of relative invariants of weight χ (i.e., of f ∈ K[x1 , . . . , xn ] such that σ(f ) = χ(σ) · f ∀σ ∈ G) is a module over I and also over A = K[f1 , . . . , fk ] for any invariants fi . If the fi satisfy the assumption made in Algorithm 8, then a generating set for Mχ as an A-module can be constructed by Algorithm 8 with χ introduced into Equation (3.1). (b) Let χ be as above and H ≤ G a p-Sylow subgroup, where p = char(K) 6= 0. Then χ|H = 1. Taking left coset representatives σ1 , . . . , σr of G/H, we have a “relative Reynolds operator” given by X 1 χ(σi−1 ) · σi (G : H) i=1 r
πχG/H =
(see Campbell et al., 1991 or Smith, 1995, p. 28). This is a projection K[x1 , . . . , xn ]H → →Mχ of I-modules. Hence we only have to calculate generators for K[x1 , . . . , xn ]H G/H as a module over A and then apply πχ to obtain Mχ . K[x1 , . . . , xn ]H itself is most conveniently calculated by applying Algorithm 8 recursively to a chain of subgroups H1 ≤ H2 ≤ · · · ≤ Hs = H with |Hi | = pi .
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(c) The set {g1 , . . . , gm } from Algorithm 8 can be made into a minimal set of secondary invariants (w.r.t. the primary invariants f1 , . . . , fn chosen once and for all) by successively deleting those gi which are contained in the module generated by the others. This test for membership amounts to the solution of a system of linear equations over K. / Let us look at an example now. Example 10. Consider the cyclic group G = Z4 with its regular representation over K = F2 . We find primary invariants f1 = x1 + x2 + x3 + x4 , f2 = x1 x3 + x2 x4 , f 3 = x 1 x 2 + x 2 x 3 + x 3 x 4 + x 4 x 1 , f 4 = x1 x 2 x 3 x 4 . With H = {ι} ≤ G, there are 16 generators h1 , . . . , h16 of K[x1 , . . . , xn ] as a module over A = K[f1 , . . . , f4 ], and the solution module M of Equation (3.1) is generated by 16 f in Algorithm 7 then has 80 generators and a Gr¨ vectors. The module M obner basis of 113 elements, which was calculated by Macaulay in a few seconds. From these, five lie in K[t1 , . . . , t4 ], and we get the secondary invariants g1 = 1, g2 = x31 + x32 + x33 + x34 , g4 = x31 x2 + x32 x3 + x33 x4 + x34 x1 ,
g3 = x21 x2 + x22 x3 + x23 x4 + x24 x1 , g5 = x31 x22 + x32 x23 + x33 x24 + x34 x21 .
The calculation can be essentially accelerated by taking H = Z2 ≤ G and applying Algorithm 8 recursively. We shall continue this example on p. 360 and show that the invariant ring is not Cohen–Macaulay. From the above secondary invariants we already see that the invariants of degree ≤ 4 do not generate I as a K-algebra, i.e., Noether’s degree bound (Noether, 1916) does not hold in the modular case. See Richman (1990) for examples of linear groups where generators of arbitrarily high degree are necessary for the invariant ring although the group order remains the same. Bertin (1967) and Benson (1993), p. 104, mention this example, but do not calculate the complete invariant ring. / Clearly, the Gr¨ obner basis calculation involved in Algorithm 7 is the most time consuming part of Algorithm 8. It will set the limit to the practical scope of the algorithm. 4. Applications This section is concerned with calculating some data associated with invariant rings, in particular free resolutions, Poincar´e series and depth. Easy algorithms are presented to decide the Cohen–Macaulay property and the property of being isomorphic to a polynomial algebra. 4.1. depth and the Cohen–Macaulay property Suppose that we have calculated primary invariants f1 , . . . , fn and secondary invariants g1 , . . . , gm by Algorithm 3 and 8. Then we can compute the module M ≤ K[x1 , . . . , xn ]m of syzygies between the gi and intersect M with Am (where as above A = K[f1 , . . . , fn ]) using Algorithm 7. We can further calculate a free resolution · · · → F2 → F1 → M ∩ Am → 0
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of M ∩ Am as an A-module, which yields a free resolution · · · → F2 → F1 → F0 → I → 0
(4.1)
of the invariant ring I. From this the Poincar´e series of I can easily be obtained (see Stanley, 1979, p. 496). If {g1 , . . . , gm } is a minimal set of secondary invariants and if (4.1) is a minimal free resolution then the homological dimension hdimA (I) is (by definition) the largest i such that Fi 6= 0. By the Auslander–Buchsbaum formula (see Auslander and Buchsbaum, 1957 or Eisenbud, 1995, Theorem 19.9) we have depthA (I) = n − hdimA (I), where depthA (I) is the maximal length of a regular sequence for I whose elements lie in A and have no constant term. But depth(I) := depthI (I) is the same as depthA (I) by the following proposition, whose proof is a word by word adaption of the proof of Corollary 19.15 in Eisenbud (1995) to the graded case: Proposition 11. Let A ≤ B be Noetherian graded algebras over a field K such that A0 = B0 = K and assume that B is finitely generated as an A-module. Then depthA (B) = depthB (B). Thus we can calculate the depth of I. Recall that I is called Cohen–Macaulay if depth(I) = n. The following proposition provides a much easier criterion to decide the Cohen–Macaulay property. This proposition seems to be part of the folklore, but since I could not find a proof in the literature I shall present one here. Proposition 12. Let f1 , . . . , fn ∈ I be a system of primary invariants of degrees d1 , . . . , dQ n . Then the invariant ring I is Cohen–Macaulay if and only if it can be generated by n ( i=1 di )/ |G| secondary invariants as a module over A = K[f1 , . . . , fn ]. Otherwise, more secondary invariants are necessary. Proof. The invariant ring K[x1 , . . . , xn ] of the trivial group is Cohen–Macaulay and Qn generated by i=1 di polynomials over A (see, for Q example, Sturmfels, 1993, Proposin tion 2.3.6). Hence [K(x1 , . . . , xn ) : K(f1 , . . . , fn )] = i=1 di , and by Galois theory n ´. ³Y [K(x1 , . . . , xn )G : K(f1 , . . . , fn )] = |G| =: d. di i=1
Thus there are at least d invariants necessary to generate I over A, and more than d will be linearly dependent. 2 There is not very much known about the question of Cohen–Macaulayness of modular invariant rings. Good references are Smith (1995) and Landweber and Strong (1987). A remarkable result of Ellingsrud and Skjelbred (1980) states that if G is a cyclic p-group with p = char(K) 6= 0 then depth(I) = min(n, n − m + 2), where m is the dimension of the K-vector space generated by all σ(xi ) − xi (σ ∈ G, i = 1, . . . , n). For general G there is the inequality depth(I) ≥ min(n, n − m + 2). It is time now to take another look at Example 10 Example 13. Let G = Z4 with its regular representation over K = F2 . In Example 10
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we calculated primary and secondary invariants for I and saw that 5 secondary ¢ ¡Q4 invariants were needed. Hence by Proposition 12, I is not Cohen–Macaulay, since i=1 deg(fi ) / |G| = 4. Calculating syzygies between the secondary invariants and applying Algorithm 7, we obtain the following relation: ¡ ¡ ¡ ¢ ¢ ¢ f1 f14 + f22 + f2 f3 · g1 + f12 + f2 + f3 · g2 + f12 + f2 · g3 + f1 · g4 = 0, which generates the module of syzygies. Thus the Poincar´e series is P (I, t) =
t5 t4 + 2t3 + 1 t5 + t4 + 2t3 + 1 − = , (1 − t)(1 − t2 )2 (1 − t4 ) (1 − t)(1 − t2 )2 (1 − t4 ) (1 − t)(1 − t2 )2 (1 − t4 )
and hdim(I) = 1, so depth(I) = 3 in accordance with the result of Ellingsrud and Skjelbred quoted above. Since G is a permutation group the Poincar´e series does not depend on the ground field (see, for example, Smith, 1995, Proposition 4.3.4) and can thus be calculated by Molien’s formula, which leads to a confirmation of the above formula. / It is natural to ask for reduction principles for the question of Cohen–Macaulayness. We obtain the following result. Proposition 14. Let 0 6= p be the characteristic of K and let H ≤ G be a p-Sylow subgroup of G. Then we have depth(K[x1 , . . . , xn ]G ) ≥ depth(K[x1 , . . . , xn ]H ). Proof. By Proposition 11, depth(K[x1 , . . . , xn ]H ) = depthK[x1 ,...,xn ]G (K[x1 , . . . , xn ]H ). Hence there exists a maximal regular sequence f1 , . . . , fm for K[x1 , . . . , xn ]H such that all fi ∈ I = K[x1 , . . . , xn ]G . We claim that f1 , . . . , fm is regular for I, too, so we have to prove that fi is not a zero divisor in I/(f1 , . . . , fi−1 ) for i = 1, . . . , m. Let gi fi = g1 f1 + · · · + gi−1 fi−1 with g1 , . . . , gi ∈ I. Since f1 , . . . , fm is regular for K[x1 , . . . , xn ]H , there are h1 , . . . , hi−1 ∈ K[x1 , . . . , xn ]H such that gi = h1 f1 + · · · + hi−1 fi−1 . Applying the relative Reynolds operator π G/H : K[x1 , . . . , xn ]H → I (see Remark 9(b)) to this equation yields gi = π G/H (h1 ) · f1 + · · · + π G/H (hi−1 ) · fi−1 , hence gi is zero in I/(f1 , . . . , fi−1 ).
2
This proposition implies that if K[x1 , . . . , xn ]H is Cohen–Macaulay (with the notation from the proposition), then so is K[x1 , . . . , xn ]G . This result already appeared in Campbell et al. (1991) and Smith (1995), Proposition 8.3.1. Together with Ellingsrud and Skjelbred’s result this implies that I is Cohen–Macaulay if n ≤ 3, since a linear representation of a p-group H always has a fixed vector. The above proof carries over to the case of relative invariants (see Remark 9(a)). The converse of Proposition 14 does not hold. Consider the example given by Campbell et al. (1991) of the regular representation of the cyclic group H = Zp , p a prime, over a field K of characteristic p. H occurs as p-Sylow subgroup of the symmetric group G = Sp whose invariant ring is polynomial and in particular Cohen–Macaulay. But if p > 3, then K[x1 , . . . , xp ]H is not Cohen–Macaulay by Ellingsrud and Skjelbred (1980). The next proposition concerns the relation between Cohen–Macaulayness for I and for the invariant ring of subrepresentations.
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Pk Proposition 15. Let G act on the first k and last n − k variables, i.e., let i=1 Kxi Pn and i=k+1 Kxi be G-stable vector spaces. Then if K[x1 , . . . , xn ]G is Cohen–Macaulay, so are K[x1 , . . . , xk ]G and K[xk+1 , . . . , xn ]G . Proof. Let f1 , . . . , fk and fk+1 , . . . , fn be primary invariants for K[x1 , . . . , xk ]G and K[xk+1 , . . . , xn ]G , respectively. Then f1 , . . . , fn are primary invariants for I = K[x1 , . . . , xn ]G . By assumption, all minimal systems of homogeneous generators for I as a module over A := K[f1 , . . . , fn ] are linearly independent over A since they all have the same cardinality dimK (K ⊗A I) (see p. 353). Let g1 , . . . , gr be a minimal system of homogeneous generators for K[x1 , . . . , xk ]G over K[f1 , . . . , fk ] and choose homogeneous gr+1 , . . . , gm ∈ I which together with g1 , . . . , gr generate I over A, with m minimal. The proposition is proved if we can show that this is a minimal generating set. So suppose that this is not true. Then one of the first r generators, say g1 satisfies a relation g1 =
m X
hi g i
(4.2)
i=2
with hi ∈ A, where we can assume all hi to be homogeneous and deg(gi ) < deg(g1 ) for those i > r with hi 6= 0 by the minimal choice of m. Setting xk+1 , . . . , xn = 0 in (4.2) yields r m X X g1 = hi g i + hi g i i=2
i=r+1
with hi ∈ K[f1 , . . . , fk ] and gi ∈ K[x1 , . . . , xk ]G , so g1 is a K[f1 , . . . , fk ]-linear combination of g2 , . . . , gr and other invariants in K[x1 , . . . , xk ]G of strictly lower degree than g1 , which is a contradiction to the minimal choice of g1 , . . . , gr . 2 Once again, the converse of Proposition 15 does not hold: By the result of Ellingsrud and Skjelbred (1980), a counter example is given for each prime number p by the cyclic group G generated by the block matrix 1 1 1 ∈ GL5 (Fp ). 1 1 1 1 1 Calculating concretely for the case p = 2 by means of the algorithms 3 and 8 yields primary invariants of degrees 1, 1, 2, 2, 4 and secondary invariants of degrees 0, 2, 3, 3, 4, hence indeed the invariant ring is not Cohen–Macaulay by Proposition 12 since |G| = 2. In fact, there is one relation of degree 4 between the secondary invariants. 4.2. polynomial rings In the modular case, it still holds that if I is a polynomial ring, then G must be a reflection group, but the converse is no longer true in general. It is thus an interesting question to assess the exact scope of validity of this converse. Examples of groups whose invariant rings are polynomial rings are, to name a few, the
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general and special linear groups GLn (Fq ) and SLn (Fq ) (see Dickson, 1911, or Wilkerson, 1983), the group U ≤ GLn (q) of unipotent upper triangular matrices (Wilkerson, 1983), the orthogonal and unitary groups On (q) and Un (q 2 ) for n ≤ 3 and n ≤ 2, respectively (Nakajima, 1979; Kemper, 1994), and the complex reflection groups G29 and G31 of Shephard and Todd (1954) reduced modulo 5 (Xu, 1994). Counter examples are the Weyl group W (F4 ) of the root system F4 (Toda, 1972), the orthogonal and unitary groups for n ≥ 4 and n ≥ 3, respectively, and a reflection subgroup of the unipotent upper triangular matrices which we shall inspect in Example 20 below (Nakajima, 1979). The following proposition leads to an easy algorithm to check if the invariant ring is a polynomial ring. The case char(K) 6 | |G| is Proposition 7.4.2 of Smith (1995). Proposition 16. Let f1 , . . . , fn ∈ I be homogeneous invariants of degrees d1 , . . . , dn . Then the following statements are equivalent: (a) I = K[f1 , . . . , fn ], Qn (b) The fi are algebraically independent over K and i=1 di = |G|. Qn (c) The Jacobian determinant J = det(∂fi /∂xj ) is non-zero and i=1 di = |G|. Remark 17. This proposition appeared in Xu (1994), Proposition 2.1. In the proof, Xu concludes directly from the equality of the transcendence degrees that K[x1 , . . . , xn ] is integral over K[f1 , . . . , fn ]. This conclusion is by no means clear and in fact false in general (take n = 2 and f1 = x1 , f2 = x1 x2 ). Hence it appears appropriate to provide a proof here. / Proof. The implication “(a) ⇒ (b)” is well known and follows from Proposition 12. For the reverse implication, it suffices by Smith (1995), Proposition 5.5.5, to show that the ¯ where K ¯ is an algebraic system f1 = · · · = fn = 0 has no projective zero in Pn−1 (K), closure of K. e of We take additional indeterminates t1 , . . . , tn and x0 and an algebraic closure K ¯ K(t1 , . . . , tn ) which contains K. By B´ezout’s theorem (see Fulton, 1984, Example 12.3.7, where there is no assumption made on the dimension of the zero manifold), the projective e given by algebraic set V ⊂ Pn (K) (4.3) f1 (x1 , . . . , xn ) − t1 xd01 = · · · = fn (x1 , . . . , xn ) − tn xd0n = 0 Qn has at most i=1 di = |G| irreducible components. So we have to show that there are at least |G| components of V with x0 6= 0. K(x1 , . . . , xn ) is a finite extension of K(f1 , . . . , fn ), so each xi has a minimal polye n is a solution of nomial over K(f1 , . . . , fn ): gi (xi , f1 , . . . , fn ) = 0. If (ξ1 , . . . , ξn ) ∈ K f1 (x1 , . . . , xn ) − t1 = · · · = fn (x1 , . . . , xn ) − tn = 0, then gi (ξi , t1 , . . . , tn ) = 0, hence there are at most finitely many such solutions, and each constitutes a component of V with x0 6= 0. We shall complete the proof by giving |G| such solutions. Via the homomorphism K(f1 , . . . , fn ) → K(t1 , . . . , tn ), fi 7→ ti , form L = K(t1 , . . . , tn ) ⊗K(f1 ,...,fn ) K(x1 , . . . , xn ), e Take a which is a finite field extension of K(t1 , . . . , tn ) and can be assumed to lie in K. e then σ ∈ G and ξi := 1 ⊗ σ(xi ) ∈ K, fi (ξ1 , . . . , ξn ) = 1 ⊗ fi (σ(x1 ), . . . , σ(xn )) = 1 ⊗ σ(fi ) = 1 ⊗ fi = ti ⊗ 1.
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Hence the elements of G give rise to |G| distinct solutions of Equation (4.3) with x0 6= 0, and “(b) ⇒ (a)” is proved. Finally, J 6= 0 if and only if K(x1 , . . . , xn ) is a finite separable field extension of K(f1 , . . . , fn ) (see, for example, Benson, 1993, Proposition 5.4.2). Hence (a) and (b) imply (c) since K(x1 , . . . , xn ) is Galois over the field of fractions Quot(I) with group G, and (c) implies (b). 2 We obtain the following algorithm, which reduces the question whether the invariant ring of a given group is a polynomial ring to pure linear algebra. Algorithm 18. (Check if I is a polynomial ring) Input: A set of generators of G. The ground field K is assumed to be perfect. Output: False if I is not a polynomial ring, otherwise generators f1 , . . . , fn of I as an algebra over K. Begin Calculate the group order |G|. Set i := 0, m := 1, R := K and d := 1. While i < n Do { Calculate a K-basis of the space of invariants of degree d. Select a maximal subset {b1 , . . . , bmd } of this basis which is linearly independent modulo the homogeneous degree-d part Rd of R. If md > 0 And d 6 | |G| Then Return(False). For j = 1, . . . , md set fi+j := bj . Set m := m · dmd , i := i + md and R := K[f1 , . . . , fi ]. If i > n Or m > |G| Then Return(False). Set d := d + 1. } (End While) If m < |G| Then Return(False). Calculate the Jacobian determinant J = det(∂fi /∂xj ). If J = 0 Then Return(False) Else Return(f1 , . . . , fn ). End. We finish with two examples. Example 19. The complex reflection group G23 from Shephard and Todd (1954) is isomorphic to {±1} × A5 . It is generated by the matrices −1 0 0 0 0 −1 1 −1 α 0 −1 0 , −1 0 0 and 0 α −α 0 0 −1 0 1 0 0 α −1 √ with α = (1 + 5)/2. Algorithm 18 yields invariants f2 , f6 and f10 of degrees 2, 6 and 10, respectively, whose Jacobian determinant does not vanish. The degrees of the fi are of course classically known from Shephard and Todd (1954). Reducing the above matrices modulo a maximal ideal in Z[α] containing p for p = 3 or p = 5, we obtain a linear group G defined over K = F9 or F5 , respectively, which remains isomorphic to {±1} × A5 . Reducing the Jacobian determinant yields non-zero polynomials, hence I is a polynomial ring.
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Reducing modulo the maximal ideal containing p = 2 (which sends α to a root of X 2 + X + 1) yields a group G ≤ GL3 (F4 ) which is isomorphic to A5 . Here we find an additional invariant f5 of degree 5. The Jacobian determinant of f2 , f5 and f6 does not vanish modulo 2, hence these three invariants generate I. f6 is a non-split group extension of f6 , where A Similarly, we can take G27 ∼ = {±1} × A A6 with kernel Z3 , and reduce this modulo a maximal ideal containing 2 to obtain a f6 . Here Algorithm 18 produces invariants group G ≤ GL3 (F4 ) which is isomorphic to A of degrees 6, 12 and 15 with non-vanishing Jacobian determinant, hence I again is a polynomial ring. The calculations for this example where done with the computer programs mentioned at the end of the introduction. The resulting invariants are too long to be reprinted here. / Finally, we present a simple example of a modular reflection group whose invariant ring is not a polynomial ring. Example 20. The group G of all 1 0 0 0
matrices 0 1 0 0
a b b c ∈ GL4 (Fq ) 1 0 0 1
with a, b, c ∈ K := Fq is a reflection group of order q 3 . We assume that I is a polynomial ring. There are two invariants of degree 1, namely f1 = x1 and f2 = x2 . If f3 and f4 are the two remaining generators with deg(f3 ) ≤ deg(f4 ), then deg(f4 ) > q, and both degrees are powers of p, where p is the prime number dividing q. Hence any invariant of degree q + 1 must have the form g1 · f3 + g2 with g1 , g2 ∈ K[x1 , x2 ]. But it is easily verified that xq1 x3 − x1 xq3 + xq2 x4 − x2 xq4 is an invariant which is not of the above form. This contradiction shows that I is not a polynomial ring. Nakajima (1979) stated this result without proof. / Acknowledgements I would like to express my thanks to Prof. Matzat, who raised my interest in invariant theory, and to the referees for their valuable comments on the first version of this paper. References Adem, A., Milgram, R. J. (1994). Cohomology of Finite Groups. Berlin, Heidelberg, New York: SpringerVerlag. Auslander, M., Buchsbaum, D. A. (1957). Homological Dimension in Local Rings. Trans. of the Amer. Math. Soc. 85, 390–405. Becker, T., Weispfenning, V. (1993). Gr¨ obner Bases. Berlin, Heidelberg, New York: Springer-Verlag. Benson, D. J. (1993). Polynomial Invariants of Finite Groups. Lond. Math. Soc. Lecture Note Ser. 190. Cambridge: Cambridge Univ. Press. Bertin, M.-J. (1967). Anneaux d’invariants d’anneaux de polynomes, en caract´eristique p. Comptes Rendus Acad. Sci. Paris (S´ erie A) 264, 653–656. Campbell, H. E. A., Hughes, I., Pollack, R. D. (1991). Rings of Invariants and p-Sylow Subgroups. Canad. Math. Bull. 34(1), 42–47.
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Char, B., Geddes, K., Gonnet, G., Monagan, M., Watt, S. (1990). Maple Reference Manual. Waterloo, Ontario: Waterloo Maple Publishing. Dickson, L. E. (1911). A Fundamental System of Invariants of the General Modular Linear Group with a Solution of the Form Problem. Trans. Amer. Math. Soc. 12, 75–98. Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. New York: Springer-Verlag. Ellingsrud, G., Skjelbred, T. (1980). Profondeur d’anneaux d’invariants en caract´eristique p. Compos. Math. 41, 233–244. Fulton, W. (1984). Intersection Theory. Berlin, Heidelberg, New York: Springer-Verlag. Gianni, P., Trager, B., Zacharias, G. (1988). Gr¨ obner Bases and Primary Decomposition of Polynomial Ideals. J. of Symbolic Computation 6, 149–267. G¨ obel, M. (1995). Computing Bases for Rings of Permutation-invariant Polynomials. J. Symbolic Computation 19, 285–291. ¨ Hilbert, D. (1893). Uber die vollen Invariantensysteme. Math. Ann. 42, 313–370. Kemper, G. (1993). The Invar Package for Calculating Rings of Invariants. IWR Preprint 93-34. Heidelberg. Kemper, G. (1994). Das Noethersche Problem und generische Polynome. PhD Thesis. University of Heidelberg. Also available as: IWR Preprint 94-49, Heidelberg 1994. Landweber, P. S., Stong, R. E. (1987). The Depth of Rings of Invariants over Finite Fields. In: Proc. New York Number Theory Seminar, 1984. Lecture Notes in Math. 1240. New York: Springer-Verlag. McShane, J. M. (1992). Computation of Polynomial Invariants of Finite Groups. PhD Thesis. University of Arizona. M¨ oller, H. M., Mora, F. (1986). New Constructive Methods in Classical Ideal Theory. J. of Algebra 100, 138–178. Nakajima, H. (1979). Invariants of Finite Groups Generated by Pseudo-Reflections in Positive Characteristic. Tsukuba J. Math. 3, 109–122. Noether, E. (1916). Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann. 77, 89–92. Noether, E. (1926). Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charkteristik p. Nachr. Ges. Wiss. G¨ ottingen pp. 28–35. Richman, D. R. (1990). On Vector Invariants over Finite Fields. Adv. in Math. 81, 30–65. Shephard, G. C., Todd, J. A. (1954). Finite Unitary Reflection Groups. Canad. J. Math. 6, 274–304. Sloane, N. J. A. (1977). Error-Correcting Codes and Invariant Theory: New Applications of a NineteenthCentury Technique. Amer. Math. Monthly 84, 82–107. Smith, L. (1995). Polynomial Invariants of Finite Groups. Wellesley, Mass.: A. K. Peters. Stanley, R. P. (1979). Invariants of Finite Groups and their Applications to Combinatorics. Bull. Amer. Math. Soc. 1(3), 475–511. Stillman, M., Stillman, M., Bayer, D. (1989). Macaulay User Manual. Available by anonymous ftp from various sites. Sturmfels, B. (1993). Algorithms in Invariant Theory. Wien, New York: Springer-Verlag. Toda, H. (1972). Cohomology mod 3 of the Classifying Space BF4 of the Exceptional Group F4 . J. Math. Kyoto Univ. 13, 97–115. Weber, H. (1899). Lehrbuch der Algebra. Braunschweig: Viehweg Verlag. Wilkerson, C. (1983). A Primer on the Dickson Invariants. Amer. Math. Soc. Contemp. Math. Series 19, 421–434. Xu, C. (1994). Computing Invariant Polynomials of p-adic Reflection Groups. In: Proc. of Symposia in Appl. Math. 48. Providence, RI: Amer. Math. Soc.
