J. Symbolic Computation (1996) 21, 351–366

Calculating Invariant Rings of Finite Groupsover Arbitrary Fields

GREGOR KEMPER

IWR, Universitat 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 lineargroups over an arbitrary field K. Up to now, such algorithms have been available onlyfor 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–Macaulayor 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 variouseffective methods to calculate the invariant ring I of G, i.e., to find a finite system ofgenerators 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 for-mula, the Cohen–Macaulay property of invariant rings, and of Noether’s degree boundin the case char(K) > |G| (Noether, 1916).

If, on the other hand, |G| is a multiple of char(K) (which we shall call the modularcase), none of these techniques are available since they all involve divisions by |G|. Infact, G is not a linearly reductive group in this case. Nevertheless, the invariant ring isfinitely generated as a K-algebra (Noether, 1926). If G is a permutation group there is avery nice algorithm by Gobel (1995) which calculates generators over any commutativeground ring and which implies a degree bound for the generators. But for the case offinite linear groups there is no algorithm available at the moment to construct a systemof 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 purposeof this paper is to fill this gap.

With the knowledge of generators for the invariant ring, it is quite easy to calcu-late its depth and its Poincare series (see Section 4.1). It is even easier to check theCohen–Macaulay property. So the algorithm presented here could be useful to gain someexperience 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: Gregor.Kemper@iwr.uni-heidelberg.de

0747–7171/96/030351 + 16 $18.00/0 c© 1996 Academic Press Limited

352 G. Kemper

construct primary invariants is proposed. Section 3 contains the main algorithm whichdoes the step from the primary invariants to the full invariant ring for arbitrary groundfields. The last section is devoted to some applications. Here we probe into the questionof the Cohen–Macaulay property mentioned above, and give a simple algorithm to decidewhether the invariant ring of a given group is isomorphic to a polynomial ring. The lastquestion has its background in another “defect” of modular linear groups, that theirinvariant 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 versionof 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 polynomialring K[x1, . . . , xn] by linear transformations of the indeterminates xi. We write I for theinvariant 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’snormalization lemma (see, for example, Benson, 1993, Theorem 2.2.7), there exist ho-mogeneous elements f1, . . . , fm ∈ R such that R is finitely generated as a module overA := K[f1, . . . , fm].

If R is the invariant ring I, then m = n and f1, . . . , fn are called primary invari-ants. In this case, f1, . . . , fn are also primary invariants for any subgroup H ≤ G sinceK[x1, . . . , xn] is integral over I. The following proposition is the key to the calculationof primary invariants:

Proposition 1. A set {f1, . . . , fi} ⊂ I of homogeneous invariants can be extended to asystem 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 andonly if k = n and

VK(f1, . . . , fk) = {0}, (2.1)

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 pri-mary 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 ofthe 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.

Calculating Invariant Rings 353

so dim(f1, . . . , fn) = 0 by Becker and Weispfenning (1993), Theorem 6.5.4. This com-pletes the proof. 2

Primary invariants are far from being uniquely determined by G. But the choice ofprimary invariants f1, . . . , fn determines the minimum number mf1,...,fn of generatorsfor 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 subsequentcalculations that mf1,...,fn be as low as possible, as this is most likely to be achieved ifthe degrees of the primary invariants are chosen as small as possible, as is illustrated byProposition 12 and Example 5(b). Hence I will call an algorithm that constructs primaryinvariants good if it is likely to produce primary invariants of small degrees. So the bestalgorithm 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), Algo-rithm 2.5.8. He proposed that as a first step one should collect invariants of increasingdegrees until arriving at a set {f1, . . . , fk} having the property (2.1). This set is obtainedby successively calculating K-bases for the vector spaces of homogeneous invariants ofdegree 1, 2, . . . and including a basis element into the set if it does not lie in the radical ofthe ideal spanned be the fi gotten so far. In the next step, one tries to delete elements fifrom the set while retaining (2.1). Experience shows that in most cases this will lead toa set of n elements, i.e., a system of primary invariants. If it does not, we are left with kpolynomials fi, k > n, which satisfy (2.1), and must apply a third step which consists offirst converting all fi to invariants of the same degree by taking suitable powers and thenchoosing random n×k-matrices (ai,j) and checking condition (2.1) for fi =

∑kj=1 ai,j ·fj

(i = 1, . . . , n), until the fi 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. Buteven in the majority of cases when this step is unnecessary, the algorithm often yieldsprimary invariants whose degrees are not the lowest possible. Let us look at a typicalexample for this.

Example 2. Let

G = 〈(

i 00 −i

)〉 ≤ GL2(C).

Then the above algorithm would in the first step produce the invariants

f1 = x1x2, f2 = x41 and f3 = x4

2

and omit f1 in the second step, yielding primary invariants of degrees 4 and 4. But thereare primaries of degrees 2 and 4, namely f1 = x1x2 and f2 = x4

1 +x42, which the algorithm

would miss. /

Another algorithm due to E. Dade (see Stanley, 1979) constructs primary invariantsby taking products over G-orbits of suitable linear forms. This algorithm is very quickbut also has the disadvantage that it tends to produce primary invariants of too highdegrees (as it would in Example 2).

While the algorithm of Sturmfels always tries to find systems of primary invariants asa whole, Proposition 1 suggests a strategy of successively adding primary invariants to

354 G. Kemper

the system, the condition for a new primary invariant fi+1 being that it decreases thedimension of the ideal (f1, . . . , fi), or, equivalently, that it lies in none of the associatedprime 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 un-known coefficients, applying the generators of G and solving the resulting systemof linear equations for the unknown coefficients.)Form Id(t1, . . . , tmd) :=

∑mdj=1 tj · bj with indeterminates tj .

For k = 1, . . . , r calculate the complete residues rk(t1, . . . , tmd) of Id w.r.t. theGrobner basis Pk.

If there exist α1, . . . , αmd ∈ K with rk(α1, . . . , αmd) 6= 0 ∀k = 1, . . . , rThen {(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).Let P1, . . . , Pr be Grobner bases of the associated primes w.r.t. any mono-

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 associatedprime 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 describeproper subspaces of Kmd , so in the case of an infinite K there are always αi satisfying thecondition. It is easy to write an algorithm that specifies the αi successively to produce anon-solution of all these equations in this case. If on the other hand K is finite (and allrk non-zero), we can simply search the elements of Kmd 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 Algo-rithm 3 is f1 = x1x2, yielding associated primes (x1) and (x2). Passing to degree 4, we

Calculating Invariant Rings 355

getI4 = t1 · x4

1 + t2 · x21x

22 + t3 · x4

2

with complete residuesr1 = t3 · x4

2 and r2 = t1 · x41.

So (α1, α2, α3) = (1, 0, 1) is a non-solution, leading to f2 = x41 +x4

2 as the second primaryinvariant. /

According to my experience Algorithm 3 is very likely to produce a system of primaryinvariants of smallest possible degrees, but it is not guaranteed to do so, as there arecounter examples:

Example 5.

(a) Consider the permutation group G = 〈(123)(456), (12)(45)〉 ∼= S3 acting on theindeterminates 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 threevariables. But by an unlucky choice Algorithm 3 might pick f1 = x1 + x2 + x3,f2 = x4 + x5 + x6, f3 = x1x2 + x1x3 + x2x3, f4 = x1x4 + x2x5 + x3x6, f5 = x4x5x6

as the first five primary invariants. Then x1 = x2 = x3 = x4 = 0, x5 = −x6 isa zero of these fi and also of all invariants of degree 3, hence there is no furtherprimary invariant of this degree. So Algorithm 3 would proceed to the next degreeand 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

G = 〈

0 1 0 01 1 0 00 0 1 10 0 1 0

,

1 0 0 00 1 0 01 0 1 00 1 0 1

1 0 1 00 1 0 10 0 1 00 0 0 1

〉 ≤ GL4(F2).

We only sketch the results here. Adem and Milgram’s primary invariants are ofdegrees 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 , seep. 353) are of degrees 3, 5, 8 and 12. They were also obtained by Algorithm 3together with some degree of human intervention. With these primary invariantsthere are 24 module-generators for I of degree at most 24 needed, while Adem andMilgram’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 primeideals of (f1, . . . , fi−1). Algorithms which perform this over any field K that is finitelygenerated over its prime field are given in Gianni et al. (1988). Here we can assumeK to be finitely generated over its prime field by the matrix entries of the elements ofG. These algorithms automatically yield Grobner bases for the associated primes. Forthe characteristic zero case see also Becker and Weispfenning (1993), Table 8.10. Thecomplete computation of the associated primes can in most cases be circumvented byusing the “Grobner factorization algorithm”, which is likely to produce the associatedprimes since in our case (f1, . . . , fi−1) has no embedded primes. This algorithm is imple-mented as the function gsolve in Maple (Char et al., 1990). Since this algorithm always

356 G. Kemper

yields ideals which lie between (f1, . . . , fi−1) and an associated prime, the condition thatrk(α1, . . . , αmd) 6= 0 ∀k in Algorithm 3 will be necessary for the existence of anotherprimary invariant of degree d, if the Pk are the output of the Grobner factorization al-gorithm. After having found a candidate fi = Id(α1, . . . , αmd), we have to check if thisreally decreases the dimension of the ideal by a Grobner basis method which is muchsimpler 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 primesrigorously.

In the non-modular case, i.e., if char(K) |6 |G|, we have Molien’s formula to calculatethe Poincare series of the invariant ring. From this, we can make good guesses at thedegrees of the primary invariants (see, for example, Sloane, 1977). So another variant ofAlgorithm 3 would be to incorporate guesses for complete systems of primary invariantsby 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 performsreasonably well and is, in the author’s opinion, the best algorithm available at the momentfor 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 invari-ants. In the non-modular case, Molien’s formula provides complete information aboutthe number and degrees of the secondary invariants. This reduces their calculation to asimple exercise of filling up homogeneous subspaces of invariants (see McShane, 1992, orKemper, 1994). The idea for the general case now is to calculate secondary invariantsg1, . . . , 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 thenlead to a system of linear equations with coefficients in K[x1, . . . , xn], for which we haveto calculate the solutions whose components lie in A. So we need an algorithm whichintersects a submodule of K[x1, . . . , xn]r with Ar, where Rr denotes the free module ofrank r over a ring R.

Lemma 6. Let R = K[x1, . . . , xn], M =∑si=1R · bi ≤ Rr a submodule and A =

K[f1, . . . , fk] ≤ R the subalgebra generated by elements f1, . . . , fk ∈ R. Take the polyno-mial rings S = K[x1, . . . , xn, t1, . . . , tk] and T = K[t1, . . . , tk] with further indeterminatest1, . . . , tk and form

M =s∑i=1

S · bi +k∑j=1

(tj − fj) · Sr ≤ Sr

and

MT = M ∩ T r.

Then with Φ: T r → Ar, tj 7→ fj we have

Φ(MT ) = M ∩Ar.

Calculating Invariant Rings 357

Proof. We consider the homomorphism Ψ: Sr → Rr/M obtained by substitutingtj 7→ fj , and claim ker(Ψ) = M .

By construction, M clearly lies in the kernel. Let P (t1, . . . , tk) be a product of tj ’sand ei ∈ Sr a vector of the standard basis. We show by induction on deg(P ) that(P (t1, . . . , tk)− P (f1, . . . , fk)

)· ei ∈ M . There is nothing to show for deg(P ) = 0. If

deg(P ) > 0, then P (t1, . . . , tk) = tj ·P ′(t1, . . . , tk) with deg(P ′) < deg(P ), so by induction(P ′(t1, . . . , tk)− P ′(f1, . . . , fk)

)· ei ∈ M , and we conclude(

P (t1, . . . , tk)− P (f1, . . . , fk))· ei =(

(tj − fj) · P ′(t1, . . . , tk) + fj ·(P ′(t1, . . . , tk)− P ′(f1, . . . , fk)

) )· ei ∈ M.

It follows that for (g1, . . . , gr) ∈ Sr we have

(g1, . . . , gr)− (g1

∣∣tj=fj , . . . , gr

∣∣tj=fj ) ∈ M.

Now if Ψ(g1, . . . , gr) = 0, then (g1

∣∣tj=fj , . . . , gr

∣∣tj=fj ) ∈M ⊂ M and hence (g1, . . . , gr)

∈ M , which proves the claim.

It follows that ker(Ψ |Tr

) = MT , and due to

Ψ(T r) = (Ar +M)/M ∼= Ar /(M ∩Ar)

we haveT r/MT∼= 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].Form the submodule M ≤ Sr generated by the bi (i = 1, . . . , s) and by the

(tj − fj) · ei (j = 1, . . . , k, i = 1, . . . , r).Calculate a Grobner basis B of M w.r.t. a term order ≺ with the property that

every xi is greater than any monomial in the tj .(See below for Grobner 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 Grobner bases of modules. These were introducedby Moller and Mora (1986), who also extended Buchberger’s algorithm to this case. Thisalgorithm is implemented in Macaulay (see Stillman et al., 1989). The eliminationproperty for these Grobner bases, which was used in the algorithm, follows easily andcan be found in (loc. cit.).

358 G. Kemper

Also implemented in Macaulay and described by Moller and Mora (1986) are algo-rithms to find syzygy modules, i.e., solution modules for systems of linear equations overa polynomial ring. These constitute the second computational ingredient of the generalalgorithm 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 in-variants of small degrees).

Output: Secondary invariants g1, . . . , gm.Begin

Choose a subgroup H ≤ G (for example, H = {ι}) and calculate homogeneousgenerators 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 withr∑i=1

(σ(hi)− hi) · pi = 0 for all σ ∈ S. (3.1)

(This is the calculation of a syzygy module.)Calculate M ∩Ar =

∑mi=1A · ci using Algorithm 7.

Set gi :=∑rj=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 =∑ri=1 ai · hi ∈ I with ai ∈ A we

have by Equation (3.1)

(a1, . . . , ar) ∈M ∩Ar ⇒ (a1, . . . , ar) =m∑i=1

ai · ci with ai ∈ A ⇒ f =m∑i=1

ai · gi,

hence I =∑mi=1A · 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 ofweight χ (i.e., of f ∈ K[x1, . . . , xn] such that σ(f) = χ(σ) · f ∀σ ∈ G) is a moduleover I and also over A = K[f1, . . . , fk] for any invariants fi. If the fi satisfy theassumption made in Algorithm 8, then a generating set for Mχ as an A-module canbe 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. Thenχ|H = 1. Taking left coset representatives σ1, . . . , σr of G/H, we have a “relativeReynolds operator” given by

πG/Hχ =1

(G : H)

r∑i=1

χ(σ−1i ) · σi

(see Campbell et al., 1991 or Smith, 1995, p. 28). This is a projectionK[x1, . . . , xn]H

→→Mχ of I-modules. Hence we only have to calculate generators for K[x1, . . . , xn]H

as a module over A and then apply πG/Hχ to obtain Mχ. K[x1, . . . , xn]H itself is

most conveniently calculated by applying Algorithm 8 recursively to a chain ofsubgroups H1 ≤ H2 ≤ · · · ≤ Hs = H with |Hi| = pi.

Calculating Invariant Rings 359

(c) The set {g1, . . . , gm} from Algorithm 8 can be made into a minimal set of sec-ondary 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 bythe others. This test for membership amounts to the solution of a system of linearequations over K. /

Let us look at an example now.

Example 10. Consider the cyclic group G = Z4 with its regular representation overK = F2. We find primary invariants

f1 = x1 + x2 + x3 + x4, f2 = x1x3 + x2x4,

f3 = x1x2 + x2x3 + x3x4 + x4x1, f4 = x1x2x3x4.

With H = {ι} ≤ G, there are 16 generators h1, . . . , h16 of K[x1, . . . , xn] as a moduleover A = K[f1, . . . , f4], and the solution module M of Equation (3.1) is generated by 16vectors. The module M in Algorithm 7 then has 80 generators and a Grobner basis of113 elements, which was calculated by Macaulay in a few seconds. From these, five liein K[t1, . . . , t4], and we get the secondary invariants

g1 = 1, g2 = x31 + x3

2 + x33 + x3

4, g3 = x21x2 + x2

2x3 + x23x4 + x2

4x1,

g4 = x31x2 + x3

2x3 + x33x4 + x3

4x1, g5 = x31x

22 + x3

2x23 + x3

3x24 + x3

4x21.

The calculation can be essentially accelerated by taking H = Z2 ≤ G and applyingAlgorithm 8 recursively. We shall continue this example on p. 360 and show that theinvariant ring is not Cohen–Macaulay. From the above secondary invariants we alreadysee that the invariants of degree ≤ 4 do not generate I as a K-algebra, i.e., Noether’sdegree 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 necessaryfor the invariant ring although the group order remains the same.

Bertin (1967) and Benson (1993), p. 104, mention this example, but do not calculatethe complete invariant ring. /

Clearly, the Grobner basis calculation involved in Algorithm 7 is the most time con-suming 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, Poincare series and depth. Easy algorithms are presentedto decide the Cohen–Macaulay property and the property of being isomorphic to a poly-nomial algebra.

4.1. depth and the Cohen–Macaulay property

Suppose that we have calculated primary invariants f1, . . . , fn and secondary invariantsg1, . . . , 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

360 G. Kemper

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 Poincare series of I can easily be obtained (seeStanley, 1979, p. 496).

If {g1, . . . , gm} is a minimal set of secondary invariants and if (4.1) is a minimal freeresolution then the homological dimension hdimA(I) is (by definition) the largest i suchthat 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 liein 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 ofCorollary 19.15 in Eisenbud (1995) to the graded case:

Proposition 11. Let A ≤ B be Noetherian graded algebras over a field K such thatA0 = 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 ifdepth(I) = n. The following proposition provides a much easier criterion to decide theCohen–Macaulay property. This proposition seems to be part of the folklore, but since Icould 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, . . . ,dn. Then the invariant ring I is Cohen–Macaulay if and only if it can be generated by(∏ni=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 andgenerated by

∏ni=1 di polynomials over A (see, for example, Sturmfels, 1993, Proposi-

tion 2.3.6). Hence [K(x1, . . . , xn) : K(f1, . . . , fn)] =∏ni=1 di, and by Galois theory

[K(x1, . . . , xn)G : K(f1, . . . , fn)] =( n∏i=1

di

)/|G| =: d.

Thus there are at least d invariants necessary to generate I over A, and more than d willbe linearly dependent. 2

There is not very much known about the question of Cohen–Macaulayness of modularinvariant rings. Good references are Smith (1995) and Landweber and Strong (1987). Aremarkable result of Ellingsrud and Skjelbred (1980) states that if G is a cyclic p-groupwith p = char(K) 6= 0 then depth(I) = min(n, n−m+ 2), where m is the dimension ofthe K-vector space generated by all σ(xi)−xi (σ ∈ G, i = 1, . . . , n). For general G thereis 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

Calculating Invariant Rings 361

we calculated primary and secondary invariants for I and saw that 5 secondary invariantswere needed. Hence by Proposition 12, I is not Cohen–Macaulay, since

(∏4i=1 deg(fi)

)/

|G| = 4. Calculating syzygies between the secondary invariants and applying Algorithm 7,we obtain the following relation:

f1

(f4

1 + f22 + f2f3

)· g1 +

(f2

1 + f2 + f3

)· g2 +

(f2

1 + f2

)· g3 + f1 · g4 = 0,

which generates the module of syzygies. Thus the Poincare series is

P (I, t) =t5 + t4 + 2t3 + 1

(1− t)(1− t2)2(1− t4)− t5

(1− t)(1− t2)2(1− t4)=

t4 + 2t3 + 1(1− t)(1− t2)2(1− t4)

,

and hdim(I) = 1, so depth(I) = 3 in accordance with the result of Ellingsrud andSkjelbred quoted above. Since G is a permutation group the Poincare series does notdepend on the ground field (see, for example, Smith, 1995, Proposition 4.3.4) and can thusbe 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-Sylowsubgroup 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 thatall fi ∈ I = K[x1, . . . , xn]G. We claim that f1, . . . , fm is regular for I, too, so we have toprove that fi is not a zero divisor in I/(f1, . . . , fi−1) for i = 1, . . . ,m.

Let gifi = g1f1 + · · · + gi−1fi−1 with g1, . . . , gi ∈ I. Since f1, . . . , fm is regular forK[x1, . . . , xn]H , there are h1, . . . , hi−1 ∈ K[x1, . . . , xn]H such that gi = h1f1 + · · · +hi−1fi−1. Applying the relative Reynolds operator πG/H : K[x1, . . . , xn]H → I (see Re-mark 9(b)) to this equation yields

gi = πG/H(h1) · f1 + · · ·+ πG/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 nota-tion from the proposition), then so is K[x1, . . . , xn]G. This result already appeared inCampbell et al. (1991) and Smith (1995), Proposition 8.3.1. Together with Ellingsrudand Skjelbred’s result this implies that I is Cohen–Macaulay if n ≤ 3, since a linearrepresentation of a p-group H always has a fixed vector. The above proof carries over tothe case of relative invariants (see Remark 9(a)).

The converse of Proposition 14 does not hold. Consider the example given by Campbellet al. (1991) of the regular representation of the cyclic group H = Zp, p a prime, over afield K of characteristic p. H occurs as p-Sylow subgroup of the symmetric group G = Spwhose invariant ring is polynomial and in particular Cohen–Macaulay. But if p > 3, thenK[x1, . . . , xp]H is not Cohen–Macaulay by Ellingsrud and Skjelbred (1980).

The next proposition concerns the relation between Cohen–Macaulayness for I and forthe invariant ring of subrepresentations.

362 G. Kemper

Proposition 15. Let G act on the first k and last n − k variables, i.e., let∑ki=1Kxi

and∑ni=k+1Kxi 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 andK[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 moduleover A := K[f1, . . . , fn] are linearly independent over A since they all have the samecardinality dimK(K ⊗A I) (see p. 353).

Let g1, . . . , gr be a minimal system of homogeneous generators for K[x1, . . . , xk]G overK[f1, . . . , fk] and choose homogeneous gr+1, . . . , gm ∈ I which together with g1, . . . , grgenerate I over A, with m minimal. The proposition is proved if we can show that thisis a minimal generating set. So suppose that this is not true. Then one of the first rgenerators, say g1 satisfies a relation

g1 =m∑i=2

higi (4.2)

with hi ∈ A, where we can assume all hi to be homogeneous and deg(gi) < deg(g1) forthose i > r with hi 6= 0 by the minimal choice of m. Setting xk+1, . . . , xn = 0 in (4.2)yields

g1 =r∑i=2

higi +m∑

i=r+1

higi

with hi ∈ K[f1, . . . , fk] and gi ∈ K[x1, . . . , xk]G, so g1 is a K[f1, . . . , fk]-linear combina-tion 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 Ellingsrudand Skjelbred (1980), a counter example is given for each prime number p by the cyclicgroup G generated by the block matrix

11 1

1 111 1

∈ GL5(Fp).

Calculating concretely for the case p = 2 by means of the algorithms 3 and 8 yieldsprimary 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 areflection group, but the converse is no longer true in general. It is thus an interestingquestion 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

Calculating Invariant Rings 363

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(q2) for n ≤ 3 and n ≤ 2, respectively(Nakajima, 1979; Kemper, 1994), and the complex reflection groups G29 and G31 ofShephard and Todd (1954) reduced modulo 5 (Xu, 1994). Counter examples are theWeyl group W (F4) of the root system F4 (Toda, 1972), the orthogonal and unitarygroups for n ≥ 4 and n ≥ 3, respectively, and a reflection subgroup of the unipotentupper 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 apolynomial 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],(b) The fi are algebraically independent over K and

∏ni=1 di = |G|.

(c) The Jacobian determinant J = det(∂fi/∂xj) is non-zero and∏ni=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 ingeneral (take n = 2 and f1 = x1, f2 = x1x2). Hence it appears appropriate to provide aproof here. /

Proof. The implication “(a)⇒ (b)” is well known and follows from Proposition 12. Forthe reverse implication, it suffices by Smith (1995), Proposition 5.5.5, to show that thesystem f1 = · · · = fn = 0 has no projective zero in Pn−1(K), where K is an algebraicclosure of K.

We take additional indeterminates t1, . . . , tn and x0 and an algebraic closure K ofK(t1, . . . , tn) which contains K. By Bezout’s theorem (see Fulton, 1984, Example 12.3.7,where there is no assumption made on the dimension of the zero manifold), the projectivealgebraic set V ⊂ Pn(K) given by

f1(x1, . . . , xn)− t1xd10 = · · · = fn(x1, . . . , xn)− tnxdn0 = 0 (4.3)

has at most∏ni=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 poly-

nomial over K(f1, . . . , fn): gi(xi, f1, . . . , fn) = 0. If (ξ1, . . . , ξn) ∈ Kn is a solution off1(x1, . . . , xn) − t1 = · · · = fn(x1, . . . , xn) − tn = 0, then gi(ξi, t1, . . . , tn) = 0, hencethere are at most finitely many such solutions, and each constitutes a component of Vwith 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),

which is a finite field extension of K(t1, . . . , tn) and can be assumed to lie in K. Take aσ ∈ G and ξi := 1⊗ σ(xi) ∈ K, then

fi(ξ1, . . . , ξn) = 1⊗ fi(σ(x1), . . . , σ(xn)) = 1⊗ σ(fi) = 1⊗ fi = ti ⊗ 1.

364 G. Kemper

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 ofK(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 invariantring 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) isisomorphic to {±1} ×A5. It is generated by the matrices −1 0 0

0 −1 00 0 −1

,

0 0 −1−1 0 00 1 0

and

1 −1 α0 α −α0 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 ofcourse classically known from Shephard and Todd (1954). Reducing the above matricesmodulo a maximal ideal in Z[α] containing p for p = 3 or p = 5, we obtain a linear groupG 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 polynomialring.

Calculating Invariant Rings 365

Reducing modulo the maximal ideal containing p = 2 (which sends α to a root ofX2 + X + 1) yields a group G ≤ GL3(F4) which is isomorphic to A5. Here we find anadditional invariant f5 of degree 5. The Jacobian determinant of f2, f5 and f6 does notvanish modulo 2, hence these three invariants generate I.

Similarly, we can take G27∼= {±1} × A6, where A6 is a non-split group extension of

A6 with kernel Z3, and reduce this modulo a maximal ideal containing 2 to obtain agroup G ≤ GL3(F4) which is isomorphic to A6. Here Algorithm 18 produces invariantsof degrees 6, 12 and 15 with non-vanishing Jacobian determinant, hence I again is apolynomial ring.

The calculations for this example where done with the computer programs mentionedat 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 invariantring is not a polynomial ring.

Example 20. The group G of all matrices1 0 a b0 1 b c0 0 1 00 0 0 1

∈ GL4(Fq)

with a, b, c ∈ K := Fq is a reflection group of order q3. We assume that I is a polynomialring. 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 bothdegrees are powers of p, where p is the prime number dividing q. Hence any invariantof degree q + 1 must have the form g1 · f3 + g2 with g1, g2 ∈ K[x1, x2]. But it is easilyverified that

xq1x3 − x1xq3 + xq2x4 − x2x

q4

is an invariant which is not of the above form. This contradiction shows that I is not apolynomial ring. Nakajima (1979) stated this result without proof. /

Acknowledgements

I would like to express my thanks to Prof. Matzat, who raised my interest in invarianttheory, and to the referees for their valuable comments on the first version of this paper.

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