J. Symbolic Computation, to appear.

Calculating Invariant Rings of Finite Groups over Arbitrary Fields^{†}

GREGOR KEMPER

IWR, Universität Heidelberg, Im Neuenheimer Feld 368, 69 120 Heidelberg, Germany

                    email Gregor.Kemper@iwr.uni-heidelberg.de

(Received 21 November 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.

Introduction

If G is a finite linear group over a field K such that char(K) ∤ |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 and 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 Göbel (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 Poincaré series (see Section 3.1). 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.

† This paper is to appear in the Journal of Symbolic Computation.

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 construct primary invariants is proposed. Section 2 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 is planning to incorporate the algorithms presented in this paper into a new version of the Invar-package† for Maple (Kemper, 1993). A preliminary version can already be obtained from the author.

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.

Let us fix some notation. Throughout this paper, K is an arbitrary field and G ≤ GL_{n}(K) is a finite matrix group of degree n over K. G operates on the polynomial ring K[x_{1}, …, x_{n}] by linear transformations of the indeterminates x_{i}. We write I for the invariant ring:

I = K[x_{1}, …, x_{n}]^{G} = {f ∈ K[x_{1}, …, x_{n}] | σ(f) = f ∀σ ∈ G}.

This is a graded subalgebra of K[x_{1}, …, x_{n}].

1. Calculating Primary Invariants

Let R be any graded K-algebra of Krull dimension m with R_{0} = K, then by Noether’s normalization lemma (see, for example, Benson, 1993, Theorem 2.2.7), there exist homogeneous elements f_{1}, …, f_{m} ∈ R such that R is finitely generated as a module over A := K[f_{1}, …, f_{m}].

If R is the invariant ring I, then m = n and f_{1}, …, f_{n} are called primary invariants. In this case, f_{1}, …, f_{n} are also primary invariants for any subgroup H ≤ G since K[x_{1}, …, x_{n}] is integral over I. The following proposition is the key to the calculation of primary invariants:

Proposition 1. A set {f_{1}, …, f_{i}} ⊂ I of homogeneous invariants can be extended to a system of primary invariants if and only if

dim(f_{1}, …, f_{i}) = n − i.

In particular, homogeneous f_{1}, …, f_{k} ∈ I form a system of primary invariants if and only if k = n and

VK̄(f_{1}, …, f_{k}) = {0},  (1.1)

where VK̄ denotes the set of zeros over the algebraic closure of K.

† This package is part of Maple V.3 and beyond. For earlier versions of Maple, it can be obtained by anonymous ftp from daisy.uwaterloo.ca.

Proof. By Smith (1995), Prop. 5.3.7, homogeneous invariants f_{1}, …, f_{n} are primary invariants if and only if dim(f_{1}, …, f_{n}) = 0. By Krull’s Principal Ideal Theorem (see, for example, Eisenbud, 1995, Theorem 10.1), each f_{i} can diminish the dimension of the ideal by at most one, hence dim(f_{1}, …, f_{i}) = n − i if f_{1}, …, f_{n} are primary invariants.

Conversely, if dim(f_{1}, …, f_{i}) = n − i there exist homogeneous \overline{f_{i+1}}, …, \overline{f_{n}} ∈ R := I/(f_{1}, …, f_{i}) such that R is finitely generated as a module over K[\overline{f_{i+1}}, …, \overline{f_{n}}]. Hence

dim_{K} (K[x_{1}, …, x_{n}]/(f_{1}, …, f_{n})) = dim_{K} (R/(\overline{f_{i+1}}, …, \overline{f_{n}})) < ∞,

so dim(f_{1}, …, f_{n}) = 0 by Becker and Weispfenning (1993), Theorem 6.5.4. This completes the proof.

Primary invariants are far from being uniquely determined by G. But the choice of primary invariants f_{1}, …, f_{n} determines the minimum number m_{f_{1},... ,f_{n}} of generators for I as a module over A = K[f_{1}, …, f_{n}], which is m_{f_{1},... ,f_{n}} = dim_{K}(K ⊗_{A} I) by Smith (1995), Corollary 5.2.5. It is of crucial importance for the effectiveness of subsequent calculations that m_{f_{1},... ,f_{n}} 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 {f_{1}, …, f_{k}} having the property (1.1). 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 f_{i} gotten so far. In the next step, one tries to delete elements f_{i} from the set while retaining (1.1). 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 f_{i}, k > n, which satisfy (1.1), and must apply a third step which consists of first converting all f_{i} to invariants of the same degree by taking suitable powers and then choosing random n×k-matrices (a_{i,j}) and checking condition (1.1) for f̃_{i} = ∑_{j=1}^{k} a_{i,j} ·f_{j} (i = 1, …, n), until the f̃_{i} 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

G = ⟨(\begin{matrix}i & 0 \\ 0 & −i \end{matrix})⟩ ≤ GL_{2}(ℂ).

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

f_{1} = x_{1}x_{2}, f_{2} = x_{1}^{4} and f_{3} = x_{2}^{4}

and omit f_{1} in the second step, yielding primary invariants of degrees 4 and 4. But there are primaries of degrees 2 and 4, namely f_{1} = x_{1}x_{2} and f_{2} = x_{1}^{4}+x_{2}^{4}, 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 the system, the condition for a new primary invariant f_{i+1} being that it decreases the dimension of the ideal (f_{1}, …, f_{i}), or, equivalently, that it lies in none of the associated prime ideals of (f_{1}, …, f_{i}). This brings us to propose the following algorithm:

Algorithm 3 (Constructing primary invariants). Input: A set of generators of G. Output: Primary invariants f_{1}, …, f_{n}. Begin

Set i := 1, d := 1, r := 1 and P_{1} := {}.

While i ≤ n Do {

Calculate a K-basis {b_{1}, …, b_{m_{d}}} 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.) Form I_{d}(t_{1}, …, t_{m_{d}}) := ∑_{j=1}^{m_{d}} t_{j} · b_{j} with indeterminates t_{j}. For k = 1, …, r calculate the complete residues r_{k}(t_{1}, …, t_{m_{d}}) of I_{d} w.r.t.

the Gröbner basis P_{k}. If there exist α_{1}, …, α_{m_{d}} ∈ K with r_{k}(α_{1}, …, α_{m_{d}}) ≠ 0 ∀k = 1, …, r Then { (See below for how to decide this.) Set f_{i} := I_{d}(α_{1}, …, α_{m_{d}}). Calculate the associated prime ideals of (f_{1}, …, f_{i}). (See below for algorithms which perform this.) Set r :=(number of associated primes). Let P_{1}, …, P_{r} be Gröbner 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 r_{k} are the complete residues of I_{d} w.r.t. the P_{k}, the condition r_{k}(α_{1}, …, α_{m_{d}}) ≠ 0 ∀k is equivalent to the requirement that I_{d}(α_{1}, …, α_{m_{d}}) lies in none of the associated prime ideals of (f_{1}, …, f_{i}). If any of the r_{k} is zero, then this condition is certainly false. If, on the other hand, all r_{k} are non-zero, then the equations r_{k}(α_{1}, …, α_{m_{d}}) = 0 describe proper subspaces of K^{m_{d}}, so in the case of an infinite K there are always α_{i} satisfying the condition. It is easy to write an algorithm that specifies the α_{i} successively to produce a non-solution of all these equations in this case. If on the other hand K is finite (and all r_{k} non-zero), we can simply search the elements of K^{m_{d}} 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 f_{1} = x_{1}x_{2}, yielding associated primes (x_{1}) and (x_{2}). Passing to degree 4, we get

I_{4} = t_{1} · x_{1}^{4} + t_{2} · x_{1}^{2}x_{2}^{2} + t_{3} · x_{2}^{4}

with complete residues

r_{1} = t_{3} · x_{2}^{4} and r_{2} = t_{1} · x_{1}^{4}.

So (α_{1}, α_{2}, α_{3}) = (1, 0, 1) is a non-solution, leading to f_{2} = x_{1}^{4} +x_{2}^{4} 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 = ⟨(123)(456), (12)(45)⟩ ≅ S_{3} acting on the indeterminates of ℂ[x_{1}, …, x_{6}]. 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 f_{1} = x_{1} + x_{2} + x_{3}, f_{2} = x_{4} + x_{5} + x_{6}, f_{3} = x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3}, f_{4} = x_{1}x_{4} + x_{2}x_{5} + x_{3}x_{6}, f_{5} = x_{4}x_{5}x_{6} as the first five primary invariants. Then x_{1} = x_{2} = x_{3} = x_{4} = 0, x_{5} = −x_{6} is a zero of these f_{i} 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 A_{5} in SL_{4}(𝔽_{2})” of Adem and Milgram (1994), p. 116, given by

G = ⟨(\begin{matrix}0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}) , (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})⟩ ≤ GL_{4}(𝔽_{2}).

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 m_{f_{1},... ,f_{n}}, see p. 3) 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 (f_{1}, …, f_{i−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 Gröbner 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 “Gröbner factorization algorithm”, which is likely to produce the associated primes since in our case (f_{1}, …, f_{i−1}) has no embedded primes. This algorithm is implemented as the function gsolve in Maple (Char et al., 1990). Since this algorithm always yields ideals which lie between (f_{1}, …, f_{i−1}) and an associated prime, the condition that r_{k}(α_{1}, …, α_{m_{d}}) ≠ 0 ∀k in Algorithm 3 will be necessary for the existence of another primary invariant of degree d, if the P_{k} are the output of the Gröbner factorization algorithm. After having found a candidate f_{i} = I_{d}(α_{1}, …, α_{m_{d}}), we have to check if this really decreases the dimension of the ideal by a Gröbner basis method which is much simpler than the calculation of associated primes (see Becker and Weispfenning, 1993, Table 9.6). Only if f_{i} does not qualify, we will have to calculate the associated primes rigorously.

In the non-modular case, i.e., if char(K) ∤ |G|, we have Molien’s formula to calculate the Poincaré 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 (1.1) 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).

2. Calculating Secondary Invariants

The next task is to find a system of generators of I as a module over A = K[f_{1}, …, f_{n}], where f_{1}, …, f_{n} 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 g_{1}, …, g_{r} for a subgroup H ≤ G with char(K) ∤ |H| (the trivial group will always do) first. Imposing G-invariance conditions on a general element of K[x_{1}, …, x_{n}]^{H} will then lead to a system of linear equations with coefficients in K[x_{1}, …, x_{n}], for which we have to calculate the solutions whose components lie in A. So we need an algorithm which intersects a submodule of K[x_{1}, …, x_{n}]^{r} with A^{r}, where R^{r} denotes the free module of rank r over a ring R.

Lemma 6. Let R = K[x_{1}, …, x_{n}], M = ∑_{i=1}^{s} R · b_{i} ≤ R^{r} a submodule and A = K[f_{1}, …, f_{k}] ≤ R the subalgebra generated by elements f_{1}, …, f_{k} ∈ R. Take the polynomial rings S = K[x_{1}, …, x_{n}, t_{1}, …, t_{k}] and T = K[t_{1}, …, t_{k}] with further indeterminates t_{1}, …, t_{k} and form

M̃ = ∑_{i=1}^{s} S · b_{i} + ∑_{j=1}^{k}(t_{j} − f_{j}) · S^{r} ≤ S^{r}

and

M̃_{T} = M̃ ∩ T^{r}.

Then with Φ: T^{r} → A^{r}, t_{j} ↦ f_{j} we have

Φ(M̃_{T}) = M ∩ A^{r}.

Proof. We consider the homomorphism Ψ: S^{r} → R^{r}/ M obtained by substituting t_{j} ↦ f_{j}, and claim ker(Ψ) = M̃.

By construction, M̃ clearly lies in the kernel. Let P (t_{1}, …, t_{k}) be a product of t_{j}’s and e_{i} ∈ S^{r} a vector of the standard basis. We show by induction on deg(P) that (P (t_{1}, …, t_{k}) − P (f_{1}, …, f_{k})) · e_{i} ∈ M̃. There is nothing to show for deg(P) = 0. If deg(P) > 0, then P (t_{1}, …, t_{k}) = t_{j} · P′(t_{1}, …, t_{k}) with deg(P′) < deg(P), so by induction (P′(t_{1}, …, t_{k}) − P′(f_{1}, …, f_{k})) · e_{i} ∈ M̃, and we conclude

(P (t_{1}, …, t_{k}) − P (f_{1}, …, f_{k})) · e_{i} =
((t_{j} − f_{j}) · P′(t_{1}, …, t_{k}) + f_{j} · (P′(t_{1}, …, t_{k}) − P′(f_{1}, …, f_{k})) ) · e_{i} ∈ M̃.

It follows that for (g_{1}, …, g_{r}) ∈ S^{r} we have

(g_{1}, …, g_{r}) − (g_{1} |_{t_{j}=f_{j}} , …, g_{r} |_{t_{j}=f_{j}}) ∈ M̃.

Now if Ψ(g_{1}, …, g_{r}) = 0, then (g_{1} |_{t_{j}=f_{j}} , …, g_{r} |_{t_{j}=f_{j}}) ∈ M ⊂ M̃ and hence (g_{1}, …, g_{r}) ∈ M̃, which proves the claim.

It follows that ker(Ψ |_{T^{r}} ) = M̃_{T}, and due to

Ψ(T^{r}) = (A^{r} + M)/ M ≅ A^{r} /(M ∩ A^{r})

we have

T^{r} /M̃_{T} ≅ A^{r} /(M ∩ A^{r})

with an isomorphism induced by Φ. This completes the proof.

We obtain the following algorithm:

Algorithm 7 (Intersecting a module with a subalgebra). Input: Generators b_{1}, …, b_{s} of a submodule M ≤ R^{r}, where R = K[x_{1}, …, x_{n}], and

generators f_{1}, …, f_{k} of a subalgebra A = K[f_{1}, …, f_{k}] ≤ R. Output: Generators c_{1}, …, c_{m} ∈ A^{r} of M ∩ A^{r} as a module over A. Begin

Take additional indeterminates t_{1}, …, t_{k} and set S = K[x_{1}, …, x_{n}, t_{1}, …, t_{k}]. Form the submodule M̃ ≤ S^{r} generated by the b_{i} (i = 1, …, s) and by the

(t_{j} − f_{j}) · e_{i} (j = 1, …, k, i = 1, …, r).

Calculate a Gröbner basis B of M̃ w.r.t. a term order ≺ with the property that

every x_{i} is greater than any monomial in the t_{j}. (See below for Gröbner bases of modules.) Form B ∩ (K[t_{1}, …, t_{k}])^{r} and substitute t_{j} ↦ f_{j} in its elements. Let {c_{1}, …, c_{m}} be the resulting set.

End.

In the above algorithm we are using Gröbner bases of modules. These were introduced by Möller 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 Gröbner bases, which was used in the algorithm, follows easily and can be found in (loc. cit.).

Also implemented in Macaulay and described by Möller 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 f_{1}, …, f_{k} ∈ I such that I is finitely

generated as a module over A = K[f_{1}, …, f_{k}] (optimally, a system of primary invariants of small degrees).

Output: Secondary invariants g_{1}, …, g_{m}. Begin

Choose a subgroup H ≤ G (for example, H = {ι}) and calculate homogeneous

generators h_{1}, …, h_{r} of K[x_{1}, …, x_{n}]^{H} as a module over A. Calculate the module M ≤ K[x_{1}, …, x_{n}]^{r} of all (p_{1}, …, p_{r}) ∈ K[x_{1}, …, x_{n}]^{r} with

∑_{i=1}^{r} (σ(h_{i}) − h_{i}) · p_{i} = 0 for all σ ∈ S.  (2.1)

(This is the calculation of a syzygy module.)

Calculate M ∩ A^{r} = ∑_{i=1}^{m} A · c_{i} using Algorithm 7.

Set g_{i} := ∑_{j=1}^{r} c_{i,j} · h_{j} (i = 1, …, m), where c_{i} = (c_{i,1}, …, c_{i,r}) ∈ A^{r}. End.

The g_{i} lie in I by construction. Conversely, for f = ∑_{i=1}^{r} a_{i} · h_{i} ∈ I with a_{i} ∈ A we have by Equation (2.1)

(a_{1}, …, a_{r}) ∈ M ∩ A^{r} ⇒ (a_{1}, …, a_{r}) = ∑_{i=1}^{m} ã_{i} · c_{i} with ã_{i} ∈ A ⇒ f = ∑_{i=1}^{m} ã_{i} · g_{i},

hence I = ∑_{i=1}^{m} A · g_{i}, which proves the correctness of Algorithm 8.

Remark 9.

(a) If χ: G → K^{×} is a linear character, then the set M_{χ} of relative invariants of weight χ (i.e., of f ∈ K[x_{1}, …, x_{n}] such that σ(f) = χ(σ) · f ∀σ ∈ G) is a module over I and also over A = K[f_{1}, …, f_{k}] for any invariants f_{i}. If the f_{i} satisfy the assumption made in Algorithm 8, then a generating set for M_{χ} as an A-module can be constructed by Algorithm 8 with χ introduced into Equation (2.1).

(b) Let χ be as above and H ≤ G a p-Sylow subgroup, where p = char(K) ≠ 0. Then χ|_{H} = 1. Taking left coset representatives σ_{1}, …, σ_{r} of G/H, we have a “relative Reynolds operator” given by

π_{χ}^{G/H} = \frac{1}{(G : H)} ∑_{i=1}^{r} χ(σ_{i}^{−1}) · σ_{i}

(see Campbell et al., 1991 or Smith, 1995, p. 28). This is a projection K[x_{1}, …, x_{n}]^{H} ↠ M_{χ} of I-modules. Hence we only have to calculate generators for K[x_{1}, …, x_{n}]^{H} as a module over A and then apply π_{χ}^{G/H} to obtain M_{χ}.

K[x_{1}, …, x_{n}]^{H} itself is most conveniently calculated by applying Algorithm 8 recursively to a chain of subgroups H_{1} ≤ H_{2} ≤ … ≤ H_{s} = H with |H_{i}| = p^{i}.

(c) The set {g_{1}, …, g_{m}} from Algorithm 8 can be made into a minimal set of secondary invariants (w.r.t. the primary invariants f_{1}, …, f_{n} chosen once and for all) by successively deleting those g_{i} 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 = Z_{4} with its regular representation over K = 𝔽_{2}. We find primary invariants

f_{1} = x_{1} + x_{2} + x_{3} + x_{4}, f_{2} = x_{1}x_{3} + x_{2}x_{4},
f_{3} = x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{4} + x_{4}x_{1}, f_{4} = x_{1}x_{2}x_{3}x_{4}.

With H = {ι} ≤ G, there are 16 generators h_{1}, …, h_{16} of K[x_{1}, …, x_{n}] as a module over A = K[f_{1}, …, f_{4}], and the solution module M of Equation (2.1) is generated by 16 vectors. The module M̃ in Algorithm 7 then has 80 generators and a Gröbner basis of 113 elements, which was calculated by Macaulay in a few seconds. From these, five lie in K[t_{1}, …, t_{4}], and we get the secondary invariants

g_{1} = 1, g_{2} = x_{1}^{3} + x_{2}^{3} + x_{3}^{3} + x_{4}^{3}, g_{3} = x_{1}^{2}x_{2} + x_{2}^{2}x_{3} + x_{3}^{2}x_{4} + x_{4}^{2}x_{1},
g_{4} = x_{1}^{3}x_{2} + x_{2}^{3}x_{3} + x_{3}^{3}x_{4} + x_{4}^{3}x_{1}, g_{5} = x_{1}^{3}x_{2}^{2} + x_{2}^{3}x_{3}^{2} + x_{3}^{3}x_{4}^{2} + x_{4}^{3}x_{1}^{2}.

The calculation can be essentially accelerated by taking H = Z_{2} ≤ G and applying Algorithm 8 recursively. We shall continue this example on p. 11 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 Gröbner 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.

3. Applications

This section is concerned with calculating some data associated with invariant rings, in particular free resolutions, Poincaré series and depth. Easy algorithms are presented to decide the Cohen-Macaulay property and the property of being isomorphic to a polynomial algebra.

3.1. Depth and the Cohen-Macaulay Property

Suppose that we have calculated primary invariants f_{1}, …, f_{n} and secondary invariants g_{1}, …, g_{m} by Algorithm 3 and 8. Then we can compute the module M ≤ K[x_{1}, …, x_{n}]^{m} of syzygies between the g_{i} and intersect M with A^{m} (where as above A = K[f_{1}, …, f_{n}]) using Algorithm 7. We can further calculate a free resolution

⋯ → F_{2} → F_{1} → M ∩ A^{m} → 0

of M ∩ A^{m} as an A-module, which yields a free resolution

⋯ → F_{2} → F_{1} → F_{0} → I → 0  (3.1)

of the invariant ring I. From this the Poincaré series of I can easily be obtained (see Stanley, 1979, p. 496).

If {g_{1}, …, g_{m}} is a minimal set of secondary invariants and if (3.1) is a minimal free resolution then the homological dimension hdim_{A}(I) is (by definition) the largest i such that F_{i} ≠ 0. By the Auslander-Buchsbaum formula (see Auslander and Buchsbaum, 1957 or Eisenbud, 1995, Theorem 19.9) we have

depth_{A}(I) = n − hdim_{A}(I),

where depth_{A}(I) is the maximal length of a regular sequence for I whose elements lie in A and have no constant term. But depth(I) := depth_{I}(I) is the same as depth_{A}(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 A_{0} = B_{0} = K and assume that B is finitely generated as an A-module. Then

depth_{A}(B) = depth_{B}(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 f_{1}, …, f_{n} ∈ I be a system of primary invariants of degrees d_{1}, …, d_{n}. Then the invariant ring I is Cohen-Macaulay if and only if it can be generated by (∏_{i=1}^{n} d_{i})/ |G| secondary invariants as a module over A = K[f_{1}, …, f_{n}]. Otherwise, more secondary invariants are necessary.

Proof. The invariant ring K[x_{1}, …, x_{n}] of the trivial group is Cohen-Macaulay and generated by ∏_{i=1}^{n} d_{i} polynomials over A (see, for example, Sturmfels, 1993, Prop. 2.3.6). Hence [K(x_{1}, …, x_{n}) : K(f_{1}, …, f_{n})] = ∏_{i=1}^{n} d_{i}, and by Galois theory

[K(x_{1}, …, x_{n})^{G} : K(f_{1}, …, f_{n})] = (∏_{i=1}^{n}d_{i})/ |G| =: d.

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

There is not very much known about the question of Cohen-Macaulayness of modular invariant rings. Good references are Smith (1995) and Landweber and Stong (1987). A remarkable result of Ellingsrud and Skjelbred (1980) states that if G is a cyclic p-group with p = char(K) ≠ 0 then depth(I) = min(n, n − m + 2), where m is the dimension of the K-vector space generated by all σ(x_{i}) − x_{i} (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 = Z_{4} with its regular representation over K = 𝔽_{2}. In Example 10 we calculated primary and secondary invariants for I and saw that 5 secondary invariants were needed. Hence by Proposition 12, I is not Cohen-Macaulay, since (∏_{i=1}^{4} deg(f_{i}))/ |G| = 4. Calculating syzygies between the secondary invariants and applying Algorithm 7, we obtain the following relation:

f_{1} (f_{1}^{4} + f_{2}^{2} + f_{2}f_{3}) · g_{1} + (f_{1}^{2} + f_{2} + f_{3}) · g_{2} + (f_{1}^{2} + f_{2}) · g_{3} + f_{1} · g_{4} = 0,

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

P (I, t) = \frac{t^{5} + t^{4} + 2t^{3} + 1}{(1 − t)(1 − t^{2})^{2}(1 − t^{4})} − \frac{t^{5}}{(1 − t)(1 − t^{2})^{2}(1 − t^{4})} = \frac{t^{4} + 2t^{3} + 1}{(1 − t)(1 − t^{2})^{2}(1 − t^{4})},

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 Poincaré series does not depend on the ground field (see, for example, Smith, 1995, Prop. 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 ≠ p be the characteristic of K and let H ≤ G be a p-Sylow subgroup of G. Then we have

depth(K[x_{1}, …, x_{n}]^{G}) ≥ depth(K[x_{1}, …, x_{n}]^{H}).

Proof. By Proposition 11, depth(K[x_{1}, …, x_{n}]^{H}) = depth_{K[x_{1},,... ,,x_{n}]^{G}}(K[x_{1}, …, x_{n}]^{H}). Hence there exists a maximal regular sequence f_{1}, …, f_{m} for K[x_{1}, …, x_{n}]^{H} such that all f_{i} ∈ I = K[x_{1}, …, x_{n}]^{G}. We claim that f_{1}, …, f_{m} is regular for I, too, so we have to prove that f_{i} is not a zero divisor in I/(f_{1}, …, f_{i−1}) for i = 1, …, m.

Let g_{i}f_{i} = g_{1}f_{1} + ⋯ + g_{i−1}f_{i−1} with g_{1}, …, g_{i} ∈ I. Since f_{1}, …, f_{m} is regular for K[x_{1}, …, x_{n}]^{H}, there are h_{1}, …, h_{i−1} ∈ K[x_{1}, …, x_{n}]^{H} such that g_{i} = h_{1}f_{1} + ⋯ + h_{i−1}f_{i−1}. Applying the relative Reynolds operator π^{G/H}: K[x_{1}, …, x_{n}]^{H} → I (see Remark 9(b)) to this equation yields

g_{i} = π^{G/H}(h_{1}) · f_{1} + ⋯ + π^{G/H}(h_{i−1}) · f_{i−1},

hence g_{i} is zero in I/(f_{1}, …, f_{i−1}).

This proposition implies that if K[x_{1}, …, x_{n}]^{H} is Cohen-Macaulay (with the notation from the proposition), then so is K[x_{1}, …, x_{n}]^{G}. This result already appeared in Campbell et al. (1991) and Smith (1995), Prop. 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 = Z_{p}, p a prime, over a field K of characteristic p. H occurs as p-Sylow subgroup of the symmetric group G = S_{p} whose invariant ring is polynomial and in particular Cohen-Macaulay. But if p > 3, then K[x_{1}, …, x_{p}]^{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.

Proposition 15. Let G act on the first k and last n − k variables, i.e., let ∑_{i=1}^{k} Kx_{i} and ∑_{i=k+1}^{n} Kx_{i} be G-stable vector spaces. Then if K[x_{1}, …, x_{n}]^{G} is Cohen-Macaulay, so are K[x_{1}, …, x_{k}]^{G} and K[x_{k+1}, …, x_{n}]^{G}.

Proof. Let f_{1}, …, f_{k} and f_{k+1}, …, f_{n} be primary invariants for K[x_{1}, …, x_{k}]^{G} and K[x_{k+1}, …, x_{n}]^{G}, respectively. Then f_{1}, …, f_{n} are primary invariants for I = K[x_{1}, …, x_{n}]^{G}. By assumption, all minimal systems of homogeneous generators for I as a module over A := K[f_{1}, …, f_{n}] are linearly independent over A since they all have the same cardinality dim_{K}(K ⊗_{A} I) (see p. 3).

Let g_{1}, …, g_{r} be a minimal system of homogeneous generators for K[x_{1}, …, x_{k}]^{G} over K[f_{1}, …, f_{k}] and choose homogeneous g_{r+1}, …, g_{m} ∈ I which together with g_{1}, …, g_{r} 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 g_{1} satisfies a relation

g_{1} = ∑_{i=2}^{m} h_{i}g_{i}  (3.2)

with h_{i} ∈ A, where we can assume all h_{i} to be homogeneous and deg(g_{i}) < deg(g_{1}) for those i > r with h_{i} ≠ 0 by the minimal choice of m. Setting x_{k+1}, …, x_{n} = 0 in (3.2) yields

g_{1} = ∑_{i=2}^{r} \overline{h_{i}}g_{i} + ∑_{i=r+1}^{m} \overline{h_{i}}\overline{g_{i}}

with \overline{h_{i}} ∈ K[f_{1}, …, f_{k}] and \overline{g_{i}} ∈ K[x_{1}, …, x_{k}]^{G}, so g_{1} is a K[f_{1}, …, f_{k}]-linear combination of g_{2}, …, g_{r} and other invariants in K[x_{1}, …, x_{k}]^{G} of strictly lower degree than g_{1}, which is a contradiction to the minimal choice of g_{1}, …, g_{r}.

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

( \frac{1 1 1 1 1}{1 1 1} ) ∈ GL_{5}(𝔽_{p}).

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.

3.2. 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 general and special linear groups GL_{n}(𝔽_{q}) and SL_{n}(𝔽_{q}) (see Dickson, 1911, or Wilkerson, 1983), the group U ≤ GL_{n}(q) of unipotent upper triangular matrices (Wilkerson, 1983), the orthogonal and unitary groups O_{n}(q) and U_{n}(q^{2}) for n ≤ 3 and n ≤ 2, respectively (Nakajima, 1979, Kemper, 1994), and the complex reflection groups G_{29} and G_{31} of Shephard and Todd (1954) reduced modulo 5 (Xu, 1994). Counter examples are the Weyl group W (F_{4}) of the root system F_{4} (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 19 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) ∤ |G| is Proposition 7.4.2 of Smith (1995).

Proposition 16. Let f_{1}, …, f_{n} ∈ I be homogeneous invariants of degrees d_{1}, …, d_{n}. Then the following statements are equivalent:

(a) I = K[f_{1}, …, f_{n}],

(b) The f_{i} are algebraically independent over K and ∏_{i=1}^{n} d_{i} = |G|.

(c) The Jacobian determinant 𝔍 = det(∂f_{i}/∂x_{j}) is non-zero and ∏_{i=1}^{n} d_{i} = |G|.

Remark. This proposition appeared in Xu (1994), Prop. 2.1. In the proof, Xu concludes directly from the equality of the transcendence degrees that K[x_{1}, …, x_{n}] is integral over K[f_{1}, …, f_{n}]. This conclusion is by no means clear and in fact false in general (take n = 2 and f_{1} = x_{1}, f_{2} = x_{1}x_{2}). 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), Prop. 5.5.5, to show that the system f_{1} = … = f_{n} = 0 has no projective zero in ℙ^{n−1}(K̄), where K̄ is an algebraic closure of K.

We take additional indeterminates t_{1}, …, t_{n} and x_{0} and an algebraic closure K̃ of K(t_{1}, …, t_{n}) which contains K̄. By Bézout’s theorem (see Fulton, 1984, Example 12.3.7, where there is no assumption made on the dimension of the zero manifold), the projective algebraic set V ⊂ ℙ^{n}(K̃) given by

f_{1}(x_{1}, …, x_{n}) − t_{1}x_{0}^{d_{1}} = … = f_{n}(x_{1}, …, x_{n}) − t_{n}x_{0}^{d_{n}} = 0  (3.3)

has at most ∏_{i=1}^{n} d_{i} = |G| irreducible components. So we have to show that there are at least |G| components of V with x_{0} ≠ 0.

K(x_{1}, …, x_{n}) is a finite extension of K(f_{1}, …, f_{n}), so each x_{i} has a minimal polynomial over K(f_{1}, …, f_{n}): g_{i}(x_{i}, f_{1}, …, f_{n}) = 0. If (ξ_{1}, …, ξ_{n}) ∈ K̃^{n} is a solution of f_{1}(x_{1}, …, x_{n}) − t_{1} = … = f_{n}(x_{1}, …, x_{n}) − t_{n} = 0, then g_{i}(ξ_{i}, t_{1}, …, t_{n}) = 0, hence there are at most finitely many such solutions, and each constitutes a component of V with x_{0} ≠ 0. We shall complete the proof by giving |G| such solutions.

Via the homomorphism K(f_{1}, …, f_{n}) → K(t_{1}, …, t_{n}), f_{i} ↦ t_{i}, form

L = K(t_{1}, …, t_{n}) ⊗_{K(f_{1},... ,f_{n})} K(x_{1}, …, x_{n}),

which is a finite field extension of K(t_{1}, …, t_{n}) and can be assumed to lie in K̃. Take a σ ∈ G and ξ_{i} := 1 ⊗ σ(x_{i}) ∈ K̃, then

f_{i}(ξ_{1}, …, ξ_{n}) = 1 ⊗ f_{i}(σ(x_{1}), …, σ(x_{n})) = 1 ⊗ σ(f_{i}) = 1 ⊗ f_{i} = t_{i} ⊗ 1.

Hence the elements of G give rise to |G| distinct solutions of Equation (3.3) with x_{0} ≠ 0, and “(b)⇒(a)” is proved.

Finally, 𝔍 ≠ 0 if and only if K(x_{1}, …, x_{n}) is a finite separable field extension of K(f_{1}, …, f_{n}) (see, for example, Benson, 1993, Prop. 5.4.2). Hence (a) and (b) imply (c) since K(x_{1}, …, x_{n}) is Galois over the field of fractions Quot(I) with group G, and (c) implies (b).

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 17 (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 f_{1}, …, f_{n} 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 {b_{1}, …, b_{m_{d}}} of this basis which is linearly independent

modulo the homogeneous degree-d part R_{d} of R. If m_{d} > 0 And d ∤ |G| Then Return(False). For j = 1, …, m_{d} set f_{i+j} := b_{j}. Set m := m · d^{m_{d}}, i := i + m_{d} and R := K[f_{1}, …, f_{i}]. 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(∂f_{i}/∂x_{j}).

If 𝔍 = 0 Then Return(False)

Else Return(f_{1}, …, f_{n}). End.

We finish with two examples.

Example 18. The complex reflection group G_{23} from Shephard and Todd (1954) is isomorphic to {±1} × A_{5}. It is generated by the matrices

(\begin{matrix}−1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix}) , (\begin{matrix}0 & 0 & −1 \\ −1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) and (\begin{matrix}1 & −1 & α \\ 0 & α & −α \\ 0 & α & −1 \end{matrix})

with α = (1+\sqrt{5})/2. Algorithm 17 yields invariants f_{2}, f_{6} and f_{10} of degrees 2, 6 and 10, respectively, whose Jacobian determinant does not vanish. The degrees of the f_{i} 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 = 𝔽_{9} or 𝔽_{5}, respectively, which remains isomorphic to {±1} × A_{5}. Reducing the Jacobian determinant yields non-zero polynomials, hence I is a polynomial ring.

Reducing modulo the maximal ideal containing p = 2 (which sends α to a root of X^{2} + X + 1) yields a group G ≤ GL_{3}(𝔽_{4}) which is isomorphic to A_{5}. Here we find an additional invariant f_{5} of degree 5. The Jacobian determinant of f_{2}, f_{5} and f_{6} does not vanish modulo 2, hence these three invariants generate I.

Similarly, we can take G_{27} ≅ {±1} × Ã_{6}, where Ã_{6} is a non-split group extension of A_{6} with kernel Z_{3}, and reduce this modulo a maximal ideal containing 2 to obtain a group G ≤ GL_{3}(𝔽_{4}) which is isomorphic to Ã_{6}. Here Algorithm 17 produces invariants 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 19. The group G of all matrices

(\begin{matrix}1 & 0 & a & b \\ 0 & 1 & b & c \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) ∈ GL_{4}(𝔽_{q})

with a, b, c ∈ K := 𝔽_{q} is a reflection group of order q^{3}. We assume that I is a polynomial ring. There are two invariants of degree 1, namely f_{1} = x_{1} and f_{2} = x_{2}. If f_{3} and f_{4} are the two remaining generators with deg(f_{3}) ≤ deg(f_{4}), then deg(f_{4}) > 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 g_{1} · f_{3} + g_{2} with g_{1}, g_{2} ∈ K[x_{1}, x_{2}]. But it is easily verified that

x_{1}^{q}x_{3} − x_{1}x_{3}^{q} + x_{2}^{q}x_{4} − x_{2}x_{4}^{q}

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. ◁

References

Adem, A., Milgram, R. J. (1994). Cohomology of Finite Groups. Berlin, Heidelberg, New York: Springer-

Verlag. 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). Gröbner 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 caractéristique p. Comptes Rendus

Acad. Sci. Paris (Série 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. 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 caractéristique p. Compos.

Math. 41, 233–244. Fulton, W. (1984). Intersection Theory. Berlin, Heidelberg, New York: Springer-Verlag. Gianni, P., Trager, B., Zacharias, G. (1988). Gröbner Bases and Primary Decomposition of Polynomial

Ideals. J. of Symbolic Computation 6, 149–267. Göbel, M. (1995). Computing Bases for Rings of Permutation-invariant Polynomials. J. Symbolic Com-

putation 19, 285–291. Hilbert, D. (1893). Über die vollen Invariantensysteme. Math. Ann. 42, 313–370. Kemper, G. (1993). The Invar Package for Calculating Rings of Invariants. IWR Preprint 93-34. Heidel-

berg. Kemper, G. (1994). Das Noethersche Problem und generische Polynome. Dissertation. University of Hei-

delberg. 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. Dissertation. University

of Arizona. Möller, 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 Character-

istic. 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. Göttingen 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 Nineteenth-

Century 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 BF_{4} of the Exceptional Group F_{4}. 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.