IWR-Preprint 97-45, Heidelberg 1997

Hilbert Series and Degree Bounds in Invariant Theory

Gregor Kemper

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

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

November 20, 1997

Abstract

The Hilbert series and degree bounds play significant roles in computational invariant theory. In the modular case, neither of these tools is available in general. In this article three results are obtained, which provide partial remedies for these shortcomings. First, it is shown that the so-called extended Hilbert series, which can always be calculated by a Molien type formula, yields strong constraints on the degrees of primary invariants. Then it is shown that for a trivial source module the (ordinary) Hilbert series coincides with that of a lift to characteristic 0 and can hence be calculated by Molien’s formula. The last result is a generalization of Göbel’s degree bound to the case of monomial representations.

Introduction

In order to study structural properties of an invariant ring k[V]^{G} (see Kemper and Malle [13]), it is usually necessary to calculate generating invariants for k[V]^{G}. For finite groups G, there exists quite an elaborate machinery for this purpose. References can be found in Sturmfels [22], Kemper [9,10], and Kemper and Steel [14]. However, when the examples become larger, the difficulty of the calculations increases quickly, and it becomes essential that any additional information on the invariant ring be exploited. Let us consider the situation somewhat closer. By Noether’s normalization lemma, the problem is split into two parts, which are of roughly equal difficulty. First one has to calculate a system of homogeneous parameters (in the sequel abbreviated by hsop), which by definition is a system f_{1}, …, f_{n} of homogeneous invariants, algebraically independent over the ground field k, with the property that the invariant ring k[V]^{G} is finitely generated as a module over the subalgebra k[f_{1}, …, f_{n}] generated by the hsop. The members of an hsop are often called primary invariants. After an hsop has been found, the second task is to calculate the (finitely many) generators of k[V]^{G} as a module over k[f_{1}, …, f_{n}]. Such generators are usually called secondary invariants. Together with the primary invariants, they form the desired system of generators for the invariant ring (as a k-algebra).

Now the knowledge of the Hilbert series H(k[V]^{G}, t) (for the definition, see Section 1) can facilitate the computations considerably at various points:

• If f_{1}, …, f_{n} is an hsop of k[V]^{G} with degrees d_{i} = deg(f_{i}), then H(k[V]^{G}, t) can be written in the form

H(k[V]^{G}, t) = \frac{f(t)}{(1 − t^{d_{1}}) ⋯ (1 − t^{d_{n}})}  (1)

with f(t) ∈ ℤ[t] a polynomial with integer coefficients. This imposes a strong constraint on the degree vectors (d_{1}, …, d_{n}) which can occur as the degrees of an hsop. Since it is very important for the calculation of secondary invariants that the d_{i} be chosen as small as possible (see [10]), constraints of this kind are crucial for a smooth operation of the algorithm from [10].

• If k[V]^{G} is Cohen-Macaulay (see [13]), then the polynomial f(t) in Equation (1) is of the form

f(t) = t^{e_{1}} + ⋯ + t^{e_{m}},

where the e_{i} are the degrees of the secondary invariants. This reduces the task of finding secondary invariants to pure linear algebra (see [14]). Even if k[V]^{G} is not Cohen-Macaulay, a Hilbert-driven approach can speed up the usual algorithm for the calculation of secondary invariants given in [9] or [14].

• With the knowledge of H(k[V]^{G}, t), one can get a very simple algorithm for the calculation of k[V]^{G}, which does not use primary and secondary invariants. In fact, for a tentative set g_{1}, …, g_{r} of generators one can calculate the algebraic relations between the g_{i} by [22, Subroutine 2.5.3] and use the algorithm given by Bayer and Stillman [1] to calculate the Hilbert series of the subalgebra k[g_{1}, …, g_{r}]. If this Hilbert series coincides with H(k[V]^{G}, t), then g_{1}, …, g_{r} generate k[V]^{G}. Now the tentative generators can be taken to be k-bases for the vector spaces of all invariants of degree ≤ d, for increasing values of d.

In the non-modular case (i.e., when the characteristic of k is not a divisor of the order of G), the Hilbert series can be calculated without touching a single invariant by Molien’s formula (see (2) on the facing page). Unfortunately, this formula is not available in the modular case. In Section 1, a variant of the Hilbert series is considered, and it is shown how it can be calculated by a generalization of Molien’s formula. Then it is shown that Equation (1) carries over for this variant of the Hilbert series, which demonstrates its significance. The kG-module V is called a trivial source module if it is a direct summand of a permutation module. In Section 2, we show that in this case another generalization of Molien’s formula yields the ordinary Hilbert series.

If char(k) > |G|, the famous “Noether bound” says that k[V]^{G} can be generated by invariants of degrees ≤ |G| (Noether [16]). In the modular case, no general bound on the generators is known and it has in fact been shown that there exists no bound which only depends on the group order (see Richman [17] or Kemper [12]). Degree bounds are very important for computational purposes, since taking a k-basis for the vector space of invariants of degree smaller than or equal to such a bound yields generators for k[V]^{G}, which can then be minimalized by linear algebra. In the case that V is a permutation module, Göbel [6] was able to prove that the invariant ring has generators of degrees bounded by max{n, n(n − 1)/2}, where n = dim_{k}(V). In Section 3, we give a generalization to the case that V is a monomial representation. We also obtain a slight simplification of Göbel’s arguments and of his rewriting algorithm.

I thank Jürgen Müller for suggesting to me that one could generalize Molien’s formula to the case of trivial source modules.

1 Molien’s formula in modular invariant theory

Let G be a finite group and k an algebraically closed field. Consider a linear representation of G, i.e., a finite dimensional kG-module V. Then G also acts on the symmetric powers of V, so the symmetric algebra S(V) becomes a kG-module. We write k[V] for the symmetric algebra, which is a polynomial ring k[x_{1}, …, x_{n}] in the vectors x_{1}, …, x_{n} of a basis of V. We are interested in the invariant ring k[V]^{G} consisting of all polynomials invariant under the action of G. More generally, if S is a simple kG-module, we can consider the largest submodule of k[V] consisting of a direct sum of copies of S. We denote this subspace by k[V]^{S}. By Smith [19, p. 7] or Benson [3, p. 2], k[V]^{S} is isomorphic to S ⊗_{End_{kG}(S)} Hom_{kG}(S, k[V]). If S is of dimension 1, then k[V]^{S} consists of the relative invariants with respect to a linear character, and k[V]^{G} = k[V]^{S} with S the trivial module. Clearly k[V]^{S} is a graded vector space, i.e., it is the direct sum of its graded components k[V]_{d}^{S} consisting of the polynomials in k[V]^{S} of total degree d, and 0. The Hilbert series of k[V]^{S} is the formal power series

H(k[V]^{S}, t) = ∑_{d≥0} dim_{k}(k[V]_{d}^{S}) · t^{d}.

If the characteristic of k is zero, then the Hilbert series can easily be calculated by Molien’s celebrated formula

H(k[V]^{S}, t) = \frac{dim_{k}(S)}{|G|} ∑_{σ∈G} \frac{χ_{S}(σ^{−1})}{det_{V} (1 − tσ)},  (2)

where χ_{S}(σ) = Tr_{S}(σ) is the character associated to S and det_{V} is the determinant of the action on V (see Benson [3, Theorem 2.5.3]; the slight difference between our formula and Benson’s arises from the fact that by Benson’s convention k[V] = S(V^{∗})). Molien’s formula can easily be generalized to the case that the characteristic of k is positive but not a divisor of |G| by taking Brauer lifts. This case will be covered by the considerations below as well as by Section 2.

Definition 1. Let G, k and S be as above. For a graded kG-module M = ⊕_{d≥0}M_{d} with finite dimensional graded parts M_{d}, write m_{M_{d}}^{S} ∈ ℕ_{0} for the number of occurrences of S in a composition series of M_{d} as a kG-module. Then the formal power series

H̃(M, S, t) = dim_{k}(S) · ∑_{d≥0} m_{M_{d}}^{S} · t^{d}

will be called the extended Hilbert series of M with respect to S.

We are interested in H̃(k[V], S, t), which is much more accessible to calculations than the ordinary Hilbert series. If char(k) ∤ |G|, then both types of Hilbert series coincide, but in general they differ. We will first explain how H̃(k[V], S, t) can be calculated, and then why it is a useful function.

Let χ: G → ℂ be the projective indecomposable character of S (see Goldschmidt [7, p. 6-11]), which is the Brauer character of the projective cover (principle indecomposable module) of S. This can easily be calculated if the Brauer character table X of G is known, the main step being an inversion of X. Furthermore, let ψ: G_{p′} → ℂ be the Brauer character associated to the module V (see, for example, Curtis and Reiner [5, p. 420]). Here G_{p′} is the set of all elements of G whose order is not divisible by the characteristic p of k. Note that ψ depends on the choice of a primitive m-th root of unity in ℂ and in k, where exp(G) = m · p^{a} with p ∤ m. We now Brauer-lift the expression det_{V} (1 − tσ) in Molien’s formula (2) as follows: let s_{i} ∈ ℂ[X_{1}, …, X_{n}] be the i-th elementary symmetric polynomial and p_{i} = ∑_{j=1}^{n} X_{j}^{i} the i-th power sum, where n = dim_{k}(V). By Newton’s formulae (see Curtis and Reiner [5, p. 314]), the s_{i} can be expressed as polynomials in the p_{i}, so there is a polynomial F ∈ ℂ[X_{1}, …, X_{n}; t] such that

∏_{i=1}^{n}(1 − tX_{i}) = F (p_{1}, …, p_{n}; t).

Then for σ ∈ G_{p′}, F (ψ(σ), ψ(σ^{2}), …, ψ(σ^{n}); t) will be the Brauer lift of det_{V} (1 − tσ). The following theorem, which was already hinted at by Benson [3, p. 22] and Smith [20, p. 218], gives the appropriate generalization of Molien’s formula to the modular case.

Theorem 2. With the above notation, we have

H̃(k[V], S, t) = \frac{dim_{k}(S)}{|G|} · ∑_{σ∈G_{p′}} \frac{χ(σ^{−1})}{F (ψ(σ), ψ(σ^{2}), …, ψ(σ^{n}); t)}.

Proof. If S′ is a simple kG-module with Brauer character φ_{S′}, then 1/|G| · ∑_{σ∈G_{p′}} φ_{S′} (σ) · χ(σ^{−1}) is 1 if S ≅ S′ and 0 otherwise by Goldschmidt [7, Theorem 6.10]. If W is any finite dimensional kGmodule and S_{1}, …, S_{r} the composition factors (with repetitions) of W, then the Brauer character associated to W is φ_{W} = ∑_{i=1}^{r} φ_{S_{i}}. Hence the multiplicity of S as a composition factor of W is

m_{W}^{S} = \frac{1}{|G|} · ∑_{σ∈G_{p′}} φ_{W} (σ) · χ(σ^{−1}).

We obtain the formula

H̃(k[V], S, t) = \frac{dim_{k}(S)}{|G|} · ∑_{σ∈G_{p′}} (∑_{d≥0} φ_{k[V ]_{d}} (σ) · t^{d}) χ(σ^{−1}).

Hence we must show that for σ ∈ G_{p′} the identity ∑_{d≥0} φ_{k[V ]_{d}} (σ) · t^{d} = 1/F (ψ(σ), ψ(σ^{2}), …, ψ(σ^{n}); t) holds. Indeed, if ψ(σ) = λ_{1} + ⋯ + λ_{n} with m-th roots of unity λ_{i}, we have

F (ψ(σ), ψ(σ^{2}), …, ψ(σ^{n}); t) = ∏_{i=1}^{n}(1 − tλ_{i}),

and

φ_{k[V ]_{d}} (σ) = ∑_{\begin{matrix}j_{1},... ,j_{n}≥0, \\ j_{1}+···+j_{n}=d \end{matrix}} ∏_{i=1}^{n} λ_{i}^{j_{i}},

hence

∑_{d≥0} φ_{k[V ]_{d}} (σ) · t^{d} = ∑_{j_{1},... ,j_{n}≥0} ∏_{i=1}^{n}(λ_{i}t)^{j_{i}} = ∏_{i=1}^{n} ∑_{j≥0}(λ_{i}t)^{j} = ∏_{i=1}^{n}(1 − tλ_{i})^{−1}.

We now explain how the knowledge of the extended Hilbert series can be used for computational purposes. The following theorem poses severe restrictions on the degrees that can occur in an hsop of k[V]^{G}. It is well known (and often used) for the ordinary Hilbert series (see the Introduction).

Theorem 3. Suppose that f_{1}, …, f_{n} ∈ k[V]^{G} is an hsop of k[V]^{G} with d_{i} = deg(f_{i}) (i = 1, …, n). Then for every simple kG-module S we can write H̃(k[V], S, t) in the form

H̃(k[V], S, t) = \frac{f_{S}(t)}{(1 − t^{d_{1}}) ⋯ (1 − t^{d_{n}})}  (3)

with f_{S}(t) ∈ ℤ[t].

Proof. We write A = k[f_{1}, …, f_{n}]. Then k[V] is a finitely generated graded module over the group ring AG. We will prove the theorem for an arbitrary finitely generated graded AG-module M. Since the graded components M_{d} of M are finitely generated kG-modules, the subspace M^{S} and the extended Hilbert series H̃(M, S, t) are well defined. Now observe that M^{S} is a graded submodule of M over AG. Hence the sum of all M^{S′} for S′ any simple kG-module is also a graded submodule, which we denote by Soc(M). Now we can recursively define M^{(0)} = M and M^{(i+1)} = M^{(i)}/ Soc(M^{(i)}) for i > 0. There are canonical maps M → M^{(i)}, whose kernels we denote by Soc^{i}(M). Then Soc^{1}(M) ⊆ Soc^{2}(M) ⊆ … ⊆ M is an ascending sequence of A-submodules of M, hence by Noetherianity Soc^{N+1}(M) = Soc^{N}(M) for some N, so M^{(i)} = 0 for i > N. We form M̃ = ⨁_{i≥0} M^{(i)}, which is a finitely generated graded AG-module. Now by the definition of the extended Hilbert series we have

H̃(M, S, t) = H(M̃^{S}, t),

where the right hand side is the ordinary Hilbert series ∑_{d≥0} dim_{k}(M̃_{d}^{S}) · t^{d} of M̃^{S}. By Hilbert’s syzygy theorem (see, for example, Smith [19, Theorem 6.3.1]) there exists a finite free resolution of M̃^{S} as an A-module:

0 ⟶ F_{r} ⟶ ⋯ ⟶ F_{1} ⟶ F_{0} ⟶ M̃^{S} ⟶ 0.

With the right choices of degrees for the free generators of the F_{i}, all maps are degree-preserving, and we obtain

H(M̃^{S}, t) = ∑_{i=0}^{r}(−1)^{i}H(F_{i}, t) = ∑_{i=0}^{r}(−1)^{i}f_{i}(t)H(A, t)

with f_{i}(t) ∈ ℕ_{0}[t] encoding the degrees of the generators of F_{i}. But H(A, t) = ∏_{i=1}^{n}(1 − t^{d_{i}})^{−1} (see, for example Sturmfels [22, Lemma 2.2.3]), which completes the proof.

Example 4. We consider two representations of the group A_{5} in characteristic 2. In this example, the Brauer characters are taken from the Atlas of Brauer Characters (Jansen et al. [8]).

(a) First let V be one of the irreducible two-dimensional modules defined over k = 𝔽_{4}. We calculate the extended Hilbert series of k[V] with respect to the trivial module S and obtain

H̃(k[V], S, t) = \frac{t^{14} + t^{12} + t^{10} + 2t^{9} + 2t^{7} + 2t^{5} + t^{4} + t^{2} + 1}{(1 − t^{6})(1 − t^{10})}

as the representation in the form (3) with the smallest possible degrees d_{i}. It turns out that the smallest degrees which are possible for an hsop are d_{1} = 5 and d_{2} = 12 here, which corresponds to the representation

H̃(k[V], S, t) = \frac{t^{15} + t^{13} + t^{11} + t^{10} + t^{9} + t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{2} + 1}{(1 − t^{5})(1 − t^{12})}

with the second-lowest degrees. So it can be said that H̃(k[V], S, t) contains very strong restrictions on the degrees of an hsop.

(b) Now we take V as the irreducible module of dimension 4, which occurs as a submodule of the natural permutation module. Here again the representation of H̃(k[V], S, t) with the second-smallest degrees d_{i} actually gives the degrees of an optimal hsop. It is

H̃(k[V], S, t) = \frac{t^{10} + t^{8} + 3t^{7} + 4t^{6} + 6t^{5} + 4t^{4} + 3t^{3} + t^{2} + 1}{(1 − t^{2})(1 − t^{3})(1 − t^{4})(1 − t^{5})}.  (4)

2 The Hilbert series for trivial source modules

We now turn to the quest for generalizations of Molien’s formula which will yield the ordinary Hilbert function H(k[V]^{G}, t). It has been noted that this is easily possible in the case where G acts as a permutation group on a basis of V, since due to the purely combinatorial description of permutation-invariant polynomials, H(k[V]^{G}, t) does not depend on the ground field k (see Smith [19, Proposition 4.3.4]). We will now generalize this observation to the case where V is a trivial source module. We assume the following situation:

Let V be a trivial source module for G over the finite field k of characteristic p, and let (K, R, k) be a p-modular system (see, for example, Curtis and Reiner [5, p. 402]). By Benson [2, Corollary 3.11.4], V lifts to an RG-module V̂ which is also a trivial source module. Now Ṽ := K ⊗_{R} V̂ is a KG-module. Moreover, let S be a simple trivial source module over kG and define Ŝ and S̃ as above. The most important special case is that S is the trivial module. Then Ṽ and S̃ are defined over K, which is of characteristic 0, so the Hilbert series H(K[Ṽ]S̃, t) can be calculated by Molien’s formula. Now we have:

Theorem 5. In the above situation the formula

H(k[V]^{S}, t) = H(K[Ṽ]S̃, t)

holds. In particular,

H(k[V]^{G}, t) = H(K[Ṽ]^{G}, t).

Proof. By assumption, there exist permutation modules P and P̂ over kG and RG, respectively, such that

P = V ⊕ W and P̂ = V̂ ⊕ Ŵ ,

where W and Ŵ are modules over kG and RG, respectively. For d ∈ ℕ, the d-th symmetric power is

S^{d}(P) ≅ ⨁_{i=0}^{d} (S^{i}(V) ⊗_{k} S^{d−i}(W))

with the corresponding formula for S^{d}(P̂). Hence S^{d}(V) is a trivial source module, and S^{d}(V̂) is a lifting. So by Benson [2, Corollary 3.11.4],

Hom_{RG}(Ŝ, S^{d}(V̂))/𝔭 Hom_{RG}(Ŝ, S^{d}(V̂)) ≅ Hom_{kG}(S, S^{d}(V)),

where 𝔭 ◁ R is the maximal ideal. The dimension of the left hand side is equal to

rank (Hom_{RG}(Ŝ, S^{d}(V̂))) = dim_{K} (K ⊗_{R} Hom_{RG}(Ŝ, S^{d}(V̂)))
= dim_{K} (Hom_{KG}(S̃, S^{d}(Ṽ)))

by Landrock [15, Chapter I, Lemma 14.5]. We conclude

dim_{k} (Hom_{kG}(S, S^{d}(V))) = dim_{K} (Hom_{KG}(S̃, S^{d}(Ṽ))) ,

which completes the proof.

If the order of G is not divisible by p (for which we say that G is a p′-group), then every kGmodule is a trivial source module by the theorem of Maschke. Furthermore, if W is a trivial source module for a subgroup H ≤ G, then the induced module Ind_{H}^{G}(W) is also a trivial source module (see Thévenaz [23, Corollary 27.4]). Hence every module which is induced from a p′-subgroup is a trivial source module. An important class of examples are monomial representations, i.e., modules induced from a one-dimensional module of a subgroup. Examples of trivial source modules which are not induced from p′-subgroups can be obtained by taking a permutation module P = ⊕_{i=1}^{n}ke_{i} with p ∤ n and V = {α_{1}e_{1} + ⋯ + α_{n}e_{n} ∈ P | α_{1} + ⋯ + α_{n} = 0}. Then the sequence

0 ⟶ V ⟶ P ⟶ k ⟶ 0

splits, so V is a trivial source module. Example 6. We considered a representation of the above type in Example 4(b), where the group was A_{5}. Lifting the representation to characteristic 0 and applying Molien’s formula yields

H(k[V]^{G}, t) = \frac{t^{10} + 1}{(1 − t^{2})(1 − t^{3})(1 − t^{4})(1 − t^{5})}.

A comparison with Equation (4) illustrates the differences and similarities between the ordinary and the extended Hilbert series.

Corollary 7. Suppose that V is a trivial source module. Then the degree of H(k[V]^{G}, t) as a rational function in t is bounded from above by −n, where n = dim(V).

Proof. This follows from Theorem 5, since Molien’s formula holds for K[Ṽ]^{G}, and the degree of each summand in Molien’s formula is −n.

In general, invariant rings of trivial source modules do not have any nice properties such as the Cohen-Macaulay property or the property that they obey Noether’s degree bound (see Kemper [11, Corollary 2.4] and [12]). However, we do get some degree bound for the invariants of trivial source modules. We write

β(k[V]^{G}) = min{d ∈ ℕ | k[V]^{G} can be generated by invariants of degrees ≤ d}.

Theorem 8. Suppose that V is a trivial source module, embedded as a direct summand in a permutation module of dimension m. Then

β(k[V]^{G}) ≤ max\{m, \frac{m(m − 1)}{2}\}.

Proof. We have a split epimorphism P ↠ V of kG-modules, where P is the permutation module of dimension m. This induces an epimorphism k[P] ↠ k[V] of the symmetric algebras, which is also split. Hence the induced map k[P]^{G} → k[V]^{G} is surjective, and the inequality

β(k[V]^{G}) ≤ β(k[P]^{G})

follows. But by Göbel [6], β(k[P]^{G}) is bounded from above by max{m, m(m − 1)/2}, which yields the assertion.

Suppose that V is a monomial representation of dimension n, induced from a linear character χ whose values are m-th roots of unity. Then V is a direct summand of a permutation module P of dimension nm, and Theorem 8 yields

β(k[V]^{G}) ≤ max\{mn, \frac{mn(mn − 1)}{2}\}.

In the next section we will show that we can obtain a much better degree bound.

3 A degree bound for monomial representations

In this section we assume the following situation: Let R be a commutative ring with unity and μ ≤ R^{×} a finite subgroup of exponent m of the units in R. Further, for n ∈ ℕ, let

Ŝ_{n} = \{π · (\begin{matrix}α_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & α_{n} \end{matrix}) | π is an n × n permutation matrix, α_{i} ∈ μ\} ≤ GL_{n}(R)

and G ≤ Ŝ_{n} a subgroup. G acts on the module V = R^{n} and also on the symmetric algebra R[V], which is the polynomial ring R[x_{1}, …, x_{n}] in the standard basis vectors x_{i} of V. Our goal is to establish the degree bound

β(R[V]^{G}) ≤ max\{mn, m · \frac{n(n + 1)}{2} − n\}

for the invariant ring. We introduce some notation. We write ŝ_{i} = s_{i}(x_{1}^{m}, …, x_{n}^{m}), where s_{i} is the i-th elementary symmetric polynomial (i = 1, …, n), and

M = {x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}} | e_{i} ∈ ℕ_{0}}, T = {a · t | 0 ≠ a ∈ R, t ∈ M}

for the set of monomials and terms, respectively. For f ∈ R[V] with f = ∑_{t∈M} a_{t} · t let T (f) = {a_{t} · t | a_{t} ≠ 0} ⊂ T. For t ∈ T, set

orb_{G}(t) = ∑_{t′∈{σ(t)| σ∈G}} t′ ∈ R[V]^{G}.

Note that every element of the orbit enters only once into the sum.

The following lemma shows that the ŝ_{i} take the same role of a “universal hsop” for subgroups of Ŝ_{n} as do the elementary symmetric polynomials for permutation groups.

Lemma 9. R[V]Ŝ_{n} = R[ŝ_{1}, …, ŝ_{n}]. In fact, the ŝ_{i} form a SAGBI-basis of R[V]Ŝ_{n} (see, for example, Robbiano and Sweedler [18]).

Proof. We obviously have R[V]Ŝ_{n} = R[x_{1}^{m}, …, x_{n}^{m}]^{S_{n}}. Thus the lemma follows from the corresponding property of the elementary symmetric polynomials.

The lemma gives us a simple algorithm to represent any invariant of Ŝ_{n} as a polynomial in the ŝ_{i}. We now come to a generalization of Göbel’s concept of “special” terms.

Definition 10. Let t = a · x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}} ∈ T.

(a) We call t special if there is a k ∈ ℕ_{0} such that

{[e_{1}/m], …, [e_{n}/m]} = {0, 1, …, k},

where [z] denotes the greatest integer ≤ z. The set of all special monomials in M is denoted by M_{spec}.

(b) Generally, obtain a special term t̃ = a · xẽ_{1} ; 1 ⋯ xẽ_{n} ; n from t by applying the following step iteratively: Let k ∈ ℕ_{0} be minimal with k ∉ {[e_{1}/m], …, [e_{n}/m]}, and lower all e_{i} with [e_{i}/m] > k by m. Repeat this step until there are no such e_{i}. Then we write t̃= Red(t).

Remark 11. With the notation of Definition 10, Red(t) is special and

ẽ_{i} ≤ ẽ_{j} ⟺ e_{i} ≤ e_{j}.

The (total) degree of a special term t is bounded by

deg(t) ≤ (m − 1) + (2m − 1) + ⋯ + (nm − 1) = m · \frac{n(n + 1)}{2} − n.

It follows that M_{spec} is a finite set.

Lemma 12. For t ∈ T and σ ∈ Ŝ_{n} we have:

(a) σ(Red(t)) = Red(σ(t)),

(b) σ(Red(t)) = Red(t) ⟺ σ(t) = t.

Proof.

(a) Write σ = π · diag(α_{1}, …, α_{n}) with π a permutation matrix and α_{i} ∈ μ. Since the exponents of t and Red(t) are congruent modulo m, diag(α_{1}, …, α_{n}) commutes with Red. The same is true for π by the definition of Red(t).

(b) By (a) we have σ(Red(t)) = Red(σ(t)). Hence if σ(t) = t, then σ(Red(t)) = Red(t). If, on the other hand, σ(Red(t)) = Red(t), then Red(σ(t)) = Red(t), so σ(t) and t have the same coefficient a. As the application of Red does not change the ordering of the exponents (Remark 11), we conclude σ(t) = t.

Definition 13. We define a relation ≻ on T as follows: For t = ax_{1}^{e_{1}} ⋯ x_{n}^{e_{n}} and t′ = a′x_{1}^{e′_{1}} ⋯ x_{n}^{e′_{n}} choose permutations π and π′ ∈ S_{n} such that e_{π(1)} ≥ e_{π(2)} ≥ … ≥ e_{π(n)} and e′_{π′(1)} ≥ … ≥ e′_{π′(n)}. Then we say that t ≻ t′ if there exists a k ∈ {1, …, n} with

e_{π(i)} = e′_{π′(i)} for i < k and e_{π(k)} > e′_{π′(k)}.

We write t ⪯ t′ if t ≻ t′ does not hold.

Remark 14. Clearly, the relations ≻ and ⪯ are transitive, and for t ∈ T there exists no infinite sequence s_{1}, s_{2}, … ∈ T with

t ≻ s_{1} ≻ s_{2} ≻ ….

But ⪯ is not an ordering even when restricted to M, since t ⪯ t′ and t′ ⪯ t fail to imply t = t′.

We can come to the central lemma now (compare Lemma 3.10 in Göbel [6]).

Lemma 15. Let t ∈ T be a term and set u = t/ Red(t) ∈ M. Then we have

(a) t ≻ s for all s ∈ T (orb_{S_{n}}(u) · Red(t) − t),

(b) t ≻ s for all s ∈ T (orb_{S_{n}}(u) · orb_{G}(Red(t)) − orb_{G}(t)).

Proof.

(a) Obviously t lies in T (orb_{S_{n}}(u) · Red(t)). We have to show that t ≻ s for all other s ∈ T (orb_{S_{n}}(u) · Red(t)). So take such an s and assume that t ⪯ s. Clearly s = π(u) · Red(t) with π ∈ S_{n}. If we can show that π(u) = u, then s = t and (a) is proved.

Let t = ax_{1}^{e_{1}} ⋯ x_{n}^{e_{n}}, Red(t) = axẽ_{1} ; 1 ⋯ xẽ_{n} ; n and u = x_{1}^{d_{1}} ⋯ x_{n}^{d_{n}}, where we may assume by

renaming the x_{i} that e_{1} ≥ e_{2} ≥ … ≥ e_{n}. Then the construction of Red(t) and Remark 11 imply ẽ_{1} ≥ … ≥ ẽ_{n} and d_{1} ≥ … ≥ d_{n}. The exponents of s are ẽ_{1} + d_{π(1)}, …, ẽ_{n} + d_{π(n)}. Let k be maximal with d_{1} = d_{2} = … = d_{k}. Then t ⪯ s implies that π must permute the set {x_{1}, …, x_{k}}, since the greatest exponents of t and s must coincide. Proceeding analogously for the second-highest degrees d_{k+1}, …, d_{l}, we conclude that π also permutes {x_{k+1}, …, x_{l}}, and finally arrive at π(u) = u.

(b) For σ ∈ G we have by (a)

σ(t) ≻ s for all s ∈ T (orb_{S_{n}} (\frac{σ(t)}{Red(σ(t))}) · Red(σ(t)) − σ(t)) ,

hence also t ≻ s for these s. But by Lemma 12(a)

orb_{S_{n}} (\frac{σ(t)}{Red(σ(t))}) · Red(σ(t)) = orb_{S_{n}}(σ(u)) · σ(Red(t)) = orb_{S_{n}}(u) · σ(Red(t)),

where the last equation follows since σ acts on u by permuting the x_{i} because the exponents of u are divisible by m. We obtain

t ≻ s for all s ∈ T (orb_{S_{n}}(u) · σ(Red(t)) − σ(t)) .

Now by Lemma 12(b) the σ ∈ G fixing t are the same that fix Red(t), so summation over coset representatives σ of Stab(t) ≤ G yields the result.

We obtain the reduction algorithm listed on the next page. It is clear that this algorithm is correct if it terminates. For the proof of its termination we need

the following observation.

Input: An invariant f ∈ R[V]^{G}.

Output: For each t ∈ M_{spec} a polynomial p_{t} in n indeterminates such that

f = ∑_{t∈M_{spec}} p_{t}(ŝ_{1}, …, ŝ_{n}) · orb_{G}(t).

Begin

for t ∈ M_{spec} set p_{t} := 0;
while f ≠ 0 do

choose t ∈ T (f) that is maximal w.r.t. ≻; compute Red(t) =: a · t̃ with t̃∈ M_{spec}; set u := t/ Red(t); calculate a polynomial p such that orb_{S_{n}}(u) = p(ŝ_{1}, …, ŝ_{n}); set pt̃:= pt̃ + a · p; set f := f − orb_{S_{n}}(u) · orb_{G}(Red(t))

end while

end.

Algorithm 1: Write an invariant in terms of special orbit sums and ŝ_{i}

Lemma 16. If f ∈ R[V]^{G} is an invariant and t ∈ T (f), then T (orb_{G}(t)) ⊂ T (f).

Proof. This follows from the fact that G acts on T.

Proposition 17. Algorithm 1 terminates after a finite number of steps.

Proof. Each pass through the while-loop replaces f by

f − orb_{G}(t) + (orb_{G}(t) − orb_{S_{n}}(u) · orb_{G}(Red(t))) .

By Lemma 16, all terms in orb_{G}(t) are also terms in f, and by Lemma 15(b), all terms in (orb_{G}(t)− orb_{S_{n}}(u)·orb_{G}(Red(t))) are strictly smaller than t. So there are some maximal terms removed from f, and only strictly smaller terms are added. Hence after a finite number of steps the maximum of T (f) w.r.t. ≻ decreases strictly, and iterating further eventually yields f = 0 by Remark 14.

Theorem 18. The following degree bounds hold independently of the choice of R:

(a) The invariant ring R[V]^{G} is generated as a module over A = R[ŝ_{1}, …, ŝ_{n}] by homogeneous invariants g_{1}, …, g_{r} satisfying

deg(g_{i}) ≤ m · \frac{n(n + 1)}{2} − n.

(b) R[V]^{G} is generated as an algebra over R by invariants whose degrees are bounded from above

by

max\{mn, m · \frac{n(n + 1)}{2} − n\}.

Proof. Part (a) follows from the algorithm and the degree estimate in Remark 11, and (b) follows since deg(ŝ_{i}) ≤ mn.

The following remark shows that the bound from Theorem 18(a) is very natural and in many cases sharp.

Remark 19. Consider the special case that R is a field k whose characteristic does not divide the group order |G|. Then k[V]^{G} is Cohen-Macaulay. Let e_{1}, …, e_{r} be the degrees of the secondary invariants. Using Molien’s formula for the Hilbert series of k[V]^{G}, Stanley obtained the equation [21, Proposition 3.8]

max\{e_{1}, …, e_{r}\} = ∑_{i=1}^{n}(deg(ŝ_{i}) − 1) − l = m · \frac{n(n + 1)}{2} − n − l,

where l is the least degree of a homogeneous f ∈ k[V] with σ(f) = det(σ)^{−1} · f for all σ ∈ G. This implies the bound from Theorem 18(a), and we see that in our special case the bound is sharp if and only if G ≤ SL_{n}(k). Hence what we have done is the generalization of a long-known result of non-modular invariant theory to the case of arbitrary ground rings.

4 Some conjectures

Since faithful monomial representations are induced from cyclic subgroups of order not divisible by the characteristic k, it is natural to ask for a further generalization to the case of a representation which is induced from any p′-subgroup. More precisely, suppose that H is a finite group, k a field of characteristic p not dividing |H|, and let W be a kH-module of dimension m. Fix n ∈ ℕ, then the wreath product Ŝ_{n} = H ≀ S_{n} has a natural action on V = ⊕_{i=1}^{n}W_{i} with W_{i} ≅ W. Now we consider subgroups G ≤ Ŝ_{n}. Let x_{1}, …, x_{m} be a basis of W and x_{i,1}, …, x_{i,n} the corresponding basis of W_{i}. Fix an hsop f_{1}(x_{1}, …, x_{m}), …, f_{m}(x_{1}, …, x_{m}) of k[W]^{H} and write f_{i,j} = f_{j}(x_{i,1}, …, x_{i,m}) ∈ k[W_{i}]^{H}. Then, as in the previous section, we obtain a “universal hsop” for all subgroups G ≤ Ŝ_{n} by taking

f̂_{i,j} = s_{i} (f_{1,j}, …, f_{n,j}) (i = 1, …, n, j = 1, …, m),

where s_{i} are the elementary symmetric functions. In the previous section, it was essential that G acted by permutations on the set of terms. This feature can be generalized to the situation considered here by assigning to x_{i,j} the multi-degree (0, …, 0, 1, 0, …, 0) with 1 in the i-th position. This yields a direct sum decomposition of k[V] into multi-homogeneous components with the property that every element of G maps each component into another one. The author has been unsuccessful in his efforts to find an appropriate definition of special multi-degrees and to generalize the arguments from Section 3. The goal would be to prove

Conjecture 20. With the above notation, let G ≤ Ŝ_{n} and take the f̂_{i,j} as primary invariants for G. Then for the secondary invariants g_{i} of k[V]^{G} we have

deg(g_{i}) ≤ \frac{n(n + 1)}{2} · (deg(f_{1}) + ⋯ + deg(f_{n})) − nm.

This conjecture would be an instance of a much more general conjecture made by Harm Derksen at the Workshop on Computational Invariant Theory held in April/May 1996 as part of the DFG-Schwerpunkt:

Conjecture 21. Suppose that G is a finite group, k a field and V a finitely generated kG-module. If f_{1}, …, f_{n} is an hsop of k[V]^{G} then the degrees of the secondary invariants are bounded from above by ∑_{i=1}^{n}(deg(f_{i}) − 1).

There is a lot of computational evidence for this conjecture. For the case where k[V]^{G} is Cohen-Macaulay, it would be a consequence of yet another conjecture on the degree of the Hilbert series:

Conjecture 22. Under the hypotheses of Conjecture 21, we have the bound

deg_{t} (H(k[V]^{G}, t)) ≤ − dim_{k}(V)

for the degree of the Hilbert series as a rational function in t.

For the special case that V is a trivial source module, this is Corollary 7, and if the Hilbert series is replaced by the extended Hilbert series, the conjecture follows from Theorem 2. The degree of the Hilbert series is also called the a-invariant and has some connection to local cohomology (see Bruns and Herzog [4, p. 169]).

References

[1] Dave Bayer, Mike Stillman, Computation of Hilbert Functions, J. Symbolic Computation 14 (1992), 31–50.

[2] David J. Benson, Representations and Cohomology I, Cambridge studies in advanced mathematics 30, Cambridge Univ. Press, Cambridge 1991.

[3] David J. Benson, Polynomial Invariants of Finite Groups, Lond. Math. Soc. Lecture Note Ser. 190, Cambridge Univ. Press, Cambridge 1993.

[4] Winfried Bruns, Jürgen Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge 1993.

[5] Charles W. Curtis, Irving Reiner, Methods of Representation Theory, vol. 1, J. Wiley & Sons, New York 1981.

[6] Manfred Göbel, Computing Bases for Rings of Permutation-invariant Polynomials, J. Symbolic Computation 19 (1995), 285–291.

[7] David M. Goldschmidt, Lectures on Character Theory, Publish or Perish, Inc., Berkeley 1980.

[8] Christoph Jansen, Klaus Lux, Richard Parker, Robert Wilson, An Atlas of Brauer Characters, Clarendon Press, Oxford 1995.

[9] Gregor Kemper, Calculating Invariant Rings of Finite Groups over Arbitrary Fields, J. Symbolic Computation 21 (1996), 351–366.

[10] Gregor Kemper, Calculating Optimal Homogeneous Systems of Parameters, IWR Preprint 97-08, Heidelberg 1997.

[11] Gregor Kemper, On the Cohen-Macaulay Property of Modular Invariant Rings, IWR Preprint 97-38, Heidelberg 1997.

[12] Gregor Kemper, Lower Degree Bounds for Modular Invariants and a Question of I. Hughes, IWR Preprint 97-40, Heidelberg 1997.

[13] Gregor Kemper, Gunter Malle, Invariant rings and fields of finite groups, in preparation.

[14] Gregor Kemper, Allan Steel, Some Algorithms in Invariant Theory of Finite Groups, Preprint, Sydney 1997.

[15] Peter Landrock, Finite Group Algebras and their Modules, Lond. Math. Soc. Lecture Note Ser. 84, Cambridge Univ. Press, Cambridge 1983.

[16] Emmy Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1916), 89–92.

[17] David R. Richman, Invariants of Finite Groups over Fields of Characteristic p, Adv. in Math. 124 (1996), 25–48.

[18] Lorenzo Robbiano, Moss Sweedler, Subalgebra Bases, in: W. Bruns, A. Simis, eds., Commutative Algebra, Lecture Notes in Math. 1430, pp. 61–87, Springer-Verlag, New York 1990.

[19] Larry Smith, Polynomial Invariants of Finite Groups, A. K. Peters, Wellesley, Mass. 1995.

[20] Larry Smith, Polynomial Invariants of Finite Groups: A Survey of Recent Developments, Bull. Amer. Math. Soc. 34 (1997), 211–250.

[21] Richard P. Stanley, Invariants of Finite Groups and their Applications to Combinatorics, Bull. Amer. Math. Soc. 1(3) (1979), 475–511.

[22] Bernd Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, Wien, New York 1993.

[23] Jacques Thévenaz, G-Algebras and Modular Representation Theory, Clarendon Press, Oxford