Unspecified Journal Volume 00, Number 0, Pages 000–000 S ????-????(XX)0000-0

DEGREE BOUNDS AND THE REGULAR REPRESENTATION

DEDICATED TO THE MEMORY OF ANDERS J. FRANKILD [1971-2007].

MARA D. NEUSEL

Abstract. Let ρ : G ↪ GL(V) be a permutation representation of a finite group G. We describe universal algebra generating set for the associated ring of polynomial invariants over any field as long as the invariants are Cohen-Macaulay. This has many consequences for permutation invariants as well as for invariants of arbitrary representations. In particular, it implies Killius’ Conjecture for permutation invariants and Schmid’s Inequality for the general nonmodular case.

1. Introduction

Let ρ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group G over a field 𝔽. The representation ρ induces an action of G on the vector space V = 𝔽^{n} of dimension n and hence on the ring of polynomial functions 𝔽[V] = 𝔽[x_{1}, …, x_{n}], where we chose a basis x_{1}, …, x_{n} of the dual space V^{∗}. Our interest is focused on the subring of invariants

𝔽[V]^{G} = {f ∈ 𝔽[V]^{G}|gf = f ∀g ∈ G},

which is a graded connected Noetherian commutative algebra, see [11] for more information on rings of polynomial invariants of finite groups.

We denote by β(𝔽[V]^{G}) the maximal degree of an 𝔽-algebra generator of 𝔽[V]^{G} in a minimal homogeneous generating set. By Noether’s Bound we have that

β(𝔽[V]^{G}) ≤ |G|

in the nonmodular case (where |G| ∈ 𝔽^{×}), see Theorem 2.3.3 in [11]. Many refinements of this statement have been proven over the past years as well as many results valid in the modular case (where |G| = 0 ∈ 𝔽), see [8] for a survey on these matters. This paper has been motivated by the question whether there is a “worst” representation.

In [12] Schmid showed that

β(𝔽[V]^{G}) ≤ β(𝔽[𝔽G]^{G})

whenever the ground field 𝔽 has characteristic zero. It is known that this inequality does not remain valid in the modular case, where char(𝔽) = p||G|, see Example 1 of Section 3.2 in [11] for a counterexample. However, it was a long standing conjecture

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

Received by the editors February 12, 2007.

2000 Mathematics Subject Classification. Primary 13A50, Secondary.

Key words and phrases. Invariant Theory of Finite Permutation Groups, Permutation Representation, Cohen-Macaulay, Regular Representation.

I would like to thank the referee for many helpful comments.

that Schmid’s inequality holds in the general nonmodular case. Indeed, it was generalized to the strong nonmodular case (i.e., char(𝔽) = 0 or char(𝔽) > |G|) by Smith, see [15], and for abelian groups G in the general nonmodular case by Sezer, see [13]. Furthermore, Knop proved this result for nonmodular representations where the characteristic of the ground field is zero or p > 3/8|G| + 1, see Corollary 7.5 in [5].

In this paper we prove Schmid’s inequality for the general nonmodular case. Indeed, the proof follows easily from Corollary 2.12. Section 2 of this paper is devoted to the proof of that result as well as of Theorem 2.11 which gives us precise information on the set of algebra generators. In Section 3 we deduce several results on degree bounds for polynomial invariants. In particular we prove Killius’ Conjecture in Theorem 3.1 for permutation representations and we prove Schmid’s Inequality in Theorem 3.2 for the general nonmodular case.

2. Rings of Invariants of Permutation Representations

Let ρ : G ↪ GL(n, 𝕂) be a permutation representation of a finite group G. We note that ρ factorizes through the prime field 𝔽 = 𝔽_{p}, resp. 𝔽 = ℚ:

ρ : G ↪ GL(n, 𝕂)
↘  ↑
GL(n, 𝔽).

We obtain for the respective rings of invariants

𝕂[V]^{G} = 𝕂 ⊗_{𝔽} 𝔽[V]^{G}.

Thus a minimal set of 𝔽-algebra generators of 𝔽[V]^{G} minimally generates 𝕂[V]^{G} as an 𝕂-algebra as well. Therefore we can assume in what follows that the fields involved are prime fields.

Furthermore, any permutation representation ρ : G ↪ GL(n, ℚ) factorizes through GL(n, ℤ)

ρ : G ↪ GL(n, ℚ)
↘  ↑
GL(n, ℤ).

Taking coefficients mod p we obtain a faithful representation G ↪ GL(n, 𝔽_{p}) which we will call ρ also. Our goal is to explicitly describe a common algebra generating set for both rings, ℚ[V]^{G} and 𝔽_{p}[V]^{G}, in the case that both rings are Cohen-Macaulay. (Of course, ℚ[V]^{G} is always Cohen-Macaulay, see Theorem 5.5.2 in [11].)

We note that any element in a ring of permutation invariants is a sum of orbit sums. Therefore, we can consider any invariant f ∈ 𝔽_{p}[V]^{G} as an element in ℤ[V]^{G} ⊆ ℚ[V]^{G}. On the other hand, after clearing denominators we can consider an invariant f ∈ ℚ[V]^{G} as a (possibly zero) element f ∈ 𝔽_{p}[V]^{G}.

Let s_{1}, …, s_{m} ∈ ℤ[V] ⊆ ℚ[V]. We note that s_{1}, …, s_{m} are algebraically independent in ℤ[V] if and only if they are algebraically independent in ℚ[V]. We have the following lemma.

Lemma 2.1. Let s_{1}, …, s_{m} ∈ ℤ[V] ⊆ ℚ[V]. If s_{1}, …, s_{m} are algebraically independent in 𝔽_{p}[V], then they are algebraically independent in ℚ[V].

Proof. Let s_{1}, …, s_{m} ∈ 𝔽_{p}[V] be algebraically independent. Assume there is a polynomial P (X_{1}, …, X_{m}) ∈ ℚ[X_{1}, …, X_{m}] such that

P (s_{1}, …, s_{m}) = 0.

By clearing denominators we can assume that P (X_{1}, …, X_{m}) ∈ ℤ[X_{1}, …, X_{m}]. Without loss of generality we assume that the greatest common divisor of the coefficients of P (X_{1}, …, X_{m}) is 1 (or P (X_{1}, …, X_{m}) = 0). Since the s_{j}’s are algebraically independent over 𝔽_{p} we have that P (X_{1}, …, X_{m}) ≡ 0 mod p. This implies that the coefficients of P (X_{1}, …, X_{m}) are divisible by p > 1. Hence P (X_{1}, …, X_{m}) = 0 ∈ ℤ[X_{1}, …, X_{m}]. □

We note that a ring of polynomial invariants 𝔽[V]^{G} of a permutation representation is a finite module over the ring of invariants of the full symmetric group in n letters, Σ_{n}, in its defining representation, i.e., the extension

𝔽[V]^{Σ_{n}} ↪ 𝔽[V]^{G}

is finite, see Exercise (1) in Section 12.6 of [9]. Denote by e_{1}, …, e_{n} the elementary symmetric functions in x_{1}, …, x_{n}. Then

𝔽[V]^{Σ_{n}} = 𝔽[e_{1}, …, e_{n}]

is a Noether normalization for 𝔽[V]^{G}. We note that this is true for any ground field, i.e., the elementary symmetric functions form a universal system of parameters for any ring of permutation invariants over any field 𝔽. Furthermore we have

𝔽[V]^{G} = ∑_{i=1}^{k} 𝔽[e_{1}, …, e_{n}]f_{i}

for some homogeneous invariants f_{1}, …, f_{k}. We note that this sum is direct if and only if 𝔽[V]^{G} is Cohen-Macaulay, see Theorem 5.1.3 in [11].

Next we show that we can choose also a universal set of secondary invariants across characteristics as long as the rings of invariants involved remain Cohen-Macaulay.

Proposition 2.2. Assume that 𝔽_{p}[V]^{G} is Cohen-Macaulay. Let s_{1}, …, s_{n}, f_{1}, …, f_{k} be homogeneous invariant polynomials of positive degree. If

(•)  𝔽_{p}[V]^{G} = ⨁_{i=1}^{k} 𝔽_{p}[s_{1}, …, s_{n}]f_{i},

then

(∘)  ℚ[V]^{G} = ⨁_{i=1}^{k} ℚ[s_{1}, …, s_{n}]f_{i}.

Proof. Assume that 𝔽_{p}[V]^{G} = ⨁_{i=1}^{k} 𝔽_{p}[s_{1}, …, s_{n}]f_{i}. Consider the extension

∑_{i=1}^{k} ℚ[s_{1}, …, s_{n}]f_{i} ⊆ ℚ[V]^{G}.

We note that ℚ[s_{1}, …, s_{n}] remains a polynomial algebra by Lemma 2.1. Since Poincarè series of rings of permutation invariants are independent of the characteristic of the ground field (see Proposition 3.2.7 in [11]), it is enough to show that this sum is direct.

We proceed by induction on k. Let k = 1 and assume that

(1)  α_{1}f_{1} = 0

for some α_{1} = α_{1}(s_{1}, …, s_{n}) ∈ ℚ[s_{1}, …, s_{n}]. By clearing denominators we can assume that

α_{1}(X_{1}, …, X_{n}) ∈ ℤ[X_{1}, …, X_{n}].

Without loss of generality we assume that p |̸α_{1}(X_{1}, …, X_{n}). From Equation (1) we have that α_{1}f_{1} ≡ 0 mod p. By assumption it follows that α_{1}(s_{1}, …, s_{n}) ≡ 0 mod p. Since s_{1}, …, s_{n} are algebraically independent in 𝔽_{p}[V]^{G} it follows that α_{1}(X_{1}, …, X_{n}) ∈ 𝔽_{p}[X_{1}, …, X_{n}] is the zero polynomial. Since p does not divide α_{1}(X_{1}, …, X_{n}) we have that α_{1}(X_{1}, …, X_{n}) = 0 and hence α_{1}(s_{1}, …, s_{n}) = 0 ∈ ℤ[s_{1}, …, s_{n}] ⊆ ℚ[s_{1}, …, s_{n}] as claimed.

Next let k > 1 and assume there is a relation

(2)  ∑_{i=1}^{k} α_{i}f_{i} = 0

for some α_{i} = α_{i}(s_{1}, …, s_{n}) ∈ ℚ[s_{1}, …, s_{n}], i = 1, …, k. By clearing denominators we assume that the polynomials α_{i}(X_{1}, …, X_{n}) have integer coefficients. By Equation (2) it follows that ∑_{i=1}^{k} α_{i}f_{i} ≡ 0 mod p. Since the sum (•) is direct it follows that

α_{i}(s_{1}, …, s_{n}) ≡ 0 mod p.

Since s_{1}, …, s_{n} are algebraically independent in 𝔽_{p}[V]^{G}, it follows that the polynomials α_{i}(X_{1}, …, X_{n}) ∈ 𝔽_{p}[X_{1}, …, X_{n}] are zero. Thus

α_{i}(X_{1}, …, X_{n}) = p^{d_{i}}α′_{i}(X_{1}, …, X_{n}) ∈ ℤ[X_{1}, …, X_{n}]

for some d_{i} ∈ ℕ with p |̸α′_{i}(X_{1}, …, X_{n}). Without loss of generality set d_{k} = min{d_{1}, …, d_{k}}. Dividing the Equation (2) by p^{d_{k}} leads to

∑_{i=1}^{k} α′′_{i}f_{i} = 0

where p does not divide α′′_{k} = α′_{k}(s_{1}, …, s_{n}). Hence α_{k}(s_{1}, …, s_{n}) = 0 and we are done by induction. □

Remark 2.3. The preceding result shows that a set of invariants s_{1}, …, s_{n} forming a homogeneous system of parameters for 𝔽_{p}[V]^{G}, forms also a system of parameters for ℚ[V]^{G}. Furthermore once we have chosen a universal system of parameters we obtain a universal set of homogeneous secondary generators f_{1}, …, f_{k} across characteristics as long as the ring of invariants remains Cohen-Macaulay.

Remark 2.4. The converse of the preceding result remains true if and only if s_{1}, …, s_{n} remain algebraically independent as elements in 𝔽[V].

Remark 2.5. If the system of parameters s_{1}, …, s_{n} ∈ 𝔽_{p}[V]^{G} is not homogeneous, but none of the s_{i}’s has a constant term, then the preceding result remains true with some slight modification: By assumption the system of parameters s_{1}, …, s_{n} is contained in the irrelevant ideal 𝔪 ⊂ 𝔽_{p}[V]^{G}. Thus by localizing we obtain

𝔽_{p}[V]_{𝔪}^{G} = ⨁_{i=1}^{k} 𝔽_{p}[s_{1}, …, s_{n}]_{𝔫}f_{i}

where 𝔫 = 𝔪 ∩ 𝔽_{p}[s_{1}, …, s_{n}], see Proposition 2.2.11 in [2]. The preceding proof then shows that

ℚ[V]_{𝔪′}^{G} = ⨁_{i=1}^{k} ℚ[s_{1}, …, s_{n}]_{𝔫′}f_{i}

for 𝔪′ ⊆ ℚ[V]^{G} the irrelevant ideal and 𝔫′ = 𝔪′ ∩ ℚ[s_{1}, …, s_{n}] its contraction.

Thus we have found a common set of primary and secondary invariants. Of course it follows that

𝔽_{p}[V]^{G} = 𝔽[s_{1}, …, s_{n}, f_{1}, …, f_{k}]

and

ℚ[V]^{G} = ℚ[s_{1}, …, s_{n}, f_{1}, …, f_{k}].

However, in both cases the set of algebra generators given should not be minimal. Our next goal is to show that in both rings the same secondary invariants can be deleted from the list of algebra generators.

Proposition 2.6. We continue with the same notation as above. Assume that the invariants s_{1}, …, s_{n}, f_{1}, …, f_{k} are homogeneous. Then

f_{k} ∈ 𝔽_{p}[s_{1}, …, s_{n}, f_{1}, …, f_{k−1}]

if and only if

f_{k} ∈ ℚ[s_{1}, …, s_{n}, f_{1}, …, f_{k−1}].

Proof. Assume that

f_{k} ∈ ℚ[s_{1}, …, s_{n}, f_{1}, …, f_{k−1}].

We write

(⋆)  f_{k} = ∑_{I} p_{I}(s_{1}, …, s_{n})f^{I}

where f^{I} = f_{1}^{i_{1}} ⋯ f_{k−1}^{i_{k−1}} and p_{I} are polynomials in the elements of the system of parameters. For degree reasons we have for each I appearing in the sum (⋆)

(∗)  f^{I} = λ_{I}f_{k} + ∑_{i=1}^{k−1} q_{Ii}(s_{1}, …, s_{n})f_{i}

for some coefficients λ_{I} ∈ ℚ. By linear independence we obtain

∑_{I} λ_{I} = 1.

Thus for all I_{0} such that λ_{I_{0}} ≠ 0 we have

f_{k} = λ_{I_{0}}^{−1}f^{I_{0}} + ∑_{i=1}^{k−1} λ_{I_{0}}^{−1}q_{I_{0},i}f_{i}

and hence

ℚ[V]^{G} = ⨁_{i=1}^{k−1} ℚ[s_{1}, …, s_{n}]f_{i} ⊕ ℚ[s_{1}, …, s_{n}]f^{I_{0}}

whenever λ_{I_{0}} ≠ 0. We claim that there exists at least one I_{0} so that we can do the same over the prime field 𝔽_{p}. For that we multiply Equation (∗) by a minimal μ ∈ ℤ such that

μf^{I_{0}} = (μλ_{I_{0}})f_{k} + ∑_{i=1}^{k−1} μq_{I_{0},i}f_{i}

is an equation with integer coefficients. We reduce modulo p. If μ ≡ 0 mod p then

μλ_{I_{0}} ≡ 0 mod p

as well as

μq_{I_{0},i} ≡ 0 mod p

because the sum (•) is direct. This contradicts that μ was chosen to be minimal. Thus μ is not divisible by p. We find

μf^{I_{0}} = (μλ_{I_{0}})f_{k} + ∑_{i=1}^{k−1} μq_{I_{0},i}f_{i} ≠ 0 mod p.

Since ∑ λ_{I_{0}} = 1 and no denominator of Equation (∗) is divisible by p (recall μ ≠ 0 mod p), there exists an index I_{0} such that μλ_{I_{0}} ≠ 0 mod p. Therefore

f_{k} ≡ (μλ_{I_{0}})^{−1}μf^{I_{0}} − ∑_{i=1}^{k−1}(μλ_{I_{0}})^{−1}μq_{I_{0},i}f_{i} mod p,

and

𝔽_{p}[V]^{G} = ⨁_{i=1}^{k−1} 𝔽_{p}[s_{1}, …, s_{n}]f_{i} ⊕ 𝔽_{p}[s_{1}, …, s_{n}]f^{I_{0}}

as desired.

We prove the converse and assume that

f_{k} ≡ ∑_{I} p_{I}(s_{1}, …, s_{n})f^{I} mod p,

for some polynomials p_{I}(s_{1}, …, s_{n}) ∈ 𝔽_{p}[s_{1}, …, s_{n}] and I = (i_{1}, …, i_{k−1}). For every multi-index I appearing in this sum we have

f^{I} ≡ λ_{I}f_{k} + ∑_{i=1}^{k−1} q_{I,i}(s_{1}, …, s_{n})f_{i} mod p

with ∑_{I} λ_{I} ≡ 1 mod p. Thus for all I_{0} such that λ_{I_{0}} ≠ 0 mod p we can write

f_{k} = λ_{I_{0}}^{−1}f^{I_{0}} − λ_{I_{0}}^{−1} ∑_{i=1}^{k−1} q_{I_{0},i}(s_{1}, …, s_{n})f_{i} mod p.

Lifting to ℤ[V]^{G} gives the equation

f^{I_{0}} = λ_{I_{0}}f_{k} + ∑_{i=1}^{k−1} q_{I_{0},i}(s_{1}, …, s_{n})f_{i} + pH

for some invariant H ∈ ℤ[V]^{G}. Thus as a rational invariant we can write H as

H = μf_{k} + ∑_{i=1}^{k−1} q_{H,i}(s_{1}, …, s_{n})f_{i}

for suitable μ ∈ ℚ and q_{H,i}(s_{1}, …, s_{n}) ∈ ℚ[s_{1}, …, s_{n}]. Thus we have

f^{I_{0}} = (λ_{I_{0}} + pμ)f_{k} + ∑_{i=1}^{k−1} q_{I_{0},i}f_{i} + ∑_{i=1}^{k−1} pq_{H,i}f_{i}.

If λ_{I_{0}} + pμ ≠ 0 for some index I_{0} we are done. Otherwise we have μ = −\frac{λ_{I_{0}}}{p} and thus

f^{I_{0}} = ∑_{i=1}^{k−1} q_{I_{0},i}f_{i} + ∑_{i=1}^{k−1} pq_{H,i}f_{i}

for all I_{0}. We multiply this equation with the minimal α ∈ ℕ such that

αf^{I_{0}} = ∑_{i=1}^{k−1} αq_{I_{0},i}f_{i} + ∑_{i=1}^{k−1} αpq_{H,i}f_{i}

has integer coefficients. Modulo p this relation has to become trivial (for otherwise f^{I_{0}} ∈ ⊕_{i=1}^{k−1}𝔽_{p}[s_{1}, …, s_{n}]f_{i}) and thus α ≡ 0 mod p. Let α = p^{s}α′ such that p |̸α′. Then p^{s+1} is the maximal p-power dividing a denominator in q_{H,i}. Thus there is an i_{0} such that αpq_{H,i_{0}} is not zero modulo p. This is a contradiction, because the sum (•) is direct. □

Remark 2.7. The preceding result actually shows that a secondary invariant f_{k} that is redundant as an algebra generator can be replaced by some f^{I_{0}} = f_{1}^{i_{1}} ⋯ f_{k−1}^{i_{k−1}}. Thus inductively we find

𝔽_{p}[V]^{G} = 𝔽_{p}[s_{1}, …, s_{n}, f_{1}, …, f_{l}]

if and only if

ℚ[V]^{G} = ℚ[s_{1}, …, s_{n}, f_{1}, …, f_{l}].

Remark 2.8. Assume that the invariants s_{1}, …, s_{n}, f_{1}, …, f_{k} are inhomogeneous, but have no constant terms. Then they are contained in the maximal ideal generated by all homogeneous invariants of positive degree. By localizing at that maximal ideal, the preceding proof extends to that case, cf. Remark 2.5.

We are now prepared to prove that a minimal algebra generating set of 𝔽_{p}[V]^{G} generates also ℚ[V]^{G}.

Proposition 2.9. Let 𝔽_{p}[V]^{G} = 𝔽_{p}[h_{1}, …, h_{m}]. Then

ℚ[V]^{G} = ℚ[h_{1}, …, h_{m}].

Proof. By assumption we have

𝔽_{p}[V]^{G} = ⨁_{i=1}^{k} 𝔽_{p}[s_{1}, …, s_{n}]f_{i} = 𝔽_{p}[h_{1}, …, h_{m}].

Without loss of generality we assume that none of the h_{i}’s is redundant. Since this ring of invariants is Cohen-Macaulay there are 𝔽_{p}-linear combinations of the h_{i}’s

L_{j}(h_{1}, …, h_{m}) j = 1, …, n

such that L_{1}, …, L_{n} ∈ 𝔽_{p}[V]^{G} forms a system of parameters. If the L_{i}’s are not homogeneous we need to localize at the irrelevant ideal in order to be able to proceed. To simplify notation we assume they are homogeneous and write

𝔽_{p}[V]^{G} = 𝔽_{p}[L_{1}, …, L_{n}, h_{n+1}, …, h_{m}] = ⨁_{i=n+1}^{m} 𝔽_{p}[L_{1}, …, L_{n}]h_{i}⊕⨁_{i=1}^{l} 𝔽_{p}[L_{1}, …, L_{n}]F_{i}

for suitable invariants F_{i}. By Proposition 2.2 we have

ℚ[V]^{G} = ⨁_{i=n+1}^{m} ℚ[L_{1}, …, L_{n}]h_{i} ⊕ ⨁_{i=1}^{l} ℚ[L_{1}, …, L_{n}]F_{i}.

Thus by Lemma 2.6 we have

ℚ[V]^{G} = ℚ[L_{1}, …, L_{n}, h_{n+1}, …, h_{m}]

as desired. □

Remark 2.10. We could try to reverse the preceding proof, and construct linear combinations of the algebra generators L_{1}, …, L_{n} ∈ ℚ[V]^{G}. The proof then goes through if and only if L_{1}, …, L_{n} remain algebraically independent in 𝔽_{p}[V]^{G}.

Let

{x_{1}, …, x_{n}} = B_{1} ⊔ ⋯ ⊔ B_{r}

be a decomposition into disjoint G-orbits. We denote by e_{ij} the jth elementary symmetric function in the elements of B_{i}. Then the e_{ij}’s form a homogeneous system of parameters for 𝔽[V]^{G} over any field 𝔽. Thus the preceding results can be combined to the following:

Theorem 2.11. Let 𝔽 and 𝕂 be fields. Let ρ : G ↪ GL(n, 𝔽) be a permutation representation of a finite group G and consider the “same” representation over the field 𝕂. Assume both rings of invariants, 𝔽[V]^{G} and 𝕂[V]^{G}, are Cohen-Macaulay. Then

𝔽[V]^{G} = 𝔽[s_{11}, …, s_{r|B_{r}|}, f_{1}, …, f_{l}]

if and only if

𝕂[V]^{G} = 𝕂[s_{11}, …, s_{r|B_{r}|}, f_{1}, …, f_{l}].
Proof.  □

From this description we find the following degree bound for the algebra generators.

Corollary 2.12. Assume that G acts by permutations and 𝔽_{p}[V]^{G} is Cohen-Macaulay. Then

β(𝔽_{p}[V]^{G}) = max{β(ℚ[V]^{G}), δ_{V} },

where δ_{V} is the maximal degree of an elementary symmetric function in the orbit elements of B_{1}, …, B_{r} that is not redundant as algebra generator.

Proof. This is immediate from the description given in Theorem 2.11. □

3. Applications to general invariant theory

We apply the results of the preceding sections to several problems of degree bounds in invariant theory of arbitrary representations of finite groups.

First we note that we can prove Killius’ Conjecture for permutation representations, see [4] or Problem 7.3 in [8].

Theorem 3.1 (Killius’ Conjecture for Permutation Invariants). Let ρ : G ↪ GL(n, 𝔽) be a permutation representation of a finite group G. Assume that 𝔽[V]^{G} is Cohen-Macaulay. Then

β(𝔽[V]^{G}) ≤ |G|.

Proof. Since the length of an orbit B_{i} does not exceed the group order this result follows from Corollary 2.12. □

Theorem 3.2. Let ρ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group. Let V = 𝔽^{n} be a projective 𝔽G-module and let 𝔽[n𝔽G]^{G} be Cohen-Macaulay. Then

β(𝔽[V]^{G}) ≤ β(𝔽[𝔽G]^{G}).

Proof. Since V is a projective 𝔽G-module, the Noether map

η_{G}^{G} : 𝔽[n𝔽G]^{G} ⟶ 𝔽[V]^{G}

is surjective, see Proposition 3.1 in [10]. Thus β(𝔽[n𝔽G]^{G}) ≥ β(𝔽[V]^{G}), cf. proof of Proposition 4.1 in [10].

By Corollary 2.12 we have

β(𝔽[n𝔽G]^{G}) = max{β(ℚ[nℚG]^{G}), δ_{n𝔽G}} == max{β(ℂ[nℂG]^{G}), δ_{n𝔽G}}.

By Weyl’s Theorem the invariants of the n-fold regular representation are generated by polarizations and hence

β(ℂ[nℂG]^{G}) = β(ℂ[ℂG]^{G}).

Since 𝔽G ↪ n𝔽G is a direct summand we have by Theorem 1.2 in [7] that

CM − defect(𝔽[𝔽G]^{G}) ≤ CM − defect(𝔽[n𝔽G]^{G}) = 0.

Thus 𝔽[𝔽G]^{G} is Cohen-Macaulay, and a second application of Corollary 2.12 gives

β(𝔽[𝔽G]^{G}) = max{β(ℚ[ℚG]^{G}), δ_{𝔽G}} = max{β(ℂ[ℂG]^{G}), δ_{𝔽G}}.

Combining these inequalities gives

β(𝔽[V]^{G}) ≤ β(𝔽[n𝔽G]^{G}) = max{β(ℂ[nℂG]^{G}), δ_{n𝔽G}}

= max{β(ℂ[ℂG]^{G}), δ_{n𝔽G}}

≤^{(1)} max{β(ℂ[ℂG]^{G}), δ_{𝔽G}} = β(𝔽[𝔽G]^{G}),

where (1) follows since an elementary symmetric function is irredundant as an algebra generator for 𝔽[𝔽G]^{G} if it is so for 𝔽[n𝔽G]^{G}. □

Theorem 3.3 (Schmid’s Inequality). Let ρ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group G. Assume that the ground field 𝔽 has characteristic zero or p |̸|G|. Then β(𝔽[V]^{G}) ≤ β(𝔽[𝔽G]^{G}).

Proof. Since ρ is a nonmodular representation the 𝔽G-module V = 𝔽^{n} is projective, and 𝔽[n𝔽G]^{G} is Cohen-Macaulay. Thus the Theorem 3.2 applies. □

Remark 3.4. We note that the argument used in the preceding proof shows that rings of invariants of sums of permutation representations that are Cohen-Macaulay are generated by polarizations independent of the ground field, see [3], [5], and [14] for recent progress on polarizations.

Remark 3.5. Finally, note that the preceding results yield also that the relative Noether bound holds for Cohen-Macaulay permutation invariants, cf. [8].

References

1. Michael F. Atiyah and Ian G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading MA 1969.

2. Winfried Bruns and Jurgen Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge 1993.

3. Frank D. Grosshans, Vector Invariants in Arbitrary Characteristic, preprint 2006.

4. Nelson Killius, talk at the Dagstuhl Workshop on Computational Invariant Theory, Dagstuhl 1996.

5. Friedrich Knop, On Noether’s and Weyl’s Bound in Positive Characteristic, pp. 175-188 in: CRM Conference Proceedings Volume 35, edited by Eddy Campbell and David Wehlau, AMS, Providence RI 2004.

6. M. D. Neusel, Invariants of some Abelian p-Groups in Characteristic p, Proceedings of the AMS 125 (1997) 1921–1931.

7. Mara D. Neusel, Comparing the Depths of Rings of Invariants, pp. 189-192 in: CRM Conference Proceedings Volume 35, edited by Eddy Campbell and David Wehlau, AMS, Providence RI 2004.

8. Mara D. Neusel, Degree Bounds - An Invitation to Postmodern Invariant Theory -, Topology and its Applications 154 (2007), 792-814.

9. Mara D. Neusel, Invariant Theory, Student Mathematical Library Volume 36, AMS, Providence RI 2007.

10. Mara D. Neusel and Müfit Sezer, The Noether Map I, Forum Mathematicum, to appear.

11. Mara D. Neusel and Larry Smith, Invariant Theory of Finite Groups, Mathematical Surveys and Monographs Volume 94, AMS, Providence RI 2002.

12. Barbara Schmid, Finite Groups and Invariant Theory, pp. 35-66 in: Séminaire d’Algèbre P. Dubreil et M.-P. Malliavin 1989-1990, Lecture Notes in Mathematics 1478, Springer-Verlag, Heidelberg, Berlin 1991.

13. Müfit Sezer, Sharpening the Generalized Noether Bound in the Invariant Theory of Finite Groups, Journal of Algebra 254 (2002), 252-263.

14. Gerald W. Schwarz, When Polarizations Generate, preprint 2006.

15. Larry Smith, On a Theorem of Barbara Schmid, Proceedings of the AMS, 128 (2000), 2199-2201.

Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lubbock, Texas 79409

E-mail address: Mara.D.Neusel@ttu.edu