IWR-Preprint 97-40, Heidelberg 1997

Lower Degree Bounds for Modular Invariants and a Question of I. Hughes

Gregor Kemper

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

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

September 18, 1997

Abstract

We prove two statements. The first one is a conjecture of Ian Hughes which states that if f_{1}, …, f_{n} are primary invariants of a finite linear group G, then the least common multiple of the degrees of the f_{i} is a multiple of the exponent of G.

The second statement is about vector invariants: If G is a permutation group and K a field of positive characteristic p such that p divides |G|, then the invariant ring K[V^{m}]^{G} of m copies of the permutation module V over K requires a generator of degree m(p − 1). This improves a bound given by Richman [6], and implies that there exists no degree bound for the invariants of G which is independent of the representation.

Introduction

In a talk given in May 1996, Ian Hughes proved the following statement: Let G ≤ GL(V) be a finite linear group acting on a finite dimensional vector space V over a field K such that char(K) ∤ |G|, and let f_{1}, …, f_{n} ∈ K[V]^{G} be a homogeneous system of parameters for the ring of invariants (also called primary invariants). Then lcm(deg(f_{1}), …, deg(f_{n})) is a multiple of the exponent exp(G) of the group. This result is now contained in Campbell et al. [2]. He posed the question if this is also true in the modular case, i.e., if the assumption that |G| is invertible in K is dropped. In Section 1 of this paper I shall prove that this is indeed the case (Theorem 2). The main tool is the introduction of a weight function on the monomials which provides a necessary condition for a monomial to occur in an invariant.

In Section 2, these techniques are extended to the case of vector invariants. We obtain an upper degree bound for vector invariants of permutation groups which improves the (more general) bound given by Richman [6]. A consequence is that if G is a finite group and R is a commutative ring such that |G| is not a unit in R, then there exists no upper bound for the degrees of a generating set of R[V]^{G} which holds for all finitely generated RG-modules V. Thus in this case Noether’s degree bound is “arbitrarily wrong”.

1 A Property of Invariants of Cyclic Groups

This section concerns the question of Hughes mentioned above. The proof essentially repeats Hughes’ argument and inserts the following lemma at the right place. This lemma is actually the main part of the paper and of some interest in itself.

Lemma 1. Let K[x_{0}, …, x_{n}] be a polynomial algebra over a field K of characteristic p ≠ 0, and let σ be the automorphism given by

σ: x_{i} ↦ x_{i} + x_{i+1} (i = 0, …, n − 1), x_{n} ↦ x_{n}.

Then any monomial t = x_{0}^{e_{0}} ⋯ x_{n}^{e_{n}} ≠ 1 occurring in an invariant f ∈ K[x_{0}, …, x_{n}]^{σ} must satisfy

w(t) := ∑_{i=0}^{n} i · e_{i} + p -gcd(e_{0}, …, e_{n}) > n.

Here p -gcd means the p-power part of the greatest common divisor.

Proof. Let ∑_{\underline{e}=(e_{0},... ,e_{n})} a\underline{e} · \underline{x}\underline{e} ∈ K[x_{0}, …, x_{n}]^{σ} with a\underline{e} ∈ K be a homogeneous invariant of degree

d > 0, where \underline{x}\underline{e} signifies x_{0}^{e_{0}} ⋯ x_{n}^{e_{n}}. We have to consider the equation

∑_{\underline{e}} a\underline{e} · (σ(\underline{x}\underline{e}) − \underline{x}\underline{e}) = 0.  (1)

By looking at certain coefficients of this equation in a certain order, we shall show that a\underline{e} = 0 if w(\underline{x}\underline{e}) ≤ n. All monomials considered in this proof are of degree d.

We first make the following observation: If \underline{x}\underline{e} is a monomial, then the monomials \underline{x}\underline{f} occurring (with nonzero coefficient) in

σ(\underline{x}\underline{e}) − \underline{x}\underline{e} = (x_{0} + x_{1})^{e_{0}} ⋯ (x_{n−1} + x_{n})^{e_{n−1}} · x_{n}^{e_{n}} − x_{0}^{e_{0}} ⋯ x_{n}^{e_{n}}

can be characterized as follows:

From the first factor (x_{0} + x_{1})^{e_{0}} = ∑_{i=0}^{e_{0}} \binom{e_{0}}{i}x_{0}^{i}x_{1}^{e_{0}−i}, the summand \binom{e_{0}}{f_{0}}x_{0}^{f_{0}}x_{1}^{e_{0}−f_{0}} must be chosen in order to obtain the right power of x_{0}. Setting d_{1} = e_{0} − f_{0}, we must have 0 ≤ d_{1} ≤ f_{1}. Furthermore, the summand \binom{e_{1}}{f_{1}−d_{1}}x_{1}^{f_{1}−d_{1}}x_{2}^{e_{1}−f_{1}+d_{1}} must be chosen from (x_{1} + x_{2})^{e_{1}}, hence 0 ≤ d_{2} := e_{1} − f_{1} + d_{1} ≤ f_{2}. Proceeding in this way, we obtain 0 ≤ d_{i} := e_{i−1} − f_{i−1} + d_{i−1} ≤ f_{i} for 1 ≤ i ≤ n and finally f_{n} = d_{n} + e_{n}. For all i, it follows that e_{i} = f_{i} + d_{i+1} − d_{i}, where we set d_{0} = d_{n+1} = 0. Some d_{i} must be nonzero since otherwise the monomial \underline{x}\underline{f} will be cancelled, and all \binom{e_{i}}{d_{i+1}} must be nonzero as elements of K. Reversing our point of view we get the following

Observation. If a\underline{e} occurs (with nonzero coefficient) in the \underline{x}\underline{f}-coefficient of Equation (1), then there exist d_{0}, …, d_{n+1} ∈ ℤ with 0 ≤ d_{i} ≤ f_{i}, d_{0} = d_{n+1} = 0 and \binom{e_{i}}{d_{i+1}} ≠ 0, such that

e_{i} = f_{i} + d_{i+1} − d_{i}.

In particular, ord_{p}(d_{i+1}) ≥ ord_{p}(e_{i}), where ord_{p}(x) denotes the greatest k such that p^{k} divides x.

For a monomial t = \underline{x}\underline{e} with w(t) ≤ n we define the “killer monomial” Z(t) as follows: Let p^{k} = p -gcd(e_{0}, …, e_{n}), and let j be maximal with ord_{p}(e_{j}) = k. Then Z(t) is obtained by decreasing the exponent of x_{j} by p^{k} and increasing the exponent of x_{j+1} by p^{k}. If for example p = 2, then Z(x_{0}^{2}x_{1}^{6}x_{2}^{4}) = x_{0}^{2}x_{1}^{4}x_{2}^{6}. Note that j < j + p^{k} ≤ w(t) ≤ n, so x_{j+1} exists as a variable. In terms of the above observation, t can be obtained from Z(t) by taking d_{j+1} = p^{k} and all other d_{i} = 0. So a\underline{e} does occur in the Z(t)-coefficient of Equation (1).

We shall prove the assertion that a\underline{e} = 0 for w(\underline{x}\underline{e}) ≤ n by induction, so we must first introduce an ordering on the monomials. For a monomial u = \underline{x}\underline{f}, set s(u) = ∑_{i=0}^{n} i · f_{i}. We remark that if w(t) ≤ n, then w(t) = s(Z(t)). If u and u′ are two monomials, we say that u′ ≺ u if either s(u′) < s(u), or s(u′) = s(u) and u′ >_{lex} u, where >_{lex} denotes the lexicographical term order with x_{0} >_{lex} x_{1} >_{lex} … >_{lex} x_{n}. In other words, ≺ is a graded reverse lexicographical ordering with the grading given by the function s.

Now take any monomial t = \underline{x}\underline{e} with w(t) ≤ n and assume that a\underline{e}′ = 0 has been proved for all \underline{e}′ with w(\underline{x}^{\underline{e}′}) ≤ n such that Z(\underline{x}^{\underline{e}′}) ≺ Z(t). Note that a monomial is uniquely determined by its killer monomial. Hence if t is the monomial with the smallest killer monomial Z(t), then the above assumption is empty. Thus we are done by induction if we can prove the following Claim. If t′ = \underline{x}^{\underline{e}′} ≠ t is any monomial such that a\underline{e}′ occurs in the Z(t)-coefficient of Equation (1), then w(t′) ≤ n and Z(t′) ≺ Z(t).

Let t′ = \underline{x}^{\underline{e}′} be such a monomial and set u = \underline{x}\underline{f} = Z(t), so by the above observation we have e′_{i} = f_{i} + d_{i+1} − d_{i} with 0 ≤ d_{i} ≤ f_{i}, d_{0} = d_{n+1} = 0, ∑_{i=0}^{n} d_{i} > 0 and ord_{p}(d_{i+1}) ≥ ord_{p}(e′_{i}). It follows that

s(u) − s(t′) = ∑_{i=0}^{n} i · (d_{i} − d_{i+1}) = ∑_{i=0}^{n} d_{i}.

As in the definition of Z(t) let p^{k} = p -gcd(e_{0}, …, e_{n}), and set p^{k′} = p -gcd(d_{0}, …, d_{n}). We distinguish two cases:

Case I: k′ < k. Let j′ be maximal with ord_{p}(d_{j′}) = k′. Then ord_{p}(e′_{j′}) = ord_{p}(f_{j′} +d_{j′+1}−d_{j′}) = k′, and ord_{p}(e′_{i}) ≥ k′ for all i, hence p -gcd(e′_{0}, …, e′_{n}) = p^{k′}. We obtain

w(t′) = s(t′) + p^{k′} = s(u) − ∑_{i=0}^{n} d_{i} + p^{k′} ≤_{(∗)} s(u) − p^{k′} + p^{k′} = w(t) ≤ n,  (2)

where equality at (∗) holds if and only if d_{i} = 0 for i ≠ j′ and d_{j′} = p^{k′}. This proves that w(t′) ≤ n, and u′ = \underline{x}^{\underline{f}′} := Z(t′) exists. We have w(t′) = s(u′), so the inequality also proves that Z(t′) ≺ Z(t) if it is strict at (∗). Hence we only have to show that u′ >_{lex} u in the case that d_{j′} = p^{k′} and all other d_{i} are zero. But then we have

\underline{e}′ = (f_{0}, …, f_{j′−2}, f_{j′−1} + p^{k′}, f_{j′} − p^{k′}, f_{j′+1}, …, f_{n}).

As ord_{p}(f_{j′}) ≥ k > k′, it follows ord_{p}(f_{j′} − p^{k′}) = k′, so

\underline{f}′ = (f_{0}, …, f_{j′−2}, f_{j′−1} + p^{k′}, f_{j′} − 2p^{k′}, f_{j′+1} + p^{k′}, f_{j′+2}, …, f_{n}),

which is lexicographically greater than \underline{f}.

Case II: k′ ≥ k. This means that all d_{i} are divisible by p^{k}, and the same follows for all e′_{i}. Let j be maximal with ord_{p}(e_{j}) = k. By the definition of Z(t), we have ord_{p}(f_{j+1}) = k. Now assume that ord_{p}(e′_{j+1}) > k. Then ord_{p}(d_{j+2}) ≥ ord_{p}(e′_{j+1}) > k, hence ord_{p}(d_{j+1}) = ord_{p}(f_{j+1} + d_{j+2} − e′_{j+1}) must be k. This implies ord_{p}(e′_{j}) ≤ ord_{p}(d_{j+1}) = k. So we have shown that p -gcd(e′_{0}, …, e′_{n}) = p^{k}. We get the same chain of inequalities (2) with k′ replaced by k, which proves that w(t′) ≤ n and thus the existence of u′ = \underline{x}^{\underline{f}′} := Z(t′), and also the inequality Z(t′) ≺ Z(t) if (∗) in (2) is strict. So we only have to consider the case that only one of the d_{i}, say d_{j′}, is nonzero, and d_{j′} = p^{k}. As above,

\underline{e}′ = (f_{0}, …, f_{j′−2}, f_{j′−1} + p^{k}, f_{j′} − p^{k}, f_{j′+1}, …, f_{n}).

If ord_{p}(e′_{i}) = k for an i ≥ j′, then the (j′ − 1)st entry of \underline{e}′ is preserved when forming \underline{f}′ from \underline{e}′, so \underline{f}′ >_{lex} \underline{f}. On the other hand we have k ≤ ord_{p}(e′_{j′−1}) ≤ ord_{p}(d_{j′}) = k, hence if there is no such e′_{i} we obtain \underline{f}′ = \underline{f} and thus \underline{e}′ = \underline{e}. This settles Case II.

We have proved the above claim for both cases and conclude by induction that a\underline{e} = 0. But \underline{e} was chosen arbitrarily with w(\underline{x}\underline{e}) ≤ n, so the lemma is proved.

Remark. The converse of Lemma 1 does not hold in general, i.e., not all monomials t with w(t) > n occur in an invariant. As an example, take K = 𝔽_{2} and n = 2. Then by the proof the x_{1}x_{2}-coefficient of Equation (1) reads a_{1,0,1} + a_{1,1,0} = 0. But w(x_{0}x_{1}) = 2, so a_{1,1,0} = 0, and it follows a_{1,0,1} = 0. Hence t = x_{0}x_{2} occurs in no invariant, although w(t) = 3 > n. In fact, the invariants of degree 2 are

x_{2}^{2}, x_{1}^{2} + x_{1}x_{2}, and x_{1}^{2} + x_{1}x_{2} + x_{2}^{2}.

We can now prove some corollaries from Lemma 1. The first one is the promised answer to Ian Hughes’ question.

Theorem 2. Let G ≤ GL(V) be a finite linear group acting on a finite dimensional vector space V over a field K. If f_{1}, …, f_{m} ∈ K[V]^{G} are homogeneous invariants such that K[V]^{G} is finitely generated as a module over the algebra K[f_{1}, …, f_{m}], then lcm(deg(f_{1}), …, deg(f_{m})) is a multiple of the exponent exp(G). The f_{i} can be a homogeneous system of parameters or a system of fundamental invariants for K[V]^{G}, for example.

Proof. We repeat the argument given by Hughes (see Campbell et al. [2, Theorem 5.4]) and fill in Lemma 1 at the proper place.

By extending K we may assume that the polynomial X^{exp(G)} − 1 splits into linear factors over K. Let p be a prime number and k = ord_{p}(exp(G)) > 0. Then there exists a σ ∈ G such that the cyclic group H := ⟨σ⟩ has order p^{k}, and a maximal indecomposable KH-submodule W ≤ V on which H acts faithfully. We can choose any basis x_{0}, …, x_{n} of the dual W^{∗} and extend the x_{i} to linear forms on V. (Here I follow the convention that K[V] is the symmetric algebra S(V^{∗}).) The assumptions imply that also K[V] is finitely generated as a module over K[f_{1}, …, f_{m}]. In particular, there is a j > 0 such that

x_{0}^{j} ∈ (f_{1}, …, f_{m}), i.e., x_{0}^{j} = ∑_{i=1}^{m} g_{i} · f_{i}

with g_{i} ∈ K[V]. Hence one of the f_{i} must contain a power of x_{0}, say x_{0}^{e}, as a monomial. But f_{i} restricted to W lies in the invariant ring K[W]^{H}. We now distinguish two cases:

If p ≠ char(K), then W is of dimension 1 and σ(x_{0}) = ζ · x_{0} with a primitive p^{k}-th root of unity ζ. From the H-invariance of f_{i} (restricted to W) it then follows that p^{k} divides e = deg(f_{i}).

If on the other hand p = char(K), then with a suitable choice of the x_{i}, σ acts as in Lemma 1, and n ≥ p^{k−1} since H acts faithfully on W (see, for example, Alperin [1, p. 24]). Since t = x_{0}^{e} occurs in an invariant, it follows by Lemma 1 that p^{ord_{p}(e)} = w(t) > n ≥ p^{k−1}, hence p^{k} divides e.

In both cases we found an f_{i} whose degree is a multiple of p^{k}. Form this, the assertion follows.

If G is a cyclic group and V indecomposable, we get a slightly stronger result, which generalizes the lower degree bound |G| for the generators of the invariant ring which is well-known in the non-modular case.

Corollary 3. With the notation of Theorem 2 let G be cyclic and V indecomposable as a KGmodule. Then the degree of one of the f_{i} is a multiple of |G|.

Proof. Write |G| = r · p^{k} with p = char(K) and r not divisible by p. As in the previous proof, we can extend K and choose a basis x_{0}, …, x_{n} of V^{∗} such that a generating element σ of G acts by

σ: x_{i} ↦ ζ · x_{i} + x_{i+1} (i = 0, …, n − 1), x_{n} ↦ ζ · x_{n},

where ζ ∈ K is a primitive r-th root of unity and n ≥ p^{k−1}. For j = 1, 2, … we have σ^{j}(x_{i}) = ζ^{j}x_{i} + j · ζ^{j−1}x_{i+1}+ (higher x’s), in particular σ^{r}(x_{i}) = x_{i} + r · ζ^{−1}x_{i+1}+ (higher x’s). So (σ^{r} − 1) has rank n, and by yet another choice of variables x′_{0}, …, x′_{n}, σ^{r} acts as the σ of Lemma 1. We also have σ^{p^{k}}(x_{i}) = ζ^{p^{k}}x_{i}, hence σ^{p^{k}} is multiplication by ζ^{p^{k}} on the new variables, too.

As in the proof of Theorem 2, an f_{i} must contain a power x′_{0}^{e} as a monomial. Then e must be divisible by r since ζ^{p^{k}} is of order r, and by Lemma 1 e must also be divisible by p^{k}. This completes the proof.

For a graded, commutative ring R = ⊕_{d=0}^{∞}R_{d} we write

β(R) = min{k ∈ ℕ_{0} ∪ {∞} | ⊕_{d=0}^{k}R_{d} generates R as an algebra over R_{0}}.

An element of R_{d} is called irreducible if it cannot be represented as a polynomial over R_{0} in elements of degrees strictly smaller than d. We now turn back to the situation of Lemma 1 and mention a special application.

Corollary 4. With the notation of Lemma 1, let t be a monomial occurring in a homogeneous invariant f ∈ K[x_{0}, …, x_{n}]^{σ} of degree d, and suppose that each non-constant strict divisor t′ of t with w(t′) > n satisfies w(t/t′) ≤ n. Then f is irreducible, and in particular β(K[x_{0}, …, x_{n}]^{σ}) ≥ d.

Proof. This is a trivial consequence of Lemma 1: If it were possible to represent f as a polynomial in invariants of degrees strictly smaller than d, then t would occur as a product of monomials t′ and t/t′ of degrees < d, which would both appear as monomials in some invariants. But by the assumption and by Lemma 1, this cannot be the case.

Example. Take K = 𝔽_{2} and n = 3. Then σ acts as a cyclic permutation of the new variables

y_{0} = x_{0}, y_{1} = x_{0} + x_{1}, y_{2} = x_{0} + x_{2}, y_{3} = x_{0} + x_{1} + x_{2} + x_{3},

and the sum over the orbit of y_{0}^{2} · y_{1}^{3} is an invariant

f = x_{0}^{2}(x_{0} + x_{1})^{3} + (x_{0} + x_{1})^{2}(x_{0} + x_{2})^{3} + (x_{0} + x_{2})^{2}(x_{0} + x_{1} + x_{2} + x_{3})^{3} + (x_{0} + x_{1} + x_{2} + x_{3})^{2}x_{0}^{3}.

The monomial t = x_{0}^{2} · x_{2}^{3} occurs in f, and every non-constant divisor t′ of t with w(t′) > 3 must have an exponent at x_{2} of at least 2, hence w(t/t′) ≤ 3. So we have a proof involving almost no computation that the group G generated by σ violates Noether’s degree bound.

For a computation of the full invariant ring of G, see Kemper [4]. ◁

2 Vector Invariants

Lemma 1 can be generalized to the case of vector invariants.

Lemma 5. Let σ be the automorphism of K[x_{i,j} | 1 ≤ i ≤ m, 0 ≤ j ≤ n] given by

σ: x_{i,j} ↦ x_{i,j} + x_{i,j+1} (1 ≤ i ≤ m, 0 ≤ j ≤ n − 1), x_{i,n} ↦ x_{i,n} (1 ≤ i ≤ m).

Let p = char(K) ≠ 0 and define for a monomial t = ∏_{i=1}^{m} ∏_{j=0}^{n} x_{i,j}^{e_{i,j}}

w(t) = ∑_{i=1}^{m} ∑_{j=0}^{n} j · e_{i,j} + p -gcd(e_{i,j} | 1 ≤ i ≤ m, 0 ≤ j ≤ n).

Then any monomial t occurring in an invariant must have w(t) > n.

Proof. The proof is completely analogous to that of Lemma 1. The vector (d_{0}, …, d_{n+1}) occurring in the Observation (page 2) is substituted by a matrix (d_{i,j})_{1≤i≤m,0≤j≤n+1}. We put a lexicographical ordering on the index pairs (i, j) by saying (i, j) > (i′, j′) if i > i′ or i = i′ and j > j′, and then define the killer monomial of t = ∏_{i=1}^{m} ∏_{j=0}^{n} x_{i,j}^{e_{i,j}} by the maximal (i, j) such that ord_{p}(e_{i,j}) = k, where p^{k} is the p -gcd of the e_{i,j}. Finally, we define an ordering ≺ on the monomials by using a lexicographical ordering >_{lex} with x_{i,j} >_{lex} x_{i′,j′} if (i, j) < (i′, j′). Now all conclusions of the proof of Lemma 1 carry over to this situation.

In the paper [5], Richman proved that for the two dimensional indecomposable module V of the cyclic group G = Z_{p} over a field of characteristic p the vector invariants have the property β (K[V^{m}]^{G}) ≥ m(p − 1), where V^{m} = ⊕_{i=1}^{m}V. He proved similar results for another class of groups, which contains the general and special linear groups over a finite field. In [6], which was not published until very recently, he proved that for any group G ≤ GL(V) whose order is divisible by p = char(K)

β (K[V^{m}]^{G}) ≥ m · \frac{p − 1}{p^{|G|−1} − 1}.

Here we obtain a stronger lower bound for the case that G is a permutation group:

Theorem 6. Let K be a field of characteristic p, G ≤ S_{n} a permutation group which contains an element σ of order q = p^{k}, k ∈ ℕ, and V the corresponding permutation module. Moreover, let m ∈ ℕ and 0 < d ≤ m(q − 1) with p ∤ d in the case m = 1. Then any system S of fundamental invariants generating the invariant ring K[V^{m}]^{G} must contain at least one invariant of degree d. In particular,

β (K[V^{m}]^{G}) ≥ m(q − 1).

Moreover, S must contain at least q^{m}(1 − 1/p^{m}) elements.

Proof. Let b_{i,j} be a basis of ⊕_{i=1}^{m}V (1 ≤ i ≤ m, 1 ≤ j ≤ n) with σ(b_{i,j}) = b_{i,σ(j)} for σ ∈ G, and let y_{i,j} be the dual basis. Choose d_{1}, …, d_{m} with 0 ≤ d_{i} < q, such that not all d_{i} are divisible by p and d_{1} + ⋯ + d_{m} = d. There is a total of q^{m}(1 − 1/p^{m}) choices of such d_{1}, …, d_{m} for the various d = 1, …, m(q − 1). An invariant is given by

f = ∑_{j=1}^{n} ∏_{i=1}^{m} y_{i,j}^{d_{i}}.

We shall prove that f is irreducible. By renumbering the y_{i,j} we may assume that p ∤ d_{m} and that σ acts as a cyclic permutation on each of the sets {y_{i,1}, …, y_{i,q}}. We restrict f to the subspace W ≤ ⊕_{i=1}^{m}V spanned by {b_{i,j} | 1 ≤ i ≤ m, 1 ≤ j ≤ q} and obtain the σ-invariant

f|_{W} = ∑_{j=1}^{q} ∏_{i=1}^{m} y_{i,j}^{d_{i}}.

There exist basis vectors x_{i,j} (1 ≤ i ≤ m, 0 ≤ j ≤ q − 1) of W^{∗} on which σ acts as in Lemma 5. We can choose x_{i,0} such that x_{i,0} = y_{i,1}, and it follows

f|_{W} = ∑_{j=0}^{q−1} ∏_{i=1}^{m}(σ^{j}(x_{i,0}))^{d_{i}}.  (3)

We claim that the monomial t = x_{1,0}^{d_{1}} ⋯ x_{m−1,0}^{d_{m−1}} · x_{m,0}^{d_{m}−1}x_{m,q−1} occurs in f|_{W}. Indeed, σ^{j}(x_{i,0}) = ∑_{ν=0}^{q−1} \binom{j}{ν}x_{i,ν}, so x_{m,q−1} occurs in only the last summand of Equation (3), and this summand contains t with coefficient d_{m} ≠ 0.

Now each non-constant strict divisor t′ of t with w(t′) > q − 1 must contain x_{m,q−1}, thus w(t/t′) < q. By Corollary 4, f|_{W} cannot be represented in terms of σ-invariants of smaller degree. But if it were possible to represent f as a polynomial in invariants of smaller degree, then restriction to W would yield such a representation.

I thank Peter Fleischmann for making me think about how the lower bound m given in an earlier version of Theorem 6 could be improved to m(q − 1).

Remark. In [3] Fleischmann proved the exact equation

β (K[V^{m}]^{G}) = max{p^{k}, m(p^{k} − 1)}

for n = p^{k} and G = S_{n} in the situation of Theorem 6. So in this case the lower bound from Theorem 6 is sharp if m > 1. ◁

Let G be a finite group and R a commutative ring (with unity). Then we write

β(G, R) = max{β(R[V]^{G}) | V is a finitely generated RG-module} ∈ ℕ_{0} ∪ {∞}.

In this terminology, Noether’s degree bound states that if |G|! is a unit in R, then β(G, R) ≤ |G|. (The proof in Smith [7, Theorem 2.4.2] is valid over a ring instead of a field as well.) On the other end of the line, we have:

Corollary 7. Let G be a finite group and let R be a commutative ring in which |G| is not a unit. Then β(G, R) = ∞.

Proof. By assumption, there exists a maximal ideal 𝔪 in R which contains |G|. Hence the characteristic p of the field K = R/𝔪 is a divisor of |G|, and there is an element of order p contained in G. Let V be a faithful permutation module of G over R. Suppose now that for some m ∈ ℕ we had

β(R[V^{m}]^{G}) < m(p − 1).

Since V^{m} is again a permutation module, every invariant from K[K ⊗_{R} V^{m}]^{G} lifts to an invariant in R[V^{m}]^{G} and could then be represented by invariants of degree smaller than m(p − 1). Hence it would follow that

β(K[K ⊗_{R} V^{m}]^{G}) < m(p − 1),

in contradiction to Theorem 6. So we must have β(R[V^{m}]^{G}) ≥ m(p − 1), which tends to infinity with m.

This corollary also follows from Richman [6], but it was obtained here by a much shorter proof.

References

[1] J. P. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics 11, Cambridge University Press, Cambridge 1986.

[2] H. E. A. Campbell, A. V. Geramita, I. P. Hughes, G. G. Smith, D. L. Wehlau, Hilbert Functions of Graded Algebras, Preprint, Kingston, Ontario 1997.

[3] Peter Fleischmann, A New Degree Bound for Vector Invariants Of Symmetric Groups, Trans. of the Amer. Math. Soc. (to appear).

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

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