A NEW DEGREE BOUND FOR VECTOR INVARIANTS OF SYMMETRIC GROUPS

P. FLEISCHMANN

Abstract. Let R be a commutative ring, V a finitely generated free Rmodule and G leGL_{R}(V) a finite group acting naturally on the graded symmetric algebra A = S(V). Let β(V, G) denote the minimal number m, such that the ring A^{G} of inva riants can be generated by finitely many elements of degree at most m. For G = Σ_{n} and V (n, k), the k - fold direct sum of the natural permutat ion module, one knows that β(V (n, k), Σ_{n}) ≤ n, provided that n! is invertible in R. This was used by E.Noether to prove β(V, G) ≤ |G| if |G|! ∈ R^{∗}. In this paper we prove β(V (n, k), Σ_{n}) ≤ max{n, k(n − 1)} for arbitrary commutative rings R and show equality for n = p^{s} a prime power and R = ℤ or any ring with n · 1_{R} = 0. Our results imply β(V, G) ≤ max{|G|, rank(V)(|G| − 1)} for any ring with |G| ∈ R^{∗}.

1. Introduction

Let R be a commutative ring, V a finitely generated free R - module and G ≤ GL_{R}(V) a finite group. Then G acts naturally on the symmetric graded algebra

A := S_{R}(V) = ⊕_{i=0}^{∞}S_{R}(V)_{i}

where S_{R}(V)_{i} denotes the i- th symmetric power of V. Notice that the action of G preserves degrees, hence the algebra A^{G} of invariants inherits the grading from A. It has been a classical problem of 19^{th} century algebra to construct minimal sets of generators for the ring of invariants A^{G} or sets of generators with minimal degree. In [3] (1916) E. Noether gave two different constructive proofs for the fact that A^{G} is finitely generated by invariants of degree at most |G|, if ℚ ⊆ R. Both proofs can be made to work over any ring R having the property that the factorial |G|! of the group order is invertible. (e.g. see [7], [2]).

In [4] (1926), Noether presented a proof for the fact that A^{G} is finitely generated whenever R is a noetherian ring. No assumption whatsoever on invertibility of |G| is needed but the price one has to pay for this generality is, that the proof is not any more constructive. It essentially uses the fact that submodules of noetherian modules are finitely generated. The first proof in [3] uses a classical result of H. Weyl [8] on vector invariants of the symmetric groups. To state this and to show how it is applied, we have to introduce some notation:

Let V := V (n) = R\underline{n} be the free R - module with the natural permutation action of Σ_{n} on it. For k ∈ ℕ we define V (n, k) := V^{⊕k} = V ⊕ ⋯ ⊕ V to be the k - fold direct sum with diagonal action of Σ_{n} and A_{R}(n, k) := S(V (n, k)) to be the symmetric R - algebra over V (n, k). Let {x_{1}, ..., x_{n}} be a basis of V (n) then for k ∈ ℕ the set {x_{11}, ..., x_{n1}, ..., x_{1k}, ..., x_{nk}} is a basis of V^{⊕k}. A monomial X := x_{11}^{a_{11}} · ... · x_{n1}^{a_{n1}} · ... · x_{1k}^{a_{1k}} · ... · x_{nk}^{a_{nk}} ∈ S(V^{⊕k}) is of multidegree md(X) := (d_{1}, ..., d_{k}) ∈ ℕ^{k} if d_{j} = ∑_{i=1}^{n} a_{ij} and of (total) degree d(X) := ∑_{j=1}^{k} d_{j}. Clearly A_{R}(n, k) = ⊕_{\underline{m}∈ℕ^{k}}A_{R}(n, k)\underline{m} where A_{R}(n, k)\underline{m} is the set of R - linear combinations of monomials with multidegree \underline{m} and the A_{R}(n, k)\underline{m}’s are Σ_{n} - submodules. An element of A_{R}(n, k)\underline{a} is called homogeneous of multidegree \underline{m}.

Now let G = {e = g_{1}, g_{2}, …, g_{n}} be an arbitrary finite group of order n and let W = Rw_{1} ⊕ ⋯ ⊕ Rw_{k} be an RG - module of rank k. Consider the Cayley - embedding G ↪ Σ_{n}, g ↦(g_{i} ↦g_{j} := gg_{i}). Then

ν : A_{R}(n, k) → S_{R}(W), x_{iℓ} ↦g_{i}(w_{ℓ})

is a G - equivariant algebra epimorphism which preserves degrees. In fact

ν(g(x_{iℓ})) = ν(x_{jℓ}) = g_{j}(w_{ℓ}) = gg_{i}(w_{ℓ}) = gν(x_{iℓ}).

Now, for completeness, we repeat Noether’s original 1916 - argument showing that the restriction

ν : A_{R}(n, k)^{Σ_{n}} → S_{R}(W)^{G}

is surjective, whenever n is invertible in R: In this case, for any f = f(w_{1}, …, w_{k}) ∈ S_{R}(V)^{G} we can define

F := \frac{1}{n}(f(x_{11}, x_{12}, …, x_{1k}) + ⋯ + f(x_{n1}, x_{n2}, …, x_{nk})) ∈ A_{R}(n, k)^{Σ_{n}} and get

ν(F) = \frac{1}{n}(f(g_{1}(w_{1}), g_{1}(w_{2}), …, g_{1}(w_{k})) + ⋯ + f(g_{n}(w_{1}), g_{n}(w_{2}), …, g_{n}(w_{k})) = \frac{1}{n}(g_{1}f(w_{1}, w_{2}, …, w_{k}) + ⋯ + g_{n}f(w_{1}, w_{2}, …, w_{k}) = f.

Let β(W, G) denote the minimum number m such that S_{R}(W)^{G} is generated by finitely many invariants of degree at most m and β_{R}(G) := max{β(W, G)}, where W ranges through all RG - modules that are finitely generated and free as R - modules. We see immediately that |G| ∈ R^{∗} implies

β(W, G) ≤ β(V (n, k), Σ_{n}).

For any f = f(x_{1}, …, x_{n}) ∈ S(V (n)) and \underline{m} = (m_{1}, …, m_{k}) ∈ ℕ^{k} we define P ol(f)\underline{m} to be the A_{R}(n, k)\underline{m} - part of the element f(x_{11} + ⋯ + x_{1k}, …, x_{n1} + ⋯ + x_{nk}). Notice that P ol(f)\underline{m} ∈ A_{R}(n, k)^{G} if f ∈ S(V (n))^{G} for any G ≤ Σ_{n}; moreover P ol(f)\underline{m} = 0 whenever ∑_{i} m_{i} exceedes the degree of f ∈ S(V (n)). Now Weyl’s theorem states that, if R = K is a field containing ℚ, then

A_{K}(n, k)^{Σ_{n}} = K[P ol(e_{i})\underline{m} | i = 1, ..., n, \underline{m} ∈ ℕ^{k}],

where e_{i} is the i -th elementary - symmetric polynomial. Since (e_{i})\underline{m} = 0 if \underline{m} = (m_{1}, ..., m_{k}) and ∑_{j} m_{j} > i, A_{K}(n, k)^{Σ_{n}} is generated by a finite number of elements of degree ≤ n.

Recently, new proofs for this result have been given by D. Richman and others, which work over any commutative ring R with n! ∈ R^{∗} (see [2]), In this case Weyls theorem implies β(V (n, k), Σ_{n}) ≤ n = |G| hence β_{R}(G) ≤ |G|. But, as we will see, Weyl’s theorem does no longer hold in positive characteristic ≤ n. Hence although the restriction ν : A_{R}(n, k)^{Σ_{n}} → S_{R}(V)^{G} is surjective in the more general situation where only |G| is invertible in R, a proof of β_{R}(G) ≤ |G| in this generality is not known at the moment of this writing (see [2] and [6] for proofs in the case of solvable groups).

It is the aim of this paper to analyze vector invariants of Σ_{n} in the situation where R is an arbitrary commutative ring like for example the ring of rational integers ℤ.

In [1], Campbell, Hughes and Pollack prove β(V (n, k), Σ_{n}) ≤ max\{|G|, k · ( \begin{matrix}n \\ 2 \end{matrix} )\}

and thus obtain β(V, G) ≤ max\{|G|, rank(V) · ( \begin{matrix}|G| \\ 2 \end{matrix} )\} for any commutative ring

R with |G| ∈ R^{∗}.

I will prove that β(V (n, k), Σ_{n}) ≤ max{n, k(n − 1)} for arbitrary commutative rings R, with equality if n = p^{s} a prime power and R = ℤ or p · 1_{R} = 0. This will imply

β(V, G) ≤ max{|G|, rank(V)(|G| − 1)}

if |G| is invertible in R. It also implies for the ’global bounds’ over R = ℤ or R = K, a field of characteristic p:

β_{R}(Σ_{p^{s}}) = ∞.

Hence in this case there is no finite global bound at all, even for the smallest nontrivial group Σ_{2}.

2. Multiplication in invariant rings

Let R, V, A and G be as in the introduction and U ≤ G a subgroup. Then we have the following homomorphism of R - modules:

T_{U}^{G} : A^{U} → A^{G}, a ↦ ∑_{g∈[G:U]} ^{g}a,

where [G : U] denotes an arbitrary cross section of (left) U - cosets in G, i.e. G = ⊎_{g∈[G:U]}gU. This is called the (relative) trace homomorphism. It preserves (multi) degrees and if the index |[G : U]| is invertible in R, then T_{U}^{G} is an epimorphism. (Since a ∈ A^{U}, its relative trace T_{U}^{G}(a) does not depend on the choice of coset representatives).

We will give a ’Mackey formula’ for the product of relative traces which is particularly useful in the case of invariant rings of permutation modules.

Let U, V ≤ G be subgroups and suppose we are given a a cross section [U : G : V] of U, V double - cosets with G = ⊎_{g∈[U:G:V]}UgV and for each g ∈ [U : G : V] a cross section [UgV : V] ⊆ U of V - cosets with UgV = ⊎_{t∈[UgV:V]}tgV. Then [UgV : V] is also a cross section [U : U ∩ ^{g}V] of U ∩ ^{g}V cosets in U, i.e. U = ⊎_{t∈[UgV:V]}tU ∩ ^{g}V. Here, as in the following ^{g}V denotes gV g^{−1}.

Proposition 2.1. Let U, V ≤ G be subgroups and [U : G : V] be an arbitrary cross section of U − V - double cosets in G. Then we have for each a ∈ A^{U} and b ∈ A^{V}:

T_{U}^{G}(a) · T_{V}^{G}(b) = ∑_{g∈[U:G:V]} T_{U∩ ^{g}V}^{G} (agb).

Proof. First notice that the summands on the right hand side do not depend on the choice of double coset representatives. Indeed if u ∈ U and v ∈ V, then we have

T_{U∩ ^{ugv}V}^{G} (a(ugvb)) = T_{U∩ ^{ug}V}^{G} ((ua)(ugb)) = T_{U∩ ^{ug}V}^{G} (u(agb)) = T_{U∩ ^{g}V}^{G} (agb).

Now we choose [U : G : V] and [UgV : V] such that G = ⊎_{g∈[U:G:V],t∈[U:U∩ ^{g}V]}tgV and conclude T_{U}^{G}(a) · T_{V}^{G}(b) = T_{U}^{G}(a · T_{V}^{G}(b)) = ∑_{d∈[G:U],h∈[G:V]} d[a(hb)] = ∑_{g∈[U:G:V],t∈[U:U∩ ^{g}V],d∈[G:U]} d[a(tgb)] = ∑_{g∈[U:G:V],t∈[U:U∩ ^{g}V],d∈[G:U]} dt[a(gb)] = ∑_{g∈[U:G:V]} T_{U∩ ^{g}V}^{G} (a(gb)). ∘

From now on we suppose that G ≤ Σ_{n} and V := V (n), i.e. that V is a permutation module of G. It is well known that in this case the orbit sums of monomials form a free R - basis of A^{G} so the formulae above, applied to momomials, completely describe multiplication in A^{G}. More precisely, for any monomial a ∈ A let G_{a} denote the stablizer G_{a} := {g ∈ G| ^{g}a = a} and a_{G}^{+} denote the G - orbit sum ∑_{g∈[G:G_{a}]} ^{g}a. (We write a^{+} if G is clear from the context). Then we have

Corollary 2.2.

a_{G}^{+} · b_{G}^{+} = ∑_{g∈[G_{a}:G:G_{b}]} \frac{|G_{agb}|}{|G_{a} ∩ ^{g}G_{b}|}(ag(b))_{G}^{+}.

3. The ’row - lex’ order of matrices

Let the set of k - vectors over ℕ be ordered lexicographically, i.e. by the rule \underline{a} = (a_{1}, a_{2}, ..., a_{k}) < \underline{b} = (b_{1}, b_{2}, ..., b_{k}) if and only if there is 1 ≤ j_{0} ≤ k with a_{j} = b_{j} for j < j_{0} and a_{j_{0}} < b_{j_{0}}. We say that \underline{b} (strictly) dominates \underline{a} iff a_{i} ≤ b_{i} for all i = 1, …, k (with at least one a_{i} < b_{i}). Notice that \underline{a} < \underline{b} whenever \underline{b} strictly dominates \underline{a} but \underline{a} < \underline{b} does not imply that \underline{a} is dominated by \underline{b} (e.g. (1, 2, 2) < (2, 1, 1) ). We will identify the set M := M(n, k) of monomials in A := A_{R}(n, k) with the set Mat_{n,k}(ℕ_{0}) of nonnegative n, k - matrices using the bijection

c : M(n, k) → Mat_{n,k}(ℕ_{0}), x_{11}^{a_{11}} · ... · x_{n1}^{a_{n1}} · ... · x_{1k}^{a_{1k}} · ... · x_{nk}^{a_{nk}} ↦(a_{ij}).

Now we define a total order on the set M := M(n, k) of monomials in the following way (we call this the ’row - lex’ order) : a = (a_{ij}) < b = (b_{ij}) if and only if d(a) < d(b) or d(a) = d(b) and md(a) < md(b) or md(a) = md(b) and there is a row index i_{0} ∈ ℕ such that the row vectors a_{i} = b_{i} for all i < i_{0} and a_{i_{0}} < b_{i_{0}}. Hence the matrices are ordered first by total degree, then multidegree and, if these coincide by the ’row - lexicographic’ order. This amounts to the total order of variables x_{ij} given by x_{11} > x_{21} > ⋯ > x_{n1} > x_{12} > x_{22} ⋯ > x_{(n−1)n} > x_{nn}. Notice that the action of Σ_{n} on M corresponds to the permutation of rows of a = (a_{ij}) i.e. ^{σ}a := (a_{σ^{−1}(i),j}) = σ̂ ∘ (a_{ij}) where σ̂ := ^{σ}(δ_{ij}) is the permutation matrix corresponding to σ ∈ Σ_{n}. Moreover the product of monomials is given by componentwise addition of matrix entries

ab = c = (c_{ij}) with c_{ij} = a_{ij} + b_{ij}

hence md(ab) = md(a)+md(b). For a, b, c ∈ M we have ab ≥ a with equality if and only if b = 1 (the zero matrix) and a < b ⟺ ac < bc. For f = ∑_{a∈M} r_{a}a ∈ A we define HT (f) to be the highest monomial a with respect to ”<”, that appears in f with nonzero coefficient r_{a}. Now we get a preorder (i.e. a transitive relation) on A by defining f to be less than g ∈ A (f ≺ g) if and only if HT (f) < HT (g). We also define f ⪯ g ⟺ f = g or f ≺ g. For a ∈ M the orbit sum a^{+} will usually be written by displaying its highest element a = max a^{G}. We get immediately:

Lemma 3.1. Let a, b ∈ M with a := max a^{G} and b := max b^{G}. Then G_{ab} = G_{a} ∩ G_{b} and

(ab)^{+} − a^{+}b^{+} ≺ (ab)^{+}.

Proof. First note that G_{a}∩G_{b} ⊆ G_{ab} so we only have to prove the opposite inclusion. Suppose then that g ∈ G but g ∉ G_{a} ∩ G_{b}. Then without loss of generality we may assume g ∉ G_{a}. Hence ga ≺ a and g(ab) = g(a)g(b) ≺ ab. Therefore g ∉ G_{ab} and so G_{ab} = G_{a} ∩ G_{b}. Obviously the lead term of a^{+}b^{+} is ab; to finish the proof of the lemma it suffices to show that ab is also the lead term of (ab)^{+}. Suppose g ∈ G with g ∉ G_{ab}, then we can assume that g ∉ G_{b}, hence g(ab) = g(a)g(b) ⪯ a(g(b)) ≺ ab. ∘ Remark 3.2. Notice that, by 2.2 we have in fact

(ab)^{+} − a^{+}b^{+} = −\{ ∑_{g∈[G_{a}:G:G_{b}]\G_{a}·G_{b}} \frac{|G_{agb}|}{|G_{a} ∩ ^{g}G_{b}|}(ag(b))^{+}\}.

For j = 1, …, k let \underline{1}^{(j)} ∈ M denote the matrix whose entries are all one in the j - th column and zero elsewhere. Let a ∈ M be such that for some column index j and all i = 1, …, n the entries a_{ij} are positive. Then

a^{+} = ( \begin{matrix}a_{11} & … & a_{1j} − 1 & … & a_{1k} \\ a_{21} & … & a_{2j} − 1 & … & a_{2k} \\ … & … & … & . . . . . . \\ a_{n1} & … & a_{nj} − 1 & … & a_{nk} \end{matrix} )^{+} · ( \begin{matrix}0 & … & 1 & … & 0 \\ 0 & … & 1 & … & 0 \\ … & … & … & . . . . . . \\ 0 & … & 1 & … & 0 \end{matrix} )^{+} .

Hence A^{G} can be generated by the \underline{1}^{(i)}’s and orbit sums a^{+} such that each column of a ∈ M has a zero entry. If G = Σ_{n} and ã ∈ M, then max ã^{G} has a particularly easy shape: let (a_{1} ≥

a_{2} ≥ ⋯ ≥ a_{n}) be the family of row vectors of ã and consider b := ( \begin{matrix}b_{1} \\ b_{2} \\ … \\ b_{n} \end{matrix} ) ≠

a := ( \begin{matrix}a_{1} \\ a_{2} \\ … \\ a_{n} \end{matrix} ), where the unordered families (a_{i}) and (b_{i}) coincide. Let i_{0} :=

min{i | a_{i} ≠ b_{i}} then the families (a_{i})_{1≤i<i_{0}} and (b_{i})_{1≤i<i_{0}} coincide and b_{i_{0}} must be one of the a_{i}’s with i > i_{0}, hence b_{i_{0}} < a_{i_{0}} and b < a. We see that a is the maximal element in the orbit a^{Σ_{n}} = ã^{Σ_{n}}.

4. Indecomposable invariants

The following definition as well as the lemma make sense for arbitrary invariant rings of finite groups:

Definition 4.1. For d ∈ ℕ let R(d) := R[b ∈ A^{G} | d(b) < d]. An element a ∈ A^{G} is called indecomposable if and only if a ∉ R(d(a)) and decomposable if a ∈ R(d(a)). (In particular 0 is decomposable.)

Lemma 4.2. Let S ⊆ A^{G} then the following are equivalent:

(i): A^{G} = R[S];

(ii): For each a ∈ M with a^{+} ∈ A^{G}\S there is f ∈ R[S] with f = 0 or d(f) ≤ d(a) such that a^{+} − f is decomposable.

Proof. ’(i) ⇒ (ii)’: We can take f = a^{+}. ’(ii) ⇐ (i)’ : Let m ∈ M be of minimal degree d such that m^{+} ∈ A^{G}\R[S]. Then m^{+} ∉ S, so c := m^{+} + f is decomposable for some f ∈ R[S] of degree ≤ d or f = 0. In any case d(c) ≤ d and c ∈ R(d), so it is a linear combination of products of elements b^{+} of degree < d. By minimality of d, each of these b^{+}’s lies in R[S], hence c and m^{+} lie in R[S], a contradiction. So R[S] = A^{G}. ∘

Remark 4.3.

β(V (n, k), G) = max{d | R(d) < A^{G}} = min{d | R(d + 1) = A^{G}}.

From not on let G := Σ_{n} and A := A(n, k).

Definition 4.4. Let B^{1} denote the R - module spanned by those elements b^{+} ∈ A^{G} with b ∈ M such that the first column vector \underline{b}^{1} of b lies in \{0, 1\}\underline{n} and all rows of b starting with 1 in the first column are of the form (1, 0, …, 0).

Lemma 4.5. For any a ∈ M we have:

a^{+} ∈ (B^{1} ∩ A_{md(a)}^{G}) + R(d(a^{+})).

This implies that if md(a) = (a_{1}, a_{2}, …, a_{k}) with a_{1} > n or a_{1} = n and md(a) ≠ (n, 0, 0, …, 0) then a^{+} is decomposable and if md(a) = (n, 0, 0, …, 0) then a^{+}−r\underline{1}^{(1)} is decomposable for some r ∈ R.

Proof. We use induction with respect to the ≺ relation. Let a = max a^{G} with d(a^{+}) = d and assume that all b^{+} with b^{+} ≺ a^{+} (and md(b) = md(a) = (a_{1}, …, a_{k})) lie already in B^{1} ∩ A_{md(a)}^{G} + R(d). We can assume that at least one zero appears in the first column of a. Now there are three cases to consider: the first column has three distinct entries, two distinct entries with the non-zero entry bigger than one, and two distinct entries with the non-zero entry equal to one.

So we assume first that a_{11} ≥ a_{21} ≥ … a_{s1} > c = a_{(s+1)1} = ⋯ = a_{t1} > 0 = a_{(t+1)1} = ⋯ = a_{n1}. Consider

a^{+} = ( \begin{matrix}a_{11} & \underline{b}_{1} \\ a_{21} & \underline{b}_{2} \\ … & … \\ a_{s1} & \underline{b}_{s} \\ c & \underline{b}_{s+1} \\ c & \underline{b}_{s+2} \\ … & … \\ c & \underline{b}_{t} \\ 0 & \underline{b}_{t+1} \\ … & … \\ 0 & \underline{b}_{n} \end{matrix} )^{+}

where the \underline{b}_{i} are appropriate nonnegative k − 1 vectors satisfying \underline{b}_{t} ≤ ⋯ ≤ \underline{b}_{s+1}

and \underline{b}_{n} ≤ ⋯ ≤ \underline{b}_{t+1}. Let a′ = ( \begin{matrix}a_{11} − 1 & \underline{b}_{1} \\ a_{21} − 1 & \underline{b}_{2} \\ … & … \\ a_{s1} − 1 & \underline{b}_{s} \\ c & \underline{0} \\ c & \underline{0} \\ … & … \\ c & \underline{0} \\ 0 & \underline{b}_{t+1} \\ … & … \\ 0 & \underline{b}_{n} \end{matrix} ) and e = ( \begin{matrix}1 & \underline{0} \\ 1 & \underline{0} \\ … & … \\ 1 & \underline{0} \\ 0 & \underline{b}_{s+1} \\ 0 & \underline{b}_{s+2} \\ … & … \\ 0 & \underline{b}_{t} \\ 0 & \underline{0} \\ … & … \\ 0 & \underline{0} \end{matrix} ) then

a′ = max a′^{G} and e = max e^{G} and we get a^{+} − a′i^{+}e^{+} = (a′e)^{+} − a′^{+}e^{+} ≺ a^{+}. Hence a^{+} − a′^{+}e^{+} ∈ B^{1} ∩ A_{md(a)}^{G} + R(d). Obviously a′^{+}e^{+} ∈ R(d) so we conclude

a^{+} ∈ B^{1} ∩ AG ; \underline{d} + R(d).

So we can assume that a = ( \begin{matrix}c & \underline{b}_{1} \\ c & \underline{b}_{2} \\ … & … \\ c & \underline{b}_{s} \\ 0 & \underline{b}_{s+1} \\ 0 & \underline{b}_{s+2} \\ … & … \\ 0 & \underline{b}_{n} \end{matrix} ) with c > 1 and \underline{b}_{n} ≤ ⋯ ≤ \underline{b}_{s+1}.

Now let a′ = ( \begin{matrix}c − 1 & \underline{b}_{1} \\ c − 1 & \underline{b}_{2} \\ … & … \\ c − 1 & \underline{b}_{s} \\ 0 & \underline{0} \\ 0 & \underline{0} \\ … & … \\ 0 & \underline{0} \end{matrix} ) and e = ( \begin{matrix}1 & \underline{0} \\ 1 & \underline{0} \\ … & … \\ 1 & \underline{0} \\ 0 & \underline{b}_{s+1} \\ 0 & \underline{b}_{s+2} \\ … & … \\ 0 & \underline{b}_{n} \end{matrix} ). Again a′ = max a′^{G} and

e = max e^{G} and we conclude as above.

Finally we can assume that a = ( \begin{matrix}1 & \underline{b}_{1} \\ 1 & \underline{b}_{2} \\ … & … \\ 1 & \underline{b}_{s} \\ 0 & \underline{b}_{s+1} \\ 0 & \underline{b}_{s+2} \\ … & … \\ 0 & \underline{b}_{n} \end{matrix} ) with \underline{b}_{s} ≤ ⋯ ≤ \underline{b}_{1} and \underline{b}_{n} ≤ ⋯ ≤

\underline{b}_{s+1}.

Now let a′ = ( \begin{matrix}1 & \underline{0} \\ 1 & \underline{0} \\ … & … \\ 1 & \underline{0} \\ 0 & \underline{b}_{s+1} \\ 0 & \underline{b}_{s+2} \\ … & … \\ 0 & \underline{b}_{n} \end{matrix} ) and e = ( \begin{matrix}0 & \underline{b}_{1} \\ 0 & \underline{b}_{2} \\ … & … \\ 0 & \underline{b}_{s} \\ 0 & \underline{0} \\ 0 & \underline{0} \\ … & … \\ 0 & \underline{0} \end{matrix} ). Again a′ = max a′^{G} and

e = max e^{G} and we conclude as above. If a_{1} > n or a_{1} = n and md(a) ≠ (n, 0, …, 0) then, by definition, B^{1} ∩ A_{md(a)}^{G} = 0 and a^{+} must be decomposable. If md(a) = (n, 0, …, 0) then B^{1} ∩ A_{md(a)}^{G} = R\underline{1}^{(1)}. This finishes the proof. ∘

Notice that we also have a right Σ_{k} - action on A by permuting ’columns’ of monomials. On M this action is described by right multiplication with k × n - permutation matrices. Hence A is a RΣ_{n} - RΣ_{k} - bimodule; in particular Σ_{k} also acts on the invariant ring A^{G}. Clearly the total degree of a monomial is not changed by column permutations; hence if a, b ∈ M are largest in their Σ_{n} - orbits and a and b lie in the same Σ_{k} - orbit, then a^{+} is decomposable if and only if b^{+} is decomposable. This observation leads to the following

Theorem 4.6. For j = 1, …, k let \underline{1}^{(j)} ∈ M denote the matrix whose entries are all one in the j - th column and zero elsewhere and define

M_{n} := \{a^{+} ∈ A^{Σ_{n}} | md(a) is dominated by (n−1, …, n−1)\}∪\{\underline{1}^{(j)} | 1 ≤ j ≤ k\}.

Then A^{Σ_{n}} = R[M_{n}]. In particular

β(V (n, k), Σ_{n}) ≤ max{n, k(n − 1)},

i.e. A^{Σ_{n}} can be generated by elements of degree ≤ max{n, k(n − 1)}.

Proof. By 4.2 it suffices to show that for each element a ∈ M with a^{+} not in M_{n}, there is a suitable f ∈ R[M_{n}] with d(f) ≤ d(a) or f = 0 such that a^{+} − f is decomposable. So suppose that a^{+} has multidegree (a_{1}, …, a_{n}) with total degree d and a_{i} ≥ n. Since M_{n} and R(d) are stable under column permutations, we can assume that i = 1. Now 4.5 tells us that a^{+} ∈ (B^{1} ∩ A_{md(a)}^{G}) + R(d) and that if a_{1} > n or a_{1} = n and md(a) ≠ (n, 0, …, 0) a^{+} must be decomposable. If md(a) = (n, 0, …, 0) then a^{+} − r · \underline{1}^{(1)} is decomposable for some r ∈ R. ∘

As mentioned in the introduction, we will now show that our new degree bound is ’best possible’ in the following sense:

Theorem 4.7. Let n = p^{s} be the power of a prime p and R be ℤ or a ring with p · 1_{R} = 0. Then

β(V (n, k), Σ_{n}) = max{n, k(n − 1)}.

Proof. Let G = Σ_{n} and A_{R} denote A if it is defined over R. Since V is a permutation module, A_{R}^{Σ_{n}} is a free R - module, spanned by orbit sums of monomials. Hence A_{R}^{Σ_{n}} = A_{ℤ}^{Σ_{n}} ⊗_{ℤ} R and we have β(V_{R}(n, k), Σ_{n}) ≤ β(V_{ℤ}(n, k), Σ_{n}). Hence it suffices to show that for each k ≥ 1 and R with p · 1_{R} = 0 the element a_{k}^{+} with

a_{k} := ( \begin{matrix}1 & \underline{0} \\ 1 & \underline{0} \\ … & … \\ 1 & \underline{0} \\ 0 & \underline{b}_{n−1} \end{matrix} )

and \underline{b}_{n−1} = (n−1, …, n−1) ∈ ℕ_{0}^{k−1} is indecomposable in A(n, k)^{G}. We use induction on k. We have a_{1}^{+} = e_{n−1}; notice that A_{R}(n, 1)^{Σ_{n}} = R[e_{1}, …, e_{n}] is a polynomial ring generated by the elementary symmetric polynomials e_{i}, i = 1, …, n, which are algebraically independent; hence the indecomposability of a_{1}^{+} is a consequence of the main theorem on symmetric functions. Assume now that a_{k−1}^{+} is indecomposable but a_{k}^{+} is not. Then

a_{k}^{+} ∈ ∑_{\underline{0}≠\underline{d}_{k}} A(n, k)_{\underline{d}_{k}}^{G} · A(n, k)_{\underline{n−1}_{k}−\underline{d}_{k}}^{G},

where \underline{n − 1}_{k} = (n−1, …, n−1) ∈ ℕ^{k} and the \underline{d}_{k} are strictly dominated by \underline{n − 1}_{k}. Now we consider the G - equivariant ring epimorphism s : A_{R}(n, k) → A_{R}(n, k − 1) which is the identity on the x_{iℓ} for ℓ < k and maps x_{ik} to 1 for i = 1, …, n. Notice that for b ∈ M, s(b^{+}) = \frac{|G_{s(b)}|}{|G_{b}|}(s(b))_{G}^{+}. In particular s(a_{k}^{+}) = \frac{|G_{a_{k−1}} |}{|G_{a_{k}} |}a_{k−1}^{+} = a_{k−1}^{+}.

It is easy to see from 4.5 that each h^{+} ∈ A_{(0,...,0,i)}^{G} with 0 < i ≤ n − 1 is a linear

combination of products of elements of the form b_{j}^{+} with b_{j} = ( \begin{matrix}\underline{0} & 1 \\ \underline{0} & 1 \\ … & … \\ \underline{0} & 0 \\ \underline{0} & 0 \end{matrix} ) with

0 < j < n ones and n − j zeros in the last column. We get G_{s(b_{j})} = G and hence

\frac{|G_{s(b_{j})}|}{|G_{b_{j}} |} = |b_{j}^{G}| = ( \begin{matrix}n \\ j \end{matrix} ) ≡ 0 mod(p) by Lucas’ congruences, hence s(h^{+}) = 0 for

each h^{+} ∈ A_{(0,...,0,i)}^{G} with 0 < i < n. Clearly we have s(A(n, k)_{\underline{d}_{k}}^{G}) ⊆ A(n, k − 1)_{\underline{d}_{k−1}}^{G}, hence

a_{k−1}^{+} = s(a_{k}^{+}) ∈ ∑_{\underline{0}≠\underline{d}_{k−1}} A(n, k − 1)_{\underline{d}_{k−1}}^{G} · A(n, k − 1)_{\underline{n−1}_{k−1}−\underline{d}_{k−1}}^{G}.

Since s(A(n, k)_{(0,...,0,i)}^{G}) = 0 for 0 < i ≤ n − 1, all ’surviving’ summands must have \underline{d}_{k−1} = (d_{1}, …, d_{k−1}) different from (0, …, 0) and strictly dominated by (n − 1, …, n − 1). But this implies that a_{k−1}^{+} is decomposable, which is a contradiction. ∘

Acknowledgement: I thank the referee for carefully reading the manuscript and for valuable suggestions, which improved the exposition of the paper.

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Current address: Institute for Experimental Mathematics University of Essen Ellernstr. 29, 45326 Essen, Germany

  E-mail address: peter @exp-math.uni-essen.de