Symmetric Powers of Modular Representations, Hilbert Series and Degree Bounds

Ian Hughes^{∗} and Gregor Kemper

March 5, 1999

Abstract

Let G = Z_{p} be a cyclic group of prime order p with a representation G → GL(V) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers of V in the case that V is indecomposable. From these they derived formulas for the Hilbert series of the invariant ring K[V]^{G}. Following Almkvist and Fossum in broad outline, we start by giving a shorter, self-contained proof of their results. We extend their work to modules which are not necessarily indecomposable. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our results generalize to groups of the type Z_{p} × H with |H| coprime to p. Moreover, we prove that for any finite group G whose order is divisible by p but not by p^{2}, the invariant ring K[V]^{G} is generated by homogeneous invariants of degrees at most dim(V) · (|G| − 1).

Contents

Introduction 1

1 Symmetrization and λ-structure 3 1.1 Symmetrization by modules of symmetric groups . . . . . . . . . . . . . . . . . . . . 3 1.2 Multiplication rules for exterior powers . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Representations of the cyclic group of order p 11 2.1 A presentation for the representation ring . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Exterior powers and symmetric powers of degree < p . . . . . . . . . . . . . . . . . . 13 2.3 The periodicity of the symmetric powers . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Degree bounds for invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 The Hilbert series 20 3.1 Species, averaging operators, and Hilbert series for Z_{p}-modules . . . . . . . . . . . . 20 3.2 The quasi-Gorenstein property and some examples . . . . . . . . . . . . . . . . . . . 23 3.3 Hilbert series for the group Z_{p} × H with p ∤ |H| . . . . . . . . . . . . . . . . . . . . . 26

4 Some remarks on Z_{p^{m}} 27 4.1 The representation ring of Z_{p^{m}} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Exterior and symmetric powers, and the Periodicity Conjecture . . . . . . . . . . . . 30

References 31

∗This work was done during a visit of the first author to the University of Heidelberg.

Introduction

The easiest case of modular representation theory is the study of modules of a cyclic group G = Z_{p} of prime order p over a field K of characteristic p. In fact, there are only a finite number of indecomposable KG-modules, which are well known. It is also quite easy to compute the decompositions of tensor products of KG-modules. However, matters become much more subtle if we ask for decompositions of exterior and symmetric powers. Of particular interest is the Hilbert series

H(K[V]^{G}, t) := ∑_{r=0}^{∞} dim (S^{r}(V)^{G}) t^{r}

of the invariant ring, where S^{r}(V) denotes the r-th symmetric power and S^{r}(V)^{G} is the subspace of elements fixed by G. In a long paper [3], Almkvist and Fossum obtained complete information about the decomposition of exterior and symmetric powers, and derived formulas for H(K[V]^{G}, t) in the case that V is an indecomposable KG-module [3, pp. 36–45, p. 81, 85].

This paper originated in an attempt to read and understand Almkvist and Fossum’s groundbreaking paper. We discovered that their arguments can be presented in a much shorter way, and we obtained extensions in three directions:

(1) The restriction to indecomposable modules V is lifted.

(2) We also obtain formulas giving generating functions whose coefficients are the multiplicities of indecomposable modules in the decomposition of the symmetric powers.

(3) All results extend to the group Z_{p} × H, where H is a finite group of order coprime to p.

For the cyclic group of order p, formulas giving the Hilbert series of the invariant ring and the generating functions mentioned above can be found in Theorem 3.4. Along the way, we obtain a degree bound for homogeneous generators of the invariant ring of a finite group whose order is not divisible by p^{2}. To our knowledge, no degree bound for this class of groups was previously known. Degree bounds for generators of invariant rings play a central role in constructive invariant theory, since they give a means to compute the generators explicitly.

The paper is organized as follows. The first section is devoted to symmetrization. In particular, we set up a (well-known) machinery which translates an identity of symmetric polynomials into an isomorphism of KG-modules for any group G. This is used to obtain a relation between exterior and symmetric powers, which are special cases of symmetrization. In Section 1.2 we derive a rule to compute exterior powers of a tensor product U ⊗ V from the exterior powers of U and V. Here we roughly follow Benson’s paper [5]. The first section is the most difficult of the paper.

In the second section we specialize G to be the cyclic group of order p. Here we follow Almkvist and Fossum’s paper [3]. We obtain complete information about exterior powers, and information about symmetric powers up to degree p − 1. This is complemented by a periodicity property of symmetric powers. Apart from its utility for the investigation of Hilbert series, the periodicity leads to degree bounds for the generators of the invariant rings of cyclic groups of order p. These can be extended to the case of groups whose order is divisible by p but not by p^{2}. The results from Section 1 are heavily used in this section.

In the third section we return to G the cyclic group of order p. We introduce averaging operators which yield the Hilbert series H(K[V]^{G}, t) and the power series encoding the decompositions of symmetric powers. This is the end result and, indeed, raison d’être of the development in the first two sections. Our use of averaging operators was inspired by Almkvist [1], who used a different form of averaging operators to re-prove the formula for H(K[V]^{G}, t) in the case that V is indecomposable of odd dimension. He remarked that one could probably get the formula for V even-dimensional by similar techniques. The resulting formulas bear a remarkable resemblance to Molien’s formula. Indeed, the Hilbert series coincides with the Hilbert series of the invariant of a representation over ℂ if the dimensions of all indecomposables occurring in V are odd. In general, however, there is no invariant ring over ℂ which has the same Hilbert series. We study some properties of the Hilbert series, such as their degree and symmetry behavior, and generalize our results to groups Z_{p} × H with p ∤ |H|.

Finally, we make a few remarks about possible extensions of our method to the cyclic group of order p^{m}. We derive a presentation (folklore) of the representation ring and obtain information about exterior and symmetric powers up to degree p − 1. Then we point out the obstacles on the way to obtain formulas for the Hilbert series in that case.

In this paper K is always a field, G is a group, and KG is the group ring. All modules over KG will be left modules (unless stated otherwise). Moreover, all modules over KG and vector spaces over K will be assumed to be finite dimensional, and tensor products will be over K, unless stated otherwise. ℕ_{0} denotes the set of non-negative integers.

1 Symmetrization and λ-structure

Our main interest will lie in symmetric powers of modules over a group ring, and we will also look at exterior powers. Both symmetric and exterior powers can be obtained by the process called symmetrization, where representations of the symmetric group act as functors. In this section, S_{r} denotes the symmetric group in r symbols.

1.1 Symmetrization by modules of symmetric groups

Definition 1.1. Let W be a KS_{r}-module and assume r! ∈ K^{×} (i.e., char(K) = 0 or char(K) > r). Then for a K-space V define

Sym_{W} (V) := (V^{⊗r} ⊗ W)^{S_{r}},

where V^{⊗r} denotes the r-fold tensor product of V with itself, on which S_{r} acts by permutations of the tensor factors. Sym_{W} (V) is called the symmetrization of V by W. Sym_{W} becomes a functor from the category of finitely generated K-spaces into itself by

Sym_{W} (ϕ) = (ϕ^{⊗r} ⊗ id_{W})|_{Sym_{W} (V_{1})}

for ϕ: V_{1} → V_{2}. If r = 0 (in which case S_{r} = {id}), Sym_{W} (V) = W and Sym_{W} (ϕ) = id.

Example 1.2. If W is the trivial module then

Sym_{W} (V) = S^{r}(V),

the r-th symmetric power of V. On the other hand, if W = K with πα = sgn(π)·α is the alternating module, then

Sym_{W} (V) = Λ^{r}(V),

the r-th exterior power.

If V is a K-space then V^{⊗r} becomes a right KS_{r}-module by fπ := π^{−1}f. The next proposition shows that we could have defined Sym_{W} (V) as V^{⊗r} ⊗_{KS_{r}} W and Sym_{W} (ϕ) as ϕ^{⊗r} ⊗_{KS_{r}} W.

Lemma 1.3. Let G be a finite group with |G| ∈ K^{×} and V, W modules over KG. Make V into a right KG-module by vσ := σ^{−1}v for v ∈ V, σ ∈ G. Then

V ⊗_{KG} W ≅ (V ⊗ W)^{G}.

Under the above isomorphism, a tensor product ϕ ⊗_{KG} ψ of homomorphisms ϕ: V_{1} → V_{2}, ψ: W_{1} → W_{2} of KG-modules goes to the restriction of ϕ ⊗ ψ to (V_{1} ⊗ W_{1})^{G}.

Proof. We have to show that (V ⊗W)^{G} fulfills the universal property of the tensor product V ⊗_{KG}W. For this, we must give a balanced map Φ: V × W → (V ⊗ W)^{G}, for which we take

(v, w) ↦ \frac{1}{|G|} · ∑_{σ∈G} σ(v) ⊗ σ(w).

Now let ψ: V × W → M be a balanced map into a K-space M. Define a map η: (V ⊗ W)^{G} → M by restricting the map V ⊗ W → M, v ⊗ w ↦ ψ(v, w) to the invariants. Then

η(Φ(v, w)) = \frac{1}{|G|} ∑_{σ∈G} ψ(σ(v), σ(w)) = \frac{1}{|G|} ∑_{σ∈G} ψ(vσ^{−1}, σ(w)) = ψ(v, w).

Furthermore, η with this property is uniquely determined since the image of Φ generates (V ⊗ W)^{G} as a K-space. The result on ϕ ⊗_{KG} ψ follows from the definition of η.

If G is a group and V a KG-module, then Sym_{W} (V) also becomes a KG-module by functoriality. This makes Sym_{W} into a functor from the category of KG-modules into itself. If W ≅ W′ are isomorphic KS_{r}-modules, then by Lemma 1.3 Sym_{W} and Sym_{W′} become equivalent functors. In particular, the KG-modules Sym_{W} (V) and Sym_{W′}(V) are isomorphic.

We can now also prove some rules for symmetrization.

Proposition 1.4. Let V be a K-space and r ∈ ℕ_{0} with r! ∈ K^{×}.

(a) Let W_{1} and W_{2} be KS_{r}-modules. Then

Sym_{W_{1}⊕W_{2}}(V) ≅ Sym_{W_{1}}(V) ⊕ Sym_{W_{2}}(V).

(b) Let W_{1} be a KS_{s}-module and W_{2} a KS_{t}-module with s + t = r, and let W := KS_{r} ⊗_{K(S_{s}×S_{t})} (W_{1} ⊗ W_{2}) be the induced module of W_{1} ⊗ W_{2}. Then

Sym_{W} (V) ≅ Sym_{W_{1}}(V) ⊗ Sym_{W_{2}}(V).

Proof. Part (a) follows directly from Definition 1.1 and Lemma 1.3. As for (b), we use Lemma 1.3 twice and obtain

Sym_{W} (V) ≅ V^{⊗r} ⊗_{K(S_{s}×S_{t})} (W_{1} ⊗ W_{2}) ≅ (V^{⊗s} ⊗ W_{1} ⊗ V^{⊗t} ⊗ W_{2})^{S_{s}×S_{t}} ≅ Sym_{W_{1}}(V) ⊗ Sym_{W_{2}}(V).

We will now associate a polynomial to a KS_{r}-module W. Let V be an n-dimensional K-space with basis v_{1}, …, v_{n}. For e_{1}, …, e_{n} ∈ ℕ_{0} with e_{1} + ⋯ + e_{n} = r let ⟨v_{1}^{⊗e_{1}} ⊗ ⋯ ⊗ v_{n}^{⊗e_{n}}⟩_{KS_{r}} be the KS_{r}-submodule of V^{⊗r} generated by v_{1}^{⊗e_{1}} ⊗ ⋯ ⊗ v_{n}^{⊗e_{n}}. Set

m_{e_{1},... ,e_{n}} = dim ((⟨v_{1}^{⊗e_{1}} ⊗ ⋯ ⊗ v_{n}^{⊗e_{n}}⟩_{KS_{r}} ⊗ W)^{S_{r}}) .

The number m_{e_{1},... ,e_{n}} has the following interpretation: If ϕ ∈ End_{K}(V) is given by ϕ(v_{i}) = ξ_{i}v_{i} with ξ_{i} ∈ K such that the monomials of degree r in the ξ_{i} are all distinct, then ⟨v_{1}^{⊗e_{1}} ⊗⋯⊗v_{n}^{⊗e_{n}}⟩_{KS_{r}} ⊗W is the eigenspace of ϕ^{⊗r} ⊗ id_{W} w.r.t. the eigenvalue ξ_{1}^{e_{1}} ⋯ ξ_{n}^{e_{n}}. Hence m_{e_{1},... ,e_{n}} is the multiplicity of the eigenvalue ξ_{1}^{e_{1}} ⋯ ξ_{n}^{e_{n}} of Sym_{W} (ϕ). Moreover, ϕ^{⊗r} ⊗id_{W} is diagonalizable with all eigenvalues monomials of degree r in the ξ_{i}, hence the same holds for Sym_{W} (ϕ). Now we take x_{1}, …, x_{n} to be indeterminates and form the polynomial

g_{W,n} = ∑_{e_{1}+···+e_{n}=r} m_{e_{1},... ,e_{n}}x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}} ∈ ℤ[x_{1}, …, x_{n}],

where the summation is over all e_{1}, …, e_{n} ∈ ℕ_{0} with e_{1} + ⋯ + e_{n} = r. In the light of the above interpretation we have trace(Sym_{W} (ϕ)) = g_{W,n}(ξ_{1}, …, ξ_{n}). If π ∈ S_{n}, then

⟨v_{1}^{⊗e_{π(1)}} ⊗ ⋯ ⊗ v_{n}^{⊗e_{π(n)}}⟩_{KS_{r}} ≅ ⟨v_{1}^{⊗e_{1}} ⊗ ⋯ ⊗ v_{n}^{⊗e_{n}}⟩_{KS_{r}}

as KS_{r}-modules. Hence g_{W,n} is invariant under the permutation action of S_{n} on the x_{i}, and we can write

g_{W,n} = f_{W,n}(a_{n,1}, …, a_{n,r}),

where the a_{n,i} is the i-th elementary symmetric polynomial in the variables x_{1}, …, x_{n} and f_{W,n} ∈ ℤ[a_{1}, …, a_{r}] is a polynomial of degree r in new variables a_{i} with deg(a_{i}) = i. If n ≥ r, then the a_{n,i} (for i = 1, …, r) are algebraically independent and f_{W,n} is uniquely determined. Now we show that f_{W,n} only depends on W. Take n′ > n, then m_{e_{1},... ,e_{n}} = m_{e_{1},... ,e_{n},0,... ,0}, where on the right-hand side we appended n′ − n zeros. Hence g_{W,n}(x_{1}, …, x_{n}) = g_{W,n′}(x_{1}, …, x_{n}, 0, …, 0), and f_{W,n} = f_{W,n′} follows since a_{n,i}(x_{1}, …, x_{n}) = a_{n′,i}(x_{1}, …, x_{n}, 0, …, 0). Thus we have assigned a polynomial f_{W} ∈ ℤ[a_{1}, …, a_{r}] of degree r to W. We summarize this in the following definition.

Definition 1.5. By A = ℤ[a_{1}, a_{2}, …] we denote the graded polynomial ring over ℤ in countably many indeterminates a_{i} with deg(a_{i}) = i. For r ∈ ℕ_{0} let A_{r} denote the homogeneous part of degree r of A. If r! ∈ K^{×} and W is a KS_{r}-module, the polynomial f_{W} ∈ A_{r} defined above is called the polynomial associated to W. Moreover,

A_{r}^{+} = {f_{W} | W is a KS_{r}-module}

is called the positive cone of A_{r}.

Example 1.6. If W is the alternating module, then Sym_{W} (V) = Λ^{r}(V) (see Example 1.2), hence by the interpretation of g_{W,n}(ξ_{1}, …, ξ_{n}) as the trace of Sym_{W} (ϕ) for ϕ = diag(ξ_{1}, …, ξ_{n}) we get g_{W,n} = a_{n,r} and thus f_{W} = a_{r}. If, on the other hand, W is the trivial module, then

g_{W,n} = ∑_{deg(t)=r} t =: h_{n,r},

where the summation is over all monomials of degree r in the x_{i}. We will see in the proof of Theorem 1.11 how h_{n,r} can be expressed in terms of the a_{n,i}, which gives the polynomial f_{W} = h_{r} associated to the trivial module W.

Proposition 1.7. Let r ∈ ℕ_{0} with r! ∈ K^{×}.

(a) Let W_{1} and W_{2} be KS_{r}-modules. Then

f_{W_{1}⊕W_{2}} = f_{W_{1}} + f_{W_{2}}.

(b) Let W_{1} be a KS_{s}-module and W_{2} a KS_{t}-module with s + t = r, and let W := KS_{r} ⊗_{K(S_{s}×S_{t})} (W_{1} ⊗ W_{2}) be the induced module of W_{1} ⊗ W_{2}. Then

f_{W} = f_{W_{1}} · f_{W_{2}}.

Proof. Both parts follow from Proposition 1.4 and the interpretation of the numbers m_{e_{1},... ,e_{n}} as multiplicities of eigenvalues.

Definition 1.8. Let G be a finite group and K a field. The representation ring of KG is defined to be the ring generated by the isomorphism classes [V] of KG-modules with the relations

[V] + [W] = [V ⊕ W] and [V][W] = [V ⊗ W].

We denote the representation ring by R_{KG}, which is a commutative ring with unity. By abuse of notation we will often simply write V for the class [V] in R_{KG}, and V − W for [V] − [W].

ℤ is embedded into R_{KG} by direct sums of copies of the trivial module, which makes R_{KG} into a ℤ-algebra. As a ℤ-module, R_{KG} is free with the indecomposable KG-modules as a basis.

Theorem 1.9. Suppose r ∈ ℕ_{0} with r! ∈ K^{×}. Then the map

R_{KS_{r}} → A_{r}, [V] ↦ f_{V}

(with the notation given in Definition 1.5) is an isomorphism (of ℤ-modules).

Proof. Proposition 1.7(a) implies that the map is well-defined and a homomorphism of additive groups and thus of ℤ-modules. By Example 1.6 and Proposition 1.7(b) all monomials of degree r in the a_{i} lie in the image, hence the map is surjective. Now by non-modular representation theory the rank of R_{KS_{r}} is equal to the number of conjugacy classes of S_{r}, and this equals the number of partitions of r. On the other hand, the rank of A_{r} is equal to the number of monomials of degree r in the a_{i}, and this also equals the number of partitions of r. Hence we have an epimorphism between two free modules of equal rank, which must hence be an isomorphism.

Given an f ∈ A_{r}^{+} in the positive cone, we can choose a KS_{r}-module W with f_{W} = f. By Theorem 1.9 W is unique up to isomorphism. Thus Sym_{W} is defined by f up to equivalence of functors. In particular, if V is a KG-module for any group G, then Sym_{W} (V) is defined by f up to isomorphism of KG-modules. By a slight abuse of notation we will write Sym_{f} for Sym_{W}. Then by Propositions 1.4 and 1.7 we have the rules

Sym_{f+g}(V) ≅ Sym_{f}(V) ⊕ Sym_{g}(V) and Sym_{f·g}(V) ≅ Sym_{f}(V) ⊗ Sym_{g}(V). (1.1)

We have now set up a machinery which from an identity of symmetric polynomials constructs an isomorphism of KG-modules for any group G. For example, the identity

a_{n,1}^{2} = h_{n,2} + a_{n,2}

(which for n = 2 reads (x_{1} + x_{2})^{2} = (x_{1}^{2} + x_{1}x_{2} + x_{2}^{2}) + x_{1}x_{2}, and similarly for other n) yields the isomorphism

V^{⊗2} ≅ S^{2}(V) ⊕ Λ^{2}(V),

which holds for any KG-module V if char(K) ≠ 2. Indeed, this follows from Sym_{h_{2}}(V) = S^{2}(V) and Sym_{a_{2}}(V) = Λ^{2}(V) (see Examples 1.2 and 1.6). The above isomorphism is a special case of the following theorem. In order to state it, we need one more piece of notation.

Definition 1.10. Let V be a KG-module. Then we define

σ_{t}(V) = ∑_{r=0}^{∞} S^{r}(V) · t^{r} and λ_{t}(V) = ∑_{r=0}^{dim(V)} Λ^{r}(V) · t^{r}

as elements of the formal power series ring R_{KG}[[t]] and the polynomial ring R_{KG}[t], respectively.

Theorem 1.11. Let G be a group, K a field and V a KG-module. Then we have:

(a) If char(K) = 0, then

σ_{t}(V) · λ_{−t}(V) = 1.

(b) If char(K) = p is positive, then

σ_{t}(V) · λ_{−t}(V) ≡ 1 mod t^{p}.

Proof. We start with an identity in indeterminates x_{1}, …, x_{n}, t:

1 = ∏_{j=1}^{n}(1+x_{j}t)^{−1}· ∏_{j=1}^{n}(1+x_{j}t) = ∏_{j=1}^{n} (∑_{i=0}^{∞}(−x_{j}t)^{i})·(∑_{i=0}^{n} a_{n,i}t^{i}) = (∑_{i=0}^{∞} h_{n,i}(−t)^{i})·(∑_{i=0}^{∞} a_{n,i}t^{i}) ,

with the h_{n,i} as introduced in Example 1.6 and a_{n,0} := h_{n,0} := 1. Hence for r > 0 we have

∑_{i=0}^{r}(−1)^{i}h_{n,i}a_{n,r−i} = 0.  (1.2)

If we define polynomials h_{r} ∈ A_{r} recursively by a_{0} = h_{0} = 1 and by the equations ∑_{i=0}^{r}(−1)^{i}h_{i}a_{r−i} = 0, then (1.2) implies h_{r}(a_{n,1}, …, a_{n,r}) = h_{n,r}. Thus by Example 1.6 h_{r} is associated to the trivial KS_{r}-module if r! ∈ K^{×}, and then by Example 1.2

Sym_{h_{r}}(V) ≅ S^{r}(V).

Similarly,

Sym_{a_{r}}(V) ≅ Λ^{r}(V).

Now from (1.2) and the rules (1.1) we obtain

∑_{i=0}^{r}(−1)^{i}S^{i}(V)Λ^{r−i}(V) = 0  (1.3)

in R_{KG} (for 0 < r, r! ∈ K^{×}). The statements of the theorem are just a compact way of expressing these equalities.

Remark. If in the situation of Theorem 1.11 we have that if char(K) ∤ |G|, then the equality statement (a) always holds. In fact, one can lift the KG-module V to characteristic 0 in this case, then use Theorem 1.11(a) and reduce to characteristic p again. This works since the functors Λ^{i} and S^{i} commute with reduction modulo p.

Even in the case that p = char(K) divides |G|, more is true than stated in Theorem 1.11. In fact, we have

σ_{t}(V) · λ_{−t}(V) ∈ R_{KG}[[t^{p}]],

i.e., equality holds in degrees r not divisible by p and not only in degrees < p. A proof for this can be found in Fossum [15, Proposition 5.2], where the isomorphism (1.3) is made more explicit by considering an exact sequence coming from the Koszul-complex, which splits if p ∤ r.

In Example 2.8 we will see that the equality in Theorem 1.11(a) does in general not hold in the modular case.

1.2 Multiplication rules for exterior powers

Theorem 1.11 reduces the computation of symmetric powers to that of exterior powers, at least in degrees less than p = char(K). In order to obtain exterior powers of KG-modules for G the cyclic group of order p, a major step will be to find a rule which relates the exterior powers of a tensor product U ⊗ V to the exterior powers of U and of V. This rule will be given in Theorem 1.16, and like Theorem 1.11 it only holds in degrees less than p. In this subsection we will largely follow Benson’s paper [5].

First we define Adams operations ψ^{r} for r! ∈ K^{×}. Let G be a group and K a field. In order to define the Adams operations, we have to extend R_{KG} temporarily by forming { \atop R_{KG}} = ℂ ⊗_{ℤ} R_{KG}. Moreover, assume that K contains enough roots of unity. For r ∈ ℕ_{0} with r! ∈ K^{×} fix primitive r-th roots of unity ζ and γ in K and in ℂ, respectively. Let Z_{r} ≤ S_{r} be the cyclic subgroup generated by the permutation σ = (1, …, r). For i ∈ ℤ we have a one-dimensional KZ_{r}-module K_{i} afforded by σ ↦ ζ^{i}. Let W_{i} = KS_{r} ⊗_{KZ_{r}} K_{i} be the induced module. Now for a KG-module V we define

ψ^{r}(V) = ∑_{i=1}^{r} γ^{i} · Sym_{W_{i}}(V) ∈ { \atop R_{KG}}.

The ψ^{r} are called the Adams operations. From Lemma 1.3 we get that

Sym_{W_{i}}(V) = V^{⊗r} ⊗_{KS_{r}} KS_{r} ⊗_{KZ_{r}} ⊗K_{i} = (V^{⊗r} ⊗ K_{i})^{Z_{r}}.

Whenever the orders of σ^{i} and σ^{j} coincide, we have (V^{⊗r} ⊗K_{i})^{Z_{r}} ≅ (V^{⊗r} ⊗K_{j})^{Z_{r}} as KG-modules, because then K_{i} and K_{j} coincide up to an automorphism of Z_{r}. From this it follows that

ψ^{r}(V) = ∑_{d|r} μ(d) Sym_{W_{r/d}}(V) ∈ R_{KG},  (1.4)

where μ(d) = ∑_{i∈(ℤ/dℤ)^{×}} γ^{ri/d} ∈ ℤ (see Benson [5, Proposition 2]).

Proposition 1.12. For KG-modules V_{1} and V_{2} we have

ψ^{r}(V_{1} ⊕ V_{2}) = ψ^{r}(V_{1}) + ψ^{r}(V_{2}) and ψ^{r}(V_{1} ⊗ V_{2}) = ψ^{r}(V_{1}) · ψ^{r}(V_{2}).

Proof. (V_{1} ⊕ V_{2})^{⊗r} is the direct sum of all products of the form V_{i_{1}} ⊗ ⋯ ⊗ V_{i_{r}} with i_{j} ∈ {1, 2}. Take such a product U = V_{i_{1}} ⊗ ⋯ ⊗ V_{i_{r}} and let Z_{d} ≤ Z_{r} be the subgroup of those σ^{i} which map U into itself, d|r. If W ≤ (V_{1} ⊕ V_{2})^{⊗r} is the submodule spanned by the Z_{r}-images of U, then (W ⊗ K_{i})^{Z_{r}} ≅ (U ⊗ K_{i})^{Z_{d}}. But (U ⊗ K_{i})^{Z_{d}} ≅ (U ⊗ K_{j})^{Z_{d}} (as KG-modules) if i ≡ j mod d, hence

∑_{i=1}^{r} γ^{i}(W ⊗ K_{i})^{Z_{r}} = ∑_{i=1}^{d} ( ∑_{j=0}^{r/d−1} γ^{dj}) γ^{i} · (U ⊗ K_{i})^{Z_{d}},

which is 0 if d < r. Hence we are left with

ψ^{r}(V_{1} + V_{2}) = ∑_{i=1}^{r} γ^{i} · ((V_{1}^{⊗r} ⊗ K_{i})^{Z_{r}} ⊕ (V_{2}^{⊗r} ⊗ K_{i})^{Z_{r}}) = ψ^{r}(V_{1}) + ψ^{r}(V_{2}).

To get the rule for tensor products, first observe that for any KZ_{r}-module V we have a decomposition V = (V ⊗ K_{1})^{Z_{r}} ⊕ ⋯ ⊕ (V ⊗ K_{r})^{Z_{r}} into eigenspaces of σ, and the eigenspaces are G-modules if V has a G-action which commutes with the S_{r}-action. Hence for two such modules U, V we get

(U ⊗ V ⊗ K_{i})^{Z_{r}} = ⨁_{j=1}^{r}(U ⊗ K_{j})^{Z_{r}} ⊗ (V ⊗ K_{i−j})^{Z_{r}}.

In particular,

ψ^{r}(V_{1} ⊗ V_{2}) = ∑_{i,j=1}^{r} γ^{i}(V_{1}^{⊗r} ⊗ K_{j})^{Z_{r}} ⊗ (V_{2}^{⊗r} ⊗ K_{i−j})^{Z_{r}} = ψ^{r}(V_{1}) · ψ^{r}(V_{2}).

Tensoring the isomorphism R_{KS_{r}} → A_{r} from Theorem 1.9 with ℚ we obtain an isomorphism

{ \atop R_{KS_{r}}} := ℚ ⊗_{ℤ} R_{KS_{r}} → ℚ ⊗_{ℤ} A_{r} = ℚ[a_{1}, a_{2}, …]_{r}. Given an f ∈ ℚ[a_{1}, a_{2}, …]_{r}, we can write

f = α_{1}f_{1} + ⋯ + α_{k}f_{k} as a ℚ-linear combination of elements f_{i} ∈ A_{r}^{+} of the positive cone, and set Sym_{f} = α_{1} Sym_{f_{1}} + ⋯+α_{k} Sym_{f_{k}}, using the notation introduced just before (1.1) on page 6. Sym_{f} is a mapping from the KG-modules into { \atop R_{KG}}, and it is easily seen that Sym_{f} does not depend on the choice of the linear combination. We have additivity and multiplicativity as in (1.1). Our maps ψ^{r} are by (1.4) of the type Sym_{f}, and we will now find out which f they come from.

Proposition 1.13. Define the polynomial p_{r} ∈ A_{r} by

x_{1}^{r} + ⋯ + x_{n}^{r} = p_{r}(a_{n,1}, …, a_{n,r}).

(In other words, p_{r} comes from the Newton formulae.) Then

Sym_{p_{r}} = ψ^{r}.

Proof. Let ϕ = diag(ξ_{1}, …, ξ_{n}) ∈ End_{K}(V). We must check that

trace(ψ^{r}(ϕ)) = ξ_{1}^{r} + ⋯ + ξ_{n}^{r}.

By Proposition 1.12 we can assume n = 1. But then V is one-dimensional and Z_{r} acts trivially on V^{⊗r}, hence (V^{⊗r} ⊗ K_{i})^{Z_{r}} is non-zero only for r|i. Thus ψ^{r}(V) = V^{⊗r} and indeed trace(ψ^{r}(ϕ)) = ξ_{1}^{r}.

Now we construct new mappings λ^{r} as follows. Define polynomials F_{r} ∈ ℚ[T_{1}, …, T_{r}] by

a_{n,r} = F_{r}(x_{1} + ⋯ + x_{n}, x_{1}^{2} + ⋯ + x_{n}^{2}, …, x_{1}^{r} + ⋯ + x_{n}^{r}).

Thus the F_{r} also come from Newton’s formulae. Then set

λ^{r}(V) = F_{r} (ψ^{1}(V), …, ψ^{r}(V)) ∈ { \atop R_{KG}}  (1.5)

for V a KG-module.

Proposition 1.14. For r ∈ ℕ_{0} with r! ∈ K^{×} and V a KG-module, λ^{r}(V) is also a KG-module, and we have

λ^{r}(V) ≅ Λ^{r}(V).

Proof. By Proposition 1.13 and by the rules (1.1) we have

λ^{r} = Sym_{F_{r}(p_{1},... ,p_{r})} .

Hence we only have to show that F_{r}(p_{1}, …, p_{r}) = a_{r}. But this is true by the definition of the p_{i} and F_{r}.

Since we know from Proposition 1.12 what ψ^{r}(U ⊗ V) is, we can use this to derive a rule for Λ^{r}(U ⊗ V). This will be given in terms of the polynomials λ_{t}. Before we can state it, we need a new type of product of two such polynomials.

Definition 1.15. Let x_{1}, x_{2}, …, y_{1}, y_{2}, …, t be indeterminates over ℤ. Define polynomials P_{r} ∈ ℤ[a_{1}, …, a_{r}; b_{1}, …, b_{r}] (with a_{i}, b_{i} indeterminates) by

∏_{i=1}^{m} ∏_{j=1}^{n}(1 + x_{i}y_{j}t) = ∑_{r=0}^{mn} P_{r}(a_{m,1}, …, a_{m,r}; b_{n,1}, …, b_{n,r}) · t^{r},

where a_{m,i} and b_{n,i} is the i-th elementary symmetric polynomial in x_{1}, …, x_{m} and y_{1}, …, y_{n}, respectively. Such a representation exists since the left-hand side of the above equation is invariant under the action on S_{m} × S_{n} on the x_{i} and y_{i}. Also observe that P_{r} is uniquely determined if m, n ≥ r, and does only depend on r.

Let R be a commutative ring with unity and f = 1+∑_{i=1}^{∞} α_{i}t^{i}, g = 1+∑_{j=1}^{∞} β_{i}t^{i} ∈ R[[t]] formal power series (or polynomials) over R. Then we set

f ⊗ g = ∑_{r=0}^{∞} P_{r}(α_{1}, …, α_{r},; β_{1}, …, β_{r}) · t^{r}.

In particular, if f = ∏_{i=1}^{m}(1 + ξ_{i}t) and g = ∏_{j=1}^{n}(1 + η_{j}t) with ξ_{i}, η_{j} ∈ R, then

f ⊗ g = ∏_{i=1}^{m} ∏_{j=1}^{n}(1 + ξ_{i}η_{j}t).  (1.6)

One interpretation of the tensor product of polynomials is this: if V and W are finite dimensional K-spaces and ϕ ∈ End_{K}(V), ψ ∈ End_{K}(W), then we have the following relation for the characteristic polynomials:

det(id_{V ⊗W} +t(ϕ ⊗ ψ)) = det(id_{V} +tϕ) ⊗ det(id_{W} +tψ).

Theorem 1.16. Let G be a group, K a field and U, V KG-modules. Then with the notation of Definitions 1.10 and 1.15 we have:

(a) If char(K) = 0, then

λ_{t}(U ⊗ V) = λ_{t}(U) ⊗ λ_{t}(V).

(b) If char(K) = p is positive, then

λ_{t}(U ⊗ V) ≡ λ_{t}(U) ⊗ λ_{t}(V) mod t^{p}.

Proof. Take r ∈ ℕ_{0} with r! ∈ K^{×}. We have to show that

Λ^{r}(U ⊗ V) = P_{r} (Λ^{1}(U), …, Λ^{r}(U); Λ^{1}(V), …, Λ^{r}(V)) .  (1.7)

Let x_{1}, …, x_{r}; y_{1}, …, y_{r} be indeterminates over ℤ and set p_{r,k} = x_{1}^{k} + ⋯ + x_{r}^{k}, q_{r,k} = y_{1}^{k} + ⋯ + y_{r}^{k} (1 ≤ k ≤ r). Then p_{r,k}q_{r,k} = ∑_{i,j=1}^{r}(x_{i}y_{j})^{k}, so by the definition of F_{r} we see that F_{r}(p_{r,1}q_{r,1}, …, p_{r,r}q_{r,r}) is the r-th elementary symmetric polynomial in the r^{2} arguments x_{i}y_{j}. We can write this as

F_{r}(p_{r,1}q_{r,1}, …, p_{r,r}q_{r,r}) = ∑_{\begin{matrix}I⊆\{1,... ,r\}×\{1,... ,r\} \\ with |I|=r \end{matrix}} ∏_{(i,j)∈I} x_{i}y_{j}.  (1.8)

On the other hand,

P_{r} (F_{1}(p_{r,1}), …, F_{r}(p_{r,1}, …, p_{r,r}); F_{1}(q_{r,1}), …, F_{r}(q_{r,1}, …, q_{r,r})) = P_{r}(a_{r,1}, …, a_{r,r}; b_{r,1}, …, b_{r,r})

with a_{r,i}, b_{r,i} as in Definition 1.15. The right-hand side is the coefficient of t^{r} in ∏_{i=1}^{r} ∏_{j=1}^{r}(1+x_{i}y_{j}t), so

P_{r} (F_{1}(p_{r,1}), …, F_{r}(p_{r,1}, …, p_{r,r}); F_{1}(q_{r,1}), …, F_{r}(q_{r,1}, …, q_{r,r})) = ∑_{\begin{matrix}I⊆\{1,... ,r\}×\{1,... ,r\} \\ with |I|=r \end{matrix}} ∏_{(i,j)∈I} x_{i}y_{j}.

Comparing this with Equation (1.8) and taking into account that the p_{r,i}, q_{r,j} are algebraically independent, we obtain the identity

F_{r}(p_{1}q_{1}, …, p_{r}q_{r}) = P_{r} (F_{1}(p_{1}), …, F_{r}(p_{1}, …, p_{r}); F_{1}(q_{1}), …, F_{r}(q_{1}, …, q_{r}))

with indeterminates p_{i}, q_{j}. We can substitute p_{i} by ψ^{i}(U) and q_{j} by ψ^{j}(V) in this identity. Using (1.5), we get

F_{r} (ψ^{1}(U)ψ^{1}(V), …, ψ^{r}(U)ψ^{r}(V)) =

P_{r} (F_{1}(ψ^{1}(U)), …, F_{r}(ψ^{1}(U), …, ψ^{r}(U)); F_{1}(ψ^{1}(V)), …, F_{r}(ψ^{1}(V), …, ψ^{r}(V))) =
P_{r} (λ^{1}(U), …, λ^{r}(U); λ^{1}(V), …, λ^{r}(V)) .

Now by (1.5) and Proposition 1.12 and

λ^{r}(U ⊗ V) = F_{r} (ψ^{1}(U)ψ^{1}(V), …, ψ^{r}(U)ψ^{r}(V)) = P_{r} (λ^{1}(U), …, λ^{r}(U); λ^{1}(V), …, λ^{r}(V)) .

Using Proposition 1.14, we get the desired formula (1.7).

Remark. If in the situation of Theorem 1.16 we have that char(K) ∤ |G|, then the equality statement (a) always holds. As in the case of Theorem 1.11, this can be seen by lifting to characteristic 0, using (a) and the reducing to characteristic p again. However, we will see in Example 2.6 that we do not in general have equality.

Definition 1.15 gives a product on U_{0}(R) := 1 + tR[[t]] for a commutative ring R. Together with the usual multiplication as addition, this makes U_{0}(R) into a commutative ring! One can also define so-called λ-operations Λ^{i}: U_{0}(R) → U_{0}(R). We will not use these and refer to Almkvist and Fossum [3, Section II.3] or Knutson [22] for the definition and further details. If R has operations λ^{i}: R → R such that λ_{t}: R → U_{0}(R), x ↦ ∑_{i=0}^{∞} λ^{i}(x)t^{i} is a ring homomorphism respecting the λ-operations of both rings, then R is called a λ-ring. In the case that R = R_{KG} is a representation ring, λ-operations are given by exterior powers. We denote the exterior algebra of a KG-module V by Λ^{∗}(V) := ⨁_{r=0}^{∞} Λ^{r}(V). The obvious isomorphism

Λ^{∗}(U ⊕ V) ≅ Λ^{∗}(U) ⊗ Λ^{∗}(V),

implies

λ_{t}(U + V) = λ_{t}(U) · λ_{t}(V),  (1.9)

therefore λ_{t} is always additive. Moreover, we have shown that λ_{t} is multiplicative in the nonmodular case. In this case it turns out that λ_{t} also respects the λ-operations, so R_{KG} is a λ-ring if char(K) ∤ |G| (see Knutson [22]). The picture is completed by a remark of Kouwenhoven [23] that R_{KG} is a λ-ring only if char(K) ∤ |G|.

2 Representations of the cyclic group of order p

In this section we will specialize to the case that G = Z_{p} is the cyclic group of prime order p and K is a field of characteristic p.

2.1 A presentation for the representation ring

The first goal is to find a presentation of the representation ring R_{KG} for G = Z_{p} as an algebra over ℤ. We first have to get the indecomposable KG-modules.

Proposition 2.1. Let G = Z_{p^{m}} be a cyclic group of prime power order, and char(K) = p. Then up to isomorphism there are exactly p^{m} indecomposable KG-modules V_{1}, …, V_{p^{m}}. We have dim(V_{n}) = n, and a generator of G acts on V_{n} by a Jordan block with ones on the diagonal. Moreover, V_{p^{m}} ≅ KG is the regular module, and it is the only indecomposable module which is projective. In particular, every projective module is free.

Proof. Let σ be a generator of G. A KG-module V is determined by the action of σ, which is determined by its canonical form. But the minimal polynomial of the action must be divisible by X^{p^{m}} − 1 = (X − 1)^{p^{m}}, hence all eigenvalues are 1, so V is indecomposable if and only if the action consists of a single Jordan block with ones on the diagonal. Now the length of this Jordan block cannot exceed p^{m}, since otherwise σ^{p^{m}} ≠ id. If n < p^{m}, then there is a non-split exact sequence

0 → V_{1} → V_{n+1} → V_{n} → 0

of KG-modules, hence V_{n} is not projective. On the other hand, let v = v_{1} ∈ V_{p^{m}} generate V_{p^{m}} as a KG-module, and set v_{i} = σ^{i−1}(v). Then v_{1}, …, v_{p^{m}} form a new basis of V_{p^{m}} on which G acts as the regular representation, hence V_{p^{m}} ≅ KG.

Lemma 2.2. With the notation of Proposition 2.1 we have

V_{2} ⊗ V_{n} ≅ \begin{cases}V_{n−1} ⊕ V_{n+1} & if p ∤ n, \\ V_{n} ⊕ V_{n} & if p | n \end{cases} .

Proof. A generator σ ∈ G acts on V_{2} ⊗ V_{n} by the matrix

B = ( \begin{matrix}A & A \\ 0 & A \end{matrix} ) ,

where A is an (upper) Jordan block of length n. We must determine the Jordan canonical form of B. First we see that rank(B − 1) = 2n − 2, hence there are two Jordan blocks. Now we have

(B − 1)^{k} = ( \begin{matrix}(A − 1)^{k} & k · A(A − 1)^{k−1} \\ 0 & (A − 1)^{k} \end{matrix} ) ,

so the smallest k with (B − 1)^{k} = 0 is n + 1 if p ∤ n and n otherwise. Hence the longest Jordan block of B has length n + 1 or n, and the other one must have length n − 1 or n, respectively.

From now on G = Z_{p} is the cyclic group of prime order p. Then Lemma 2.2 already tells us that R_{KG} is generated as a ℤ-algebra by V_{2}. Now we want to get a presentation of the representation ring. For that and for later applications it is very useful to extend R_{KG} by adding an element μ with V_{2} = μ + μ^{−1}. More precisely, form the polynomial ring R_{KG}[T] and then the quotient ring

R_{KG}[μ] := R_{KG}[T]/(T^{2} − V_{2}T + 1),

and let μ be the residue class of T. Then μ has the inverse V_{2} − μ, and V_{2} = μ + μ^{−1}.

Lemma 2.3. In R_{KG}[μ] we have

V_{n} = \frac{μ^{n} − μ^{−n}}{μ − μ^{−1}} for 1 ≤ n ≤ p,

where the fraction is a short hand for ∑_{j=0}^{n−1} μ^{n−1−2j}.

Proof. This is true for n = 1, 2. For 2 ≤ n < p we use Lemma 2.2 and induction and get

V_{n+1} = V_{2}V_{n} − V_{n−1} = (μ + μ^{−1})\frac{μ^{n} − μ^{−n}}{μ − μ^{−1}} − \frac{μ^{n−1} − μ^{−(n−1)}}{μ − μ^{−1}} = \frac{μ^{n+1} − μ^{−(n+1)}}{μ − μ^{−1}} .

We can now give the presentation.

Theorem 2.4. Let G = Z_{p} be a cyclic group of prime order p = char(K), and let R_{KG}[μ] be as defined above. Then the map of ℤ-algebras given by

ℤ[X] → R_{KG}[μ], X ↦ μ

is surjective, and its kernel is the ideal generated by

F (X) := (X − 1) · \frac{X^{2p} − 1}{X + 1}.

Proof. We first prove that F is mapped to 0. Indeed, the image of F is

(μ − 1) · \frac{μ^{2p} − 1}{μ + 1} = μ(μ − 2 + μ^{−1}) · \frac{μ^{p}(μ^{p} − μ^{−p})}{μ(μ − μ^{−1})} = μ^{p}(V_{2} − 2)V_{p} = 0

by Lemmas 2.3 and 2.2. Moreover, 1 − F is divisible by X, so (1 − F)/X is mapped to μ^{−1}. But V_{2} = μ + μ^{−1} generates R_{KG}, so we have an epimorphism

ℤ[X]/(F) → R_{KG}[μ]

of ℤ-algebras. But R_{KG}[μ] is by construction and by Proposition 2.1 a free ℤ-module of rank 2p, and so is ℤ[X]/(F), hence we have an isomorphism.

We find it remarkable that it is possible to obtain a presentation of the extended representation ring without actually touching any matrices more than once, in Lemma 2.2. In Section 4.1 we will derive a presentation of the extended representation ring of Z_{p^{m}}.

2.2 Exterior powers and symmetric powers of degree < p

We can now compute the exterior powers of the indecomposable modules V_{n} of KG for G = Z_{p} cyclic of order p = char(K). They are encoded in λ_{t}(V_{n}) (see Definition 1.10).

Theorem 2.5. If G is a cyclic group of order p = char(K) and V_{n+1} is the indecomposable KGmodule of dimension n + 1 (0 ≤ n < p), then

λ_{t}(V_{n+1}) = ∏_{j=0}^{n}(1 + μ^{n−2j}t),

with μ as defined on page 12.

Proof. We have λ_{t}(V_{1}) = 1 + t, since V_{1} is one-dimensional. Furthermore,

λ_{t}(V_{2}) = 1 + V_{2}t + t^{2} = (1 + μt)(1 + μ^{−1}t),

since Λ^{2}(V_{2}) is one-dimensional. For 1 < n < p we have, using induction, Equation (1.9), Lemma 2.2, Theorem 1.16, again induction, and Equation (1.6)

λ_{t}(V_{n+1}) · ∏_{j=0}^{n−2}(1 + μ^{n−2−2j}t) = λ_{t}(V_{n+1}) · λ_{t}(V_{n−1}) = λ_{t}(V_{n−1} + V_{n+1}) =

λ_{t}(V_{2} · V_{n}) ≡ λ_{2}(V_{2}) ⊗ λ_{t}(V_{n}) = ((1 + μt)(1 + μ^{−1}t)) ⊗ ∏_{j=0}^{n−1}(1 + μ^{n−1−2j}t) =

∏_{j=0}^{n−1}(1 + μ^{n−2j}t) · ∏_{j=0}^{n−1}(1 + μ^{n−2−2j}t) mod t^{p}.

The theorem will follow if we can show that in fact equality holds in the above. First observe that the polynomial ∏_{j=0}^{n−1}(1 + μ^{n−2j}t) · ∏_{j=0}^{n−1}(1 + μ^{n−2−2j}t) on the right-hand side of the above congruence is palindromic, i.e., its (2n − j)-th coefficient is equal to its j-th coefficient. In fact it is a product of polynomials of the type (1 + μ^{i}t)(1 + μ^{−i}t), which are palindromic. On the left-hand side is a polynomial of the type λ_{t}(V) with V a KG-module of dimension 2n. But we have an isomorphism

Λ^{i}(V) → Hom_{K} (Λ^{2n−i}(V), Λ^{2n}(V)) ,

where v_{1} ∧ ⋯ ∧ v_{i} is mapped to the homomorphism that maps v_{i+1} ∧ ⋯ ∧ v_{2n} to v_{1} ∧ ⋯ ∧ v_{2n}. Hence

Λ^{i}(V) ≅ Hom_{K} (Λ^{2n−i}(V), Λ^{2n}(V)) ≅ Hom_{K} (Λ^{2n−i}(V), K) ≅ Λ^{2n−i}(V),

since Λ^{2n}(V) is the representation given by the determinant of σ ∈ G on V, which is always 1, and since all KG-modules are self-dual (as the indecomposables are determined by their dimension). Hence the left-hand side is palindromic as well, and therefore the congruence must in fact be an equality, since polynomials are of degree 2n < 2p.

Remark. Theorem 2.5 says that if we set

λ_{t}(μ^{i}) = 1 + μ^{i}t (−p < i ≤ p)

and require that λ_{t}(x + y) = λ_{t}(x)λ_{t}(y) on R_{KG}[μ], we will get the correct expressions for λ_{t}(V_{n}) by Lemma 2.3. Thus μ behaves almost as a one-dimensional representation of Z_{p}!

There are also remarkable parallels to the representation theory of G = SL_{2}(ℂ) over K = ℂ, which were exposed by Almkvist [2]. Here the indecomposable KG-modules V_{n} are also given by their dimension, and we have R_{KG} ≅ ℤ[X] with X ↦ V_{2}. Introducing a μ with μ + μ^{−1} = V_{2}, one gets V_{n} = (μ^{n} − μ^{−n})/(μ − μ^{−1}) as in Lemma 2.3, and furthermore

λ_{t}(V_{n+1}) = ∏_{j=0}^{n}(1 + μ^{n−2j}t)

as in Theorem 2.5.

Example 2.6. Let p = 2, so G = Z_{2} and char(K) = 2. We want to compare λ_{t}(V_{2} ⊗ V_{2}) and λ_{t}(V_{2}) ⊗ λ_{t}(V_{2}). First, we have

λ_{t}(V_{2}) ⊗ λ_{t}(V_{2}) = (1 + μ^{2}t)(1 + t)^{2}(1 + μ^{−2}t) = (1 + (μ^{2} + μ^{−2})t + t^{2})(1 + 2t + t^{2}),

and by Lemma 2.2

2V_{2} = V_{2} · V_{2} = μ^{2} + μ^{−2} + 2 ⟹ μ^{2} + μ^{−2} = 2(V_{2} − 1),

hence

λ_{t}(V_{2}) ⊗ λ_{t}(V_{2}) = 1 + 2V_{2}t + (4V_{2} − 2)t^{2} + 2V_{2}t^{3} + t^{4}.

On the other hand,

λ_{t}(V_{2} ⊗ V_{2}) = λ_{t}(2V_{2}) = λ_{t}(V_{2})^{2} = 1 + 2V_{2}t + 2(V_{2} + 1)t^{2} + 2V_{2}t^{3} + t^{4}.

This tells us that Theorem 1.16 cannot be sharpened, and that R_{KG} is in general not a λ-ring.

We get the following corollary from Theorems 1.11 and 2.5.

Corollary 2.7. If G is a cyclic group of order p = char(K) and V_{n+1} is the indecomposable KGmodule of dimension n + 1 (0 ≤ n < p), then

σ_{t}(V_{n+1}) ≡ ∏_{j=0}^{n}(1 − μ^{n−2j}t)^{−1} mod t^{p},

with σ_{t} as in Definition 1.10 and μ as defined on page 12.

Proof. In the multiplicative group 1 + tR_{KG}[μ][[t]] consider the subgroup N = 1 + t^{p}R_{KG}[μ][[t]]. By Theorem 1.11, σ_{t}(V_{n+1}) is equal to λ_{−t}(V_{n+1})^{−1} multiplied by an element of N. Now use Theorem 2.5.

Example 2.8. Take p = 2 and n = 1. Then

∏_{j=0}^{n}(1 − μ^{n−2j}t)^{−1} = (1 − μt)^{−1}(1 − μ^{−1}t)^{−1} =

1 + (μ + μ^{−1})t + (μ^{2} + 1 + μ^{−2})t^{2} + … = 1 + V_{2}t + (2V_{2} − 1)t^{2} + …,

hence σ_{t}(V_{2}) ≠ ∏_{j=0}^{n}(1 − μ^{n−2j}t)^{−1} since no negative coefficients can occur in S^{2}(V_{2}). It follows that in this example σ_{t}(V) · λ_{−t}(V) ≠ 1.

2.3 The periodicity of the symmetric powers

Since Corollary 2.7 only gives a congruence for σ_{t}(V_{n}), it is not sufficient for the determination of the symmetric powers or the Hilbert series of the invariant ring. Combined with the periodicity of S^{r}(V_{n}) modulo free modules, which we will prove in this subsection, Corollary 2.7 does, however, give enough information.

As above, let G = Z_{p} be a cyclic group of order p = char(K). Take an indecomposable KG-module V_{n} and choose a basis v_{1}, …, v_{n} such that a generator σ of G takes the form of an upper Jordan block, i.e., σ(v_{i}) = v_{i} + v_{i−1} for i > 1, and σ(v_{1}) = v_{1}. The symmetric algebra S^{∗}(V_{n}) = ⊕_{r=0}^{∞}S^{r}(V_{n}) is isomorphic to the polynomial ring K[v_{1}, …, v_{n}] with the v_{i} treated as indeterminates. In the sequel we will denote this algebra by K[V_{n}]. Consider the set B_{n} consisting of all elements of K[V_{n}], which, considered as polynomials in v_{n}, have degree less than p, including 0. By the special form of the G-action, B_{n} is an (infinite dimensional) KG-submodule of K[V_{n}]. Moreover, consider the invariant N = ∏_{σ∈G} σ(v_{n}) ∈ S^{p}(V_{n})^{G}. Our interest in B_{n} stems from the following observation.

Lemma 2.9. With the above notation we have a direct sum decomposition

K[V_{n}] = N · K[V_{n}] ⊕ B_{n}

of (infinite dimensional) KG-modules.

Proof. It is clear from the definitions that N · K[V_{n}] ∩ B_{n} = 0. Moreover, considering N and an f ∈ K[V_{n}] as polynomials in v_{n} and performing division with remainder of f by N yields a representation of f as a multiple of N plus an element from B_{n}. The division algorithm is applicable here since N has top coefficient 1 as a polynomial in v_{n}.

We write B_{n,r} for the subspace of B_{n} consisting of homogeneous polynomials of total degree r, including 0. This is again a KG-module.

Lemma 2.10. With the above notation, B_{n,r} is a free KG-module for r > p − n.

Proof. We first assume n = p. Set w_{i} = σ^{i−1}(v_{n}) (1 ≤ i ≤ p). Then the w_{i} form another basis of V_{p}, and G acts by permuting the w_{i}. Hence G also permutes the monomials in the w_{i}, and a decomposition of S^{r}(V_{p}) into indecomposable summands is obtained by taking the cyclic KGsubmodules ⟨t⟩ generated by monomials t of degree r in the w_{i}. Since N is a G-invariant monomial in the w_{i}, S^{r}(V_{p}) ∩ N · K[V_{p}] decomposes into indecomposable summands of the form ⟨N · t⟩ with deg(t) = r − p (it is the zero module if r < p). Therefore the sum B′ of all modules ⟨t⟩ with N ∤ t and deg(t) = r provides a direct complement of S^{r}(V_{p}) ∩ N · K[V_{p}] in S^{r}(V_{p}). If N ∤ t and deg(t) = r > 0, then t cannot be fixed by G (otherwise, t would be a power of N), hence the G-orbit of t has length p and ⟨t⟩ is isomorphic to the regular module. Therefore B′ is a free KG-module if r > 0. But by Lemma 2.9, B_{p,r} ≅ S^{r}(V_{p})/ (S^{r}(V_{p}) ∩ N · K[V_{p}]) ≅ B′, so B_{p,r} is free for r > 0, which is the claim for n = p.

We proceed by downward induction on n. So assume 1 ≤ n < p and r > p − n. Then multiplication by v_{1} gives an exact sequence

0 ⟶ B_{n+1,r−1} ⟶^{v_{1}} B_{n+1,r} ⟶ B_{n,r} ⟶ 0,

where B_{n+1,r−1} and B_{n+1,r} are free by induction. Dualizing this we obtain

0 ⟶ B_{n,r}^{∗} ⟶ B_{n+1,r}^{∗} ⟶ B_{n+1,r−1}^{∗} ⟶ 0,

which splits since B_{n+1,r−1}^{∗} is free, hence B_{n,r}^{∗} is projective and therefore free by Proposition 2.1. Thus B_{n,r} is free as well.

Theorem 2.11. If G is a cyclic group of order p = char(K) and V_{n} is the indecomposable KGmodule of dimension n, then for r ≥ 0 we have

S^{r+p}(V_{n}) ≅ S^{r}(V_{n}) ⊕ (a free KG-module).

Proof. Multiplication by N is a KG-monomorphism S^{r}(V_{n}) → S^{r+p}(V_{n}). Now combine Lemmas 2.9 and 2.10.

2.4 Degree bounds for invariants

At this point we embark on a digression from the straight path to the Hilbert series, and devote the rest of this section to the question of degree bounds for generators of the invariant ring. As usual, let G be a cyclic group of order p = char(K), and let V be any KG-module with a decomposition

V = V_{n_{1}} ⊕ ⋯ ⊕ V_{n_{k}}

into indecomposables. For each i choose a v_{i} in the i-th summand V_{n_{i}} which generates V_{n_{i}} as a KGmodule and set N_{i} to be the product over the G-orbit of v_{i}, i.e., N_{i} := ∏_{v∈{σ(v_{i})|σ∈G}} v ∈ K[V]^{G}. Let k_{1} be the number of summands V_{n_{i}} with n_{i} > 1, which is equal to the number of N_{i} with deg(N_{i}) = p. Also consider the transfer, which is given by

tr_{G}: K[V] → K[V]^{G}, f ↦ ∑_{σ∈G} σ(f).

Note that the transfer is a homomorphism of modules over the invariant ring, hence its image is an ideal in K[V]^{G}. This image has been studied by various authors, e.g. Feshbach [13], Fleischmann [14], Shank and Wehlau [26]. The following lemma is mainly a consequence of Lemmas 2.9 and 2.10.

Lemma 2.12. In the above setting, K[V]^{G} is generated as a module over the subalgebra K[N_{1}, …, N_{k}] by homogeneous invariants of degree ≤ k_{1}(p − 1) − dim(V) + k together with the image tr_{G}(K[V]) of the transfer.

Proof. Let M ⊆ K[V]^{G} be the K[N_{1}, …, N_{k}]-module generated by all homogeneous invariants of degree ≤ k_{1}(p − 1) − dim(V) + k and tr_{G}(K[V]). We will prove S^{d}(V)^{G} ⊂ M by induction on d, so we can assume d > k_{1}(p − 1) − dim(V) + k. Lemma 2.9 yields

K[V] ≅ ⨂_{i=1}^{k} K[V_{n_{i}}] ≅ ⨂_{i=1}^{k} (N_{i} · K[V_{n_{i}}] ⊕ B_{n_{i}}) ,

where we set B_{n_{i}} = K if n_{i} = 1. This leads to a direct sum decomposition of K[V] as a KG-module by “multiplying out”. Therefore any f ∈ S^{d}(V)^{G} can be written as f = ∑_{I} f_{I}, where I runs through all subsets of the index set {1, …, k}, and

f_{I} ∈ ( ⨂_{i∈I} N_{i} · K[V_{n_{i}}] ⊗ ⨂_{i∈Ī} B_{n_{i}} )^{G}

with Ī = {1, …, k} \ I. All f_{I} with I ≠ ∅ lie in M by the induction hypothesis, so we only have to prove that f_{∅} ∈ M. Since f_{∅} is of degree d, we have

f_{∅} ∈ ( ⨁_{d_{1}+···+d_{k}=d} B_{n_{1},d_{1}} ⊗ ⋯ ⊗ B_{n_{k},d_{k}} )^{G} ,

where the sum runs over all d_{1}, …, d_{k} ∈ ℕ_{0} with d_{1} + ⋯ + d_{k} = d. For each summand there is an i with n_{i} > 1 and d_{i} > p − n_{i}, or an i with n_{i} = 1 and d_{i} > 0, since otherwise one would have

d ≤ ∑_{\begin{matrix}i∈\{1,... ,k\}, \\ n_{i}>1 \end{matrix}} p − n_{i} = k_{1}p − (dim(V) − (k − k_{1})) = k_{1}(p − 1) − dim(V) + k.

If a summand has an n_{i} = 1 and d_{i} > 0 then that summand is 0 since B_{n_{i}} = K. On the other hand, if a summand has an n_{i} > 1 and d_{i} > p − n_{i}, then by Lemma 2.10 B_{n_{i},d_{i}} is free, and so by Alperin [4, Lemma 7.4, p. 47] this summand is projective and hence free (see Proposition 2.1). We conclude that f_{∅} is an invariant lying in a free KG-module. But every such invariant is in the image of the transfer, since every invariant in V_{p} is a multiple of the sum over a basis which is permuted by G.

A set {f_{1}, …, f_{n}} ⊂ K[V]^{G} of homogeneous invariants is called a set of primary invariants if the f_{i} are algebraically independent over K and K[V]^{G} is finitely generated as a module over the subalgebra K[f_{1}, …, f_{n}]. This implies n = dim(V). Primary invariants exist in any invariant ring of a finite group (see Benson [7, Theorem 2.2.7]), but they are by no means uniquely determined. After primary invariants have been chosen, it is important for subsequent computations to have an upper bound for the degrees of homogeneous generators (“secondary invariants”) of K[V]^{G} as a K[f_{1}, …, f_{n}]-module.

Proposition 2.13. With the notation of Lemma 2.12, let {f_{1}, …, f_{n}} ⊂ K[V]^{G} be a set of primary invariants which contains the invariants N_{1}, …, N_{k}. Then K[V]^{G} is generated as a module over K[f_{1}, …, f_{n}] by homogeneous invariants of degree at most

∑_{i=1}^{n} deg(f_{i}) − dim(V).

Proof. Viewing K[V] as the invariant ring of the trivial group we conclude from Smith [27, Corollary 6.8.5] that K[V] is generated as a module over A := K[f_{1}, …, f_{n}] by homogeneous polynomials of degree ≤ d := ∑_{i=1}^{n} deg(f_{i}) − dim(V). Since the transfer tr_{G} is a degree preserving homomorphism of A-modules, its image is also generated as an A-module by homogeneous invariants of degree at most d. Since k_{1}(p − 1) + k = ∑_{i=1}^{k} deg(N_{i}) ≤ ∑_{i=1}^{n} deg(f_{i}), the claim follows from Lemma 2.12.

For any finitely generated invariant ring K[V]^{G} we define

β(K[V]^{G}) = min{d | K[V]^{G} is generated by homogeneous invariants of degree at most d}.

(Here by “generated” we mean generated as a K-algebra.) Example 2.14. An interesting case is V = k · V_{2}, k a positive integer, so we are considering vector invariants in K[k · V_{2}]^{G}. These invariants were studied by Richman [24] and Campbell and Hughes [11]. We can take the first basis vectors of each of the V_{2} and the “norms” N_{1}, …, N_{k} as primary invariants. Then Proposition 2.13 yields the upper bound k(p − 1) for the degrees of the secondary invariants. Since the primary and secondary invariants generate K[V]^{G} as an algebra, we obtain β(K[k · V_{2}]^{G}) ≤ k(p − 1) if k ≥ 2. On the other hand, Richman proved that β(K[k · V_{2}]^{G}) ≥ k(p − 1) if k ≥ 3, so we have

β(K[k · V_{2}]^{G}) = k(p − 1) if k ≥ 3.

Richman gave a set of invariants (of maximal degree k(p − 1)) which he conjectured to generate K[k · V_{2}]^{G}. This conjecture was proved by Campbell and Hughes [11], so the above formula for β(K[k · V_{2}]^{G}) is a direct consequence of [11].

Corollary 2.15. Let G be a cyclic group of order p = char(K) > 2, and V an indecomposable KG-module of dimension n. Then

β(K[V]^{G}) ≤ \begin{cases}(\frac{n}{4} + 1) (p − 1) & if n is even, \\ \frac{n+3}{4}(p − 1) + 1 & if n is odd. \end{cases}

Proof. First assume that n ≥ 4. We claim that there exist primary invariants of degrees

1, \underbrace{2, …, 2}_{[\frac{n−1}{2}] times}, \underbrace{\frac{p − 1}{2}, …, \frac{p − 1}{2}}_{(n−3−[\frac{n−1}{2}]) times}, p − 1, p,

where [(n − 1)/2] is the greatest integer ≤ (n − 1)/2. We choose a basis x_{1}, …, x_{n} of V such that a generator of G acts by x_{1} ↦ x_{1} and x_{i} ↦ x_{i} + x_{i−1} for i > 1. Then x_{1} is an invariant.

Moreover, for 2 ≤ i ≤ [(n − 1)/2] + 1, Hilbert [17, p. 61] gives an invariant of degree 2 which contains x_{i}^{2} as a monomial, and all whose monomials are divisible by some x_{j} with j ≤ i. In fact, Hilbert uses a different basis of V, but it is easily seen that the property we just stated about the monomials remains true in our basis. Furthermore, if i ≤ n − 2, then Theorem 3.2 of Shank [25] yields a homogeneous invariant whose leading monomial is x_{i}^{(p−1)/2} with respect to the graded reverse lexicographical term order with x_{1} < x_{2} < ⋯ < x_{n}. From the same source we obtain a homogeneous invariant with leading monomial x_{n−1}^{p−1}. Finally, the norm ∏_{σ∈G} σ(x_{n}) provides an invariant with leading monomial x_{n}^{p}. It follows that the set in K̄^{n} (K̄ an algebraic closure) of common zeros of these n invariants consists only of the zero vector. Therefore by Smith [27, Proposition 5.3.7] we have a set of primary invariants, which have the degrees given above. Proposition 2.13 now yields the upper bound

1 + [\frac{n − 1}{2}] · 2 + (n − 3 − [\frac{n − 1}{2}]) \frac{p − 1}{2} + (p − 1) + p − n

for the degrees of the secondary invariants. This evaluates to the claimed formulas for n even or odd. The bounds become degree bounds for algebra-generators since the secondary invariants together with the primary invariants generate K[V]^{G} as a K-algebra.

If n ≤ 3, the bounds are still correct since we obtain primary invariants of degrees 1, or 1, p, or 1, 2, p if n is 1,2, or 3, respectively.

Remark. For G the cyclic group of order p, we also obtain a degree bound for KG-modules V which are not necessarily indecomposable. In the case p ≥ 5 this bound can be given as

β(K[V]^{G}) ≤ (\frac{n}{4} + dim(V^{G})) (p − 1).

Example 2.16. Shank [25] gave generators for the invariant rings K[V_{n}]^{G} for n ≤ 5 and G cyclic of order p = char(K). The maximal degree of his generators is 2p − 3 for n = 4 and 2p − 2 for n = 5. Corollary 2.15 yields the degree bounds 2p − 2 and 2p − 1 for n = 4 and 5, respectively. The reason why the degree bound differ by one is that Shank’s secondary invariant of highest degree is superfluous as an algebra generator in both cases.

Using Proposition 2.13, we obtain a degree bound for invariants of a finite group G with p^{2} ∤ |G|. Implicit in the proof is a method to obtain generators for the invariant ring.

Theorem 2.17. Let G be a finite group and K a field of characteristic p such that p^{2} ∤ |G| and |G| > 1. Then for any KG-module V with n := dim(V) > 1 we have

β(K[V]^{G}) ≤ n (|G| − 1) .

Proof. The degree bound β(K[V]^{G}) does not change if we extend the ground field K (see, for example, Kemper [19, Section 4.4]), so we can assume that K is algebraically closed. Therefore we can use Dade’s algorithm (see Stanley [28, Proposition 3.4]) to obtain primary invariants of degrees ≤ |G| for K[V]^{G}. Suppose first that p ∤ |G|. Then it follows from Smith [27, Corollary 6.8.5] that the degrees of the secondary invariants are bounded above by n(|G| − 1), hence the claim. So from now on we can assume that p divides |G|.

Let H be a Sylow p-subgroup (|H| = p) and N_{1}, …, N_{k} the H-invariants defined before Lemma 2.12. Again we can use Dade’s algorithm to extend the set {N_{1}, …, N_{k}} ⊂ K[V]^{H} to a set of primary invariants for K[V]^{H} in the following way. We extend the sequence v_{1}, …, v_{k} ∈ V of generators of the indecomposable KH-summands (see before Lemma 2.12) by vectors v_{k+1}, …, v_{n} such that

v_{i} ∉ ⋃_{σ_{1},... ,σ_{i−1}∈G} (∑_{j=1}^{i−1} K · σ_{j}(v_{j})) .

and set f_{i} = ∏_{σ∈G} σ(v_{i}) for k < i ≤ n. (Note that we take the G-orbit, which is more than necessary to obtain primary invariants for K[V]^{H}, but will yield the degree bound.) Applying Smith [27, Proposition 5.3.7] we see that N_{1}, …, N_{k} together with f_{k+1}, …, f_{n} form a set of primary invariants for K[V]^{H}, since their only common zero in K^{n} is the origin. Moreover, deg(f_{i}) = |G|. Now we form G-invariants a_{i,j} as elementary symmetric polynomials of G-orbits of the N_{i}:

∏_{σ∈G/H} (X − σ(N_{i})) = X^{[G:H]} + ∑_{j=0}^{[G:H]−1} a_{i,j}X^{j} (1 ≤ i ≤ k),

where the summation runs over a set of left coset representatives of H in G. Therefore deg(a_{i,j}) = ([G : H] − j) · deg(N_{i}) ≤ |G|. Let A ⊆ K[V]^{G} be the algebra generated by f_{k+1}, …, f_{n} and by the a_{i,j}. Then β(A) ≤ |G|.

Let B = A[N_{1}, …, N_{k}] ⊆ K[V]^{H} be the algebra generated by A and the N_{i}. Then B is generated as an A-module by the products N_{1}^{e_{1}} ⋯ N_{k}^{e_{k}} with 0 ≤ e_{i} ≤ [G : H] − 1. These products have degree at most (k_{1}p + k_{0}) · ([G : H] − 1), where k_{1} is the number of indecomposable summands V_{n_{i}} of V as a KH-module with n_{i} > 1, and k_{0} = k − k_{1}. We write

β(B/A) ≤ (k_{1}p + k_{0}) · ([G : H] − 1) = (k_{1} + k_{0}/p) · (|G| − p)

to express the fact that B is generated as an A-module by homogeneous elements of degrees at most (k_{1}p + k_{0}) · ([G : H] − 1). Proposition 2.13 yields a bound for the generators of K[V]^{H} over B, namely

β(K[V]^{H}/B) ≤ β (K[V]^{H}/K[N_{1}, …, N_{k}, f_{k+1}, …, f_{n}]) ≤ k_{1}(p − 1) + (n − k)(|G| − 1).

From these two bounds we obtain

β(K[V]^{H}/A) ≤ β(K[V]^{H}/B) + β(B/A) ≤ (k_{1} − k + n) · (|G| − 1) + k_{0}(|G|/p − 1) ≤ n(|G| − 1).

Now the “relative Reynolds operator”

π_{G/H}: K[V]^{H} → K[V]^{G}, f ↦ \frac{1}{[G : H]} ∑_{σ∈G/H} σ(f)

provides a degree-preserving epimorphism of K[V]^{G}-modules and in particular of A-modules. Therefore generators of K[V]^{H} as an A-module are mapped by π_{G/H} to generators of K[V]^{G} as an A-module, and we obtain

β(K[V]^{G}/A) ≤ n(|G| − 1).

Since β(A) ≤ |G| ≤ n(|G| − 1), this completes the proof.

Remark. Broer [9] proved that the bound n(|G| − 1) given above holds for any finite group in the case that K[V]^{G} is Cohen-Macaulay. However, in the case considered in Theorem 2.17, K[V]^{G} is “usually” not Cohen-Macaulay (see Kemper [20]). For example, if G ≅ Z_{p} is cyclic, then K[V]^{G} is Cohen-Macaulay if and only if dim(V^{G}) ≥ dim(V) − 2, which was proved by Ellingsrud and Skjelbred [12]. To our knowledge, no a priori degree bound in the case p^{2} ∤ |G| was known previously. It has been conjectured that β(K[V]^{G}) ≤ n(|G| − 1) holds for any finite group G. The first section of the above proof explains why this bound can be considered as “natural”, although it cannot be expected to be sharp in most cases.

3 The Hilbert series

3.1 Species, averaging operators, and Hilbert series for Z_{p}-modules

If G is a finite group and K is a field, then a homomorphism R_{KG} → ℂ of ℤ-algebras is called a species of KG. A typical example in the case that K = ℂ is the map ϕ_{σ} for σ ∈ G, which maps a ℂG-module V to the trace of σ on V. (In fact, it is an easy consequence of character theory that all species are of this form, so they are in one-to-one correspondence with the conjugacy classes in this case.) If χ is an irreducible character, then we obtain an “averaging operator” T_{χ} = 1/|G| · ∑_{σ∈G} χ(σ^{−1})ϕ_{σ} (a linear combination of species) with the property that T_{χ} takes the value 1 on the irreducible ℂG-module belonging to χ, and 0 on all others. We will try to mimic this process in the situation of the cyclic group of order p in characteristic p.

In the sequel, let G = Z_{p} be cyclic of order p = char(K). From the presentation of R_{KG}[μ] given in Theorem 2.4 we see that the algebra-homomorphisms R_{KG}[μ] → ℂ are exactly

ϕ_{γ}: R_{KG}[μ] → ℂ, μ ↦ γ for γ ∈ M_{2p} \ {−1}, where M_{2p} = {γ ∈ ℂ | γ^{2p} = 1}.

Since R_{KG}[μ] is an algebraic extension of R_{KG}, every homomorphism R_{KG} → ℂ can be extended to R_{KG}[μ], hence we have given all species of KG. Observe that ϕ_{γ} and ϕ_{γ^{−1}} have the same restrictions to R_{KG}, so there are exactly p species, as in the non-modular case. We will denote the map R_{KG}[[t]] → ℂ[[t]] obtained by component-wise application of ϕ_{γ} also by ϕ_{γ}. It is remarkable that the information derived in the preceding section suffices to evaluate ϕ_{γ} (σ_{t}(V_{n+1})), where V_{n+1} is the indecomposable KG-module of dimension n + 1, and σ_{t}(V_{n+1}) is as in Definition 1.10.

Proposition 3.1. With the above notation we have

ϕ_{γ} (σ_{t}(V_{n+1})) = \frac{1 − γ^{pn}t^{p}}{1 − t^{p}} · ∏_{j=0}^{n}(1 − γ^{n−2j}t)^{−1}.

Proof. We first treat the case γ = 1. By Lemma 2.3 we have ϕ_{1}(V_{m}) = m = dim(V_{m}), thus for any KG-module V we have ϕ_{1}(V) = dim(V). Therefore ϕ_{1} (σ_{t}(V_{n+1})) is the Hilbert function of K[V_{n+1}], which is (1 − t)^{−(n+1)}, as claimed.

Now assume γ ≠ 1. Then by Lemma 2.3, ϕ_{γ}(V_{p}) = (γ^{p} − γ^{−p})/(γ − γ^{−1}) = 0. Hence Theorem 2.11 implies that ϕ_{γ} (S^{r+p}(V_{n+1})) = ϕ_{γ} (S^{r}(V_{n+1})). Writing T (f) = ∑_{r=0}^{p−1} a_{r}t^{r} for the truncation of a formal power series f = ∑_{r=0}^{∞} a_{r}t^{r} ∈ ℂ[[t]], we obtain by Corollary 2.7

ϕ_{γ} (σ_{t}(V_{n+1})) = ∑_{k=0}^{∞} ∑_{r=0}^{p−1} ϕ_{γ} (S^{r}(V_{n+1})) t^{kp+r} = \frac{1}{1 − t^{p}} · T (∏_{j=0}^{n}(1 − γ^{n−2j}t)^{−1}) .  (3.1)

Now ∏_{j=0}^{n}(1−γ^{n−2j}t) = ∏_{j=0}^{n}(1−(γ^{−2})^{j} ·γ^{n}t) is a divisor of ∏_{j=0}^{p−1}(1−(γ^{−2})^{j} ·γ^{n}t), which is equal to 1−(γ^{n}t)^{p}, since γ^{−2} is a primitive p-th root of unity. Therefore 1−(γ^{n}t)^{p} = g(t)·∏_{j=0}^{n}(1−γ^{n−2j}t) with g(t) ∈ ℂ[t] of degree < p, hence

∏_{j=0}^{n}(1 − γ^{n−2j}t)^{−1} = g(t) · (1 − (γ^{n}t)^{p})^{−1} = g(t) · ∑_{k=0}^{∞}(γ^{n}t)^{pk}.

It follows that

T (∏_{j=0}^{n}(1 − γ^{n−2j}t)^{−1}) = g(t) = (1 − (γ^{n}t)^{p}) · ∏_{j=0}^{n}(1 − γ^{n−2j}t)^{−1},

which together with Equation (3.1) yields the proposition.

Now we introduce operators which play the role of the T_{χ} in representation theory over ℂ.

Proposition 3.2. With the notation preceding Proposition 3.1 define operators R_{KG}[μ] → ℂ as follows:

T_{i} = \frac{1}{2p} ∑_{γ∈M_{2p}} γ^{−i}(γ − γ^{−1}) · ϕ_{γ} (0 < i < p),

T_{p} = \frac{1}{2p} ∑_{γ∈M_{2p}} γ^{−p}(1 + γ) · ϕ_{γ},

T = \frac{1}{2p} ∑_{γ∈M_{2p}}(1 + γ) · ϕ_{γ}.

Then we have

T_{i}(V_{n}) = δ_{i,n} (1 ≤ i, n ≤ p), and T (V) = dim(V^{G}) for any KG-module V .

Proof. First note that the coefficient for γ = −1 is 0 in all these sums, hence ϕ_{−1} can formally be set to anything. Now for 0 < i < p and −p < j < p we have

T_{i}(μ^{j}) = \frac{1}{2p} ∑_{γ∈M_{2p}} γ^{j−i+1} − \frac{1}{2p} ∑_{γ∈M_{2p}} γ^{j−i−1} = δ_{i−1,j} − δ_{i+1,j},

hence for 1 ≤ n ≤ p we obtain by Lemma 2.3

T_{i}(V_{n}) = ∑_{j=0}^{n−1} T_{i}(μ^{n−1−2j}) = ∑_{j=0}^{n−1} δ_{i−1,n−1−2j} − ∑_{j=0}^{n−1} δ_{i+1,n−1−2j} = δ_{i,n}.

Similarly, T_{p}(μ^{j}) = δ_{p,j} + δ_{p−1,j} = δ_{p−1,j} for −p < j < p, and hence T_{p}(V_{n}) = δ_{p,n} by Lemma 2.3. Finally,

∑_{i=1}^{p−1} γ^{−i}(γ − γ^{−1}) + γ^{−p}(1 + γ) = 1 + γ^{−1} − γ^{−p+1} − γ^{−p} + γ^{−p} + γ^{−p+1} = 1 + γ^{−1},

hence the restriction of T to R_{KG} equals the restriction of ∑_{i=1}^{p} T_{i}. Therefore T (V_{n}) = 1 = dim(V_{n}^{G}) for 1 ≤ n ≤ p, and the last claim follows by the additivity of T.

Putting Propositions 3.1 and 3.2 together yields formulas for the Hilbert series of the invariant ring K[V_{n}]^{G}. Before we state our main result, we make a definition.

Definition 3.3. Let G be a group and K a field. For a KG-module V and an indecomposable KG-module W we write d_{W} (V) for the multiplicity of W in a decomposition of V as a direct sum of indecomposables. Moreover, we write

H_{W} (K[V], t) = ∑_{r=0}^{∞} d_{W} (S^{r}(V)) · t^{r} ∈ ℂ[[t]],

and

H(K[V]^{G}, t) = ∑_{r=0}^{∞} dim(S^{r}(V)^{G}) · t^{r} ∈ ℂ[[t]].

The last power series is the Hilbert series of the invariant ring.

Theorem 3.4. Let G = Z_{p} be a cyclic group and K a field of characteristic p. Let V be a KGmodule with a decomposition

V = V_{n_{1}+1} ⊕ ⋯ ⊕ V_{n_{k}+1}

into indecomposable modules with dim(V_{n_{i}+1}) = n_{i} + 1, 0 ≤ n_{i} < p. For γ ∈ ℂ write

D_{V} (γ, t) := ∏_{i=1}^{k} (\frac{1 − γ^{pn_{i}}t^{p}}{1 − t^{p}} · ∏_{j=0}^{n_{i}}(1 − γ^{n_{i}−2j}t)^{−1}) .

Then the Hilbert series of K[V]^{G} is

H(K[V]^{G}, t) = ∑_{γ∈M_{2p}} \frac{1 + γ}{2p} · D_{V} (γ, t),  (3.2)

where M_{2p} ⊂ ℂ denotes the set of (2p)-th roots of unity. If all n_{i} are even, the simpler formula

H(K[V]^{G}, t) = \frac{1}{p} ∑_{γ∈M_{p}} (∏_{i=1}^{k} ∏_{j=0}^{n_{i}}(1 − γ^{n_{i}−2j}t)^{−1})  (3.3)

holds. Moreover,

H_{V_{n}}(K[V], t) = ∑_{γ∈M_{2p}} \frac{γ − γ^{−1}}{2p} γ^{−n} · D_{V} (γ, t) (1 ≤ n < p),  (3.4)

and

H_{V_{p}}(K[V], t) = ∑_{γ∈M_{2p}} \frac{1 + γ}{2p} γ^{p} · D_{V} (γ, t).  (3.5)

Proof. We have σ_{t}(V) = ∏_{i=1}^{k} σ_{t}(V_{n_{i}+1}). The additivity and multiplicativity of the ϕ_{γ} and Proposition 3.1 imply

ϕ_{γ}(σ_{t}(V)) = ∏_{i=1}^{k} ϕ_{γ} (σ_{t}(V_{n_{i}+1})) = D_{V} (γ, t) for γ ∈ M_{2p} \ {−1}.

By Proposition 3.2, H(K[V]^{G}, t) = T (σ_{t}(V)) and H_{V_{n}}(K[V], t) = T_{n}(σ_{t}(V)), so all formulas except for (3.3) follow. If all n_{i} are even, then (3.3) follows easily from (3.2).

Remark. We recover the formulas of Almkvist and Fossum [3, p. 81 and 85] for the Hilbert series as the special case k = 1, where V is indecomposable, of Theorem 3.4.

Suppose that all n_{i} in Theorem 3.4 are even, i.e., V is made up of indecomposables of odd dimension. Fix a primitive p-th root of unity γ ∈ ℂ and set M_{n_{i}} = diag(γ^{n_{i}}, γ^{n_{i}−2}, …, γ^{−(n_{i}−2)}, γ^{−n_{i}}) ∈ GL_{n_{i}+1}(ℂ). Then set

M_{V} = (\begin{matrix}M_{n_{1}} & & & & & 0 \\ & M_{n_{2}} \\ & & . & . & . \\ 0 & & & & & M_{n_{k}} \end{matrix} ) ∈ GL_{dim(V)}(ℂ)

and consider the representation G → GL_{dim(V)}(ℂ) given by mapping a generator of G to M_{V}. Then evaluating Molien’s formula for the Hilbert series of the invariant ring over ℂ of this representation yields the same Hilbert series as Equation (3.3). So if all n_{i} are even there exists a representation over ℂ whose invariant ring has the same Hilbert series as our representation in characteristic p. This is very remarkable and may instigate many speculations. However Almkvist [1, p. 3] holds the opinion that we are facing a mere “combinatorial accident”. We share his opinion.

3.2 The quasi-Gorenstein property and some examples

In this subsection we study some properties of the invariant rings related to their Hilbert series. One interesting property is the Gorenstein property. A Noetherian graded algebra A of Krull dimension n is called Gorenstein if it is Cohen-Macaulay and its Hilbert series as a rational function in t satisfies the identity

H(A, t^{−1}) = (−1)^{n}t^{k} · H(A, t)  (3.6)

for some k ∈ ℤ. (Originally, the Gorenstein property is defined in terms of the canonical module by demanding that it is isomorphic to A.) A is called strongly Gorenstein if moreover k = n. For invariant rings of finite group G with char(K) ∤ |G|, Watanabe showed that K[V]^{G} is strongly Gorenstein if and only if the action of G on V has determinants 1 (see, for example, Benson [7, Theorem 4.6.2]). Let us call A quasi-Gorenstein if (3.6) holds without the hypothesis that A is Cohen-Macaulay, and strongly quasi-Gorenstein if moreover k = n. We get the following result from Theorem 3.4

Corollary 3.5. Let G = Z_{p} be a cyclic group and K a field of characteristic p. Let V be a KGmodule with a decomposition

V = V_{n_{1}+1} ⊕ ⋯ ⊕ V_{n_{k}+1}

into indecomposable modules with dim(V_{n_{i}+1}) = n_{i} + 1, 0 ≤ n_{i} < p. Then the invariant ring K[V]^{G} is strongly quasi-Gorenstein if and only if ∑_{i=1}^{k} n_{i} is even.

Proof. Using the notation of Theorem 3.4 and writing n = dim(V), we have for γ ∈ M_{2p}

D_{V} (γ, t^{−1}) = ∏_{i=1}^{k} \frac{1 − γ^{pn_{i}}t^{−p}}{1 − t^{−p}} ∏_{j=0}^{n_{i}}(1 − γ^{n_{i}−2j}t^{−1})^{−1} = ∏_{i=1}^{k} \frac{t^{p} − γ^{pn_{i}}}{t^{p} − 1} t^{n_{i}+1} ∏_{j=0}^{n_{i}}(t − γ^{n_{i}−2j})^{−1} =

t^{n} ∏_{i=1}^{k} γ^{pn_{i}} \frac{1 − γ^{pn_{i}}t^{p}}{1 − t^{p}} (−1)^{n_{i}+1} ∏_{j=0}^{n_{i}} γ^{2j−n_{i}}(1 − γ^{2j−n_{i}}t)^{−1} =

(−t)^{n}γ^{p(n−k)} ∏_{i=1}^{k} \frac{1 − γ^{pn_{i}}t^{p}}{1 − t^{p}} ∏_{j=0}^{n_{i}}(1 − γ^{n_{i}−2j}t)^{−1} = (−t)^{n}γ^{p(n−k)}D_{V} (γ, t).

Therefore, if n − k = ∑_{i=1}^{k} n_{i} is even, then it follows from Theorem 3.4 that K[V]^{G} is strongly quasi-Gorenstein. On the other hand, if n − k is odd, then

H (K[V]^{G}, t) − (−t)^{−n}H (K[V]^{G}, t^{−1}) = 2 ∑_{γ∈M_{2p}\M_{p}} \frac{1 + γ}{2p} D_{V} (γ, t).

Substituting t = 0 in the above yields the value 1, hence K[V]^{G} is not strongly quasi-Gorenstein.

For a Noetherian graded algebra A, the so-called a-invariant is the degree of H(A, t) as a rational function in t, i.e., the difference between the degree of the numerator and of the denominator. This can also be defined in terms of local cohomology (see Bruns and Herzog [10, page 142]). For an invariant ring K[V]^{G} of a finite group with char(K) ∤ |G| it follows from Molien’s formula that the a-invariant is ≤ − dim(V). It has been conjectured that this carries over to the modular case. Here we get the following result.

Corollary 3.6. In the situation of Theorem 3.4, the a-invariant of K[V]^{G} is bounded from above by − dim(V), with equality if and only if ∑_{i=1}^{k} n_{i} is even.

Proof. In Theorem 3.4, D_{v}(γ, t) has degree − dim(V), from which the upper bound follows. In general, a rational function f(t) has degree d if and only if one can substitute t = 0 in t^{d}f(t^{−1}) and obtains a (finite) non-zero value. If ∑_{i=1}^{k} n_{i} is even, then by Corollary 3.5

t^{−n}H (K[V]^{G}, t^{−1}) = (−1)^{n}H (K[V]^{G}, t)

with n = dim(V), and H(K[V]^{G}, t) specializes to dim(S^{0}(V)^{G}) = 1 for t = 0. Hence the degree is −n in that case. On the other hand, if ∑_{i=1}^{k} n_{i} is odd, then H (K[V]^{G}, t) − (−t)^{−n}H (K[V]^{G}, t^{−1}) specializes to 1 by the proof of Corollary 3.5, hence t^{−n}H(K[V]^{G}, t^{−1}) specializes to 0, and the degree cannot be −n.

Now we present a method to evaluate Equation (3.2) for the Hilbert series symbolically for given n_{1}, …, n_{k} and general p > 2. (For p = 2, all n_{i} must be 1 or 0 and we have a permutation representation, so by Smith [27, Proposition 4.3.4] we can use Molien’s formula, pretending we are in characteristic 0, to obtain the Hilbert series.) We compute a partial fraction representation of

(1 + γ) ∏_{i=1}^{k} ∏_{j=0}^{n_{i}}(1 − γ^{n_{i}−2j}t)^{−1} + (1 − γ) ∏_{i=1}^{k} (\frac{1 − (−1)^{n_{i}}t^{p}}{1 − t^{p}} ∏_{j=0}^{n_{i}}(1 − (−1)^{n_{i}}γ^{n_{i}−2j}t)^{−1})

as a rational function in γ. This yields summands of the type a/(γ − b)^{i} with a, b rational functions in \sqrt[m]{t} and ζ_{m}, where m = lcm\{n_{i} − 2j | 2j < n_{i}, i = 1, …, k\} and ζ_{m} ∈ ℂ is a primitive m-th root of unity. More precisely, b is a monomial in \sqrt[m]{t} and ζ_{m}. We have to evaluate \frac{1}{2p} ∑_{γ∈M_{p}} a/(γ − b)^{i} for all these expressions. But

\frac{1}{2p} ∑_{γ∈M_{p}} \frac{a}{γ − b} = \frac{−a}{2pb} ∑_{γ∈M_{p}} ∑_{j=0}^{∞} (\frac{γ}{b} )^{j} = \frac{−a}{2b} ∑_{j=0}^{∞} (\frac{1}{b})^{pj} = \frac{−a}{2} \frac{b^{p−1}}{b^{p} − 1},

and so

\frac{1}{2p} ∑_{γ∈M_{p}} \frac{a}{(γ − b)^{i}} = \frac{1}{(i − 1)!} (\frac{∂}{∂b})^{i−1} (\frac{1}{2p} ∑_{γ∈M_{p}} \frac{a}{γ − b}) = \frac{−a}{2(i − 1)!} (\frac{∂}{∂b})^{i−1} (\frac{b^{p−1}}{b^{p} − 1}) .

Evaluating these formulas and summing over the partial fraction decomposition yields an expression for H(K[V]^{G}, t) involving ζ_{m}^{p}. The rational function over ℚ for H(K[V]^{G}, t) therefore depends on the residue class of p modulo m. The second author has implemented this method in Magma [8]. Using this implementation, we obtain the formulas given by Almkvist and Fossum [3, p. 71] for the indecomposable modules of dimensions up to 5, and can quite easily go up to dimensions of around 8. Such formulas can be useful, and have been used by Shank [25] to compute generators for the invariant rings of the indecomposable modules of dimensions up to 5 for general p. We thank Peter Müller for the idea to use partial fractions in γ.

A particularly interesting example is the case where all n_{i} are 1, i.e., we are dealing with the ring of vector invariants in K[kV_{2}]^{G} (see Example 2.14).

Example 3.7. We give the Hilbert series of K[kV_{2}]^{G} for the first few k.

(a) k = 1:

H(K[V_{2}]^{G}, t) = \frac{1}{(1 − t)(1 − t^{p})}.

This is the expected result, since K[V_{2}]^{G} is a polynomial ring in generators of degrees 1 and p (see Campbell and Hughes [11, page 3]).

(b) k = 2:

H(K[2V_{2}]^{G}, t) = \frac{1 − t^{2p}}{(1 − t)^{2}(1 − t^{p})^{2}(1 − t^{2})}

This was also expected, since K[2V_{2}]^{G} is a hypersurface generated by invariants of degrees 1,1,2,p,p with a relation of degree 2p (see Richman [24, Proposition 11]).

(c) k = 3. This is the first case where the invariant ring is not Cohen-Macaulay (see, for example, Campbell and Hughes [11, p. 17] or Ellingsrud and Skjelbred [12]).

H(K[3V_{2}]^{G}, t) = \frac{t^{2p−1}(1 − t^{2}) (p^{2}(1 − t^{2}) − p(t^{4} + 4t + 1)) + 2(1 − t^{2p})(t^{2} + t + 1)}{2(1 − t)^{2}(1 − t^{p})^{3}(1 − t^{2})^{3}}.

We checked this formula for p ≤ 5 by explicitly computing K[3V_{2}]^{G} and the Hilbert series using the invariant theory package in Magma (see Kemper and Steel [21]). For p = 3 the above formula specializes to

H(𝔽_{3}[3V_{2}]^{Z_{3}}, t) = \frac{6t^{4} − t^{3} + 3t^{2} + 1}{(1 − t)^{3}(1 − t^{3})^{3}}.

This cannot be the Hilbert series of any invariant ring of a finite group over ℂ, due to the negative coefficient in the numerator, as can easily be shown by standard arguments (see Sturmfels [29, Proposition 2.3.6]). This amplifies the point about a “combinatorial accident” made above.

(d) k = 4:

H(K[4V_{2}]^{G}, t) =

\frac{2t^{2p−1}(1 − t^{2}) (p^{3}(1 − t^{2})^{2} − p(t^{2} + 8t + 1)(t^{2} + t + 1)) − 3(t^{4p} − 1)(t^{4} + 2t^{3} + 4t^{2} + 2t + 1)}{3(1 − t)^{2}(1 − t^{p})^{4}(1 − t^{2})^{5}}.

We checked this formula for p = 3, where it specializes to

H(𝔽_{3}[4V_{2}]^{Z_{3}}, t) = \frac{t^{8} + 6t^{6} − 4t^{5} + 21t^{4} − 4t^{3} + 6t^{2} + 1}{(1 − t)^{4}(1 − t^{3})^{4}}.

This exemplifies the quasi-Gorenstein property.

Remark. When considering a decomposable KG-module V = V^{(1)} ⊕ ⋯ ⊕ V^{(k)}, one can refine the usual grading by giving K[V] a multigrading where the homogeneous part of multi-degree (d_{1}, …, d_{k}) is S^{d_{1}}(V^{(1)}) ⊗ ⋯ ⊗ S^{d_{k}}(V^{(k)}). The corresponding Hilbert series H(K[V]^{G}; t_{1}, …, t_{k}) of the invariant ring is then a rational function in k variables. For G a cyclic group of order p, this multigraded Hilbert series can be computed by the methods of this section. Indeed, one only has to replace “t” by “t_{i}” in the definition of D_{V} (γ, t) in Theorem 3.4. The second author has implemented a Magma program which for a given sequence of non-negative integers n_{1}, …, n_{k} computes H(K[V_{n_{1}+1} ⊕ ⋯ ⊕ V_{n_{k}+1}]^{G}; t_{1}, …, t_{k}) for a general prime p > 2. For example, we obtain

H(K[V_{2} ⊕ V_{2}]^{G}; s, t) = \frac{1 − (st)^{p}}{(1 − s)(1 − t)(1 − s^{p})(1 − t^{p})(1 − st)}.

Similarly, computing H(K[3V_{2}]^{G}; t_{1}, t_{2}, t_{3}) and evaluating for p = 3 yields

H(𝔽_{3}[3V_{2}]^{Z_{3}}; t_{1}, t_{2}, t_{3}) = \frac{(t_{1}t_{2} + t_{1}t_{3} + t_{2}t_{3})^{2} − t_{1}t_{2}t_{3}(t_{1} + t_{2} + t_{3} + 1) + (t_{1}t_{2} + t_{1}t_{3} + t_{2}t_{3}) + 1}{(1 − t_{1})(1 − t_{2})(1 − t_{3})(1 − t_{1}^{3})(1 − t_{2}^{3})(1 − t_{3}^{3})}.

These formulas relate nicely to the ones obtained in Example 3.7 above and illustrate how the multigraded Hilbert series refines the ordinary one.

3.3 Hilbert series for the group Z_{p} × H with p ∤ |H|

As the headline says, we will generalize our results to the case that G is a direct product of a cyclic group Z_{p} of order p = char(K) with a finite group H of order coprime to p. By Huppert and Blackburn [18, Chapter VII, Theorem 9.15] the indecomposable KG-modules are exactly the tensor products of indecomposable modules over KZ_{p} and KH, i.e., they can be written as V_{n} ⊗ W_{χ}, where V_{n} is as in Proposition 2.1, and W_{χ} corresponds to an irreducible Brauer character χ of H. We have species

ϕ_{γ,τ}: R_{KG} → ℂ, V ⊗ W_{χ} ↦ ϕ_{γ} · χ(τ),  (3.7)

indexed by γ ∈ M_{2p}\{−1} and τ ∈ H, with ϕ_{γ} as in Section 3.1. As before, we get the Hilbert series and the generating functions H_{V_{n}⊗W_{χ}} from two components: first, we have to evaluate the species on σ_{t}(V) for a KG-module V, and then we must find averaging operators, i.e., linear combinations of the species, which send V to dim(V^{G}) and to the multiplicity of some indecomposable summand. We first look at the averaging operators.

Proposition 3.8. For 1 ≤ n ≤ p and χ ∈ IBr(H), the set of irreducible Brauer characters of H over K, define operators R_{KG} → ℂ as follows:

T_{n,χ} = \frac{1}{2|G|} ∑_{γ∈M_{2p},τ∈H} γ^{−n} · \{ \begin{matrix}γ − γ^{−1}, & n < p \\ 1 + γ, & n = p \end{matrix} \} · χ(τ^{−1}) · ϕ_{γ,τ} ,

T = \frac{1}{2|G|} ∑_{γ∈M_{2p},τ∈H}(1 + γ) · ϕ_{γ,τ} .

Then we have

T_{n,χ}(V_{n′} ⊗ W_{χ′}) = δ_{n,n′} · δ_{χ,χ′} and T (V) = dim(V^{G})

for 1 ≤ n′ ≤ p, χ′ ∈ IBr(H), and V any KG-module.

Proof. By character theory we have

\frac{1}{|H|} ∑_{τ∈H} χ(τ^{−1})χ′(τ) = δ_{χ,χ′}.

The first claim follows from this and Proposition 3.2. Suppose there exists a non-zero invariant in V_{n} ⊗ W_{χ}. This must be an invariant under H, but the restriction of V_{n} ⊗ W_{χ} is nW_{χ}, which only has invariants for χ = 1. In that case, dim(V_{n} ⊗ W_{χ})^{G} = dim(V_{n})^{Z_{p}} = 1. Hence we get a T with T (V) = dim(V^{G}) by summing over the T_{n,1}. By the proof of Proposition 3.2, the above T is correct.

We now address the evaluation of ϕ_{γ,τ} (σ_{t}(V)) for a KG-module V. Choose a lifting M_{χ}: H → GL_{χ(1)}(ℂ), τ ↦ M_{χ}(τ) of W_{χ} for each χ ∈ IBr(H). For 1 ≤ n ≤ p form

M_{n,χ}(γ, τ) = (\begin{matrix}γ^{n−1} & & & & & 0 \\ & γ^{n−3} \\ & & . & . & . \\ 0 & & & & & γ^{−(n−1)} \end{matrix}) ⊗ M_{χ}(τ) ∈ GL_{nχ(1)}(ℂ) for γ ∈ M_{2p}, τ ∈ H.

Finally, for V an arbitrary KG-module with V = ∑_{i=1}^{k} V_{n_{i}} ⊗ W_{χ_{i}}, put

M_{V} (γ, τ) = (\begin{matrix}M_{n_{1},χ_{1}}(γ, τ) & & & & & 0 \\ & M_{n_{2},χ_{2}}(γ, τ) \\ & & . & . & . \\ 0 & & & & & M_{n_{k},χ_{k}}(γ, τ) \end{matrix}) ∈ GL_{dim(V)}(ℂ)

and

M̃_{V} (γ, τ) = (\begin{matrix}γ^{p(n_{1}−1)}M_{χ_{1}}(τ^{p}) & & & & & 0 \\ & γ^{p(n_{2}−1)}M_{χ_{2}}(τ^{p}) \\ & & . & . & . \\ 0 & & & & & γ^{p(n_{k}−1)}M_{χ_{k}^{p}}(τ) \end{matrix}) .

M_{V} and M̃_{V} are both representations of the group M_{2p} × H over ℂ.

Proposition 3.9. With the above notation we have

ϕ_{γ,τ} (σ_{t}(V)) = \begin{matrix}det & (1 − t^{p}M̃_{V} (γ, τ)) \\ det & (1 − t^{p}M̃_{V} (1, τ)) \end{matrix} · det (1 − tM_{V} (γ, τ))^{−1} .

Proof. As functions of V, both sides of the equation are homomorphisms from the additive group of R_{KG} into the multiplicative group 1 + tℂ[[t]]. Hence we can assume that V is irreducible. Furthermore, ϕ_{γ,τ} can be evaluated by first applying res_{Z_{p}×⟨τ⟩}^{G}: R_{KG} → R_{K(Z_{p}×⟨τ⟩)} and then the map ϕ_{γ,τ}: R_{K(Z_{p}×⟨τ⟩)} → ℂ defined as in (3.7). Hence we can also assume that H is cyclic, so we have V = V_{n} ⊗ χ with χ a linear character. Therefore

σ_{t}(V) = ∑_{r=0}^{∞} S^{r}(V_{n} ⊗ W_{χ})t^{r} = ∑_{r=0}^{∞} (S^{r}(V_{n}) ⊗ (W_{χ})^{r}) t^{r} = σ_{W_{χ}·t}(V_{n}),

and we get, using Proposition 3.1,

ϕ_{γ,τ} (σ_{t}(V)) = \frac{1 − γ^{p(n−1)}(χ(τ)t)^{p}}{1 − (χ(τ)t)^{p}} ∏_{j=0}^{n−1}(1 − γ^{n−1−2j}χ(τ)t)^{−1}.

Evaluating the claimed formula for the particular M_{V} and M̃_{V} yields the same result.

It is now routine to put together Propositions 3.8 and 3.9 and write down formulas for the Hilbert series

H(K[V]^{G}, t) = T (σ_{t}(V)) and H_{V_{n}⊗W_{χ}}(K[V], t) = T_{n,χ}(σ_{t}(V)).

4 Some remarks on Z_{p^{m}}

In this section we try to extend our methods to the case of a cyclic group Z_{p^{m}} of prime power order. The arguments in this section are not self-contained.

4.1 The representation ring of Z_{p^{m}}

Almkvist and Fossum [3, Proposition 1.6] and Fossum [15, Proposition 2.1] give presentations of the representation ring R_{KG} for G = Z_{p^{m}}, char(K) = p, but we have not found any complete proof in the literature. The aim of this subsection is to provide a complete proof, and also give a very simple presentation of the extended representation ring.

We write V_{n} for the n-dimensional indecomposable KG-module as in Proposition 2.1 (1 ≤ n ≤ p^{m}), and formally V_{0} = 0. Furthermore, we set

χ_{i} := V_{p^{i}+1} − V_{p^{i}−1} (0 ≤ i < m).

We adjoin elements μ_{i} with χ_{i} = μ_{i} + μ_{i}^{−1}. More precisely, we form the quotient ring

R_{KG}[μ_{0}, …, μ_{m−1}] = R_{KG}[T_{0}, …, T_{m−1}]/(T_{i}^{2} − χ_{i}T_{i} + 1)

and denote the residue class of T_{i} by μ_{i}. We will use the following formulas of Green [16, Theorem 3], which hold for 0 ≤ i < m.

χ_{i}V_{r} = \begin{cases}V_{r+p^{i}} − V_{p^{i}−r} & if 1 ≤ r ≤ p^{i}, \\ V_{r+p^{i}} + V_{r−p^{i}} & if p^{i} < r < (p − 1)p^{i}, \\ V_{r−p^{i}} + 2V_{p^{i+1}} − V_{2p^{i+1}−r−p^{i}} & if (p − 1)p^{i} ≤ r ≤ p^{i+1} \end{cases} .  (4.1)

We will also use

V_{p^{i}−1}V_{r} = (r − 1)V_{p^{i}} + V_{p^{i}−r} (0 ≤ i < m, 1 ≤ r ≤ p^{i}),  (4.2)

which is in Almkvist and Fossum [3, page 8].

Lemma 4.1. For 0 ≤ i < m we have

(μ_{i} − 2(V_{p^{i}} − V_{p^{i}−1}) + μ_{i}^{−1}) \frac{μ_{i}^{p} − μ_{i}^{−p}}{μ_{i} − μ_{i}^{−1}} = 0  (4.3)

and

(V_{p^{i}} − V_{p^{i}−1})^{2} = 1.  (4.4)

Proof. We first claim that

μ_{i}^{j} + μ_{i}^{−j} = \begin{cases}V_{jp^{i}+1} − V_{jp^{i}−1} & if 0 < j < p, \\ 2 (V_{p^{i+1}} − V_{p^{i+1}−1}) & if j = p \end{cases} .  (4.5)

This is true for j = 1 by the definitions. By way of induction, assume (4.5) for 0 < j < p − 1 and multiply by χ_{i}. Then by (4.1)

μ_{i}^{j+1} + μ_{i}^{−(j+1)} + μ_{i}^{j−1} + μ_{i}^{−(j−1)} =

V_{(j+1)p^{i}+1} + V_{(j−1)p^{i}+1} − (V_{(j+1)p^{i}−1} + \{ \begin{matrix}V_{(j−1)p^{i}−1}, & j > 1 \\ −V_{1}, & j = 1 \end{matrix} \}) =

V_{(j+1)p^{i}+1} − V_{(j+1)p^{i}−1} + μ_{i}^{j−1} + μ_{i}^{−(j−1)},

from which (4.5) follows for j < p. Multiplying (4.5) for j = p − 1 by χ_{i} and using (4.1) yields (4.5) for j = p.

Summing over appropriate j, (4.5) yields

χ_{i} · \frac{μ_{i}^{p} − μ_{i}^{−p}}{μ_{i} − μ_{i}^{−1}} = 2 · \begin{cases}(∑_{j=1}^{(p−1)/2} (V_{(2j−1)p^{i}+1} − V_{(2j−1)p^{i}−1}) + V_{p^{i+1}} − V_{p^{i+1}−1}) & if p ≠ 2, \\ (V_{p^{i+1}} − V_{p^{i+1}−1} + V_{1}) & if p = 2 \end{cases} .(4.6)

We next claim that

(μ_{i}^{j} + μ_{i}^{−j})V_{p^{i}−1} = V_{(j+1)p^{i}−1} − V_{(j−1)p^{i}+1} (0 < j < p).  (4.7)

For j = 1 we have (μ_{i} +μ_{i}^{−1})V_{p^{i}−1} = χ_{i}V_{p^{i}−1} = V_{2p^{i}−1} −V_{1} by (4.1). Assume (4.7) for 0 < j < p−1, then

(μ_{i}^{j+1} + μ_{i}^{−(j+1)} + μ_{i}^{j−1} + μ_{i}^{−(j−1)}) V_{p^{i}−1} =

χ_{i}(μ_{i}^{j} + μ_{i}^{−j})V_{p^{i}−1} = χ_{i}V_{(j+1)p^{i}−1} − χ_{i}V_{(j−1)p^{i}+1} =
V_{(j+2)p^{i}−1} − V_{jp^{i}+1} + (μ_{i}^{j−1} + μ^{−(j−1)})V_{p^{i}−1},

using (4.1). This yields (4.7). Summation over (4.7) yields

V_{p^{i}−1} · \frac{μ_{i}^{p} − μ_{i}^{−p}}{μ_{i} − μ_{i}^{−1}} = \begin{cases}∑_{j=1}^{(p−1)/2} (V_{(2j−1)p^{i}−1} − V_{(2j−1)p^{i}+1}) + V_{p^{i+1}−1} & if p ≠ 2, \\ V_{p^{i+1}−1} − V_{1} & if p = 2 \end{cases} .  (4.8)

By comparing (4.6) and (4.8) we obtain

(χ_{i} + 2V_{p^{i}−1}) \frac{μ_{i}^{p} − μ_{i}^{−p}}{μ_{i} − μ_{i}^{−1}} = 2V_{p^{i+1}}.  (4.9)

Finally we prove

(μ_{i}^{j} + μ_{i}^{−j})V_{p^{i}} = V_{(j+1)p^{i}} − V_{(j−1)p^{i}} (0 < j < p)  (4.10)

using the same methods as for (4.7), and by summation obtain

V_{p^{i}} · \frac{μ_{i}^{p} − μ_{i}^{−p}}{μ_{i} − μ_{i}^{−1}} = V_{p^{i+1}}.  (4.11)

Subtracting (4.11) twice from (4.9) yields the desired formula (4.3).

To obtain (4.4), first observe that V_{p^{i}} is a free KZ_{p^{i}}-module, hence V_{p^{i}} ⊗ V_{n} is again projective over KZ_{p^{i}} for n ≤ p^{i} by Benson [6, Proposition 3.1.5], so V_{p^{i}}V_{n} = nV_{p^{i}}. We get

(V_{p^{i}} − V_{p^{i}−1})^{2} = (2 − p^{i})V_{p^{i}} + V_{p^{i}−1}^{2} = (2 − p^{i})V_{p^{i}} + (p^{i} − 2)V_{p^{i}} + V_{1} = 1,

where we used (4.2).

Theorem 4.2. With the notation introduced at the beginning of this subsection, the map of ℤalgebras given by

ℤ[X_{0}, …, X_{m−1}] → R_{KG}[μ_{0}, …, μ_{m−1}], X_{i} ↦ (V_{p^{i}} − V_{p^{i}−1}) · μ_{i}

is surjective, and its kernel is the ideal generated by the polynomials

(X_{i} − 1) · \frac{X_{i}^{2p} − 1}{X_{i} + 1} (i = 0, …, m − 1).

In particular,

R_{KG}[μ_{0}, …, μ_{m−1}] ≅ R_{KZ_{p}}[μ]^{⊗_{ℤ}m},

the m-th tensor power over ℤ of the extended representation ring of KZ_{p}.

Proof. We write F_{i}(X_{i}) := (X_{i} − 1)(X_{i}^{2p} − 1)/(X_{i} + 1) and μ̃_{i} = (V_{p^{i}} − V_{p^{i}−1}) · μ_{i}. Then μ_{i}^{2} = μ_{i}^{2} by (4.4). We have

F_{i}(μ̃_{i}) = (μ̃_{i} − 1)^{2} · \frac{μ_{i}^{2p} − 1}{μ_{i}^{2} − 1} = (μ_{i}^{2} − 2(V_{p^{i}} − V_{p^{i}−1})μ_{i} + 1) · \frac{μ_{i}^{2p} − 1}{μ_{i}^{2} − 1} =

μ_{i}^{p} (μ_{i} − 2(V_{p^{i}} − V_{p^{i}−1}) + μ_{i}^{−1}) \frac{μ_{i}^{p} − μ_{i}^{−p}}{μ_{i} − μ_{i}^{−1}} = 0

by (4.3). Moreover, 1 − F_{i} is divisible by X_{i}, so (1 − F_{i})/X_{i} is mapped to μ_{i}^{−1} and χ̃_{i} := (V_{p^{i}} − V_{p^{i}−1})·χ_{i} = μ̃_{i}+μ_{i}^{−1} lies in the image. We claim that all V_{n} for 2 ≤ n ≤ p^{m} lie in the image. Indeed, assume that every V_{r} with r < n lies in the image and choose 0 ≤ i < m such that p^{i} < n ≤ p^{i+1}. Then by (4.1) V_{n} occurs as the highest V_{r} in χ_{i}V_{n−p^{i}}. By the inductive assumption V_{p^{i}} and V_{p^{i}−1} lie in the image, hence so does χ_{i} = (V_{p^{i}} − V_{p^{i}−1}) · χ̃_{i}. This proves the surjectivity. We thus have an epimorphism

ℤ[X_{0}, …, X_{m−1}]/(F_{0}, …, F_{m−1}) → R_{KG}[μ_{0}, …, μ_{m−1}].

But these two rings have the same ℤ-rank (2p)^{m} and so the map is an isomorphism.

As a consequence we see that there are precisely p^{m} = |G| species of KG, given by (V_{p^{i}} −V_{p^{i}−1})· μ_{i} ↦ γ_{i} with γ_{i} ∈ M_{2p} \ {−1}, where substitution of γ_{i} by γ_{i}^{−1} yields the same species. It also follows from Theorem 4.2 that there exist suitable ℂ-linear combinations T_{i} of the species such that T_{i}(V_{n}) = δ_{i,n}.

4.2 Exterior and symmetric powers, and the Periodicity Conjecture

We want to explore how far we get if we try to follow the same strategy as in Section 2, which led to the decomposition rules of exterior and symmetric powers and the Hilbert series of the invariant rings in the case G = Z_{p}. As above, let G = Z_{p^{m}} and char(K) = p. We first look at what can be said about λ_{t}(V) (see Definition 1.10) for a KG-module V.

By Theorem 4.2 the set {μ_{0}^{e_{0}} ⋯ μ_{m−1}^{e_{m−1}} | −p < e_{i} ≤ p} is a basis of R_{KG}[μ_{0}, …, μ_{m−1}] as a free ℤ-module. We can thus define a homomorphism from the additive group of R_{KG}[μ_{0}, …, μ_{m−1}] into the multiplicative group 1 + tR_{KG}[μ_{0}, …, μ_{m−1}][[t]] by setting

λ̃_{t}(μ_{0}^{e_{0}} ⋯ μ_{m−1}^{e_{m−1}}) = 1 + μ_{0}^{e_{0}} ⋯ μ_{m−1}^{e_{m−1}}t (−p < e_{i} ≤ p).

The interpretation is that we look at the μ_{i} as if they were linear representations of G. It is a result of Kouwenhoven [23, Theorem 3.5] that λ_{t}(χ_{i}) = t^{2} + χ_{i}t + 1 = λ̃_{t}(χ_{i}). We use this to compare λ̃_{t} and λ_{t}, and get the following generalization of Theorem 2.5.

Theorem 4.3. For a KG-module V we have

λ̃_{t}(V) ≡ λ_{t}(V) mod t^{p},

with equality if dim(V) ≤ 2p − 1.

Proof. Both sides of the congruence are homomorphisms R_{KG} → 1 + tR_{KG}[μ_{0}, …, μ_{m−1}][[t]] of abelian groups, so we can assume that V = V_{n} is indecomposable. We have λ̃_{t}(V_{1}) = λ_{t}(V_{1}) = 1+t, so assume n > 1. Write n = kp^{i} + r with 0 ≤ i < m, 1 ≤ k < p, 0 < r ≤ p^{i}. Then Green’s formulas (4.1) imply

V_{n} = χ_{i}V_{n−p^{i}} − \begin{cases}V_{n−2p^{i}} & if k > 1, \\ −V_{p^{i}−r} & if k = 1 \end{cases} ,  (4.12)

Thus we have for k > 1

λ_{t}(V_{n}) = λ_{t}(χ_{i}V_{n−p^{i}})λ_{t}(V_{n−2p^{i}})^{−1} ≡ (λ_{t}(χ_{i}) ⊗ λ_{t}(V_{n−p^{i}})) λ_{t}(V_{n−2p^{i}})^{−1} ≡

(((1 + μ_{i}t)(1 + μ_{i}^{−1}t)) ⊗ λ̃_{t}(V_{n−p^{i}})) λ̃_{t}(V_{n−2p^{i}})^{−1} =

λ̃_{t}(χ_{i}V_{n−p^{i}})λ̃_{t}(V_{n−2p^{i}})^{−1} = λ̃_{t}(V_{n}) mod t^{p},

where we used Theorem 1.16 for the first congruence, and the result of Kouwenhoven [23] mentioned above and induction on n for the second. The case k = 1 is analogous. The equality statement follows as in the proof of Theorem 2.5, since both sides are palindromic.

The following example shows that we do not have equality in Theorem 4.3 in general. Example 4.4. Take p = 2 and m = 2. Then by (4.12)

V_{4} = χ_{1}V_{2} = μ_{1}μ_{0} + μ_{1}μ_{0}^{−1} + μ_{1}^{−1}μ_{0} + μ_{1}^{−1}μ_{0}^{−1}.

Therefore the t^{2}-coefficient of λ̃_{t}(V_{4}) is Λ̃^{2}(V_{4}) = 2 + μ_{1}^{2} + μ_{1}^{−2} + μ_{0}^{2} + μ_{0}^{−2}. From Example 2.6 we know that μ_{0}^{2} + μ_{0}^{−2} = 2(V_{2} − 1). By the definition of χ_{1} we have V_{3} = χ_{1} + V_{1} = μ_{1} + μ_{1}^{−1} + 1, and so by (4.2)

μ_{1}^{2} + μ_{1}^{−2} + 3 + 2μ_{1} + 2μ_{1}^{−1} = V_{3}^{2} = 2V_{4} + 1,

thus μ_{1}^{2} + μ_{1}^{−2} = 2(V_{4} − V_{3}), and we get

Λ̃^{2}(V_{4}) = 2(V_{2} − V_{3} + V_{4}).

This cannot be equal to Λ^{2}(V_{4}) since no negative coefficients occur in the latter.

Now also define a homomorphism σ̃_{t}: R_{KG}[μ_{0}, …, μ_{m−1}] → 1 + tR_{KG}[μ_{0}, …, μ_{m−1}][[t]] of abelian groups by setting

σ̃_{t}(μ_{0}^{e_{0}} ⋯ μ_{m−1}^{e_{m−1}}) = (1 − μ_{0}^{e_{0}} ⋯ μ_{m−1}^{e_{m−1}}t)^{−1} (−p < e_{i} ≤ p).

Then we get the following corollary to Theorem 4.3, which generalizes Corollary 2.7.

Corollary 4.5. For a KG-module V we have

σ̃_{t}(V) ≡ σ_{t}(V) mod t^{p}.

Proof. This follows from Theorems 4.3 and 1.11.

The second major step on our way towards the Hilbert series was the periodicity of S^{r}(V_{n}) up to free modules. This is no longer true in the case G = Z_{p^{m}}. Instead we make the following conjecture.

Conjecture 4.6 (Periodicity Conjecture). If G = Z_{p^{m}} is a cyclic group of order p^{m}, p = char(K), and V_{n} is the indecomposable KG-module of dimension n, then for r ≥ 0 we have

S^{r+p^{m}}(V_{n}) ≅ S^{r}(V_{n}) ⊕ (a module induced up from Z_{p^{m−1}}).

The induced modules from Z_{p^{m−1}} are exactly the direct sums of modules of the type V_{kp} (1 ≤ k ≤ p^{m−1}), so the Periodicity Conjecture is a generalization of Theorem 2.11. Using Magma, we did numerous explicit computations which support the Periodicity Conjecture, so we are almost convinced that it must be true. However, we did not manage to find a proof. If we define B_{n} as the set of all elements in K[V_{n}] whose degree in the variable v_{n} generating V_{n} as a KG-module is less than p^{m}, then we get

S^{r+p^{m}}(V_{n}) ≅ S^{r}(V_{n}) ⊕ B_{n,r+p^{m}}

as in Lemma 2.9. We can reformulate the Periodicity Conjecture in a slightly stronger way by asserting that B_{n,r} is induced for r > p^{m} − n. For n = p^{m} we can use the same arguments as in the proof of Lemma 2.10 to see that B_{p^{m},r} consists of copies of V_{p^{i}}, 1 ≤ i ≤ m, for r > 0. We also have an exact sequence

0 ⟶ B_{n+1,r−1} ⟶ B_{n+1,r} ⟶ B_{n,r} ⟶ 0,

but this is where the proof fails: whereas a quotient of two free modules is again free (as we showed in the proof of Lemma 2.10), it is not true that the quotient of two induced modules is again induced.

Even if we had a proof for the Periodicity Conjecture, there would be some further obstacles on the way of pushing through the analysis done in Section 3. First, we only know σ_{t}(V) up to degree p − 1, but the period involved in the Periodicity Conjecture is p^{m}, which does not connect properly. Second, the Periodicity Conjecture is modulo a number of indecomposable modules rather than just V_{p}, which means that the difference of S^{r+p^{m}}(V_{n}) and S^{r}(V_{n}) cannot be measured by a mere dimension count, which is what we basically did in Section 3.

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Ian Hughes  Gregor Kemper
Department of Mathematics and Statistics IWR
Queen’s University  Universität Heidelberg
Kingston  Im Neuenheimer Feld 368
Ontario K7L 3N6  69 120 Heidelberg
Canada  Germany

    hughesi@mast.queensu.ca              Gregor.Kemper@iwr.uni-heidelberg.de