DECOMPOSING SYMMETRIC POWERS OF CERTAIN MODULAR REPRESENTATIONS OF CYCLIC GROUPS

R. JAMES SHANK AND DAVID L. WEHLAU

Abstract. For a prime number p, we construct a generating set for the ring of invariants for the p + 1 dimensional indecomposable modular representation of a cyclic group of order p^{2}. We then use the constructed invariants to describe the decomposition of the symmetric algebra as a module over the group ring, confirming the Periodicity Conjecture of Ian Hughes and Gregor Kemper for this case.

1. Introduction

Suppose that V is a finite dimensional representation of a finite group G over a field F, i.e., V is a finitely generated module over the group ring FG. The action of G on V induces an action on the dual V^{∗} which extends to an action by algebra automorphisms on the symmetric algebra F[V] := S(V^{∗}). The elements of V^{∗}, and thus also the elements of F[V], represent F-valued functions on V. If {x_{1}, x_{2}, …, x_{n}} is a basis for V^{∗} then F[V] can be identified with the ring of polynomials F[x_{1}, x_{2}, …, x_{n}]. Let F[V]_{d} denote the subspace of homogeneous polynomials of degree d. Since the action of G preserves degree, F[V]_{d} is a module over FG and

F[V] = ⨁_{d=0}^{∞} F[V]_{d}

is a decomposition into a direct sum of finite dimensional FG-modules. Of course F[V]_{d} is precisely the the d^{th} symmetric power of V^{∗}. Understanding the action of G on F[V]_{d}, and hence the action on F[V], is an important problem in representation theory. The primary goal is to write F[V]_{d} as a direct sum of indecomposable FG-modules, refining the given decomposition of F[V]. This means decomposing F[V]_{d} for infinitely many d. An important aspect of the group action is the ring of invariants

F[V]^{G} := {f ∈ F[V] | g(f) = f, ∀g ∈ G},

a finitely generated subalgebra of F[V]. A fundamental problem in invariant theory is the construction of a finite generating set for F[V]^{G}. Since G is finite, F[V] is a finite module over F[V]^{G}. Thus F[V] is a module over both F[V]^{G} and FG. Perhaps the right approach is to study F[V] as a finitely generated module over the extended group ring F[V]^{G}G. Certainly, in the work of both Karagueuzian & Symonds [11] and Hughes & Kemper [10], the finite F[V]^{G}- module structure of F[V] has been used to reduce decomposing F[V] over FG to a finite problem.

Date: September 2, 2005.

1991 Mathematics Subject Classification. 13A50.

The research of the first author is supported by grants from EPSRC.

The research of the second author is supported by grants from ARP and NSERC.

For the remainder of the paper, we assume that F has characteristic p for a prime number p, and that G ≅ Z/p^{r} is a cyclic group of order p^{r}. Choose a generator σ for G. The isomorphism type of a representation of G is determined by the Jordan canonical form of σ. If m ≤ p^{r}, then the m×m matrix over F consisting of a single Jordan block with eigenvalue 1 determines an indecomposable FGmodule which we denote by V_{m}. Note that if m > p^{r}, then the matrix has order greater than p^{r} and does not determine a representation of G. It follows from the form of the matrix that V_{m} is faithful if and only if p^{r−1} < m ≤ p^{r}, and that V_{m} is a cyclic FG-module. Since a field of characteristic p has no non-trivial p^{th} roots of unity, the indecomposable FG-modules are, up to isomorphism, V_{1}, V_{2}, …, V_{p^{r}}. Furthermore, V_{1} is the unique irreducible module, V_{p^{r}} ≅ FG is the unique projective indecomposable, and an FG-module is projective if and only if it is injective (see, for example, [1, Ch. II]). Also, it is easy to see that the representation V_{m} is induced from a representation of a proper subgroup of G if and only if p divides m.

For f ∈ F[V_{n}], we define the norm of f, denoted by N^{G}(f), to be the product over the G-orbit of f. Clearly N^{G}(f) ∈ F[V_{n}]^{G}. For a subgroup L = ⟨σ^{p^{t}}⟩, we define the relative transfer Tr_{L}^{G} := ∑_{i=0}^{p^{t}−1} σ^{i} ∈ FG.

The two main results we prove in this article concern the representation V_{p+1} and are stated as Theorem 1.1 and Theorem 1.2 below.

We start by developing tools for decomposing F[V_{n}] as an FG-module. We then specialise to r = 2 and n = p + 1. In Section 3 we construct generators for F[V_{p+1}]^{Z/p^{2}}. We apply the “ladder technique” described in [15, §7], using group cohomology and a spectral sequence argument, to prove the following.

Theorem 1.1. Let G ≅ Z/p^{2} and let L ≅ Z/p denote its non-trivial proper subgroup. The ring of invariants F[V_{p+1}]^{G} is generated by N^{G}(x_{p}), N^{G}(x_{p+1}) and elements from the image of the relative transfer, Tr_{L}^{G} (F[N^{L}(x_{p+1}), x_{p}, …, x_{1}]).

In Section 4 we use the constructed generating set to describe the FZ/p^{2}-module structure of F[V_{p+1}], confirming the Periodicity Conjecture of Hughes & Kemper [10, Conjecture 4.6] in this case and proving the following.

Theorem 1.2. Let G ≅ Z/p^{2} and let d be any non-negative integer. In the decomposition of F[V_{p+1}]_{d} into a direct sum of indecomposable FG-modules there is at most one indecomposable summand V_{m} which is not induced from a representation of a proper subgroup. In particular, writing d = ap^{2} + bp + c where 0 ≤ b, c < p, there is exactly one non-induced indecomposable summand when b ≤ p − 2 and F[V_{p+1}]_{d} is an induced module when b = p − 1. Moreover, if b ≤ p − 2 then the non-induced indecomposable summand is isomorphic to V_{cp+b+1} and we may choose the decomposition of F[V_{p+1}]_{d} such that the socle of this summand, V_{cp+b+1}^{G}, is spanned by the invariant N^{G}(x_{p+1})^{a}N^{G}(x_{p})^{b}x_{1}^{c}.

The following example illustrates Theorems 1.1 and 1.2.

Example 1.3. Let F be any field of characteristic p = 3. We consider the irreducible four dimensional representation V_{4} of the cyclic group G = Z/9 of order 9. The group G contains the subgroup L of order 3. Theorem 1.1 asserts that F[V]^{G} is generated by M = N^{G}(x_{3}) = x_{3}^{3} − x_{3}x_{2}^{2} + x_{3}^{2}x_{1} + x_{3}x_{2}x_{1}, N = N^{G}(x_{4}) = x_{4}^{9} − x_{4}^{3}x_{3}^{6} + …, and elements from the image of the relative transfer, Tr_{L}^{G}(F[x_{4}^{3} − x_{4}x_{1}^{2}, x_{3}, x_{2}, x_{1}]). In fact, a Magma [3] computation shows that F[V]^{G} is minimally generated by M and N together with the following 9 invariants:

Tr_{L}^{G}(x_{3}) = x_{1},
Tr_{L}^{G}(−x_{3}^{2}) = x_{2}^{2} + x_{1}x_{3} − x_{1}x_{2} − x_{1}^{2},
Tr_{L}^{G}(−x_{2}x_{3}^{2}) = x_{2}^{3} − x_{1}x_{2}^{2} − x_{1}^{2}x_{3} + x_{1}^{3},
Tr_{L}^{G}(−x_{3}(x_{4}^{3} − x_{4}x_{1}^{2})) = x_{2}x_{3}^{3} + …,
Tr_{L}^{G}(x_{3}^{2}(x_{4}^{3} − x_{4}x_{1}^{2})) = x_{2}x_{3}^{4} + …,
Tr_{L}^{G}(x_{2}x_{3}^{2}(x_{4}^{3} − x_{4}x_{1}^{2})) = x_{2}^{2}x_{3}^{4} + …,
Tr_{L}^{G}(−x_{3}(x_{4}^{3} − x_{4}x_{1}^{2})^{2}) = x_{3}^{7} + …,
Tr_{L}^{G}(−x_{3}^{2}(x_{4}^{3} − x_{4}x_{1}^{2})^{2}) = x_{3}^{8} + …,
Tr_{L}^{G}(x_{2}x_{3}^{2}(x_{4}^{3} − x_{4}x_{1}^{2})^{2}) = x_{2}x_{3}^{8} + ….

The homogeneous components of F[V] decompose as follows:

F[V]_{0} ≅ V_{1},
F[V]_{1} ≅ V_{4},
F[V]_{2} ≅ V_{7} ⊕ V_{3},
F[V]_{3} ≅ V_{2} ⊕ V_{3} ⊕ V_{6} ⊕ V_{9},
F[V]_{4} ≅ V_{5} ⊕ 2 V_{3} ⊕ V_{6} ⊕ 2 V_{9},
F[V]_{5} ≅ V_{8} ⊕ 3 V_{3} ⊕ 2 V_{6} ⊕ 3 V_{9},
F[V]_{6} ≅  4 V_{3} ⊕ 3 V_{6} ⊕ 6 V_{9},
F[V]_{7} ≅  5 V_{3} ⊕ 4 V_{6} ⊕ 9 V_{9},
F[V]_{8} ≅  6 V_{3} ⊕ 5 V_{6} ⊕ 13 V_{9}.

The (one-dimensional) socle of the non-induced indecomposable summand in F[V]_{i} for i = 0, 1, 2, …, 5 may be chosen to contain 1, x_{1}, x_{1}^{2}, M, x_{1}M and x_{1}^{2}M respectively.

For degrees d ≥ 9, write d = 9a + c where 0 ≤ c ≤ 8. Then F[V]_{d} ≅ F[V]_{c} ⊕ α V_{3} ⊕ β V_{6} ⊕ γ V_{9} for some non-negative integers α, β and γ. Furthermore if 0 ≤ c ≤ 5 and if we denote by f a non-zero element of the socle of the noninduced summand in F[V]_{c}, then the non-induced summand in F[V]_{d} may be chosen such that its socle is spanned by N^{a}f.

We note that Symonds, in a recent paper [16] based on his joint work with Karagueuzian [11], has proven the Periodicity Conjecture of Hughes & Kemper. He goes on to prove that for p^{r−1} < n < p^{r} and d < p^{r}, the FZ/p^{r}-module F[V_{n}]_{d} is isomorphic to Ω^{−d}Λ^{d}(V_{p^{r}−n}) modulo induced modules [16, Corollary 3.11]. Here Λ^{d} denotes the d^{th} exterior power and Ω^{−d} denotes the d^{th} cokernel of a minimal injective resolution (see [2, page 30]). It is instructive to compare this with Theorem 1.2 and Example 1.3.

2. Preliminaries

Let G = ⟨σ⟩ ≅ Z/p^{r}. It will be convenient to define Δ := σ − 1 ∈ FG. It is easy to see that Δ acts as a twisted derivation on F[V_{n}], i.e., Δ(a · b) = aΔ(b) + Δ(a)σ(b). We denote the full transfer, Tr_{⟨1⟩}^{G}, by Tr^{G} and the image of the relative transfer, Tr_{L}^{G}(F[V_{n}]^{L}), by Im Tr_{L}^{G}. Clearly Im Tr_{L}^{G} is an ideal in F[V_{n}]^{G}. A simple calculation with binomial coefficients shows that Δ^{p^{t}} = σ^{p^{t}} −1 and Δ^{p^{t}−1} = (σ^{p^{t}} −1)/(σ−1) = Tr_{L}^{G}. We denote the group cohomology of G with coefficients in the FG-module W by H^{∗}(G, W). Note that H^{0}(G, W) is just the fixed subspace W^{G}. Furthermore, since G is cyclic, H^{2i−1}(G, W) = ker(Tr^{G} |_{W})/ Im(Δ|_{W}) and H^{2i}(G, W) = ker(Δ|_{W})/ Im(Tr^{G} |_{W}) for i > 0 (see, for example, [6, §2.1]). It is clear from the definition of group cohomology that H^{i}(G, P) = 0 if i > 0 and P is projective. Thus H^{1}(G, V_{p^{r}}) = H^{2}(G, V_{p^{r}}) = 0. Furthermore, if V_{m} is generated as an FG-module by e, then H^{0}(G, V_{m}) = V_{m}^{G} = span_{F}(Δ^{n−1}(e)) and {e, Δ(e), …, Δ^{n−1}(e)} is a vector space basis for V_{m}. If we identify V_{m−1} with the submodule Δ(V_{m}), then, for m < p^{r}, H^{1}(G, V_{m}) is the one dimensional vector space V_{m}/V_{m−1} and H^{2}(G, V_{m}) is the one dimensional vector space V_{m}^{G}.

Let W be any finite dimensional FG-module. Define L_{t}(W) := Δ^{t−1}(W). Clearly L_{i+1}(W) ⊆ L_{i}(W) and, since σ has order p^{r}, L_{p^{r}+1}(W) = 0. Thus we have the following filtration of W by FG-modules:

W = L_{1}(W) ⊇ L_{2}(W) ⊇ L_{3}(W) ⊇ ⋯ ⊇ L_{p^{r}}(W) ⊇ L_{p^{r}+1}(W) = 0.

This filtration of W obviously induces a filtration of the subspace W^{G}:

W^{G} = L_{1}^{G}(W) ⊇ L_{2}^{G}(W) ⊇ L_{3}^{G}(W) ⊇ ⋯ ⊇ L_{p^{r}}^{G}(W) ⊇ L_{p^{r}+1}^{G}(W) = 0

where L_{t}^{G}(W) := L_{t}(W) ∩ W^{G}.

Definition 2.1. For a non-zero f ∈ W, we define the length of f, denoted by ℓ(f), by ℓ(f) ≥ t ⟺ f ∈ L_{t}(W). Note that 1 ≤ ℓ(f) ≤ p^{r}. We will refer to the above filtration of W as the length filtration and say that a basis B for W^{G} is compatible with the length filtration if L_{t}^{G}(W) ∩ B is a basis for L_{t}^{G}(W) for all t (using the convention that the empty set is a basis for the zero vector space).

Lemma 2.2. If W is a finite dimensional FG-module, then

dim(W) = ∑_{t=1}^{p^{r}} t (dim(L_{t}^{G}(W)) − dim(L_{t+1}^{G}(W)) .

Proof. Choose a decomposition of W into indecomposable FG-modules. For each indecomposable summand, choose a basis in which σ is in Jordan canonical form. The union of these bases gives a basis for W. Intersecting this basis with W^{G} gives a basis for W^{G}, say B, which is compatible with the length filtration. It is clear that the number of elements in B ∩ (L_{t}^{G}(W) \ L_{t+1}^{G}(W)) coincides with the number of indecomposable modules in the decomposition which are isomorphic to V_{t}, giving the required formula. □

Suppose that W is a finite dimensional FG-module and B is a basis for W^{G} which is compatible with the length filtration. For each α ∈ B choose γ ∈ W with Δ^{ℓ(α)−1}(γ) = α. (The existence of a suitable γ follows from the definition of length.) Define V (α) to be the FG-module generated by γ. Note that α spans the socle of V (α) and that dim(V (α)) = ℓ(α).

Proposition 2.3.

W = ⨁_{α∈B} V (α) .

Proof. The natural homomorphism of the external direct sum of the V (α) to W is injective on the socle and is therefore injective. Thus the internal sum of the V (α) is direct. It follows from Lemma 2.2 that the dimension of W coincides with the dimension of ⊕_{α∈B}V (α), giving equality. □

The above shows how we may obtain a direct sum decomposition of W into indecomposable submodules from any basis of W^{G} which is compatible with the length filtration of W^{G}. Clearly every such decomposition arises in this way. Furthermore, an element f ∈ W^{G} has length t if and only if there is an FG decomposition W = W′ ⊕ V_{t} with f spanning V_{t}^{G}.

Note that if f, h ∈ F[V]^{G} then ℓ(fh) ≥ ℓ(f). To see this write f = Δ^{ℓ(f)−1}(F). Then fh = Δ^{ℓ(f)−1}(F h). In general it may happen that ℓ(fh) > max{ℓ(f), ℓ(h)}. Computer computations together with various results, such as Proposition 4.3, lead us to make the following conjecture.

Conjecture 2.4. Suppose f, h ∈ F[V]^{G} with ℓ(f) ≡ 0 (mod p). Then ℓ(fh) ≡ 0 (mod p).

For n ≤ p^{r}, choose an FG-module generator x_{n} for V_{n}^{∗} and define x_{i} = Δ^{n−i}(x_{n}) for i = 1, …, n − 1. Then \{x_{1}, x_{2}, …, x_{n}\} is a basis of V_{n}^{∗}. Let \overline{F} denote the algebraic closure of F and define \overline{V_{n}} := \overline{F} ⊗_{F} V_{n}. Let \{e_{1}, e_{2}, e_{3}, …, e_{n}\} denote the basis for \overline{V_{n}} dual to \{1 ⊗ x_{1}, …, 1 ⊗ x_{n}\}. Note that e_{1} generates \overline{V_{n}} as an \overline{F}G-module and that Δ(e_{n}) = 0. Using the inclusion F ⊆ \overline{F}, allows us to interpret elements of F[V_{n}] as regular functions on \overline{V_{n}}, i.e., we identify F[V_{n}] in a natural way with a subset of \overline{F}[\overline{V_{n}}]. For a subset X ⊆ \overline{F}[\overline{V_{n}}], define V(X) = \{v ∈ \overline{V_{n}} | f(v) = 0 ∀f ∈ X}.

Lemma 2.5. Suppose p^{r−1} < n ≤ p^{r} and let H denote the subgroup ⟨σ^{p^{t+1}}⟩ ≅ Z/p^{r−t−1} of G = ⟨σ⟩ ≅ Z/p^{r} where 0 ≤ t ≤ r − 1.

(1) V(Im Tr_{H}^{G}) = \overline{V_{n}}^{Z/p^{r−t}} = span\overline{F}\{e_{n−p^{t}+1}, e_{n−p^{t}+2}, …, e_{n−1}, e_{n}\}.

(2) For f ∈ F[V_{n}]^{G}, if ℓ(f) ≥ p^{t} + 1 then

f ∈ \sqrt{Im Tr_{H}^{G}} = ((x_{1}, x_{2}, …, x_{n−p^{t}})F[V_{n}]) ∩ F[V_{n}]^{G}.

Proof. The equality V_{n}^{Z/p^{r−t}} = span_{F}{e_{n−p^{t}+1}, e_{n−p^{t}+2}, …, e_{n−1}, e_{n}} is easily verified. The equality V(Im Tr_{H}^{G}) = \overline{V_{n}}^{Z/p^{r−t}} follows from [8, Proposition 12.5] (see also [5, Theorem 12]). This equality of sets may be expressed equivalently as the

equality of ideals \sqrt{Im Tr_{H}^{G}} = ((x_{1}, x_{2}, …, x_{n−p^{t}})F[V_{n}]) ∩ F[V_{n}]^{G} (see, for example, [5, Proposition 11]). Thus it only remains to show that if ℓ(f) ≥ p^{t} + 1 then

f ∈ \sqrt{Im Tr_{H}^{G}}.

To see this suppose that ℓ(f) ≥ p^{t} + 1. Then f = Δ^{p^{t}}(F) for some F ∈ F[V_{n}]. Therefore f(e_{i}) = (Δ^{p^{t}}F)(e_{i}) = ((σ − 1)^{p^{t}}F)(e_{i}) = (σ^{p^{t}}(F) − F)(e_{i}) = F (σ^{−p^{t}}(e_{i}))−F (e_{i}). Thus f(e_{i}) = 0 if e_{i} is fixed by σ^{p^{t}}, i.e., if i ≥ n−p^{t}+1. Therefore if ℓ(f) ≥ p^{t}+1 then f vanishes on the set span_{F}{e_{n−p^{t}+1}, e_{n−p^{t}+2}, …, e_{n−1}, e_{n}}.

Hence if ℓ(f) ≥ p^{t} + 1 then f ∈ \sqrt{Im Tr_{H}^{G}} □

Proposition 2.6. Suppose that f is a non-zero homogeneous element of F[V_{n}]^{G}. Then ℓ(fN^{G}(x_{n})) = ℓ(f).

Proof. Denote N^{G}(x_{n}) by N. Define t such that p^{t−1} < n ≤ p^{t} (the case n = 1 is trivial). Then the leading term of N is x_{n}^{p^{t}}. Let F[V_{n}]^{♭} denote the span of the monomials in F[V_{n}] which, as polynomials in x_{n}, have degree less than p^{t}. The fact that x_{n} ∉ Δ(F[V_{n}]) means that F[V_{n}]^{♭} is a FG-submodule of F[V_{n}]. For an arbitrary polynomial h ∈ F[V_{n}], viewing h as a polynomial in x_{n} and dividing by N^{G}(x_{n}) gives h = qN +r for unique r ∈ F[V_{n}]^{♭} and q ∈ F[V_{n}]. This gives the FGmodule decomposition F[V_{n}] = NF[V_{n}] ⊕ F[V_{n}]^{♭} (compare with [10, Lemma 2.9] and [14, § 2]). As noted above ℓ(Nf) ≥ ℓ(f). Suppose Nf = Δ^{t}(F) and write F = NF_{1}+F_{0} with F_{0} ∈ F[V_{n}]^{♭}. Then Nf = Δ^{t}(NF_{1}+F_{0}) = NΔ^{t}(F_{1})+Δ^{t}(F_{0}) = NΔ^{t}(F_{1}) and thus f = Δ^{t}(F_{1}). This shows that ℓ(f) ≥ ℓ(Nf). □

3. Computing F[V_{p+1}]^{Z/p^{2}}

In this section we use the ladder technique described in [15, §7] to prove Theorem 1.1. We use the notation

G := ⟨σ⟩ ≅ Z/p^{2}, L := ⟨σ^{p}⟩ ≅ Z/p, and Q := G/L = ⟨\overline{σ}⟩ ≅ Z/p.

Note that deg (N^{G} (x_{p})) = p, deg (N^{G} (x_{p+1})) = p^{2} and T r_{L}^{G}(x_{p}) = x_{1}.

The action of L on V_{p+1}^{∗} is given by σ^{p}(x_{p+1}) = x_{p+1} + x_{1} and σ^{p}(x_{i}) = x_{i} for i ≤ p. Thus as L-modules, F[V_{p+1}] ≅ F[V_{2} ⊕ (p − 1)V_{1}]. Therefore F[V_{p+1}]^{L} ≅ F[N^{L}(x_{p+1}), x_{p}, …, x_{1}] with N^{L}(x_{p+1}) = x_{p+1}^{p} − x_{1}^{p−1}x_{p+1}. The action of Q on F[V_{p+1}]^{L} is given by \overline{σ}(N^{L}(x_{p+1})) = N^{L}(x_{p+1}) + x_{p}^{p} − x_{1}^{p−1}x_{p} and \overline{σ}(x_{i}) = σ(x_{i}) for i = 1, 2, …, p. Define

A := F[z_{p}, …, z_{1}, X_{p}, …, X_{1}]

with deg(z_{i}) = p and deg(X_{i}) = 1. Further define an algebra homomorphism π : A → F[V_{p+1}]^{L} by π(z_{i}) = x_{i+1}^{p} − x_{1}^{p−1}x_{i+1} and π(X_{i}) = x_{i}. Note that π is a degree preserving surjection with π(z_{p}) = N^{L}(x_{p+1}). Further note that the kernel of π is the ideal

I := (z_{p−1} − (X_{p}^{p} − X_{1}^{p−1}X_{p}) , …, z_{1} − (X_{2}^{p} − X_{1}^{p−1}X_{2})) A.

Define an action, by algebra automorphisms, of Q on A by taking \overline{σ}(z_{i}) = z_{i}+z_{i−1} and \overline{σ}(X_{i}) = X_{i} + X_{i−1} for i > 1, \overline{σ}(z_{1}) = z_{1} and \overline{σ}(X_{1}) = X_{1}. Thus as FQmodules A ≅ F[2V_{p}] and π is a map of FQ-modules.

The short exact sequence of FQ-modules, 0 → I → A ⟶^{π} F[V_{p+1}]^{L} → 0, gives a long exact sequence on group cohomology

0 → I^{Q} → A^{Q} → (F[V_{p+1}]^{L})^{Q} → H^{1}(Q, I) → H^{1}(Q, A) → · · · .

We will show that the inclusion of I into A induces an injection of H^{1}(Q, I) into H^{1}(Q, A). Thus π restricts to a surjection from A^{Q} to (F[V_{p+1}]^{L})^{Q} = F[V_{p+1}]^{G}. Since 2V_{p} is a permutation representation of Q, after a suitable change of basis, Q acts on A by permuting the variables. Using the permutation basis, the orbit sums of monomials form a vector space basis for A^{Q}. Since Q ≅ Z/p, these orbits are of size p or size 1. The orbits of size p span projective FQ-module summands of F[V_{p+1}]^{L} while the orbits of size 1 span trivial summands. It easy to see that, in the original basis, the orbits of size 1 are polynomials in N^{Q}(z_{p}) and N^{Q}(X_{p}), while the orbit sums coming from orbits of size p are elements in the image of the transfer. Thus A^{Q} is generated by N^{Q}(z_{p}), N^{Q}(X_{p}) and elements from Tr^{Q}(A), giving Theorem 1.1.

Remark 3.1. A bound on the degrees of the generators coming from the image of the relative transfer can be computed using the degrees of a homogeneous system of parameters for A^{Q}. Using the system described in the proof of [10, Corollary 2.15] gives and upper bound of (p + 1)^{3}/4. The results of Fleischmann et al. [9] can be used to give an upper bound on the degrees of a minimal generating set for A^{Q}, and hence for F[V_{p+1}]^{Z/p^{2}}, of 2p^{2} − 2p − 1. Magma [3] calculations give a Noether number of p^{2} for p = 2 and p = 3.

The rest of this section is devoted to completing the proof of Theorem 1.1 by showing that the inclusion of I into A induces an injection of H^{1}(Q, I) into H^{1}(Q, A) (see Theorem 3.11 (b)). We start by describing H^{∗}(Q, A).

Proposition 3.2. (a) H^{2}(Q, A) = A^{Q}/ Tr^{Q}(A) ≅ F[N^{Q}(X_{p}), N^{Q}(z_{p})].

(b) H^{1}(Q, A) is a principal A^{Q}-module with annihilator given by Tr^{Q}(A).

Proof. It follows from the discussion preceding Remark 3.1 that, as an FQmodule, A consists of projective summands and trivial summands with the trivial summands spanned by the monomials in N^{Q}(X_{p}) and N^{Q}(z_{p}). The projective summands do not contribute to the cohomology. The trivial summands contribute non-zero classes to both the first and second cohomology. □

Note that, although H^{1}(Q, A) does not have a multiplicative structure, it is isomorphic to F[N^{Q}(X_{p}), N^{Q}(z_{p})] as an A^{Q}-module.

To compute H^{∗}(Q, I), we start by resolving I as an A − FQ-module using a Koszul resolution (see, for example, [4, §1.6]). Observe that I is generated by an A-regular sequence of length p − 1. Furthermore, these generators span the degree p homogeneous component, I_{p}, of I and, as a FQ-module, I_{p} ≅ V_{p−1}. Let μ denote the Q-equivariant map from V_{p−1} ⊗A to A given by identifying elements of V_{p−1} with elements of I_{p} and then using the multiplication in A. Let Λ^{i}(V_{p−1}) denote the i^{th} exterior power of V_{p−1}. Define ζ^{i} : Λ^{i}(V_{p−1}) → Λ^{i−1}(V_{p−1}) ⊗ V_{p−1} by

ζ^{i}(v_{1} ∧ v_{2} ∧ ⋯ ∧ v_{i}) = ∑_{j=1}^{i}(−1)^{i−j} (v_{1} ∧ ⋯ ∧ v̂_{j} ∧ ⋯ ∧ v_{i}) ⊗ v_{j}

for all v_{1}, v_{2}, …, v_{i} ∈ V_{p−1}. Define F^{−i} := Λ^{i}(V_{p−1}) ⊗ A for i = 1, 2, …, p − 1 and define ρ^{−i} : F^{−i} → F^{−i+1} to be (1_{Λ^{i−1}(V_{p−1})} ⊗ μ) ∘ (ζ^{i} ⊗ 1_{A}). This gives the following sequence of A − FQ-modules:

0 → F^{1−p} ⟶^{ρ^{1−p}} F^{2−p} ⟶^{ρ^{2−p}} ⋯ ⟶^{ρ^{−3}} F^{−2} ⟶^{ρ^{−2}} F^{−1} ⟶^{μ} I → 0.

Since the generators of I form a regular A-sequence, it follows from [4, Corollary 1.6.14] that this sequence is exact. For i > 0, define K^{−i} to be the kernel of the map ρ^{−i} : F^{−i} → F^{−i+1}. For convenience, we define K^{0} := I, K^{1} := A/I, K^{a} := 0 for a > 1, F^{1} := A/I and F^{a} := 0 for a > 1. Using the exactness of the resolution, we get a series of short exact sequences 0 → K^{−i} → F^{−i} → K^{−i+1} → 0. For each of these short exact sequences, we apply group cohomology to get a long exact sequence:

0 → H^{0}(Q, K^{−i}) → H^{0}(Q, F^{−i}) → H^{0}(Q, K^{−i+1}) → H^{1}(Q, K^{−i})
→ H^{1}(Q, F^{−i}) → H^{1}(Q, K^{−i+1}) → ….

Defining D^{a,b} := H^{b}(Q, K^{a}) and E^{a,b} := H^{b}(Q, F^{a}) gives a bigraded exact couple which leads to a spectral sequence. This is essentially the construction given at the end of [12, Ch XI, §5]. We will use this spectral sequence to describe H^{1}(Q, I). The following series of lemmas lead to a description of H^{b}(Q, F^{a}).

Lemma 3.3. The map ζ^{i+1} is an isomorphism of FQ-modules from Λ^{i+1}(V_{p−1}) to a direct summand of Λ^{i}(V_{p−1}) ⊗ V_{p−1}.

Proof. It is clear that ζ^{i+1} followed by the natural projection from Λ^{i}(V_{p−1})⊗V_{p−1} to Λ^{i+1}(V_{p−1}) is i + 1 times the identity map on Λ^{i+1}(V_{p−1}). Since i + 1 ≤ p − 1, this is an isomorphism. □

Lemma 3.4. V_{p−1} ⊗ V_{p−1} ≅ V_{1} ⊕ (p − 2)V_{p}.

Proof. If the representation ring R_{FZ/p} is extended by adjoining an element α satisfying V_{2} = α + α^{−1}, then by [10, Lemma 2.3], V_{n} = \frac{α^{n}−α^{−n}}{α−α^{−1}} . Thus, in the augmented representation ring,

V_{p−1}^{2} − V_{p} · V_{p−2} = \frac{(α^{p−1} − α^{−(p−1)})^{2} − (α^{p} − α^{−p})(α^{p−2} − α^{−(p−2)}))}{(α − α^{−1})^{2}}
= \frac{α^{2p−2} − 2 + α^{−(2p−2)} − (α^{2p−2} − α^{2} − α^{−2} + α^{−(2p−2)})}{α^{2} − 2 + α^{−2}}
= 1 = V_{1}.

Therefore, V_{p−1} ⊗ V_{p−1} ≅ V_{p} ⊗ V_{p−2} ⊕ V_{1}. It follows from [1, Ch. II §7 Lemma 4] that V_{p} ⊗ V_{i} ≅ iV_{p}. Thus V_{p−1} ⊗ V_{p−1} ≅ (p − 2)V_{p} ⊕ V_{1}, as required. □

Lemma 3.5. For i even,

Λ^{i}(V_{p−1}) ≅ V_{1} ⊕ \frac{1}{p} (\binom{p − 1}{i} − 1) V_{p}

and for i odd,

Λ^{i}(V_{p−1}) ≅ V_{p−1} ⊕ \frac{1}{p} (\binom{p − 1}{i} − (p − 1)) V_{p}.

Proof. First observe that dim_{F}(Λ^{i}(V_{p−1}) = \binom{p−1}{i} ≡ (−1)^{i} (mod p). Therefore the dimensions are correct. Thus it follows from Lemma 3.3 that Λ^{i}(V_{p−1}) is a non-projective summand of Λ^{i−1}(V_{p−1}) ⊗ V_{p−1}. Also note that the result is true for i = 0 and i = 1. We proceed by induction. Suppose the result holds for i. For i even this gives,

Λ^{i}(V_{p−1}) ⊗ V_{p−1} ≅ V_{1} ⊗ V_{p−1} ⊕ \frac{1}{p} (\binom{p − 1}{i} − 1)) V_{p} ⊗ V_{p−1}
≅ V_{p−1} ⊕ \frac{p − 1}{p} (\binom{p − 1}{i} − 1)) V_{p}

and therefore, Λ^{i}(V_{p−1}) ⊗ V_{p−1} is isomorphic to V_{p−1} plus projective modules. Hence Λ^{i+1}(V_{p−1}) is isomorphic to V_{p−1} plus projective modules. For i odd,

Λ^{i}(V_{p−1}) ⊗ V_{p−1} ≅ V_{p−1} ⊗ V_{p−1} ⊕ \frac{1}{p} (\binom{p − 1}{i} − (p − 1)) V_{p} ⊗ V_{p−1}
≅ V_{p−1} ⊗ V_{p−1} ⊕ \frac{p − 1}{p} (\binom{p − 1}{i} − (p − 1)) V_{p}

and, therefore, Λ^{i}(V_{p−1}) ⊗ V_{p−1} is isomorphic to V_{p−1} ⊗ V_{p−1} plus projective modules. From Lemma 3.4, V_{p−1} ⊗ V_{p−1} is isomorphic to V_{1} plus projective modules. Hence Λ^{i+1}(V_{p−1}) is isomorphic to V_{1} plus projective modules. □

Lemma 3.6. For a ≤ 0 and b > 0, H^{b}(Q, F^{a}) is a principal A^{Q}-module with annihilator Tr^{Q}(A).

Proof. As an FQ-module, A is a direct sum of projective summands with socles contained in Tr^{Q}(A) and one dimensional summands spanned by monomials in N^{Q}(X_{p}) and N^{Q}(z_{p}). It follows from Lemma 3.5 that Λ^{−a}(V_{p−1}) contains a a single non-projective summand. Note that projective summands do not contribute to the cohomology. Further note that for any module M and projective module P, M ⊗ P is projective. Thus H^{b}(Q, Λ^{−a}(V_{p−1})) is a one dimensional vector space and H^{b}(Q, F^{a}) ≅ H^{b}(Q, Λ^{−a}(V_{p−1}) ⊗ A) ≅ H^{b}(Q, A). The result follows from Proposition 3.2 □

The following lemma is a preliminary step in evaluating d^{a,b} : E^{a,b} → E^{a+1,b} for a < 0 and b > 0.

Lemma 3.7. The inclusion of I_{p} into A_{p} induces an injection from H^{1}(Q, I_{p}) to H^{1}(Q, A_{p}) and the zero map from H^{2}(Q, I_{p}) to H^{2}(Q, A_{p}).

Proof. To see that the inclusion induces an injection from H^{1}(Q, I_{p}) to H^{1}(Q, A_{p}), first note that Δ is a twisted derivation and that Δ(f), for f a generator of A, lies in Span_{F}(z_{p−1}, …, z_{1}, X_{p−1}, …, X_{1}). Therefore Δ(A) is contained in the ideal (z_{p−1}, …, z_{1}, X_{p−1}, …, X_{1})A. As an FQ-module, I_{p} is isomorphic to V_{p−1} with generator r := z_{p−1}−(X_{p}^{p} − X_{1}^{p−1}X_{p}). Thus H^{1}(Q, I_{p}) is a one dimensional vector space with r representing a non-zero cohomology class. Since r does not lie in the ideal (z_{p−1}, …, z_{1}, X_{p−1}, …, X_{1})A, this element does not lie in Δ(A) and, therefore, represents a non-zero class in H^{1}(Q, A_{p}).

To see that inclusion induces the zero map from H^{2}(Q, I_{p}) to H^{2}(Q, A_{p}), observe that I_{p} ≅ V_{p−1} and A_{p} is isomorphic to V_{1} plus projectives. Thus the map on cohomology is determined by a Z/p-equivariant map from V_{p−1} to V_{1}, and all such maps induce the zero map in second cohomology. □

For a < 0, the first differential in the spectral sequence is the map on cohomology d^{a,b} : H^{b}(Q, F^{a}) → H^{b}(Q, F^{a+1}) induced by ρ^{a} : F^{a} → F^{a+1}

Theorem 3.8. For b > 0 and a < 0, d^{a,b} : H^{b}(Q, F^{a}) → H^{b}(Q, F^{a+1}) is an isomorphism if a and b have the same parity and zero if a and b have different parities.

Proof. From Lemma 3.6, H^{b}(Q, F^{a}) is a principal A^{Q}-module. Since d^{a,b} is an A^{Q}-module map, it is sufficient to evaluate d^{a,b} on a generator. Thus, using the definition of ρ^{a}, we see that d^{a,b} is determined by the composition

Λ^{−a}(V_{p−1}) ⟶^{ζ^{−a}} Λ^{−a−1}(V_{p−1}) ⊗ V_{p−1} ⟶^{≅} Λ^{−a−1}(V_{p−1}) ⊗ I_{p} ⟶^{⊂} Λ^{−a−1}(V_{p−1}) ⊗ A.

It follows from Lemmas 3.3 and 3.5 that ζ^{−a} induces an isomorphism in cohomology. Thus d^{a,b} is determined by the inclusion of Λ^{−a−1}(V_{p−1}) ⊗ I_{p} into Λ^{−a−1}(V_{p−1}) ⊗ A.

For a odd, using Lemma 3.5, Λ^{−a−1}(V_{p−1}) is isomorphic to V_{1} plus projectives. Therefore, in this case, d^{a,b} is induced by the inclusion of I_{p} into A_{p}. Thus, using Lemma 3.7, if a and b are both odd then d^{a,b} is injective and if a is odd and b is even then d^{a,b} is zero.

For a even, using Lemma 3.5, Λ^{−a−1}(V_{p−1}) is isomorphic to V_{p−1} plus projectives. Therefore, in this case, d^{a,b} is induced by the inclusion of V_{p−1} ⊗ I_{p} into V_{p−1} ⊗ A_{p} ≅ V_{p−1} ⊗ V_{p−1}. By Lemma 3.4, V_{p−1} ⊗ V_{p−1} is isomorphic to V_{1} plus projectives. Since A_{p} is isomorphic to V_{1} plus projectives, V_{p−1} ⊗A_{p} is isomorphic to V_{p−1} plus projectives. Thus d^{a,b} is determined by the composition

V_{1} → V_{p−1} ⊗ I_{p} → V_{p−1} ⊗ A_{p} → V_{p−1}.

This map clearly induces the zero map from H^{1}(Q, V_{1}) to H^{1}(Q, V_{p−1}). Thus for a even and b odd, d^{a,b} = 0. To show that d^{a,b} is injective for a even and b even, we need to show that the given map from V_{1} to V_{p−1} is non-zero. It follows from Lemma 3.7, that for the purposes of computing cohomology, the inclusion of I_{p} into A_{p} is the injection of V_{p−1} into V_{1} ⊕ V_{p} taking e′ to (e′′, Δ(e)) where e, e′ and e′′ denote elements which generate the cyclic G-modules V_{p}, V_{p−1} and V_{1} respectively. The cokernel of this map is isomorphic to V_{2}. Tensoring over F is exact so we have a short exact sequence

0 → V_{p−1} ⊗ V_{p−1} → V_{p−1} ⊗ (V_{1} ⊕ V_{p}) → V_{p−1} ⊗ V_{2} → 0.

This gives rise to a long exact sequence in cohomology. Recall that V_{p−1} ⊗ V_{2} ≅ V_{p−2} ⊕ V_{p} (see, for example, [1, Ch.II § 7 Lemma 5]). Thus, modulo projectives, the sequence is V_{1} → V_{p−1} → V_{p−2}. This can only give a long exact sequence on cohomology if the the map from V_{1} to V_{p−1} is non-zero. □

Corollary 3.9. For b > 0 and a < 0, the spectral sequence satisfies

E_{2}^{a,b} = \{ \begin{matrix}F & if a = 1 − p and b odd; \\ 0 & otherwise. \end{matrix}

It follows from Theorem 3.8 that ρ^{−1} induces an isomorphism from H^{1}(Q, F^{−1}) to H^{1}(Q, A). This map factors through H^{1}(Q, I) with the first map in the factorisation induced by μ and the second induced by inclusion. Thus to complete the proof of Theorem 1.1, it is sufficient to show the following.

Lemma 3.10. The map μ induces an epimorphism from H^{1}(Q, F^{−1}) to H^{1}(Q, I).

Proof. Denote by ∂^{a,b} the connecting homomorphism from H^{b}(Q, K^{a}) to H^{b+1}(Q, K^{a−1}) and define a filtration on H^{1}(Q, I) = H^{1}(Q, K^{0}) by

F_{t} := kernel(∂^{−t,t+1} ∘ ∂^{−t+1,t} ∘ ⋯ ∘ ∂^{−2,3} ∘ ∂^{−1,2} ∘ ∂^{0,1}).

Since ∂^{1−p,p} = 0, we have F_{p−1} = H^{1}(Q, I). Using the long exact sequence in cohomology coming from 0 → K^{−1} → F^{−1} → K^{0} → 0, we see that F_{0} is the image of H^{1}(Q, F^{−1}) in H^{1}(Q, I). We will prove the lemma by showing F_{0} = F_{p−1}.

Using the definition a derived couple (see, for example, [12, Ch. XI, §5]), we have D_{t+2}^{a−t−1,b+t+1} = ∂^{a−t,b+t} ∘ ⋯ ∘ ∂^{a−1,b+1} ∘ ∂^{a,b}(D^{a,b}). If x ∈ F_{t} \ F_{t−1} then ∂^{−t+1,t} ∘ ⋯ ∘ ∂^{0,1}(x) is a non-zero element of D_{t+1}^{−t,t+1} which lifts to a non-zero element of E_{t+1}^{−(t+1),t+1}. However, it follows from Corollary 3.9 that E_{2}^{−(t+1),t+1} = 0 for t ≥ 1. Thus E_{t+1}^{−(t+1),t+1} = 0 for t ≥ 1. Therefore F_{t} = F_{0} for all t ≥ 1. □

These calculations give the following.

Theorem 3.11. (a) H^{1}(Q, I) is a principal A^{Q}-module with generator represented by z_{p−1} − (X_{p}^{p} − X_{1}^{p−1}X_{p}) and annihilator Tr^{Q}(A).

(b) The inclusion of I into A induces an A^{Q}-module monomorphism of H^{1}(Q, I) to H^{1}(Q, A) taking [z_{p−1} − (X_{p}^{p} − X_{1}^{p−1}X_{p})] to −[N^{Q}(X_{p})].

This completes the proof of Theorem 1.1.

4. Decomposing F[V_{p+1}]

The main goal of this section is to describe the FG-module decomposition of F[V_{p+1}]. We do this by considering a basis for F[V_{p+1}]_{d}^{G} which is compatible with the length filtration. By Theorem 1.1, F[V_{p+1}]^{G} is generated by M := N^{G}(x_{p}) and N := N^{G}(x_{p+1}) together with elements of the relative transfer, Tr_{L}^{G}(F[V_{p+1}]^{L}). To identify the summands occurring in the decomposition, we need to determine the lengths of the basis elements.

Suppose f ∈ F[V_{p+1}]^{G}. It follows from Lemma 2.5 that if ℓ(f) ≥ 2 then

f ∈ \sqrt{Im Tr_{L}^{G}} = (x_{1}, x_{2}, …, x_{p})F[V_{p+1}] ∩ F[V_{p+1}]^{G}, and if ℓ(f) ≥ p + 1 then

f ∈ \sqrt{Im Tr^{G}} = (x_{1})F[V_{p+1}] ∩ F[V_{p+1}]^{G}. Furthermore, since Tr_{L}^{G} = Δ^{p−1}, if ℓ(f) ≥ p then f ∈ Im Tr_{L}^{G}. Since N = N^{G}(x_{p+1}) has lead term^{1} x_{p+1}^{p^{2}}, we have that N ∉ (x_{1}, x_{2}, …, x_{p})F[V_{p+1}] and thus ℓ(N) = 1. Similarly since the lead term of M = N^{G}(x_{p}) is x_{p}^{p}, we have that M ∉ (x_{1})F[V_{p+1}] and thus ℓ(M) ≤ p.

Lemma 4.1. Let 1 ≤ q < p. Then

Δ^{qp}(x_{p+1}^{i}) = \begin{cases}0, & if i < q \\ q!x_{1}^{q}, & if i = q \\ x_{1}^{q}h for some h ∈ F[V_{p+1}], & if i ≥ q + 1. \end{cases}

In particular, Δ^{qp}(x_{p+1}^{i}) ∈ (x_{1}^{q})F[V_{p+1}] for all i ≥ 0.

Proof. We consider Δ^{qp}(x_{p+1}^{i}) using induction on q. For q = 1 we have

Δ^{p}(x_{p+1}^{i}) = (x_{p+1} + x_{1})^{i} − x_{p+1}^{i} = ∑_{j=0}^{i−1} \binom{i}{j}x_{1}^{i−j}x_{p+1}^{j}

= \begin{cases}0, & if i = 0 \\ x_{1}, & if i = 1 \\ x_{1} ∑_{j=0}^{i−1} \binom{i}{j}x_{1}^{i−j−1}x_{p+1}^{j}, & if i ≥ 2. \end{cases}

1Use any monomial order with x_{1} < x_{2} < ⋯ < x_{p+1}.

Now take q+1 ≥ 2. Then Δ^{(q+1)p}(x_{p+1}^{i}) = Δ^{qp}(Δ^{p}(x_{p+1}^{i})) = Δ^{qp}(∑_{j=0}^{i−1} \binom{i}{j}x_{1}^{i−j}x_{p+1}^{j}) = ∑_{j=0}^{i−1} \binom{i}{j}x_{1}^{i−j}Δ^{qp}(x_{p+1}^{j}). By induction this gives

Δ^{(q+1)p}(x_{p+1}^{i}) = ∑_{j=q}^{i−1} \binom{i}{j}x_{1}^{i−j}Δ^{qp}(x_{p+1}^{j})

= \begin{cases}0, & if i − 1 < q \\ \binom{q+1}{q}x_{1}Δ^{qp}(x_{p+1}^{q}), & if i − 1 = q \\ x_{1} ∑_{j=q}^{i−1} \binom{i}{j}x_{1}^{i−j−1}Δ^{qp}(x_{p+1}^{j}), & if i − 1 > q \end{cases}

= \begin{cases}0, & if i < q + 1 \\ (q + 1)x_{1}q!x_{1}^{q}, & if i = q + 1 \\ x_{1} ∑_{j=q}^{i−1} x_{1}^{q}h_{j} where h_{j} ∈ F[V_{p+1}], & if i ≥ q + 1. \end{cases}

= \begin{cases}0, & if i < q + 1 \\ (q + 1)!x_{1}^{q+1}, & if i = q + 1 \\ x_{1}^{q+1}h for some h ∈ F[V_{p+1}], & if i ≥ q + 1. \end{cases}

□

Proposition 4.2. Let f be a non-zero element of F[V_{p+1}]^{G} and let 1 ≤ q < p. Then ℓ(f) ≥ qp + 1 ⟺ f ∈ Im Δ^{qp} ⟺ f ∈ (x_{1}^{q})F[V_{p+1}]^{G}.

Proof. The first equivalence is just the definition of length. For the second equivalence, first suppose that f = Δ^{qp}(F) ∈ Im Δ^{qp}. Write F = ∑_{i=0}^{r} f_{i}x_{p+1}^{i} where each f_{i} ∈ F[x_{1}, x_{2}, …, x_{p}]. Then f = Δ^{qp}(F) = ∑_{i=0}^{r} f_{i}Δ^{qp}(x_{p+1}^{i}) ∈ (x_{1}^{q})F[V_{p+1}]^{G} by the previous lemma. Conversely, suppose that f ∈ (x_{1}^{q})F[V_{p+1}]^{G} and write f = x_{1}^{q}f′ where f′ ∈ F[V_{p+1}]^{G}. Then

Δ^{qp} (\frac{x_{p+1}^{q}f′}{q!} ) = \frac{f′}{q!} Δ^{qp}(x_{p+1}^{q}) = \frac{f′}{q!} q!x_{1}^{q} = f.

□

Proposition 4.3. Let f be a non-zero element of F[V_{p+1}]^{G} and write f = x_{1}^{q}f′ where x_{1} does not divide f′. If q ≥ p then ℓ(f) = p^{2}. Otherwise ℓ(f) = qp + ℓ(f′).

Proof. Applying Lemma 4.1 gives Δ^{(p−1)p}(x_{p+1}^{p−1}) = (p − 1)!x_{1}^{p−1} = −x_{1}^{p}. Furthermore, Δ^{p}(x_{p}) = 0. Thus

Δ^{p^{2}−1}(x_{p+1}^{p−1}x_{p}) = Δ^{p−1} ((Δ^{p})^{p−1}(x_{p+1}^{p−1}x_{p})) = Δ^{p−1}(x_{p}Δ^{(p−1)p}(x_{p+1}^{p−1}))
= −Δ^{p−1}(x_{p}x_{1}^{p−1}) = −x_{1}^{p−1}Δ^{p−1}(x_{p}) = −x_{1}^{p}.

Therefore Δ^{p^{2}−1}(−x_{p+1}^{p−1}x_{p}f′) = x_{1}^{p}f′. This implies that if q ≥ p then ℓ(f) = p^{2}. Suppose then that q < p. By Proposition 4.2, we have qp ≤ ℓ(f) − 1. Since x_{1}^{q+1} does not divide f, Proposition 4.2 also implies that ℓ(f) − 1 < (q + 1)p. Write ℓ(f) − 1 = qp + r where 0 ≤ r ≤ p − 1 and define s := ℓ(f′) − 1. Since x_{1} does not divide f′, Lemma 2.5 implies that 0 ≤ s ≤ p − 1. We will show that r = s.

Clearly there exists F ∈ F[V_{p+1}] such that f = Δ^{qp+r}(F). Therefore f = Δ^{r}(Δ^{qp}(F)) = Δ^{r}(x_{1}^{q}F′) for some F′ ∈ F[V_{p+1}]. Hence x_{1}^{q}f′ = f = x_{1}^{q}Δ^{r}(F′) and therefore f′ = Δ^{r}(F′). Hence s + 1 = ℓ(f′) ≥ r + 1.

Conversely we may write f′ = Δ^{s}(F′′) for some F′′ ∈ F[V_{p+1}]. Since s ≤ p − 1 we have Δ^{p}(F′′) = Δ^{p−s}(Δ^{s}(F′′)) = Δ^{p−s}(f′) = 0. This shows that F′′ ∈ F[V_{p+1}]^{L}. Thus Δ^{qp+s}(x_{p+1}^{q}F′′) = (Δ^{s}(Δ^{p})^{q})(x_{p+1}^{q}F′′) = Δ^{s}(q!x_{1}^{q}F′′) = q!x_{1}^{q}Δ^{s}(F′′) = q!x_{1}^{q}f′ = q!f where q! ≠ 0 since q < p. This shows that f ∈ Im Δ^{qp+s} and thus qp+r +1 = ℓ(f) ≥ qp+s+1. Therefore r = s as required. □

Proposition 4.4. Let f be a non-zero element in the image of the relative transfer, Tr_{L}^{G}(F[V_{p+1}]^{L}). Suppose that x_{1} does not divide f. Then ℓ(f) = p.

Proof. Since x_{1} does not divide f, Lemma 2.5 implies that ℓ(f) ≤ p. Conversely Tr_{L}^{G} = 1 + σ + σ^{2} + ⋯ + σ^{p−1} = Δ^{p−1}. Thus the hypothesis that f ∈ Im Tr_{L}^{G} implies that ℓ(f) ≥ p. □

Remark 4.5. Since elements in Tr^{G}(F[V_{p+1}]) have length p^{2}, it is clear that F[V_{p+1}]^{G} is generated by N, M and elements from Im Tr_{L}^{G} \ Im Tr^{G}.

Proposition 4.6. ℓ(M^{j}) = j + 1 for all j = 0, 1, …, p − 1. In particular M^{j} lies in the image of the relative transfer, Tr_{L}^{G}(F[V_{p+1}]^{L}), if and only if j ≥ p − 1.

Proof. From Theorem 1.1, F[V_{p+1}]^{G} is generated by M, N and elements from Im Tr_{L}^{G}. Note that deg(M) = p and deg(N) = p^{2}. Thus for d < p^{2}, if p does not divide d, we have F[V_{p+1}]_{d}^{G} = Tr_{L}^{G}(F[V_{p+1}]_{d}^{L}) and, if d = ip with i < p, F[V_{p+1}]_{d}^{G} = F · M^{i} + Tr_{L}^{G}(F[V_{p+1}]_{ip}^{L}). Fix j ∈ {1, 2, …, p − 1}. Choose a basis, B, for F[V_{p+1}]_{jp}^{G} so that B is compatible with the length filtration. Applying Proposition 2.3 gives a decomposition

F[V_{p+1}]_{jp} = ⨁_{α∈B} V (α) .

Suppose f ∈ L_{p}^{G}(F[V_{p+1}]) ∩ B. Then f ∈ Tr_{L}^{G}(F[V_{p+1}]^{L}). If x_{1} does not divide f, then by Proposition 4.4, ℓ(f) = p. Suppose x_{1} does divide f. Write f = x_{1}^{q}f′ where x_{1} does not divide f′. If f′ ∈ Tr_{L}^{G}(F[V_{p+1}]^{L}), then by Proposition 4.3 and Proposition 4.4, ℓ(f) is a multiple of p. If f′ ∉ Tr_{L}^{G}(F[V_{p+1}]^{L}), then f′ = cM^{i} +f′′ for some non-zero c ∈ F and some f′′ ∈ Tr_{L}^{G}(F[V_{p+1}]^{L}). Thus deg(f′) = ip and q = (j − i)p. Thus q > p and by Proposition 4.3, ℓ(f) is p^{2}. Hence, for any f ∈ L_{p}^{G}(F[V_{p+1}]) ∩ B, ℓ(f) is a multiple of p. Therefore p divides the dimension of

⨁_{α∈L_{p}^{G}(F[V_{p+1}])∩B} V (α).

Since dim F[V_{p+1}]_{pj} = \binom{p+pj}{pj}, Lucas’ Lemma (see, for example, [7]) implies that dim F[V_{p+1}]_{pj} ≡ \binom{j+1}{j}\binom{0}{0} (mod p). Thus dim F[V_{p+1}]_{pj} ≡ j + 1 (mod p). This shows that M^{j} ∉ Tr_{L}^{G}(F[V_{p+1}]^{L}) for all j ≤ p − 2 and that M^{p−1} ∈ Tr_{L}^{G}(F[V_{p+1}]^{L}). Furthermore, for 1 ≤ j < p − 1 we have ℓ(M^{j}) < p and ℓ(M^{j}) ≡ j + 1 (mod p) and thus ℓ(M^{j}) = j + 1. Since F[V_{p+1}]_{0} ≅ F, it is clear that ℓ(M^{0}) = 1. □

Define F[V_{p+1}]^{♭} to be the span of the monomials in F[V_{p+1}] which, as polynomials in x_{p+1}, have degree less than p^{2}. It follows from the proof of Proposition 2.6 that, as FG-modules, F[V_{p+1}] = NF[V_{p+1}] ⊕ F[V_{p+1}]^{♭}. Thus a decomposition of F[V_{p+1}]^{♭} gives a decomposition of F[V_{p+1}]. Therefore the following theorem implies Theorem 1.2.

Theorem 4.7. (i) For d < p^{2} − p, an FG-module decomposition of F[V_{p+1}]_{d}^{♭} includes precisely one non-induced indecomposable summand. Divide p into d to get d = bp + c with 0 ≤ c < p. The non-induced summand is isomorphic to V_{cp+b+1} and the decomposition may be chosen so that the socle of the non-induced indecomposable is spanned by x_{1}^{c}M^{b}. (ii) For d ≥ p^{2} − p, F[V_{p+1}]_{d}^{♭} is a direct sum of indecomposable induced modules.

Proof. Fix d and choose a basis, B, for (F[V_{p+1}]_{d}^{♭})^{G}, so that B is compatible with the length filtration. Applying Proposition 2.3 gives a decomposition

F[V_{p+1}]_{d}^{♭} = ⨁_{α∈B} V (α)

with V (α) ≅ V_{ℓ(α)}. Write α = x_{1}^{i}α′ where x_{1} does not divide α′. If i ≥ p, then by Proposition 4.3, ℓ(α) = p^{2} and V (α) is projective. Suppose i < p. If α′ ∈ Im Tr_{L}^{G}, then by Proposition 4.4, ℓ(α′) = p. Thus, using Proposition 4.3, ℓ(α) = ip + p and V (α) is an induced module. Suppose α′ ∉ Im Tr_{L}^{G}. Then α′ = kM^{j} + h where j < p − 1, k is a non-zero element of F and h ∈ Im Tr_{L}^{G}. It follows from Proposition 4.6 that ℓ(α′) = j + 1. Applying Proposition 4.3 gives ℓ(α) = pi+j +1. This last case is the only way in which a non-induced summand can appear in the decomposition. Note that in this case, d = pj +i with 0 ≤ i < p and j < p−1, giving d ≤ (p−2)p+(p−1) = p^{2} −p−1, i = c and j = b. Suppose, by way of contradiction, that α_{1}, α_{2} ∈ B are distinct elements both having length cp + b + 1. Then α_{1} = x_{1}^{c}(k_{1}M^{b} + h_{1}) and α_{2} = x_{1}^{c}(k_{2}M^{b} + h_{2}) with k_{i} ∈ F \ {0} and h_{i} ∈ Im Tr_{L}^{G}. From Proposition 4.3, ℓ(k_{2}α_{1} − k_{1}α_{2}) ≥ cp + p > cp + b + 1 contradicting the fact that B is compatible with the length filtration. □

Remark 4.8. The strong form of the Hughes-Kemper Periodicity Conjecture [10, Conjecture 4.6] states that for p^{m−1} < n ≤ p^{m} and d > p^{m} − n, F[V_{n}]_{d}^{♭} is induced. The preceding Theorem verifies the conjecture for n = p + 1.

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Institute of Mathematics, Statistics & Actuarial Science, University of Kent at Canterbury, CT2 7NF, UK

E-mail address: R.J.Shank@kent.ac.uk

Department of Mathematics & Computer Science, Royal Military College, Kingston, Ontario, Canada, K7K 7B4

E-mail address: wehlau@rmc.ca