CYCLIC GROUP ACTIONS ON POLYNOMIAL RINGS

PETER SYMONDS

Abstract. We consider a cyclic group of order p^{n} acting on a module incharacteristic p and show how to reduce the calculation of the symmetric algebra to that of the exterior algebra.

Consider a cyclic group of order p^{n} acting on a polynomial ring S = k[x_{1}, …, x_{r}], where k is a field of characteristic p; this is equivalent to the symmetric algebra S^{∗}(V) on the module V generated by x_{1}, …, x_{r}. We would like to know the decomposition of S into indecomposables.

This was calculated by Almkvist and Fossum in [1] in the casen = 1; see also [6]. They reduced the problem to the calculation of the exterior powers of V, and then gave a formula for these.

In this note we accomplish the first part for general n, that is to say the reduction of the calculation of the symmetric algebra to that of the exterior algebra. Many of the results extend to a group with normal cyclic Sylow p-subgroup, in particular to any finite cyclic group.

We wish to thank Dikran Karagueuzian for providing the computer calculations using Magma [4] that motivated this work.

1. Periodicity

Let G be a finite group, k a field of characteristic p and V a kG-module. Suppose that V′ < V is a kG-submodule of codimension 1, and let x ∈ V ∖ V′. Set a = Π_{g∈G/ Stab_{G}(x)}gx^{r}, for some r ∈ ℕ. Choose a basis for V′ and add x to obtain a basis of V. Let T ≤ S = S^{∗}(V) be the kG-submodule spanned by the monomials in the basis elements that are not divisible by x^{deg a}.

Lemma 1.1. As kG-modules, S ≅ k[a] ⊗_{k} T.

Proof. (cf. [7]) Since V/V′ is a simple module we must have rad V ≤ V′. Now x is an eigenvector modulo rad V, so a ≡ λx^{deg a} mod T for some 0 ≠ λ ∈ k. Since S = x^{deg a}S ⊕ T, we obtain S = aS ⊕ T, and the result follows by repeated substitution for S. □

Remark. It is easy to extend this to a multiple periodicity, with one polynomial generator for each 1-dimensional summand of V/ rad V.

Theorem 1.2. Suppose that G has a normal cyclic Sylow p-subgroup H, and that V is a non-zero kG-module that is indecomposable on restriction to H. Then S^{∗}(V) ≅ k[a] ⊗ B modulo modules projective relative to proper subgroups of H, where a is an eigenvector for G of degree |H| and B is a sum of homogeneous submodules of degree strictly less than deg a.

2000 Mathematics Subject Classification. Primary: 20C02; Secondary: 13D02.

Partially supported by a grant from the Leverhulme Trust.

Remark. This was conjectured in the case G = H by Hughes and Kemper [6].

Proof. There must be a complement F to H in G. This is because the extension H → G → G/H is classified by an element of H^{2}(G/H, H), and this group is 0 since |G/H| is prime to |H|, (or invoke the Schur-Zassenhaus Theorem).

Let dim V = r and |H| = p^{n}. We assume that V is faithful, so r > p^{n−1} and hence n ≤ r − 1.

Now L = V/ rad V is 1-dimensional ([2] II 5 8), so can be written over a finite field 𝔽_{q}. The map V → L splits over F, since |F | is not divisible by p: let x ∈ V be a non-zero element in the image of the splitting. Then V is generated over G by x, and x is an eigenvector for F. We set a = Π_{h∈H}hx, so a is an eigenvector for G of degree p^{n}.

Let W_{r}(k) denote the group of invertible upper-triangular r × r-matrices over k and U_{r}(k) its subgroup of matrices with entries 1 on the diagonal.

The projective cover of L is Ind_{F}^{G} L, so V is a quotient of this. It follows that V can be realized over 𝔽_{q}, and from now on we suppose that k = 𝔽_{q}. It also follows that we may regard G as a subgroup of W_{r}(k) and V as the restriction to G of the natural representation of W_{r}(k), with x as top basis element. Let d_{r} be the orbit product of x under U_{r}(k), so deg d_{r} = q^{r−1}.

The previous Lemma 1.1 tells us that S ≅ k[a] ⊗ T over H, where T is spanned by the monomials in the basis elements that are not divisible by x^{p^{n}}, and also S ≅ k[d_{r}] ⊗ T′ over U_{r}(k), where T′ is spanned by the monomials in the basis elements that are not divisible by x^{q^{r−1}}. In the first of these isomorphisms, the right hand side is naturally a kG-module, since a is an eigenvector for G. The natural map from right to left is G-equivariant, so the isomorphism is valid over G.

Observe that a generator of H can not be contained in any of the s-row subgroups of U_{r}(k) for s ≤ r − 2, i.e. the subgroups U_{J} < U_{r} for J ≨ I = {1, …, r − 1} consisting of elements with zero off-diagonal elements in the rows corresponding to the elements of J. This is because such a group has a fixed space on the dual module V^{∗} of dimension at least 2, and the dimension of the fixed space of H on V^{∗} is 1.

The Main Theorem of [7] as generalized to W_{r}(k) in [8] 7.1 states that, as a kW_{r}(k)-module, T′ is a sum of pieces of the form D ⊗ X̄_{J}(I) ≅ D ⊗ Ind_{W_{J}}^{W_{r}} X̄_{J}(J), (where D is 1-dimensional and W_{J} is defined analogously to U_{J}). The double coset formula (see e.g. [3] 3.3.4) and the observation above show that, as kG-modules, T′ ≅ B′ modulo summands induced from subgroups with strictly smaller Sylow p-subgroups, where B′ is the module X̄_{I}(I) of that paper, (which only occurs once in the decomposition of T′), so a sum of homogeneous pieces known to be of degree less than deg d_{r}.

It is easy to check that, over kG, T′ ≅ T ⊕ aT ⊕ ⋯ ⊕ a^{q^{r−1}p^{−n}−1}T. In particular T can only have a finite number of summands that are not induced; call their sum B. Then B′ ≅ B ⊕ aB ⊕ ⋯ ⊕ a^{q^{r−1}p^{−n}−1}B.

This shows that deg B′ = deg a^{q^{r−1}p^{−n}−1} + deg B (where deg denotes the maximum degree of the homogeneous pieces). Thus deg B = deg B′ − q^{r−1} + p^{n} and, since deg B′ < q^{r−1}, we obtain deg B < p^{n}. □

Remark. If F is abelian and k is a splitting field then all indecomposable kG-modules satisfy the condition of the theorem (see e.g. [2] II 6).

2. Working modulo induced modules

Let C_{p^{n}} denote the cyclic group of order p^{n} and let A(C_{p^{n}}) denote the Green ring of kC_{p^{n}}-modules, (where k has characteristic p).

Recall that kC_{p^{n}} ≅ k[X]/X^{p^{n}}, where X corresponds to g − 1 for some generator g ∈ C_{p^{n}}. The indecomposable representations of C_{p^{n}} are V_{t} = k[X]/X^{t}, up to isomorphism, where 1 ≤ t ≤ p^{n} (see [2], for example). For convenience we let V_{0} denote the 0 module. The decomposition of tensor products is described implicitly in [5] and explicitly in [9].

For m < n there are restriction and induction operators R_{m}^{n} : A(C_{p^{n}}) → A(C_{p^{m}}) and I_{m}^{n} : A(C_{p^{m}}) → A(C_{p^{n}}) with the properties that I_{m}^{n}V_{t} = V_{p^{n−m}t}, R_{m}^{n}I_{m}^{n} = p^{n−m} and I_{m}^{n}R_{m}^{n} = V_{p^{n−m}} ⊗ −.

For any finite dimensional kC_{p^{n}}-module V, let H^{n} = H^{n}(V) be the element of A(C_{p^{n}})[[t]] with S^{r}(V) as coefficient of t^{r}. Note that H^{n}(V ⊕ W) = H^{n}(V)H^{n}(W).

Let P^{n}(V) be the part of H^{n}(V) where we only consider indecomposables of dimension prime to p, or, equivalently, those not induced from a proper subgroup. In other words, we write S^{n}(V) as a sum of indecomposables, omit all the summands of dimension prime to p and put the remainder as the coefficient of t^{n} in P^{n}(V).

Lemma 2.1. If V is indecomposable then P^{n}(V) = Q^{n}(V)(1 − t^{p^{n}})^{−1}, where Q^{n}(V) is a polynomial of degree < p^{n}.

Proof. This is part of the periodicity result 1.2 above. □

Proposition 2.2. Suppose that we know the P^{m}(V) for m ≤ n. Then we also know the H^{n}(V), and, in fact,

H^{n} = (1 − p^{−1}V_{p})P^{n} + p^{−1}I_{n−1}^{n}((1 − p^{−1}V_{p})P^{n−1}) + p^{−2}I_{n−2}^{n}((1 − p^{−1}V_{p})P^{n−2}) + · · · .

In this formula, P^{n−r} is an abbreviation for P^{n−r}R_{n−r}^{n}V, and V_{p} is the indecomposable module of dimension p for C_{p^{n−r}}.

Proof. Write R = R_{n−1}^{n} and I = I_{n−1}^{n}. Now IRI = pI, so on inducedmodules we have p^{−1}IR = 1. Thus H^{n} −P^{n} = p^{−1}IR(H^{n} −P^{n}) = p^{−1}(IH^{n−1} −IRP^{n}) = p^{−1}(IH^{n−1} −V_{p}P^{n}), and so H^{n} = p^{−1}IH^{n−1} + (1 − p^{−1}V_{p})P^{n}.

Now use recursion in n. □

For kG-modules W ≤ V, write (V, W) for the pair. There is anobvious definition of direct sum.

Lemma 2.3. For a cyclic p-group G and indecomposable G-module V, every Res_{H}^{G}(V, W) is a sum of (V′, W′) with V′ (and W′) indecomposable and codim(V′, W′) ≥ ⌊codim(V, W)/|G : H|⌋.

Proof. By induction, we may assume that H is of index p in G. So kG ≅ k[X]/X^{p^{n}} and kH ≅ k[X^{p}]/X^{p^{n}} ≅ k[Y]/Y^{p^{n−1}}, where Y = X^{p}.

Also we may take V = k[X]/X^{v} and W = X^{v−w}k[X]/X^{v}, where v = dim V and w = dim W.

Now Res_{H}^{G} V = k[Y]/X^{v} ⊕ ⋯ ⊕ X^{p−1}k[Y]/X^{v}and Res_{H}^{G} W = X^{v−w}k[Y]/X^{v} ⊕ ⋯ ⊕ X^{v−w+p−1}k[Y]/X^{v}.

Write v − w = pa + b with 0 ≤ b ≤ p − 1. Then, with a little rearranging, Res_{H}^{G} W = Y^{a+1}k[Y]/X^{v} ⊕ ⋯ ⊕ X^{b−1}Y^{a+1}k[Y]/X^{v} ⊕ X^{b}Y^{a}k[Y]/X^{v} ⊕ ⋯ ⊕ X^{p−1}Y^{a}k[Y]/X^{v}; each of these summands is clearly a submodule of the summand of Res_{H}^{G} V labeled by the same power of X, and we need to show that the codimension is at least a.

Now X^{r}Y^{s}k[Y]/X^{v} ≤ X^{r}k[Y]/X^{v} has codimension s provided that X^{r}Y^{s−1} is not 0 in the module, which happens when v − r > p(s − 1), and in our cases s is equal to a or a + 1.

We claim that in all our cases in fact v − r > p(s − 1). In the cases when s = a + 1 we would otherwise have v − r ≤ pa = v − w − b, hence r ≥ b + w. But in these cases r ≤ b − 1, a contradiction. In the cases when s = a a similar argument gives r ≥ b + w + p, when we know that r ≤ p − 1. □

3. Symmetric Powers

Let H < G, and let B be a G-module projective relative to H. Let f : A → B be a map of G-modules.

Lemma 3.1. If f, considered as a map of H-modules, factors through a projective then it also factors through a projective as a map of G-modules. The same claim also holds for any g : B → A.

Proof. Recall that, for groups U ≤ V ≤ G and V -modules A, B, we define tr_{U}^{V} : Hom_{U}(A, B) → Hom_{V} (A, B) by tr_{U}^{V} (ϕ) = ∑_{v∈V/U} vϕv^{−1}, ϕ ∈ Hom_{U}(A, B).

The hypotheses are equivalent to f = tr_{1}^{H} f′ and Id_{B} = tr_{H}^{G} i, where i is H-equivariant (see e.g. [3] 3.6). We need to show that f is in the image of tr_{1}^{G}.

But tr_{1}^{G}(if′) = tr_{H}^{G} tr_{1}^{H}(if′) = tr_{H}^{G}(i tr_{1}^{H} f′) = tr_{H}^{G}(i)f = f.

The proof of the second claim is similar. □

Definition 3.2. A chain complex C_{∗} is called:

• acyclic if it only has homology in degree 0;

• weakly induced if each module is induced, and weakly induced except in degrees I if each C_{i}, i ∉ I, is induced; and

• separated if each Im d_{i} → C_{i} factors through a projective.

Write B_{i} for Im d_{i} ⊆ C_{i}. If the inclusion B_{i} → C_{i} factors through a projective then it factors through the injective hull of B_{i}, call it P_{i} (injective is equivalent to projective here), and P_{i} → C_{i} is injective since it is so on the socle. Thus we can write C_{i} = P_{i} ⊕ C′_{i} and B_{i} ⊆ P_{i}.

Recall that the Heller translate ΩV of a module V is defined to be the kernel of the projective cover P_{V} → V and Ω^{i}V for i ≥ 1 is Ω iterated i times. Similarly Ω^{−1}V is the cokernel of the injective hull V → I_{V} and Ω^{−i} for i ≥ 1 is its iteration. We let Ω^{0}V denote V with any projective summands removed.

In our context, projective is equivalent to injective, so Ω^{i}Ω^{j}V ≅ Ω^{i+j}V and if V is induced so is Ω^{i}V. Also (Ω^{i}V)^{∗} ≅ Ω^{−i}(V^{∗}) and Ω(V ⊗ W) ≅ (ΩV) ⊗ W modulo projectives. See [3] 2.4, 3.1.6.

Note that Ω^{2} = 1 modulo projectives for cyclic p-groups.

Proposition 3.3. Suppose that the chain complex of G-modules K_{w} → ⋯ → K_{0} is: • acyclic and H_{0}(K_{∗}) = L, say;

• weakly induced from H except in at most one degree;

• on restriction to H, K_{∗} is separated.

Then K_{∗} is separated and L ⊕ P ≅ K_{0} ⊕ Ω^{−1}K_{1} ⊕ Ω^{−2}K_{2} ⊕ ⋯, for some projective module P.

Proof. We use induction on w, so first consider the case w = 1, where we have a short exact sequence K_{1} → K_{0} → L.

Now one of K_{0}, K_{1} is induced from H, and K_{1} → K_{0} factors through a projective over H, so we can apply lemma 3.1 to see that K_{1} → K_{0} factors through a projective over G. The shifted triangle K_{0} → L → Ω^{−1}K_{1} is stably split, so L ≅ K_{0} ⊕ Ω^{−1}K_{1} stably, and thus there are projective modules P and Q such that L ⊕ P ≅ K_{0} ⊕ Ω^{−1}K_{1} ⊕ Q.

We claim that we can take Q = 0. Since L is a quotient of K_{0}, any projective summand of L is also isomorphic to a summand of K_{0}, and we can cancel, so we can assume that L has no projective summands. But then Q must be isomorphic to a summand of P and we can cancel.

Now for w ≥ 2 we break up our complex into two: K_{w} → K_{w−1}, which we shift so that K_{w−1} is in degree 0 and the homology there is Im d_{w−1}, and Im d_{w−1} → K_{w−2} → ⋯ → K_{0}.

These both satisfy the hypotheses of the proposition; in particularif a K_{i} with i ≤ w − 2 is not induced then both K_{w} and K_{w−1} are induced, hence so is Im d_{w−1} by the case w = 1.

Thus both complexes are separated, hence so is K_{∗}. Also L ⊕ P′ ≅ K_{0} ⊕ Ω^{−1}K_{1} ⊕ ⋯ ⊕ Ω^{w−2}K_{w−2} ⊕Ω^{w−1} Im d_{w−1} and Im d_{w−1} ⊕P′′ ≅ K_{w−1} ⊕Ω^{−1}K_{w}, so the last claim follows. □

Definition 3.4. For any pair (V, W) of G-modules we have a Koszul complex K(V, W):

⋯ → Λ^{2}(W) ⊗ S^{r−2}(V) → W ⊗ S^{r−1}(V) → S^{r}(V),

the boundary map is d(w_{1} ∧ ⋯ ∧ w_{j} ⊗ s) = ∑_{i=1}^{j}(−1)^{j−i}w_{1} ∧ ⋯ ∧ ŵ_{i} ∧ ⋯ ∧ w_{j} ⊗ w_{i}s, w_{i} ∈ W, s ∈ S^{∗}(V).

Lemma 3.5. K(V, W) is exact except at S^{∗}(V), where its homology is S^{∗}(V/W).

Proof. Ignoring the group action, K(V, W) ≅ K(W, W) ⊗ S^{∗}(V/W);now K(W, W) is a standard Koszul complex, so known to be exact except in degree 0, where its homology is k. □

Lemma 3.6. ([1]) K(V, V) is split exact in degrees r not divisible by p.

Proof. The splitting is given by e(λ ⊗ v_{1} ⋯ v_{j}) = r^{−1} ∑_{i=1}^{j} λ ∧ v_{i} ⊗ v_{1} ⋯ v̂_{i} ⋯ v_{j}, λ ∈ Λ^{∗}(W), v_{i} ∈ V. □

Remark. This can be used (inductively on r) to obtain a formula for S^{r}(V) in terms of Λ^{∗}(V) for r < p. For cyclic groups of order p, this fact and periodicity are all that is needed to obtain the reduction for all r (see [1]).

Lemma 3.7. For G cyclic of prime power order and H < G of index p, S^{r}(V_{pu}) is induced in degrees not divisible by p, and S^{ps}(V_{pu}) is the tensor induced module (↑_{H}^{⊗,G} S^{s}(V_{u})), modulo induced modules.

Proof. Note that V_{pu} ≅ Ind_{H}^{G} V_{u}. The first part follows from the lemma above. In general use S^{∗}(V_{u} ⊕ t_{2}V_{u} ⊕ ⋯ ⊕ t_{p}V_{u}) ≅ S^{∗}(V_{u}) ⊗ t_{2}S^{∗}(V_{u}) ⊗ ⋯ ⊗ t_{p}S^{∗}(V_{u}), where {1, t_{2}, …, t_{p}} is a set of coset representatives for G/H. □

The next lemma is clear.

Lemma 3.8. K((V_{1}, W_{1}) ⊕ (V_{2}, W_{2})) ≅ K(V_{1}, W_{1}) ⊗ K(V_{2}, W_{2}).

Lemma 3.9. If C_{∗}^{1} and C_{∗}^{2} are separated then so is C_{∗}^{1} ⊗ C_{∗}^{2}.

Proof. We omit sub- and superscripts. We have B ⊆ P ⊆ C (in the notation established after 3.2). The boundary in C⊗C has image in (B⊗C)+(C⊗B) ⊆ (P ⊗C)+(C⊗P). The + looks unpromising, but we have a left exact sequence 0 → P ⊗ P → (P ⊗ C) ⊕ (C ⊗ P) → C ⊗ C. This must split, since everything with a P in it is projective, hence injective, so the image on the right is projective. □

Theorem 3.10. Let G be a cyclic group of order p^{n}. For any 0 ≤ t ≤ p^{n} − p^{n−1}, K(V_{p^{n}}, V_{t}) is separated.

Proof. Use induction on n. First consider n = 1. Because S^{r}(V_{p}) isprojective except in degrees divisible by p, and because we do not allow t = p, all but one of the terms in K are projective, and this forces separability.

For general n, write S = k[a] ⊗ T, as in 1.1, and let K′ be the subcomplex of K defined using T instead of S. This is consistent with the boundary morphisms, because the x used in the definition of T is not in the submodule V_{t} (this is why we need an upper bound on t). Thus K = k[a] ⊗ K′, and we need only consider K′.

From 3.7 we know that S^{r}(V_{p^{n}}) is induced except whenr is divisible by p^{n}, so T^{r} is induced except when r = 0. But for r = 0 the corresponding term in K′ is either 0 or the last one in that degree, so (graded module)-degree-wise K′ is weakly induced except in the highest (complex)-degree.

Now restrict to H of order p^{n−1}. By 2.3, (V_{p^{n}}, V_{t}) will decompose as ⊕_{i=1}^{p}(V_{p^{n−1}}, W_{i}) with dim W_{i} ≤ p^{n−1} − p^{n−2}, and Res_{H}^{G} K(V_{p^{n}}, V_{t}) ≅ ⊗_{i}K(V_{p^{n−1}}, W_{i}), by 3.8.

By induction, as a complex for H, each K(V_{p^{n−1}}, W_{i}) is acyclic and separated, hence so is K(V_{p^{n}}, V_{t}), by 3.9, and thus K′(V_{p^{n}}, V_{t}). Now apply 3.3. □

Corollary 3.11. For G ≅ C_{p^{n}}, r < p^{n} and p^{n−1} ≤ t ≤ p^{n} we have S^{r}(V_{t}) ≅ Ω^{−r}Λ^{r}(V_{p^{n}−t}) modulo induced modules.

In particular, if r + t > p^{n} then S^{r}(V_{t}) is induced.

Proof. S^{∗}(V_{t}) is the homology of K(V_{p^{n}}, V_{p^{n}−t}). Use Theorem 3.10 and the last part of Proposition 3.3. As in the proof above, the only possible non-induced term in (module-) degree r is Λ^{r}(V_{p^{n}−t}) ⊗ S^{0}(V_{p^{n}}). □

We adopt the convention that S^{r} = 0 for r < 0.

Corollary 3.12. With the same conditions as above, S^{r}(V_{t}) ≅ Ω^{p^{n}−t}S^{p^{n}−t−r}(V_{t}) modulo induced modules.

Proof. S^{r}(V_{t}) ≅ Ω^{−r}Λ^{r}(V_{p^{n}−t}) ≅ Ω^{−r}Λ^{p^{n}−t−r}(V_{p^{n}−t}) ≅ Ω^{p^{n}−t−2r}S^{p^{n}−t−r}(V_{t}) modulo induced modules. □

Remark. It is thus only necessary to calculate S^{r}(V_{t}) for r ≤ (p^{n} − t)/2: the rest will follow.

Corollary 3.13. Suppose that G has normal Sylow p-subgroup H ≅ C_{p^{n}} and that V is a kG-module such that Res_{H}^{G} V ≅ V_{t}. If r < p^{n} and p^{n−1} ≤ t ≤ p^{n} then S^{r}(V) ≅ Ω^{−r}Λ^{r}(ΩV) ≅ Ω^{t−p^{n}}(S^{p^{n}−t−r}(V)^{∗}) ⊗ det(ΩV) modulo summands projective relative to proper subgroups of H (where ∗ denotes the contragredient).

Proof. As in 1.2, H has complement F, V/ rad V = L and the projective cover of V is Ind_{F}^{G} L.

We consider the Koszul complex K(Ind_{F}^{G} L, ΩV). On restriction to H this becomes K(V_{p^{n}}, V_{p^{n}−t}). The property of a map factoring through a projective depends only on the restriction to H, so we obtain the same conclusion as in 3.11, which is the first isomorphism. The second follows as in 3.12, but being more careful about the duality. Modulo induced modules, Λ^{r}(ΩV) ≅ Hom(Λ^{p^{n}−t−r}(ΩV), Λ^{p^{n}−t}(ΩV)) ≅ Λ^{p^{n}−t−r}(ΩV)^{∗} ⊗ det(ΩV), and Λ^{p^{n}−t−r}(ΩV)^{∗} ≅ (Ω^{p^{n}−t−r}S^{p^{n}−t−r}(V))^{∗} ≅ Ω^{r+t−p^{n}}(S^{p^{n}−t−r}(V)^{∗}). □

References

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School of Mathematics, University of Manchester, P.O. Box 88, Manchester M60 1QD, England

 E-mail address: Peter.Symonds@manchester.ac.uk