MODULES OF CONSTANT JORDAN TYPE WITH ONE NON-PROJECTIVE BLOCK

DAVID J. BENSON

Abstract. Let k be a field of characteristic p and G = (ℤ/p)^{r} with r ≥ 2. We prove that there cannot exist a finite dimensional kG-module of stable constant Jordan type [a] with 2 ≤ a ≤ p − 2. This is a generalisation of a conjecture of Carlson, Friedlander and Pevtsova.

1. Introduction

Let G = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} be an elementary abelian p-group, and let k be a field of characteristic p. A cyclic shifted subgroup of G is a subgroup of kG generated by an element of the form 1 + ∑_{i=1}^{r} λ_{i}(g_{i} − 1) where not all the λ_{i} are equal to zero. We say that a finite dimensional kG-module M has stable constant Jordan type [a_{1}, …, a_{r}] (with 0 ≤ a_{i} ≤ p − 1) if the restriction of M to every cyclic shifted subgroup has Jordan canonical form given by blocks of length a_{1}, …, a_{r} together with some number (possibly zero) of blocks of length p. This concept is investigated in detail in the paper of Carlson, Friedlander and Pevtsova [2]. In particular, Conjecture 9.5 of their paper states that there cannot exist a module of stable constant Jordan type [2] for an elementary abelian p-group of rank at least two with p > 3. In this paper, we prove the following strengthened form of their conjecture.

Theorem 1.1. Let G = (ℤ/p)^{r} with r ≥ 2, and let k be a field of characteristic p. If 2 ≤ a ≤ p − 2 then there does not exist a finite dimensional kG-module of stable constant Jordan type [a].

The idea of the proof is to start with a module M of stable constant Jordan type [a] and use it to produce modules of constant Jordan type [0], which by Dade’s lemma [3, Lemma 11.8] are projective modules. Their dimensions then have to be divisible by p^{2}, and this gives incompatible congruences for the dimension of M.

Unfortunately, the methods of this paper do not give information about modules of stable constant Jordan type with more than one Jordan block. So Question 9.6 of [2] is still an interesting open question. For example, it is not known whether there are modules of stable constant Jordan type [2, 2] or [3, 1] for an elementary abelian group of rank at least two with p ≥ 5.

The author thanks MSRI for its hospitality while this work was carried out.

Theorem 1.1 has the following consequence for generic Jordan type, a notion investigated by Friedlander, Pevtsova and Suslin [4].

Corollary 1.2. Let G = (ℤ/p)^{r} and let M be a finite dimensional kG-module with stable generic Jordan type [a], with 2 ≤ a ≤ p − 2. Then the locus of non-generic Jordan type of M has codimension one in affine space 𝔸^{r}(k).

2. Proof of the theorem

First we recall from [1] and [2] the facts that we need to use in the proof.

Theorem 2.1. (a) Every direct summand of a module of constant Jordan type also has constant Jordan type.

(b) If M and N have constant Jordan type then so does M ⊗_{k} N.

(c) If M has constant Jordan type then so do Λ^{r}M and S^{r}M provided r < p.

Proof. Referring to [2], (a) is proved in Theorem 3.7 and (b) in Corollary 4.3. Part (c) follows from parts (a) and (b), since for r < p the exterior and symmetric powers are summands of the corresponding tensor power. □

It follows from part (c) of the theorem that for r < p the restriction of an exterior or symmetric power to a cyclic shifted subgroup is isomorphic to the corresponding exterior or symmetric power of the restriction, because among the cyclic shifted subgroups are the actual cyclic subgroups.

We remark that since the proof of part (b) is not straightforward, another route to take would be to use the Hopf structure on k(ℤ/p)^{r} coming from regarding it as a restricted universal enveloping algebra. Then tensor products commute with restriction to cyclic shifted subgroups, which are usually called one parameter subgroups in this context. It is also necessary to observe that the proof Theorem 2.2 below works with either Hopf structure on the group algebra of the cyclic shifted subgroup, so it does not make a difference which one we use.

Theorem 2.2. As representations in characteristic p of a cyclic group of order p, S^{r}[a] is projective provided r < p and r+a > p. In particular S^{p−a+1}[a] is projective.

Proof. This is proved in Chapter III, Proposition 2.10 of Almkvist and Fossum [1], but since the proof is easy, we reproduce it here. There are obvious short exact sequences

0 → S^{r}[a] → S^{r+1}[a] → S^{r+1}[a − 1] → 0.

If a = p and r < p − 1 then both S^{r}[a] and S^{r+1}[a] are projective (and hence injective), so S^{r+1}[p − 1] is projective. Repeat the argument using downward induction on a. □

Proof of Theorem 1.1. Let M be a kG-module of stable constant Jordan type [a] with 2 ≤ a ≤ p − 2, and let dim_{k} M = np + a. Then Λ^{a+1}M is projective, so dim_{k} Λ^{a+1}M is divisible by p^{2}. This dimension is

\binom{np + a}{a + 1} = \frac{(np + a)(np + a − 1) … (np)}{a!}.

The only term in the numerator or the denominator of this fraction that is divisible by p is the last term on the top, namely np. So we deduce that n is a multiple of p.

On the other hand, by Theorem 2.2, S^{p−a+1}M is also projective, and its dimension is

\binom{np + p}{p − a + 1} = \frac{(np + p)(np + p − 1) … (np + a)}{(p − a + 1)!}.

The only term in the numerator or the denominator of this fraction that is divisible by p is the first term on the top, namely np + p. Since n is a multiple of p, this is not divisible by p^{2}. This contradicts the fact that this module is projective, and hence no such M can exist. □

Proof of Corollary 1.2. Let V ⊆ 𝔸^{r}(k) be the locus of non-generic Jordan type of M. If V has codimension at least two then there is a linear subspace W of 𝔸^{r}(k) of dimension two such that V ∩ W = {0}. Corresponding to W there is a subalgebra of kG isomorphic to k(ℤ/p)^{2}. The restriction of M to this subalgebra has constant Jordan type [a], contradicting Theorem 1.1. □

References

1. G. Almkvist and R. Fossum, Decomposition of exterior and symmetric powers of indecomposable ℤ/pℤ-modules in characteristic p and relations to invariants, Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), Lecture Notes in Mathematics, vol. 641, Springer-Verlag, Berlin/New York, 1978, pp. 1–111.

2. J. F. Carlson, E. M. Friedlander, and J. Pevtsova, Modules of constant Jordan type, J. Reine & Angew. Math. 614 (2008), 191–234.

3. E. C. Dade, Endo-permutation modules over p-groups, II, Ann. of Math. 108 (1978), 317–346.

4. E. M. Frielander, J. Pevtsova, and A. Suslin, Generic and maximal Jordan types, Invent. Math. 168 (2007), 485–522.

Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE