IRREDUCIBLE TENSOR PRODUCTS OVER ALTERNATING GROUPS

CHRISTINE BESSENRODT AND ALEXANDER S. KLESHCHEV

Abstract. We classify irreducible tensor products of modular representations of the alternating group over an algebraically closed field of characteristic p > 5.

Introduction

Let F be an algebraically closed field of characteristic p ≥ 0, and A_{n} be the alternating group on n letters. In this paper we study tensor decomposable irreducible F A_{n}-modules, i.e. irreducible modules E which can be written as E ≅ E_{1} ⊗ E_{2} for non-trivial F A_{n}-modules E_{1} and E_{2}. We refer the reader to [28, 2, 14, 3, 15] for results on the similar problem for the symmetric group Σ_{n}. For example, the following theorem is part of [3, Main Theorem].

Theorem A. Let D_{1} and D_{2} be F Σ_{n}-modules of dimensions greater than one. Then D_{1} ⊗D_{2} is reducible unless p = 2 and n is even.

Of course, the same result will follow for F A_{n}-modules E_{1} and E_{2}, which lift to the symmetric group Σ_{n}. However, some F A_{n}-modules do not lift, and we need further investigation to complete the problem.

To describe our main result we first explain how the irreducible F A_{n}-modules can be parametrised. We start from the symmetric group, referring the reader to [16] for the standard facts on its representation theory. In particular, to every p-regular partition λ of n one can associate the irreducible F Σ_{n}-module D^{λ}. Assume that p > 2 and denote by sgn the 1dimensional sign representation of Σ_{n}. Then D^{λ} ⊗ sgn is irreducible so there should exist a p-regular partition λ^{M} with

D^{λ} ⊗ sgn ≅ D^{λ^{M}}.  (1)

The bijection λ ↦ λ^{M} on the set of p-regular partitions is called the Mullineux bijection. This bijection can be described explicitly using a combinatorial algorithm suggested by Mullineux, see [22, 13, 4, 27].

Moreover, it follows easily from Clifford theory (see e.g. [12]) that the restriction D^{λ}↓_{A_{n}} is irreducible if and only if D^{λ} ⊗ sgn ≅ D̸^{λ}. If this is the case, we denote this irreducible restriction by E^{λ}. Of course, E^{λ} ≅ E^{λ^{M}}. On the other hand, if D^{λ} ⊗ sgn ≅ D^{λ}, then the restriction D^{λ}↓_{A_{n}} splits as a direct sum E_{+}^{λ} ⊕ E_{−}^{λ} of two irreducible F A_{n}-modules. Finally,

{E^{λ} | λ ≠ λ^{M}} ∪ {E_{+}^{λ}, E_{−}^{λ} | λ = λ^{M}}

is a complete set of irreducible F A_{n}-modules, and distinct modules L and M from this set are isomorphic if and only if L ≅ E^{λ}, M ≅ E^{λ^{M}} for some p-regular partition λ with λ ≠ λ^{M}. If λ = λ^{M} we say that λ is Mullineux-fixed.

Gathering together equal parts of λ we can represent it in the form λ = (l_{1}^{a_{1}}, l_{2}^{a_{2}}, …, l_{k}^{a_{k}}) where l_{1} > l_{2} > ⋯ > l_{k} > 0 and all a_{i} > 0. Then the partition λ is called a Jantzen-Seitz partition (or JS-partition for short) if it is p-regular and p divides l_{i} − l_{i+1} + a_{i} + a_{i+1} for all i with 1 ≤ i < k. These partitions are important because the restriction D^{λ}↓_{Σ_{n−1}} is irreducible if and only if λ is a JS-partition, cf. [19, 20, 11].

1991 Mathematics Subject Classification: 20C20, 20C30, 20B35, 20B20.

The second author gratefully acknowledges the support of the NSF (grant No DMS-9900134).

The number a_{1} + a_{2} + ⋯ + a_{k} is denoted h(λ) and called the height of λ.

Now we can state our main result, which describes tensor decomposable F A_{n}-modules in characteristic p > 5. The case p = 0 has been treated in [2] (see also [28]), but the cases p = 2, 3 and 5 remain open.

Main Theorem. Let p > 5 and E_{1}, E_{2} be F A_{n}-modules of dimensions greater than one. Then E_{1} ⊗ E_{2} is reducible with the only exception of E^{(n−1,1)} ⊗ E_{±}^{λ} where p |̸n and λ is a JS-partition with p |̸h(λ). In the exceptional case we have E^{(n−1,1)} ⊗ E_{±}^{λ} ≅ E^{ν}, where the Young diagram of ν is obtained from that of λ by removing the top removable node and adding the bottom addable one.

Example. (i) Let p = 7. Then E^{(28,1)} ⊗ E_{±}^{(15,3,2^{5},1)} ≅ E^{(14,3,2^{5},1^{2})}.

(ii) To emphasize that the situation is really exceptional for small p, we note using [18, p. 2] that E_{+}^{(3,2)} ⊗ E_{−}^{(3,2)} ≅ E^{(4,1)} for p = 2, and E_{+}^{(4,1^{2})} ⊗ E_{−}^{(4,1^{2})} ≅ E^{(4,2)} for p = 3. But we believe that the main theorem should still hold for p = 5, and, with few exceptions, for p = 3. We refer the reader to [14] and [15] for some exceptional phenomena in characteristic 2.

The proof of the Main Theorem is given in section 3. Note that in view of Theorem A, we only need to consider tensor products of the form E^{λ} ⊗ E_{±}^{μ} and E_{±}^{λ} ⊗ E_{±}^{μ}.

1. Preliminaries

Throughout the paper we assume that char F = p > 2.

Let G be a group. We write 1_{G} for the trivial F G-module. If M is an F G-module and D_{1}, …, D_{k} are irreducible F G-modules then the notation M = D_{1}| … |D_{k} means that M is a uniserial F G-module with composition factors D_{1}, …, D_{k} counted from bottom to top.

We record two well-known general facts (for the first one see the explanations in [5, 5.1]).

Lemma 1.1. Let G be a group, M be an F G-module and H ⊲ G be a normal subgroup of index [G : H] prime to p. Then soc(M↓_{H}) ≅ soc(M)↓_{H}.

Lemma 1.2. Let X, Y, Z be F G-modules. Assume that Z ⊆ X ⊕Y and that Z has a simple socle. Then X or Y (or both) contains an isomorphic copy of Z as a submodule.

Proof. Let f : Z → X ⊕ Y be an embedding, and π_{X}, π_{Y} be projections of X ⊕ Y to X, Y, respectively. Then π_{X} ∘ f or π_{Y} ∘ f must be an injection, since otherwise both maps annihilate the simple socle of Z, which is impossible as π_{X} ∘ f + π_{Y} ∘ f = f is injective.

Let σ ∈ Σ_{n} \ A_{n}. Then A_{n} → A_{n}, g ↦ σgσ^{−1} is an (outer) automorphism of A_{n}. If M is an F A_{n}-module we can use this automorphism to twist the action of A_{n} on M: Let

g · m := σgσ^{−1}m, g ∈ A_{n}, m ∈ M.  (2)

This defines a new F A_{n}-module denoted by ^{σ}M. The following lemma and corollary follow from Clifford theory and the definitions:

Lemma 1.3. Let λ be a p-regular partition of n with λ = λ^{M}. Then ^{σ}E_{±}^{λ} ≅ E_{∓}^{λ}.

Corollary 1.4. Let λ be a p-regular partition of n with λ = λ^{M}. Then we have isomorphisms of F A_{n}-modules:

σ Hom_{F} (E_{±}^{λ}, E_{±}^{λ}) ≅ Hom_{F} (E_{∓}^{λ}, E_{∓}^{λ}),
σ Hom_{F} (E_{±}^{λ}, E_{∓}^{λ}) ≅ Hom_{F} (E_{∓}^{λ}, E_{±}^{λ}).

Now we show that partitions λ with few parts are usually not Mullineux-fixed.

Lemma 1.5. [24, 1.9] Let n ≥ 5, and λ be a partition of n.

(i) If h(λ) ≤ 2 then h(λ^{M}) ≠ h(λ). In particular, λ ≠ λ^{M}.

(ii) If p > 3 and h(λ) = 3 then λ^{M} ≠ λ, except for the cases where p > 5 and λ is one of

the following (3, 1^{2}), (3, 2, 1), (3^{2}, 2), (3^{3}). (iii) If p > 5 and h(λ) = 4 then λ^{M} ≠ λ, except for the cases where p > 7 and λ is one of

the following: (4, 1^{3}), (4, 2, 1^{2}), (4, 3, 2, 1), (4, 3^{2}, 1), (4^{2}, 2^{2}), (4^{2}, 3, 2), (4^{3}, 3), (4^{4}).

In the next two lemmas we investigate when ‘rectangular’ and ‘near rectangular’ partitions are Mullineux-fixed:

Lemma 1.6. Let a < p and λ = (l^{a}). Then λ^{M} = λ if and only if l = a and p > 2l − 1.

Proof. Note that λ is a p-core if and only if l + a − 1 < p. In this case λ^{M} = λ^{t}, the transpose partition, see e.g. [6, 4.1] or [8, 2.1], and so λ = λ^{M} means l = a. If λ is not a p-core then the height of λ^{M} is p − a ≠ a = h(λ), so λ ≠ λ^{M}, see e.g. [8, 2.4].

Lemma 1.7. Let a_{1}, a_{2} < p and λ = ((l + 1)^{a_{1}}, l^{a_{2}}). Then λ^{M} = λ if and only if λ = (2, 1) and p > 3.

Proof. The proof is similar to that of Lemma 1.6. If λ is a p-core, we must have λ = λ^{t} by [6, 4.1] or [8, 2.1], which is only possible if λ = (2, 1) and p > 3. Now, we may assume that λ is not a p-core. If h(λ) ≥ p − 1, the result follows for example from [1, 2.2]. Otherwise the height of λ^{M} is p − h(λ) ≠ h(λ), so λ ≠ λ^{M}.

As in [16], we denote by S^{λ} the Specht module over the symmetric group Σ_{n} corresponding to a partition λ. By construction, S^{λ} is a submodule of the permutation module M^{λ}. The module M^{λ} is known to be self-dual, so (S^{λ})^{∗} is naturally a quotient module of M^{λ}. We are especially interested in two row partitions. For such a partition (n − k, k) denote by Y^{(n−k,k)} the block component of M^{(n−k,k)} containing the Specht module S^{(n−k,k)} ⊆ M^{(n−k,k)}. We will use the description of the blocks of the symmetric group known as ‘Nakayama’s conjecture’, see for example [17, 2.7.41, 6.1.21]. Now we describe the submodule structure of Y^{(n−k,k)}:

Lemma 1.8. [8, 3.3] Let k ≥ 0, p > k and n ≥ 2k. If there exists l such that 0 ≤ l < k and n ≡ k + l − 1 (mod p) then

S^{(n−k,k)} = D^{(n−l,l)}|D^{(n−k,k)},
Y^{(n−k,k)} = D^{(n−l,l)}|D^{(n−k,k)}|D^{(n−l,l)}, and
Y^{(n−k,k)}/D^{(n−l,l)} ≅ (S^{(n−k,k)})^{∗}.

Otherwise Y^{(n−k,k)} = S^{(n−k,k)} ≅ (S^{(n−k,k)})^{∗} ≅ D^{(n−k,k)}.

Lemma 1.8 immediately implies the following

Corollary 1.9. Let k ≥ 0, p > k and n ≥ 2k. If there exists l such that 0 ≤ l < k and n ≡ k + l − 1 (mod p) then

dim End_{F Σ_{n}}(Y^{(n−k,k)}) = 2,
dim Hom_{F Σ_{n}}(S^{(n−k,k)}, Y^{(n−k,k)}) = 1,
dim Hom_{F Σ_{n}}(Y^{(n−k,k)}, (S^{(n−k,k)})^{∗}) = 1,
dim Hom_{F Σ_{n}}(S^{(n−k,k)}, (S^{(n−k,k)})^{∗}) = 1.

Otherwise, all four homomorphism spaces above are 1-dimensional.

Corollary 1.10. The socles of Y^{(n−k,k)}↓_{A_{n}} and (S^{(n−k,k)}↓_{A_{n}})^{∗} are simple.

Proof. This follows from Lemmas 1.8, 1.1 and 1.5(i).

Sometimes we will denote Y^{(n−k,k)} by X_{1}^{k} and (S^{(n−k,k)})^{∗} by X_{2}^{k}. The next result will be used to prove that certain homomorphism spaces are large enough.

Lemma 1.11. Let k = k_{1} > k_{2} > ⋯ > k_{r} ≥ 0 be integers, p > k, n ≥ 2k, and M, N be F Σ_{n}-modules. Assume that for some i_{j}, i′_{j} ∈ {1, 2} the modules (X_{i_{j}}^{k_{j}})^{∗} are quotients of M and the modules X_{i′_{j}}^{k_{j}} are submodules of N, j = 1, 2, …, r. Then dim Hom_{F Σ_{n}}(M, N) ≥ r.

Proof. For r = 1 the claim follows immediately from Corollary 1.9. Assume r = 2 and n ≡ k_{1} + k_{2} − 1 (mod p). Then Lemma 1.8 implies (X_{i_{2}}^{k_{2}})^{∗} ≅ D^{(n−k_{2},k_{2})}, and (X_{i_{1}}^{k_{1}})^{∗} ≅ S := D^{(n−k_{2},k_{2})}|D^{(n−k_{1},k_{1})} or (X_{i_{1}}^{k_{1}})^{∗} ≅ Y := D^{(n−k_{2},k_{2})}|D^{(n−k_{1},k_{1})}|D^{(n−k_{2},k_{2})}. If (X_{i_{1}}^{k_{1}})^{∗} = S then (X_{i_{1}}^{k_{1}})^{∗} and (X_{i_{2}}^{k_{2}})^{∗} have different simple heads, so (X_{i_{1}}^{k_{1}})^{∗} ⊕ (X_{i_{2}}^{k_{2}})^{∗} is a quotient of M. Otherwise, (X_{i_{1}}^{k_{1}})^{∗} ≅ Y is a quotient of M. Similarly, either X_{i′_{1}}^{k_{1}} ⊕ X_{i′_{2}}^{k_{2}} or Y is a submodule of N. By Corollary 1.9 again, we conclude that dim Hom_{F Σ_{n}}(M, N) ≥ 2.

Now observe that for any 1 ≤ s < t ≤ r such that n ≢ k_{s} + k_{t} − 1 (mod p) the modules X_{i_{s}}^{k_{s}} and X_{i_{t}}^{k_{t}} are in different blocks by virtue of Nakayama’s conjecture. So the general case follows from the argument in the special cases considered above. Corollary 1.12. Let k = k_{1} > k_{2} > ⋯ > k_{r} ≥ 0 be integers, p > k, n ≥ 2k, and M, N be F A_{n}-modules. Assume that for some i_{j}, i′_{j} ∈ {1, 2} the modules (X_{i_{j}}^{k_{j}}↓_{A_{n}})^{∗} are quotients of M and the modules X_{i′_{j}}^{k_{j}}↓_{A_{n}} are submodules of N, j = 1, 2, …, r. Then dim Hom_{F A_{n}}(M, N) ≥ r.

Proof. Repeat the proof of Lemma 1.11 using Lemma 1.5 and Corollary 1.10.

The following two results verify some assumptions of Lemma 1.11 for the F Σ_{n}-module End_{F} (D^{λ}). Theorem 1.13. [24, 2.3, 2.4] Let p > k > 1, n ≥ 2k, and λ be a p-regular partition of n satisfying h(λ), h(λ^{M}) ≥ k. Then Y^{(n−k,k)} or (S^{(n−k,k)})^{∗} (or both) is a submodule of the self-dual module End_{F} (D^{λ}). Proposition 1.14. Let p > 5, n ≥ 8, and λ = (m, k) be a two row partition with k ≥ 2.

(i) If m ≢ k − 2 (mod p) and m > k then Y^{(n−3,3)} or (S^{(n−3,3)})^{∗} (or both) is a submodule of End_{F} (D^{λ}).

(ii) If m ≡ k − 2 (mod p) or m = k then Y^{(n−4,4)} or (S^{(n−4,4)})^{∗} (or both) is a submodule

of End_{F} (D^{λ}). Proof. This follows from [24, 2.4] and [8, 4.12].

In section 3 it will be convenient to assume that n is not too small, so we deal with small cases here. Lemma 1.15. Let p > 5.

(i) For n ≤ 9 the only (non-trivial) irreducible tensor products are E^{(8,1)} ⊗ E_{±}^{(3^{3})} ≅ E^{(4,3,2)}. (ii) For 10 ≤ n ≤ 16 there are no irreducible tensor products of the form E_{±}^{λ} ⊗ E_{±}^{μ}. Proof. (i) Follows for example from [9, 18], (ii) follows by dimensions using GAP.

2. Restriction and induction

In this section we deal with results on induction ind_{Σ_{n}}^{Σ_{n+1}} and restriction res_{Σ_{n−1}}^{Σ_{n}}. The list of notions defined for example in [22], [7] and used here is as follows.

λ = {(i, j) ∈ ℕ × ℕ | j ≤ λ_{i}} is the Young diagram of the partition λ = (λ_{1} ≥ λ_{2} ≥ …) (we do not distinguish between partitions and their Young diagrams);

(i, j) ∈ ℕ × ℕ is called a node;

(i, λ_{i}) ∈ λ is called a removable node (of λ) if λ_{i} > λ_{i+1};

(i, λ_{i} + 1) is called an addable node (for λ) if i = 1 or i > 1 and λ_{i} < λ_{i−1};

λ_{A} = λ \ {A} = (λ_{1}, …, λ_{i−1}, λ_{i} − 1, λ_{i+1}, …) is a partition of n − 1 obtained by removing a removable node A = (i, λ_{i}) from λ;

λ^{B} = λ ∪ {B} = (λ_{1}, …, λ_{i−1}, λ_{i} + 1, λ_{i+1}, …) is a partition of n + 1 obtained by adding an addable node B = (i, λ_{i} + 1) to λ;

res A = j − i (mod p) is the (p)-residue of a node A = (i, j).

A removable node A of λ is called normal if for every addable node B above A with res B = res A there exists a removable node C(B) strictly between A and B with res C(B) = res A, and such that B ≠ B′ implies C(B) ≠ C(B′). A removable node is called good if it is the lowest among the normal nodes of a fixed residue. An addable node B is called conormal if for every removable node A below B with res A = res B there exists an addable node C(A) strictly between B and A with res C(A) = res B, and such that A ≠ A′ implies C(A) ≠ C(A′). An addable node is called cogood if it is the highest among the conormal nodes of a fixed residue.

Theorem 2.1. Let D^{λ} be an irreducible F Σ_{n}-module.

(i) [20, Theorem D],[21, 0.6]. The restriction D^{λ}↓_{Σ_{n−1}} is irreducible if and only if λ is a JS-partition, which is equivalent to the fact that the top node A of λ is its only normal node. In this case D^{λ}↓_{Σ_{n−1}} ≅ D^{λ_{A}}.

(ii) The socle of D^{λ}↓_{Σ_{n−1}} (resp. D^{λ}↑^{Σ_{n+1}}) is isomorphic to ⊕D^{λ_{A}} (resp. ⊕D^{λ^{B}}) where the

sum is over all good (resp. cogood) nodes A (resp. B) of λ. (iii) [7, Theorem E(ii)] The induced module D^{λ}↑^{Σ_{n+1}} is semisimple if and only if all conormal

nodes of λ have different residues. In this case we have D^{λ}↑^{Σ_{n+1}} ≅ ⊕D^{λ^{B}}, where the sum is over all conormal nodes B of λ.

(iv) [7, Theorem E(iv)] For any addable node B such that λ^{B} is p-regular,

[D^{λ}↑^{Σ_{n+1}} : D^{λ^{B}}] = \{ \begin{matrix}d_{B} & if B is conormal for λ, \\ 0 & otherwise \end{matrix}

where d_{B} denotes the number of conormal nodes C below B (counting B itself) such that res C = res B.

(v) [7, Theorem E′(iv)] For any removable node A such that λ_{A} is p-regular,

[D^{λ}↓_{Σ_{n−1}} : D^{λ_{A}}] = \{ \begin{matrix}f_{A} & if A is normal for λ, \\ 0 & otherwise \end{matrix}

where f_{A} denotes the number of normal nodes D above A (counting A itself) such that res D = res A.

(vi) [7, Theorem E(v)] The dimension dim End_{F Σ_{n+1}}(D^{λ}↑^{Σ_{n+1}}) is equal to the number of conormal nodes for λ.

The following relation between the Mullineux bijection and (co)good nodes is known (see [22, §4] or [4, 4.12]):

Lemma 2.2. Let λ be a p-regular partition, and A (resp. B) be a good (resp. cogood) node for λ of residue α. Then there exists a unique good (resp. cogood) node A′ (resp. B′) for λ^{M} of residue −α such that (λ_{A})^{M} = (λ^{M})_{A′} (resp. (λ^{B})^{M} = (λ^{M})^{B′}).

Proof. We prove the result for good nodes, the proof for the cogood ones being similar. By Theorem 2.1(ii), the socle of D^{λ^{M}}↓_{Σ_{n−1}} is isomorphic to ⊕D^{(λ^{M})_{A′}} where the sum is over the good nodes A′ of λ^{M}. On the other hand, since D^{λ^{M}} ≅ D^{λ} ⊗ sgn the same socle is isomorphic to ⊕D^{(λ_{A})^{M}} where the sum is over the good nodes A of λ. By [25], the number of nodes in λ of residue α is equal to the number of nodes in λ^{M} of residue −α, and the result follows.

We make further remarks on restriction, induction and related combinatorics. The next lemma follows from the intertwining number theorem [10, 44.5], cf. [23, 4.1]:

Lemma 2.3. Let V be an F Σ_{n}-module. Then

dim End_{F Σ_{n+1}}(V ↑^{Σ_{n+1}}) = dim End_{F Σ_{n}}(V) + dim End_{F Σ_{n−1}}(V ↓_{Σ_{n−1}}).

Lemma 2.4. Let D^{λ} be an irreducible F Σ_{n}-module. Then

(i) D^{λ}↑^{Σ_{n+1}} is reducible; (ii) D^{λ}↑^{Σ_{n+1}} has at least three composition factors unless D^{λ}↓_{Σ_{n−1}} is irreducible.

Proof. This follows from Lemma 2.3, using Schur’s lemma and self-duality of irreducible modules over symmetric groups.

The following fact is proved in [24] using modular branching rules:

Proposition 2.5. [24, 3.6] Let λ = λ^{M} be a JS-partition. Then D^{λ}↓_{Σ_{n−2}} ≅ D^{ν} ⊕ D^{ν^{M}} for some ν ≠ ν^{M}.

Corollary 2.6. Let λ = λ^{M} be a JS-partition, and A be the top removable node of λ. Then dim End_{F Σ_{n}}((D^{λ}↓_{Σ_{n−1}})↑^{Σ_{n}}) = 3. In particular, λ_{A} has exactly three conormal nodes.

Proof. For the first claim apply Lemma 2.3 to the F Σ_{n−1}-module V = D^{λ}↓_{Σ_{n−1}} using Proposition 2.5. The second claim follows from the first and Theorem 2.1(i)(vi).

The next observation on conormal nodes follows immediately from the definitions.

Lemma 2.7. Let λ be a p-regular partition of n. The two bottom addable nodes of λ are conormal.

Lemma 2.8. Let λ be a JS-partition, and A be the top removable node of λ. Then A is a conormal addable node for the partition λ_{A}.

Proof. Let A = A_{1}, A_{2}, …, A_{l} (resp. B_{1}, B_{2}, …, B_{l+1}) be the removable (resp. addable) nodes of λ counted from top to bottom. By definition of JS-partitions we have res A_{i} = res B_{i−1} for i = 2, 3, … l. Moreover, res A_{1} ≠ res B_{1} = res A_{2}, as λ is p-regular. Now the result follows from the definition of conormal.

Now we prove our main result on conormal nodes of Mullineux-fixed JS-partitions.

Lemma 2.9. Let n > 3, λ = λ^{M} be a Mullineux-fixed JS-partition, A be the top removable node of λ, and B,C be the two bottom addable nodes of λ. Then res A = 0, res B = − res C, and A, B, C are the only three conormal nodes for λ_{A}.

Proof. Assume for certainty that B is below C. By Theorem 2.1(i), A is the only normal and hence the only good node of λ. Since λ = λ^{M}, Lemma 2.2 implies res A = 0. By Lemma 2.7, B and C are conormal nodes for λ. Moreover, it follows from Lemma 2.3 and Theorem 2.1(vi) that B and C are the only two conormal nodes for λ. If B and C have distinct residues then they are cogood and res B = − res C by virtue of Lemma 2.2. Otherwise C is the only cogood node of λ, and this time Lemma 2.2 implies res C = 0. Thus res B = res C = 0, and so res B = − res C anyway.

By Corollary 2.6 and Theorem 2.1(vi), λ_{A} has exactly three conormal nodes. Since A is one of them by Lemma 2.8, it remains to prove that B and C are conormal addable nodes for λ_{A}. Observe that if λ has at least three removable nodes then B and C are the bottom addable nodes of λ_{A}, and so they are conormal by Lemma 2.7. If λ = (l_{1}^{a_{1}}, l_{2}^{a_{2}}) has exactly two removable nodes then l_{1} − l_{2} > 1 by Lemma 1.7, in which case B and C are the bottom addable nodes of λ_{A}, and we apply Lemma 2.7 again. Finally, the case where λ has only one removable node is treated using Lemma 1.6.

Finally, we prove a couple of very special combinatorial facts which will be used only once.

Lemma 2.10. There are no p-regular Mullineux-fixed partitions λ which have exactly two normal nodes A_{1} and A_{2}, such that A_{2} is below A_{1}, res A_{1} = res A_{2}, λ_{A_{1}} is not p-regular, and λ_{A_{2}} is a JS-partition.

Proof. By definition, the top removable node is always normal, so it must be A_{1}. Since λ_{A_{1}} is not p-regular, we have λ = (l_{1}^{a_{1}}, l_{2}^{a_{2}}, …) where l_{1} − l_{2} = 1 and a_{2} = p − 1. So the second top removable node of λ is normal, hence it must be A_{2}. By Lemma 1.7, λ must have at least one more removable node, so let A_{3} be the third removable node from the top. If B is the node immediately above A_{2} then B is a removable node of λ_{A_{2}}. If a_{1} > 1 then B is normal, which contradicts the fact that λ_{A_{2}} is JS. Finally, if a_{1} = 1 then A_{3} is normal for λ_{A_{2}}, which again contradicts the fact that λ_{A_{2}} is JS.

Lemma 2.11. There are no p-regular Mullineux-fixed partitions λ which have exactly two normal nodes A_{1} and A_{2} such that res A_{1} ≠ res A_{2}, λ_{A_{1}} and λ_{A_{2}} are JS-partitions, and (λ_{A_{1}})_{B_{1}} = (λ_{A_{2}})_{B_{2}}, where B_{i} is the top removable node of λ_{A_{i}}.

Proof. Let A_{1} be above A_{2}, and let λ = (l_{1}^{a_{1}}, l_{2}^{a_{2}}, …, l_{k}^{a_{k}}). Then A_{1} is the top removable node of λ and of λ_{A_{2}}. So A_{1} = B_{2} and (λ_{A_{2}})_{B_{2}} = (λ_{A_{2}})_{A_{1}}. Now (λ_{A_{1}})_{B_{1}} = (λ_{A_{2}})_{B_{2}} implies that A_{2} = B_{1}, which is only possible if a_{1} = 1 and l_{1} −l_{2} = 1. By Lemma 1.7, we have k ≥ 3 so we can pick the third removable node from the top, A_{3}, say. Since λ_{A_{1}} is a JS-partition, we have res A_{3} = res A_{1}, hence A_{3} is normal for λ_{A_{2}}, which is impossible as λ_{A_{2}} is a JS-partition.

3. Proof of the main theorem

‘Split-non-split’ case. Throughout this subsection λ and μ are p-regular partitions of n satisfying λ ≠ λ^{M}, λ ≠ (n), (1^{n}), μ = μ^{M}. We are interested in tensor products of the form E^{λ} ⊗ E_{±}^{μ}. Note that in view of Lemma 1.3, E^{λ} ⊗ E_{+}^{μ} is irreducible if and only if E^{λ} ⊗ E_{−}^{μ} is irreducible.

Lemma 3.1. The product E^{λ} ⊗ E_{±}^{μ} is irreducible if and only if over Σ_{n} we have D^{λ} ⊗ D^{μ} ≅ D^{ν} ⊕ (D^{ν} ⊗ sgn) with D^{ν} ≅ D̸^{ν} ⊗ sgn. In this case, E^{λ} ⊗ E_{+}^{μ} ≅ E^{λ} ⊗ E_{−}^{μ} ≅ E^{ν}.

Proof. If E^{λ} ⊗ E_{±}^{μ} is irreducible then (D^{λ} ⊗ D^{μ})↓_{A_{n}} ≅ (E^{λ} ⊗ E_{+}^{μ}) ⊕ (E^{λ} ⊗ E_{−}^{μ}) is semisimple, and so D^{λ}⊗D^{μ} is also semisimple, thanks to Lemma 1.1. Moreover, since D^{μ}⊗sgn ≅ D^{μ}, we have D^{λ} ⊗ D^{μ} ⊗ sgn ≅ D^{λ} ⊗ D^{μ}. So either the tensor product D^{λ} ⊗ D^{μ} has one composition factor D^{κ} with D^{κ} ⊗ sgn ≅ D^{κ} or it has two composition factors D^{ν} and D^{ν} ⊗ sgn, with D^{ν} ≅ D̸^{ν} ⊗ sgn. But the former option is impossible by Theorem A. The rest is clear.

Theorem 3.2. Let p > 5, n ≥ 10, and E^{λ} ≅ 1̸_{A_{n}}, E^{(n−1,1)}. Then E^{λ} ⊗ E_{±}^{μ} is reducible.

Proof. If E^{λ} ⊗ E_{±}^{μ} is irreducible then Lemma 3.1 implies

dim Hom_{F Σ_{n}}(End_{F} (D^{λ}), End_{F} (D^{μ})) = dim End_{F Σ_{n}}(D^{λ} ⊗ D^{μ}) = 2.

We use Lemma 1.11 (and notation therein) to show that the first Hom-space above actually has a larger dimension. Indeed, by Lemma 1.11, it suffices to show that the following two conditions are satisfied:

(a) the modules (X_{i_{0}}^{0})^{∗}, (X_{i_{2}}^{2})^{∗}, and one of the modules (X_{i_{3}}^{3})^{∗}, (X_{i_{4}}^{4})^{∗} for some i_{j} ∈ {1, 2} are quotients of End_{F} (D^{λ}), and

(b) the modules X_{i′_{k}}^{k} are submodules of End_{F} (D^{μ}) for k = 0, 2, 3, 4 and some i′_{j} ∈ {1, 2}.

But (a) and (b) hold in view of Lemma 1.5(i),(ii), Theorem 1.13 and Proposition 1.14.

Tensor products involving the natural module. Now we study products of the form E^{(n−1,1)} ⊗ E_{±}^{μ} where μ is a Mullineux-fixed partition.

Theorem 3.3. The product E^{(n−1,1)} ⊗ E_{±}^{μ} is irreducible if and only if p |̸n and μ is a JSpartition with p |̸h(μ). In this case E^{(n−1,1)}⊗E_{±}^{μ} ≅ E^{μ_{A}^{B}} ≅ E^{μ_{A}^{C}}, where A is the top removable node of μ and B, C are the two bottom addable nodes for μ_{A}.

Proof. By Lemma 3.1, E^{(n−1,1)} ⊗ E_{±}^{μ} is irreducible if and only if

D^{(n−1,1)} ⊗ D^{μ} ≅ D^{ν} ⊕ D^{ν^{M}}

for some ν ≠ ν^{M}. To find when this happens, note that

M^{(n−1,1)} ⊗ D^{μ} ≅ (1_{Σ_{n−1}})↑^{Σ_{n}} ⊗ D^{μ} ≅ (D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}}.

Assume first that p |̸n. Then M^{(n−1,1)} ≅ D^{(n−1,1)} ⊕ 1_{Σ_{n}}. So

(D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}} ≅ D^{μ} ⊕ (D^{(n−1,1)} ⊗ D^{μ}).

Thus, we have to find when (D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}} is a direct sum of three irreducible modules. By Lemma 2.4, this can only happen if D^{μ}↓_{Σ_{n−1}} is irreducible. In view of Theorem 2.1(i), this means that μ is a JS-partition or, equivalently, the top removable node A of μ is its only normal node. So, in view of Corollary 2.6 and Theorem 2.1(iii), it remains to find when the three conormal nodes of μ_{A} have different residues. By Lemma 2.9 this happens if and only the bottom addable node of μ has residue different from 0. But the residue of this node is −h(μ), which implies the required result.

Now assume that p|n. Then M^{(n−1,1)} = 1_{Σ_{n}}|D^{(n−1,1)}|1_{Σ_{n}}. So (D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}} has a filtration with layers D^{μ}, D^{(n−1,1)} ⊗D^{μ}, D^{μ}. In particular, it has four composition factors, exactly two of which are isomorphic to each other. Assume that D^{(n−1,1)} ⊗ D^{μ} ≅ D^{ν} ⊕ D^{ν^{M}} for some ν ≠ ν^{M}. By Lemma 2.4(i), D^{μ}↓_{Σ_{n−1}} has at most two composition factors.

If D^{μ}↓_{Σ_{n−1}} = D^{μ_{A}} is irreducible then μ_{A} has three conormal nodes, see Lemma 2.9. If these nodes have different residues then (D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}} has three composition factors, giving a contradiction. So all three conormal nodes of λ have the same residue 0, see Lemma 2.9 again. The top of them, call it C, is cogood and so μ_{A}^{C} is p-regular and appears in (D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}} with multiplicity 3, see Theorem 2.1(ii)(iv). This contradiction shows that D^{μ}↓_{Σ_{n−1}} can not be irreducible.

Thus, D^{μ}↓_{Σ_{n−1}} has exactly two composition factors, say D_{1} and D_{2}. In view of Lemma 2.4, the restrictions D_{i}↓_{Σ_{n−2}} must be irreducible. If D_{1} ≅ D_{2} then by Theorem 2.1(i)(v), μ must have two normal nodes which satisfy the conditions of Lemma 2.10. Application of this lemma leads us to a contradiction, so D_{1} ≅ D̸_{2}. In this case Theorem 2.1 implies that μ has exactly two normal nodes, say A_{1}, A_{2}, such that res A_{1} ≠ res A_{2}, D^{μ}↓_{Σ_{n−1}} ≅ D^{μ_{A_{1}}} ⊕ D^{μ_{A_{2}}} , and we may assume that D_{i} = D^{μ_{A_{i}}}. Observe also that

dim Hom_{F Σ_{n}}(D^{μ}, (D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}}) = dim End_{F Σ_{n−1}}(D^{μ}↓_{Σ_{n−1}}) = 2

This implies that

(D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}} ≅ D^{μ} ⊕ D^{μ} ⊕ D^{ν} ⊕ D^{ν^{M}},

whence dim End_{F Σ_{n}}((D^{μ}↓_{Σ_{n−1}})↑^{Σ_{n}}) = 6. Now by Lemma 2.3, dim End_{F Σ_{n−2}}(D^{μ}↓_{Σ_{n−2}}) = 4. This shows that the irreducible restrictions D^{μ_{A_{2}}} ↓_{Σ_{n−2}} and D^{μ_{A_{1}}} ↓_{Σ_{n−2}} must be isomorphic to each other, which is impossible by Theorem 2.1 and Lemma 2.11.

‘Double-split case’. In this subsection λ and μ are Mullineux-fixed partitions of n. As our arguments for any product of the form E_{±}^{λ} ⊗ E_{±}^{μ} are similar we consider only E_{+}^{λ} ⊗ E_{+}^{μ}.

Lemma 3.4. Assume E_{+}^{λ} ⊗ E_{+}^{μ} is irreducible. Then

dim Hom_{F A_{n}}(End_{F} (E_{+}^{λ}), End_{F} (E_{+}^{μ})) = 1,
dim Hom_{F A_{n}}(End_{F} (E_{+}^{λ}), Hom_{F} (E_{+}^{μ}, E_{−}^{μ})) ≤ 1,
dim Hom_{F A_{n}}(Hom_{F} (E_{+}^{λ}, E_{−}^{λ}), Hom_{F} (E_{−}^{μ}, E_{+}^{μ})) ≤ 1,
dim Hom_{F A_{n}}(Hom_{F} (E_{+}^{λ}, E_{−}^{λ}), Hom_{F} (E_{+}^{μ}, E_{+}^{μ})) ≤ 1.

Proof. Note that

Hom_{F A_{n}}(End_{F} (E_{+}^{λ}), End_{F} (E_{+}^{μ})) ≅ Hom_{F A_{n}}(E_{+}^{λ} ⊗ E_{+}^{μ}, E_{+}^{λ} ⊗ E_{+}^{μ}),

which is 1-dimensional by assumptions and Schur’s lemma. Moreover, Hom_{F A_{n}}(End_{F} (E_{+}^{λ}), Hom_{F} (E_{+}^{μ}, E_{−}^{μ})) ≅ Hom_{F A_{n}}(E_{+}^{λ} ⊗ E_{+}^{μ}, E_{+}^{λ} ⊗ E_{−}^{μ}).

If the last space is at least 2-dimensional then the simple module E_{+}^{λ} ⊗ E_{+}^{μ} appears at least twice in the socle of E_{+}^{λ} ⊗ E_{−}^{μ}, which is impossible as dim(E_{+}^{λ} ⊗ E_{+}^{μ}) = dim(E_{+}^{λ} ⊗ E_{−}^{μ}). Next,

Hom_{F A_{n}}(Hom_{F} (E_{+}^{λ}, E_{−}^{λ}), Hom_{F} (E_{−}^{μ}, E_{+}^{μ})) ≅ Hom_{F A_{n}}(E_{−}^{λ} ⊗ E_{−}^{μ}, E_{+}^{λ} ⊗ E_{+}^{μ}),

which is at most 1-dimensional as both modules E_{+}^{λ} ⊗ E_{+}^{μ} and E_{−}^{λ} ⊗ E_{−}^{μ} ≅ ^{σ}(E_{+}^{λ} ⊗ E_{+}^{μ}) are irreducible by assumption. Finally,

Hom_{F A_{n}}(Hom_{F} (E_{+}^{λ}, E_{−}^{λ}), Hom_{F} (E_{+}^{μ}, E_{+}^{μ})) ≅ Hom_{F A_{n}}(E_{−}^{λ} ⊗ E_{+}^{μ}, E_{+}^{λ} ⊗ E_{+}^{μ}),

which is at most 1-dimensional as E_{+}^{λ}⊗E_{+}^{μ} is irreducible, and dim E_{−}^{λ}⊗E_{+}^{μ} = dim E_{+}^{λ}⊗E_{+}^{μ}.

Theorem 3.5. Let p > 5 and n > 16. Then E_{+}^{λ} ⊗ E_{+}^{μ} is reducible.

Proof. Assume E_{+}^{λ} ⊗ E_{+}^{μ} is irreducible. As n > 16, we have h(λ), h(μ) ≥ 5 by Lemma 1.5. So by Theorem 1.13 and self-duality of End_{F} (D^{λ}), the modules (X_{j_{k}}^{k})^{∗} are quotients of End_{F} (D^{λ}) and the modules X_{j′_{k}}^{k} are submodules of End_{F} (D^{μ}) for k = 0, 2, 3, 4, 5, and some j_{k}, j′_{k} ∈ {1, 2}, see the notation of Lemma 1.11. So the modules (X_{j_{k}}^{k}↓_{A_{n}})^{∗} are quotients of End_{F} (D^{λ})↓_{A_{n}} and the modules X_{j′_{k}}^{k}↓_{A_{n}} are submodules of End_{F} (D^{μ})↓_{A_{n}}. Note that

End_{F} (D^{ν})↓_{A_{n}} ≅ End_{F} (E_{+}^{ν}) ⊕ End_{F} (E_{−}^{ν}) ⊕ Hom_{F} (E_{+}^{ν}, E_{−}^{ν}) ⊕ Hom_{F} (E_{−}^{ν}, E_{+}^{ν})

for ν = λ or μ. Let us denote

M_{1}(ν) := End_{F} (E_{+}^{ν}) ⊕ End_{F} (E_{−}^{ν}), M_{2}(ν) := Hom_{F} (E_{+}^{ν}, E_{−}^{ν}) ⊕ Hom_{F} (E_{−}^{ν}, E_{+}^{ν}). By Lemma 1.2, Corollary 1.10 and the remarks above, there are numbers r, s ∈ {1, 2} and m, l ∈ {0, 2, 3, 4, 5}, m ≠ l, such that (X_{j_{m}}^{m}↓_{A_{n}})^{∗}, (X_{j_{l}}^{l}↓_{A_{n}})^{∗} are quotients of M_{r}(λ) and X_{j′_{m}}^{m}↓_{A_{n}}, X_{j′_{l}}^{l}↓_{A_{n}} are submodules of M_{s}(μ). We will show that this contradicts Lemma 3.4. Indeed, consider for example the case r = 1, s = 2, the remaining three cases being similar.

First, we claim that X_{j′_{m}}^{m}↓_{A_{n}} and X_{j′_{l}}^{l}↓_{A_{n}} are submodules of Hom_{F} (E_{+}^{μ}, E_{−}^{μ}). Indeed, by Corollary 1.4,

M_{2}(μ) = Hom_{F} (E_{+}^{μ}, E_{−}^{μ}) ⊕ Hom_{F} (E_{−}^{μ}, E_{+}^{μ}) ≅ Hom_{F} (E_{+}^{μ}, E_{−}^{μ}) ⊕^{σ} Hom_{F} (E_{+}^{μ}, E_{−}^{μ}). As X_{j′_{k}}^{k}↓_{A_{n}} ≅ ^{σ}(X_{j′_{k}}^{k}↓_{A_{n}}), we have X_{j′_{k}}^{k}↓_{A_{n}} is a submodule of Hom_{F} (E_{+}^{μ}, E_{−}^{μ}) if and only if it is a submodule of ^{σ} Hom_{F} (E_{+}^{μ}, E_{−}^{μ}). Now the claim follows from Lemma 1.2 and Corollary 1.10.

Similarly, (X_{j_{m}}^{m}↓_{A_{n}})^{∗} and (X_{j_{l}}^{l}↓_{A_{n}})^{∗} are quotients of End_{F} (E_{+}^{λ}). So, by Corollary 1.12, we have dim Hom_{F A_{n}}(End_{F} (E_{+}^{λ}), Hom_{F} (E_{+}^{μ}, E_{−}^{μ})) ≥ 2, giving the desired contradiction with Lemma 3.4.

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Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, D-39016 Magdeburg, Germany

E-mail address: bessen@mathematik.uni-magdeburg.de

Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

E-mail address: klesh@math.uoregon.edu