ON TENSOR PRODUCTS OF MODULAR REPRESENTATIONS OF SYMMETRIC GROUPS

CHRISTINE BESSENRODT AND ALEXANDER S. KLESHCHEV

Introduction

Let F be a field, and Σ_{n} be the symmetric group on n letters. In this paper we address the following question: given two irreducible F Σ_{n}-modules D_{1} and D_{2} of dimensions greater than 1, can it happen that D_{1} ⊗ D_{2} is irreducible? The answer is known to be ‘no’ if char F = 0 [12] (see also [2] for some generalizations). So we assume from now on that F has a positive characteristic p. The following conjecture was made in [4]:

Conjecture. Let D_{1} and D_{2} be two irreducible F Σ_{n}-module of dimensions > 1. Then D_{1} ⊗D_{2} is irreducible if and only if p = 2, n = 2+4l for some positive integer l, one of the modules corresponds to the partition (2l + 2, 2l) and the other corresponds to a partition of the form (n − 2j − 1, 2j + 1), 0 ≤ j < l. Moreover, in the exeptional cases one has

D^{(2l+2,2l)} ⊗ D^{(n−2j−1,2j+1)} ≅ D^{(2l+1−j,2l−j,j+1,j)}.

The main result of this paper is the following theorem which establishes a big part of the conjecture.

Main Theorem. Let D^{λ} and D^{μ} be two irreducible F Σ_{n}-modules of dimensions > 1. Assume that D^{λ} ⊗ D^{μ} is irreducible. Then p = 2, n is even, and if

λ = (λ_{1} > λ_{2} > ⋯ > λ_{r} > 0) and μ = (μ_{1} > μ_{2} > ⋯ > μ_{s} > 0)

then λ_{1} ≡ λ_{2} ≡ ⋯ ≡ λ_{r} (mod 2) or μ_{1} ≡ μ_{2} ≡ ⋯ ≡ μ_{s} (mod 2) (or both).

The result is relevant to the problem of describing maximal subgroups in finite groups of Lie type, cf. [1, 8]. We note that for the representations of groups of Lie type in defining characteristic the irreducible tensor products are not unusual, in view of Steinberg’s tensor product theorem. On the other hand, the case of groups of Lie type in a non-defining characteristic (including characteristic 0) has been considered recently by Magaard and Tiep [11]. They have shown that in most cases there are no irreducible tensor products.

In the case of the alternating groups we note that even in characteristic 0 there are (infinitely many) examples of (non-trivial) irreducible tensor products for alternating groups; however all of them are described [2].

Finally we note that the case p = 2 seems to be very interesting because it is really exceptional. For example, it was observed in [4] that in this case one has

D^{(4l+1,1)} ⊗ D^{(2l+2,2l)} ≅ D^{(2l+1,2l,1)} and D^{(7,3)} ⊗ D^{(6,4)} ≅ D^{(4,3,2,1)}

(see section 1 for notation). We refer the reader to [4] for further results on tensor products in characteristic 2 and the relations with the symplectic group.

1991 Mathematics Subject Classification: 20C20, 20C30.

The second author gratefully acknowledges the support of the NSF, grant DMS-9600124.

1. Preliminary results

If G is a group, D_{1}, …, D_{k} are irreducible and V_{1}, …, V_{m} are arbitrary F G-modules then we write M = D_{1}| … |D_{k} if M is a uniserial F G-module with composition factors D_{1}, …, D_{k} counted from bottom to top, and M ∼ V_{1}| … |V_{m} if M has a filtration with factors V_{1}, …, V_{m} counted from bottom to top. We denote by 1 = 1_{G} the trivial F Gmodule. If M is any F G-module, the space End_{F} (M) is an F G-module in a natural way, and End_{F G}(M) is the space of G-invariants of End_{F} (M).

We refer the reader to [5, 6, 7] for the standard facts and notation of the representation theory of Σ_{n}. In particular, D^{λ} is the irreducible F Σ_{n}-module corresponding to a pregular partition λ of n. Given any partition μ of n, one associates to it the Young subgroup Σ_{μ}, the Specht module S^{μ}, the permutation module M^{μ}, and the Young module Y^{μ}. For example, M^{μ} = (1_{Σ_{μ}})↑^{Σ_{n}}. The Young modules can be characterized as the indecomposable summands of the permutation modules M^{μ}. The modules D^{λ}, M^{λ}, and Y^{λ} are known to be self-dual.

We will need some information on the submodule structure of the permutation modules M^{(n−1,1)} and M^{(n−2,2)}. The proof of the next three lemmas is obtained by applying [5, 17.17, 24.15] and the ‘Nakayama Conjecture’ [7, 6.1.21, 2.7.41].

Lemma 1.1. The module M^{(n−1,1)} is isomorphic to D^{(n−1,1)} ⊕ 1 if n ≢ 0 (mod p), and M^{(n−1,1)} = 1|D^{(n−1,1)}|1 ∼ 1|(S^{(n−1,1)})^{∗} otherwise.

Lemma 1.2. Let p > 2 and n ≥ 4.

(i) If n ≢ 1, 2 (mod p) then M^{(n−2,2)} ≅ Y^{(n−2,2)} ⊕ M^{(n−1,1)} where

Y^{(n−2,2)} ≅ S^{(n−2,2)} ≅ D^{(n−2,2)}.

(ii) If n ≡ 1 (mod p) then M^{(n−2,2)} ≅ Y^{(n−2,2)} ⊕ D^{(n−1,1)} where

Y^{(n−2,2)} = 1|D^{(n−2,2)}|1 ∼ 1|(S^{(n−2,2)})^{∗}.

(iii) If n ≡ 2 (mod p) then M^{(n−2,2)} ≅ Y^{(n−2,2)} ⊕ 1 where

Y^{(n−2,2)} = D^{(n−1,1)}|D^{(n−2,2)}|D^{(n−1,1)} ∼ D^{(n−1,1)}|(S^{(n−2,2)})^{∗}.

We will only need the corresponding results in the case p = 2 when n is odd:

Lemma 1.3. Let p = 2 and n ≥ 4 be odd.

(i) If n ≡ 1 (mod 4) then M^{(n−2,2)} ≅ Y^{(n−2,2)} ⊕ D^{(n−1,1)} where

Y^{(n−2,2)} = 1|D^{(n−2,2)}|1 ∼ 1|(S^{(n−2,2)})^{∗}.

(ii) If n ≡ 3 (mod 4) then M^{(n−2,2)} ≅ Y^{(n−2,2)} ⊕ M^{(n−1,1)} where Y^{(n−2,2)} ≅ D^{(n−2,2)}.

Lemma 1.4. Let n ≥ 4, and assume that n is odd if p = 2. Then M^{(n−1,1)} is a quotient of M^{(n−2,2)}.

Proof. This follows from Lemmas 1.1, 1.2 and 1.3.

Unfortunately the result of Lemma 1.4 is not true when p = 2 and n is even.

We write Σ_{n−2,2} for the Young subgroup Σ_{(n−2,2)} ≅ Σ_{n−2} × Σ_{2}. The following two results from [10] will be crucial.

Theorem 1.5. [10] Let p > 2 and n ≥ 4. Assume that V is an F Σ_{n}-module such that the alternating group A_{n} < Σ_{n} does not act trivially on V. Then

dim End_{F Σ_{n−1}}(V ↓_{F Σ_{n−1}}) < dim End_{F Σ_{n−2,2}}(V ↓_{F Σ_{n−2,2}})

Let p = 2. If n = 2l is even we write S for the irreducible module D^{(l+1,l−1)} and if n = 2l + 1 is odd we write S for D^{(l+1,l)}. We call S the spinor representation of Σ_{n}.

Theorem 1.6. [10] Let p = 2, n ≥ 4, and D be a non-trivial irreducible F Σ_{n}-module. Then

dim End_{F Σ_{n−1}}(D↓_{F Σ_{n−1}}) < dim End_{F Σ_{n−2,2}}(D↓_{F Σ_{n−2,2}})

unless n is odd and D ≅ S is the spinor module.

Corollary 1.7. Let n ≥ 4, and D be an irreducible F Σ_{n} module with dim D > 1. Then

dim Hom_{F Σ_{n}}(M^{(n−2,2)}, End_{F} (D)) > dim Hom_{F Σ_{n}}(M^{(n−1,1)}, End_{F} (D)).

unless n is odd and D ≅ S is the spinor module.

Proof. The corollary follows immediately from Theorems 1.5, 1.6 and the isomorphism

Hom_{F Σ_{n}}(M^{ν}, End_{F} (D)) ≅ End_{F Σ_{ν}}(D↓_{F Σ_{ν}}),

which comes from the Frobenius reciprocity.

2. Main result

The following technical result turns out to be the key.

Lemma 2.1. Let n ≥ 4 and D be a simple F Σ_{n}-module with dim D > 1. If p = 2, assume additionally that n is odd and D ≅ S̸. Then either the dual Specht module (S^{(n−2,2)})^{∗} or the Young module Y^{(n−2,2)} (or both) is a submodule of End_{F} (D).

Proof. Assume first that p > 2, n ≢ 1, 2 (mod p) or p = 2, n ≡ 3 (mod 4). By Lemmas 1.2(i) and 1.3(ii), we have M^{(n−2,2)} ≅ M^{(n−1,1)} ⊕ D^{(n−2,2)}. Moreover, Y^{(n−2,2)} ≅ (S^{(n−2,2)})^{∗} ≅ D^{(n−2,2)}, and the result follows from Corollary 1.7.

Now let p > 2 and n ≡ 1 (mod p). By Lemma 1.4 and Corollary 1.7, there must exist a homomorphism θ : M^{(n−2,2)} → End_{F} (D) which does not factor through the surjection M^{(n−2,2)} → M^{(n−1,1)}. If θ is an injection then Y^{(n−2,2)} is a submodule of End_{F} (D). Otherwise, in view of Lemma 1.2(ii), the kernel of the restriction θ | Y^{(n−2,2)} is 1, but Y^{(n−2,2)}/1 ≅ (S^{(n−2,2)})^{∗}.

Finally, the cases p > 2, n ≡ 2 (mod p) and p = 2, n ≡ 1 (mod 4) are considered similarly to the case n ≡ 1 (mod p) using Lemmas 1.2(iii), 1.3(i) and Corollary 1.7.

The following result covers a major part of the Main Theorem.

Theorem 2.2. Let D^{λ}, D^{μ} be two irreducible F Σ_{n}-modules of dimensions > 1. If p = 2, assume additionally that n is odd. Then D^{λ} ⊗ D^{μ} is not irreducible.

Proof. For n ≤ 3 the result is trivial since irreducible modules have dimension at most 2. Assume that n ≥ 4.

If p = 2 and n is odd then no tensor product of the spinor module S with a non-trivial irreducible module is irreducible by [4, 3.1]. So from now on we assume that D^{λ}, D^{μ} ≅ S̸.

It is enough to prove that the space

End_{F Σ_{n}}(D^{λ} ⊗ D^{μ}) ≅ Hom_{F Σ_{n}}(End_{F} (D^{λ}), End_{F} (D^{μ}))

has dimension greater than 1.

For any irreducible F Σ_{n}-module D, the module End_{F} (D) is selfdual, with 1_{Σ_{n}} appearing exactly once in its socle and head.

Assume first that p > 2, n ≢ 1, 2 (mod p) or p = 2, n ≡ 3 (mod 4). Then it follows from Lemma 2.1 that 1_{Σ_{n}} ⊕ D^{(n−2,2)} appears in the socle of End_{F} (D^{μ}), as in this case we have Y^{(n−2,2)} ≅ (S^{(n−2,2)})^{∗} ≅ D^{(n−2,2)} by Lemmas 1.2 and 1.3 (and we have assumed that D^{μ} ≅ S̸). As End_{F} (D^{λ}) is self-dual, the same argument also shows that 1_{Σ_{n}} ⊕ D^{(n−2,2)} appears in the head of End_{F} (D^{λ}). Thus,

dim Hom_{F Σ_{n}}(End_{F} (D^{λ}), End_{F} (D^{μ})) > 1.

Set N_{1} := S^{(n−2,2)} and N_{2} := Y^{(n−2,2)}. By Lemma 2.1, either N_{1}^{∗} or N_{2} is a submodule of End_{F} (D^{μ}). By duality, either N_{1} or N_{2}^{∗} ≅ N_{2} is a quotient module of End_{F} (D^{λ}).

Now assume that p > 2 and n ≡ 2 (mod p). By Lemma 1.2(iii), the trivial module 1_{Σ_{n}} is not a composition factor of N_{i}, i = 1, 2. So a module of the form 1_{Σ_{n}} ⊕ N_{i} is a quotient module of End_{F} (D^{λ}), and a module of the form 1_{Σ_{n}} ⊕ N_{j}^{∗} is a submodule of End_{F} (D^{μ}). However,

dim Hom_{F Σ_{n}}(1_{Σ_{n}} ⊕ N_{i}, 1_{Σ_{n}} ⊕ N_{j}^{∗}) > 1,

for any i, j, which again implies dim Hom_{F Σ_{n}}(End_{F} (D^{λ}), End_{F} (D^{μ})) > 1.

Next, assume that p > 2, n ≡ 1 (mod p) or p = 2, n ≡ 1 (mod 4). By Lemmas 1.2(ii) and 1.3(i), the trivial module 1_{Σ_{n}} is not a submodule of N_{1}^{∗}. So if N_{2} is not a submodule of End_{F} (D^{μ}) then 1_{Σ_{n}} ⊕ N_{1}^{∗} is. Dually, if N_{2} is not a quotient module of End_{F} (D^{λ}) then 1_{Σ_{n}} ⊕ N_{1} is. Now, the result follows from the fact that dim Hom_{F Σ_{n}}(X, Y) > 1 where X is N_{2} or 1_{Σ_{n}} ⊕ N_{1}, and Y is N_{2} or 1_{Σ_{n}} ⊕ N_{1}^{∗}.

To consider the remaining cases of the Main Theorem we need the following

Proposition 2.3. Let D^{λ} and D^{μ} be two irreducible F Σ_{n}-modules such that the restrictions D^{λ}↓_{F Σ_{n−1}} and D^{μ}↓_{F Σ_{n−1}} are not irreducible. Then D^{λ} ⊗ D^{μ} is not irreducible.

Proof. First note that dim End_{F Σ_{n−1}}(D^{λ}↓_{F Σ_{n−1}}) > 1, since D^{λ}↓_{F Σ_{n−1}} is reducible and self-dual. The same is true for D^{μ}. As

Hom_{F Σ_{n}}(M^{(n−1,1)}, End_{F} (D)) ≅ End_{F Σ_{n−1}}(D↓_{F Σ_{n−1}}),

we conclude that

dim Hom_{F Σ_{n}}(M^{(n−1,1)}, End_{F} (D)) > 1 for D = D^{λ} or D^{μ}.  (1)

We know that End_{F} (D) is a self-dual module, and 1_{Σ_{n}} appears in its socle and head. By Lemma 1.1 and (1), we have that either M^{(n−1,1)} or (S^{(n−1,1)})^{∗} is a submodule of End_{F} (D^{μ}), and that either M^{(n−1,1)} or S^{(n−1,1)} is a quotient of End_{F} (D^{λ}). Now, as in the proof of Theorem 2.2, we may conclude that

dim End_{F Σ_{n}}(D^{λ} ⊗ D^{μ}) = dim Hom_{F Σ_{n}}(End_{F} (D^{λ}), End_{F} (D^{μ})) > 1.

Let p = 2 and

λ = (λ_{1} > λ_{2} > ⋯ > λ_{r} > 0)

be a 2-regular partition. By [9, Theorem D] (or by [3]), the restriction D^{λ}↓_{Σ_{n−1}} is irreducible if and only if λ_{1} ≡ λ_{2} ≡ ⋯ ≡ λ_{r} (mod 2). Now, the Main Theorem follows from Theorem 2.2 and Proposition 2.3.

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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