Full text of DOI:10.1112/S0024609300007098

ON TENSOR PRODUCTS OF MODULAR
REPRESENTATIONS OF SYMMETRIC GROUPS
CHRISTINE BESSENRODT and ALEXANDER S. KLESHCHEV
Introduction
Let F be a field, and let Σn be the symmetric group on n letters. In this paper
we address the following question: given two irreducible FΣn-modules D1 and D2 of
dimensions greater than 1, can it happen that D1 ⊗ D2 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 positive characteristic p. The following conjecture
was made in [4].
Conjecture. Let D1 and D2 be two irreducible FΣn-modules of dimensions
greater than 1. Then D1 ⊗ D2 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 6 j < l.
Moreover, in the exceptional cases, one has
(
2l+2,2l)
(n−2j−1,2j+1) ∼ (2l+1−j,2l−j,j+1,j)
D
⊗ D
= D
.
The main result of this paper is the following theorem, which establishes a big
part of the conjecture.
λ
µ
Main Theorem. Let D and D be two irreducible FΣn-modules of dimensions
λ
µ
greater than 1. Assume that D ⊗ D is irreducible. Then p = 2, n is even, and if
λ = (λ1 > λ2 > · · · > λr > 0) and µ = (µ1 > µ2 > · · · > µs > 0),
then λ1 ≡ λ2 ≡ · · · ≡ λr (mod 2) or µ1 ≡ µ2 ≡ · · · ≡ µs (mod 2) (or both).
The result is relevant to the problem of describing maximal subgroups in finite
groups of Lie type; compare [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].
Received 9 April 1999; revised 19 July 1999.
1
991 Mathematics Subject Classification 20C20, 20C30.
The second author gratefully acknowledges the support of the NSF, grant DMS-9600124.
Bull. London Math. Soc. 32 (2000) 292–296

tensor products of modular representations
293
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
(
4l+1,1)
(2l+2,2l) ∼ (2l+1,2l,1)
see Section 1 for notation). We refer the reader to [4] for further results on tensor
(7,3)
(6,4) ∼ (4,3,2,1)
D
⊗ D
= D
and
D
⊗ D
= D
(
products in characteristic 2, and the relations with the symplectic group.
1
. Preliminary results
If G is a group, D1, . . . , Dk are irreducible and V1, . . . , Vm are arbitrary FG-modules,
then we write M = D1| . . . |Dk if M is a uniserial FG-module with composition factors
D1, . . . , Dk counted from bottom to top, and M ∼ V1| . . . |Vm if M has a filtration
with factors V1, . . . , Vm counted from bottom to top. We denote by 1 = 1G the trivial
FG-module. If M is any FG-module, then the space EndF (M) is an FG-module in
a natural way, and EndFG(M) is the space of G-invariants of EndF (M). We denote
∗
by M the module dual to M.
We refer the reader to [5, 6, 7] for the standard facts and notation of the
λ
representation theory of Σn. In particular, D is the irreducible FΣn-module corre-
sponding to a p-regular partition λ of n. Given any partition µ of n, one associates
µ
µ
to it the Young subgroup Σµ, the Specht module S , the permutation module M ,
Σn
µ
µ
and the Young module Y . For example, M = (1Σ )↑ . The Young modules can
µ
µ
be characterized as the indecomposable summands of the permutation modules M .
The modules D , M and Y are known to be self-dual.
We shall need some information on the submodule structure of the permutation
λ
λ
λ
modules M(
n−1,1)
and M . 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].
(n−2,2)
Lemma 1.1. The module M(
n−1,1)
is isomorphic to D
) otherwise.
(n−1,1)
⊕ 1 if n
6≡ 0 (mod p),
and M(
n−1,1)
= 1|D(n−1,1)|1 ∼ 1|(S
(n−1,1) ∗
Lemma 1.2. Let p > 2 and n > 4.
(
n−2,2) ∼
(n−2,2)
(n−1,1)
(i) If n
6
= Y
⊕ M
, where
(
n−2,2) ∼ (n−2,2) ∼ (n−2,2)
Y
= S
= D
.
(
n−2,2) ∼
(n−2,2)
⊕ D(n−1,1), where
(
ii) If n ≡ 1 (mod p), then M
= Y
(
n−2,2)
(n−2,2)
(n−2,2) ∗
) .
Y
= 1|D
|1 ∼ 1|(S
(
n−2,2) ∼
(n−2,2)
(iii) If n ≡ 2 (mod p), then M
= Y
⊕ 1, where
(
n−2,2)
= D(n−1,1)|D(n−2,2)|D(n−1,1) ∼ D(n−1,1)|(S(n−2,2))∗.
Y
We shall need the corresponding results in the case p = 2 only when n is odd.
Lemma 1.3. Let p = 2, and let n > 4 be odd.
(
n−2,2) ∼
(n−2,2)
⊕ D(n−1,1), where
(
i) If n ≡ 1 (mod 4), then M
= Y
(
n−2,2)
(n−2,2)
(n−2,2) ∗
Y
= 1|D
|1 ∼ 1|(S
) .
(
n−2,2) ∼
(n−2,2)
(n−1,1)
, where
(ii) If n ≡ 3 (mod 4), then M
= Y
⊕ M
(
n−2,2) ∼ (n−2,2)
Y
= D
.

2
94
christine bessenrodt and alexander s. kleshchev
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 Σn−2,2 for the Young subgroup Σ(n−2,2) = Σn−2 × Σ2. The following two
results from [10] will be crucial.
Theorem 1.5 ([10]). Let p > 2 and n > 4. Assume that V is an FΣn-module such
that the alternating group An < Σn does not act trivially on V. Then
dim EndFΣ (V↓FΣn−1 ) < dim EndFΣ (V↓FΣn−2,2 ).
n−1
n−2,2
Let p = 2. If n = 2l is even, then we write S for the irreducible module
(l+1,l)
D(
l+1,l−1)
representation of Σn.
, and if n = 2l + 1 is odd, then we write S for D
. We call S the spinor
Theorem 1.6 ([10]). Let p = 2 and n > 4, and let D be a non-trivial irreducible
FΣn-module. Then
dim EndFΣ (D↓FΣn−1 ) < dim EndFΣ (D↓FΣn−2,2
)
n−1
n−2,2
∼
unless n is odd and D = S is the spinor module.
Corollary 1.7. Let n > 4, and let D be an irreducible FΣn module with dimen-
sion greater than 1. Then
(
n−2,2)
(n−1,1)
, EndF (D)) > dim HomFΣ (M , EndF (D))
dim HomFΣ (M
n
n
∼
unless n is odd and D = S is the spinor module.
Proof. The corollary follows immediately from Theorems 1.5 and 1.6, and the
isomorphism
ν
∼
HomFΣ (M , EndF (D)) = EndFΣ (D↓ ),
n
ν
FΣν
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 let D be a simple FΣn-module with dim D > 1. If
∼
p = 2, then assume additionally that n is odd and D
(
6
n−2,2) ∗
(n−2,2)
module (S
) or the Young module Y
Proof. Assume first that p > 2, n
By Lemmas 1.2(i) and 1.3(ii), we have M
6≡ 1, 2 (mod p) or p = 2, n ≡ 3 (mod 4).
(n−1,1) (n−2,2)
= M . Moreover,
(
n−2,2) ∼
⊕ D
(
n−2,2) ∼
(n−2,2) ∗ ∼ (n−2,2)
Y
= (S
) = D
, 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
(
n−2,2)
exist a homomorphism θ : M
surjection M
→ EndF (D) which does not factor through the
(
n−2,2)
→ M(n−1,1). If θ is an injection, then Y
(n−2,2)
is a submodule of

tensor products of modular representations
295
(
n−2,2)
EndF (D). Otherwise, in view of Lemma 1.2(ii), the kernel of the restriction θ | Y
(
n−2,2)
∼
(n−2,2) ∗
is 1, but Y
/1 = (S ) .
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) and 1.3(i) and Corollary 1.7.
The following result covers a major part of the Main Theorem.
λ
µ
Theorem 2.2. Let D , D be two irreducible FΣn-modules of dimensions greater
µ
λ
than 1. If p = 2, then assume additionally that n is odd. Then D ⊗D is not irreducible.
Proof. For n 6 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
λ
µ ∼
that D , D
6
It is enough to prove that the space
λ
µ ∼
λ
µ
EndFΣ (D ⊗ D ) = HomFΣ (EndF (D ), EndF (D ))
n
n
has dimension greater than 1.
For any irreducible FΣn-module D, the module EndF (D) is self-dual, with 1Σn
appearing exactly once in its socle and head.
Assume first that p > 2, n
6
(n−2,2)
µ
from Lemma 2.1 that 1Σ ⊕ D
appears in the socle of EndF (D ), as in this
by Lemmas 1.2 and 1.3 (and we have
n
(
n−2,2) ∼
(n−2,2) ∗ ∼
(n−2,2)
case we have Y
assumed that D
= (S
) = D
µ ∼
λ
6
(
n−2,2)
1Σn ⊕ D
appears in the head of EndF (D ). Thus
λ
dim HomFΣ (EndF (D ), EndF (D )) > 1.
n
Set N1 := S(
n−2,2)
and N2 := Y
(n−2,2)
. By Lemma 2.1, either N1 or N2 is a
∗ ∼
∗
µ
submodule of EndF (D ). By duality, either N1 or N = N2 is a quotient module of
2
λ
EndF (D ).
Now assume that p > 2 and n ≡ 2 (mod p). By Lemma 1.2(iii), the trivial module
1Σn is not a composition factor of Ni, i = 1, 2. So a module of the form 1Σ ⊕ Ni is
n
λ
∗
a quotient module of EndF (D ), and a module of the form 1Σ ⊕ N is a submodule
n
j
µ
of EndF (D ). However,
∗
dim HomFΣ (1Σ ⊕ Ni, 1Σ ⊕ N ) > 1,
for any i, j, which again implies dim HomFΣ (EndF (D ), EndF (D )) > 1.
n
n
n
j
λ
µ
n
Next, assume that p > 2, n ≡ 1 (mod p) or p = 2, n ≡ 1 (mod 4). By Lem-
∗
mas 1.2(ii) and 1.3(i), the trivial module 1Σ is not a submodule of N . So if N2
n
1
µ
∗
is not a submodule of EndF (D ), then 1Σ ⊕ N is. Dually, if N2 is not a quotient
n
1
λ
module of EndF (D ), then 1Σ ⊕ N1 is. Now the result follows from the fact that
n
∗
dim HomFΣ (X, Y ) > 1, where X is N2 or 1Σ ⊕ N1, and Y is N2 or 1Σ ⊕ N .
n
n
n
1
To consider the remaining cases of the Main Theorem, we need the following.
λ
µ
Proposition 2.3. Let D and D be two irreducible FΣn-modules such that the
λ
µ
λ
µ
restrictions D ↓
and D ↓
are not irreducible. Then D ⊗D is not irreducible.
FΣn−1
FΣn−1

2
96
christine bessenrodt and alexander s. kleshchev
λ
λ
Proof. First note that dim EndFΣ (D ↓
) > 1, since D ↓
is reducible
n−1
and self-dual. The same is true for D . As
FΣn−1
FΣn−1
µ
(
n−1,1)
∼
HomFΣ (M
, EndF (D)) = EndFΣ (D↓FΣn−1 ),
n
n−1
we conclude that
dim HomFΣ (M
(
n−1,1)
λ
, EndF (D)) > 1 for D = D or D .
µ
(∗)
We know that EndF (D) is a self-dual module, and 1Σ appears in its socle and
n
n
(
n−1,1)
(n−1,1) ∗
head. By Lemma 1.1 and (∗), we have that either M
or (S
is a quotient of EndF (D ). Now, as
in the proof of Theorem 2.2, we may conclude that
) is a submodule
µ
(n−1,1)
(n−1,1)
or S
λ
of EndF (D ), and that either M
λ
µ
λ
µ
dim EndFΣ (D ⊗ D ) = dim HomFΣ (EndF (D ), EndF (D )) > 1.
n
n
Let p = 2, and let
λ = (λ1 > λ2 > · · · > λr > 0)
λ
be a 2-regular partition. By [9, Theorem D] (or by [3]), the restriction D ↓
is
Σn−1
irreducible if and only if λ1 ≡ λ2 ≡ · · · ≡ λr (mod 2). Now the Main Theorem follows
from Theorem 2.2 and Proposition 2.3.
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Fakult ¨a t f u¨ r Mathematik
Otto-von-Guericke-Universit ¨a t
Magdeburg
D-39016 Magdeburg
Germany
Department of Mathematics
University of Oregon
Eugene, OR 97403
USA