CONNECTIONS BETWEEN THE REPRESENTATIONS OF THE SYMMETRIC GROUP AND THE SYMPLECTIC GROUP IN CHARACTERISTIC 2

RODERICK GOW AND ALEXANDER KLESHCHEV

Introduction

Let K be a field of positive characteristic p, l a positive integer and E a vector space of dimension 2l over K. Let b : E × E → K be a non-degenerate alternating bilinear form. The group of all isometries of b is the symplectic group Sp_{2l}(K). Sp_{2l}(K) acts on the exterior powers ∧^{r}E and the composition factors of these modules are known to be the fundamental irreducible modules, i.e. the irreducible modules labelled by the fundamental dominant weights of Sp_{2l}(K). If ω_{r} is a fundamental weight we denote the irreducible Sp_{2l}(K)-module with highest weight ω_{r} by L(ω_{r}). Recursive formulae for the dimensions of the fundamental irreducible modules were given by Premet and Suprunenko in [26] when p > 2. These results were generalized to a field of an arbitrary characteristic by Adamovich [1, 2]. The multiplicities of the composition factors can also be found in [1, 2, 26]. The numerical results of Premet and Suprunenko depend on certain properties of the representations, corresponding to two-part partitions, of appropriate symmetric groups.

Suppose then that K has characteristic 2. It is well known that there is an embedding of the symmetric group Σ_{2l+1} into Sp_{2l}(K). In the first section of this paper we show that each composition factor for the action of Sp_{2l}(K) on ∧^{r}E remains irreducible on restriction to Σ_{2l+1}. More precisely, we prove that the irreducible module L(ω_{r}) labelled by the fundamental weight ω_{r} (1 ≤ r ≤ l), is irreducible on restriction to Σ_{2l+1} and its restriction is isomorphic to the irreducible module D^{(2l+1−r,r)} labelled by the partition (2l + 1 − r, r) of 2l + 1. Thus the dimension problem for the fundamental irreducible modules of Sp_{2l}(K) is equivalent to the dimension problem for the irreducible modules D^{(2l+1−r,r)} of Σ_{2l+1}, which has been investigated in some detail in [10].

In view of the fact that the l fundamental irreducible modules of Sp_{2l}(K) remain irreducible on restriction to Σ_{2l+1}, it seems reasonable to ask whether there any other irreducible modules of Sp_{2l}(K) with this property. We have not found any such examples, but the question raised does lead to some further interesting investigations. Provided l ≥ 2, we actually have an embedding

Σ_{2l+1} < Σ_{2l+2} ≤ Sp_{2l}(K),

compatible with the previous embedding for Σ_{2l+1}. The restriction of L(ω_{r}) to Σ_{2l+2} is of course irreducible for 1 ≤ r ≤ l, with restriction isomorphic to the irreducible module D^{(2l+2−r,r)} labelled by the partition (2l + 2 − r, r) of 2l + 2. In section 2 of this paper, we show that if l is even, the irreducible module L(ω_{1} + ω_{2r}) with highest weight ω_{1} + ω_{2r} is irreducible on restriction to Σ_{2l+2}, with restriction isomorphic to D^{(2l+1−2r,2r,1)}. The case when r = l/2 is particularly noteworthy. It is well known that L(ω_{1} + ω_{l}) is expressible as the tensor product L(ω_{1})⊗L(ω_{l}). L(ω_{1}) is the natural module E of dimension 2l and L(ω_{l}) is the spin module of dimension 2^{l}. The restrictions of L(ω_{1}) and L(ω_{l}) to Σ_{2l+2} are the irreducible modules D^{(2l+1,1)} and D^{(l+2,l)}, respectively. We call D^{(l+2,l)} the spin module for Σ_{2l+2}. Our result referred to above implies that D^{(2l+1,1)} ⊗ D^{(l+2,l)} is irreducible and isomorphic to D^{(l+1,l,1)}. Thus we obtain an example of what seems to be an uncommon phenomenon, the tensor product of two non-trivial irreducible modules for the symmetric group that is itself irreducible. Such non-trivial irreducible tensor products cannot occur for the symmetric group when the underlying field has characteristic 0, by [30]. Moreover, by [6], non-trivial tensor products can not even be homogeneous.

1991 Mathematics Subject Classification: 20C20, 20C30, 20C33, 20G05.

The authors wish to thank Professor G. Michler for an invitation to the Institute für Experimentelle Mathematik in Essen in 1996 when this work was begun. The first author also wishes to thank Jens Rosenboom for performing computer calculations that sug gested that the results of Section 1 of this paper might be true.

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

Following on the example of an irreducible tensor product involving the spin module, in Section 3 we investigate what can be said in general about tensoring with the spin module. We show that there is a connection here between tensoring with the spin module and the double of a spin regular partition, as defined by Benson in [4]. We find further irreducible tensor products using the spin modules and certain irreducible modules labelled by twopart partitions in the case of Σ_{10} and Σ_{14}. We also find examples of homogeneous tensor products. While our conclusions are far from definitive in most cases, we feel that the topic of tensoring with the spin module is worthy of further investigation.

We note that both the problem on irreducible restrictions from symplectic groups to symmetric groups discussed in sections 1,2 and the problem on the irreducible tensor products discussed in section 3 are of importance for describing the maximal subgroups in finite classical groups, cf. [3, 18].

We complete this introduction with a conjecture.

Conjecture. Let D_{1} and D_{2} be two irreducible F Σ_{n}-module of dimensions greater than 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)}.

1. Restriction of the fundamental irreducible modules for Sp_{2l}(K) to Σ_{2l+1}

Let K be a field of characteristic 2 and let n be a positive integer. Let x_{1}, …, x_{n} be commuting algebraically independent indeterminates over K. Let r be an integer satisfying 1 ≤ r ≤ n and let P_{r} denote the set of monomials of the form x_{i_{1}}x_{i_{2}} … x_{i_{r}} in K[x_{1}, …, x_{n}], where i_{1}, i_{2}, …, i_{r} are different integers satisfying 1 ≤ i_{1}, i_{2}, …, i_{r} ≤ n. We note that, since the indeterminates commute, x_{i_{1}}x_{i_{2}} … x_{i_{r}} = x_{j_{1}}x_{j_{2}} … x_{j_{r}} if and only if the sets of indices {i_{1}, i_{2}, …, i_{r}} and {j_{1}, j_{2}, …, j_{r}} are equal.

The symmetric group Σ_{n} of degree n acts on P_{r} by the rule

σ · (x_{i_{1}}x_{i_{2}} ⋯ x_{i_{r}}) = x_{σ(i_{1})}x_{σ(i_{2})} … x_{σ(i_{r})}.

The corresponding permutation KΣ_{n}-module is denoted by M^{(n−r,r)}. We define the symmetric bilinear form f_{r} on M^{(n−r,r)} × M^{(n−r,r)} by the condition that the basis

{x_{i_{1}}x_{i_{2}} … x_{i_{r}} : 1 ≤ i_{1} < i_{2} < … < i_{r} ≤ n}

of M^{(n−r,r)} is orthonormal with respect to f_{r}.

Assume now that 1 ≤ 2r ≤ n and let a_{1}, b_{1}, …, a_{r}, b_{r} be different integers lying between 1 and n. Define a polynomial z(a_{1}, b_{1}, …, a_{r}, b_{r}) by

z(a_{1}, b_{1}, …, a_{r}, b_{r}) = (x_{a_{1}} + x_{b_{1}}) … (x_{a_{r}} + x_{b_{r}}).

Clearly, for σ ∈ Σ_{n}, we have

σ · z(a_{1}, b_{1}, …, a_{r}, b_{r}) = z(σ(a_{1}), σ(b_{1}), …, σ(a_{r}), σ(b_{r})).

Thus all such z(a_{1}, b_{1}, …, a_{r}, b_{r}) span a KΣ_{n}-submodule of M^{(n−r,r)}. This submodule is called the Specht module S^{(n−r,r)}. The following lemma is a special case of results proved by James, [14, 4.9 and 11.1].

Lemma 1.1. Suppose that 2r < n. Then, if rad S^{(n−r,r)} denotes the radical of S^{(n−r,r)} with respect to f_{r}, the quotient

D^{(n−r,r)} = S^{(n−r,r)}/rad S^{(n−r,r)}

is non-zero and irreducible.

According to [14, 17.17], M^{(n−r,r)} has a filtration with factors S^{n}, S^{(n−1,1)}, …, S^{(n−r,r)}. Since dim M^{(n−r,r)} = \binom{n}{r}, it follows that

dim S^{(n−r,r)} = \binom{n}{r} − \binom{n}{r−1}.

We define a map ∂ : K[x_{1}, …, x_{n}] → K[x_{1}, …, x_{n}] by

∂ = ∑_{k=1}^{n} \frac{∂}{∂x_{k}} .

Given different integers i_{0}, i_{1}, …, i_{r} lying between 1 and n, we set

d(i_{0}, i_{1}, …, i_{r}) = ∂(x_{i_{0}}x_{i_{1}} … x_{i_{r}}).

We observe that for any σ ∈ Σ_{n},

σ · d(i_{0}, i_{1}, …, i_{r}) = d(σ(i_{0}), σ(i_{1}), …, σ(i_{r})).  (1)

Let L(n, r) denote the span of all such elements d(i_{0}, i_{1}, …, i_{r}). Equation (1) shows that L(n, r) is a KΣ_{n}-submodule of M^{(n−r,r)}.

Lemma 1.2. Let i_{0}, i_{1}, …, i_{r} be different integers with 1 < i_{0}, i_{1}, …, i_{r} ≤ n. Then

d(i_{0}, i_{1}, …, i_{r}) = ∑_{s=0}^{r} d(1, i_{0}, …, î_{s}, …, i_{r}).  (2)

(The symbol î_{s} means that the term i_{s} is omitted in the formula.)

Proof. Note that

∑_{2≤k,j≤n} \frac{∂^{2}}{∂x_{k}∂x_{j}} (x_{1}x_{i_{0}} … x_{i_{r}}) = 0,

since K has characteristic 2. Therefore,

∑_{s=0}^{r} d(1, i_{0}, …, î_{s}, …, i_{r}) = ∑_{s=0}^{r} ∑_{k=1}^{n} \frac{∂}{∂x_{k}} (x_{1}x_{i_{0}} … x̂_{i_{s}} … x_{i_{r}})

= ∑_{k=1}^{n} \frac{∂}{∂x_{k}} ∑_{s=0}^{r}(x_{1}x_{i_{0}} … x̂_{i_{s}} … x_{i_{r}}) = ∑_{k=1}^{n} \frac{∂}{∂x_{k}} ∑_{j=2}^{n} \frac{∂}{∂x_{j}} (x_{1}x_{i_{0}} … x_{i_{r}})

= \frac{∂}{∂x_{1}} ∑_{j=2}^{n} \frac{∂}{∂x_{j}} (x_{1}x_{i_{0}} … x_{i_{r}}) + ∑_{2≤k,j≤n} \frac{∂^{2}}{∂x_{k}∂x_{j}} (x_{1}x_{i_{0}} … x_{i_{r}})

= ∑_{j=2}^{n} \frac{∂}{∂x_{j}} (x_{i_{0}}x_{i_{1}} … x_{i_{r}}) = ∑_{j=1}^{n} \frac{∂}{∂x_{j}} (x_{i_{0}}x_{i_{1}} … x_{i_{r}}) = d(i_{0}, i_{1}, …, i_{r}),

as demanded.

Corollary 1.3. The subset

B_{r} = {d(1, i_{1}, …, i_{r}) : 1 < i_{1} < … < i_{r} ≤ n}

is a basis for L(n, r).

Proof. It follows from equation (2) of Lemma 1.2 that B_{r} generates L(n, r). To prove the linear independence of the elements of B_{r}, it suffices to note that the only element of B_{r} involving the term x_{i_{1}}x_{i_{2}} … x_{i_{r}} is d(1, i_{1}, …, i_{r}).

The module M^{(n−1,1)} is the natural permutation module for KΣ_{n}. The Specht module S^{(n−1,1)} is the submodule of M^{(n−1,1)} which consists of sums α_{1}x_{1} + ⋯ + α_{n}x_{n}, where the α_{i} ∈ K and α_{1} + ⋯ + α_{n} = 0. We will denote the element

(x_{i_{1}} + x_{i_{0}}) ∧ (x_{i_{2}} + x_{i_{0}}) ∧ … ∧ (x_{i_{r}} + x_{i_{0}}) ∈ ∧^{r}S^{(n−1,1)}

by v(i_{0}, i_{1}, …, i_{r}). Here, the integers i_{0}, i_{1}, …, i_{r} are all different and satisfy 1 ≤ i_{0}, i_{1}, …, i_{r} ≤ n. Note that the exterior power ∧^{r}S^{(n−1,1)} is a KΣ_{n}-module in a natural way, and

σ · v(i_{0}, i_{1}, …, i_{r}) = v(σ(i_{0}), σ(i_{1}), …, σ(i_{r})).  (3)

Moreover,

v(i_{0}, i_{1}, …, i_{r}) = ∑_{s=0}^{r} v(1, i_{0}, …, î_{s}, …, i_{r}).  (4)

Indeed, this follows from the equality

(x_{i_{1}} + x_{i_{0}}) ∧ … ∧ (x_{i_{r}} + x_{i_{0}})
= ((x_{i_{1}} + x_{1}) + (x_{i_{0}} + x_{1})) ∧ … ∧ ((x_{i_{r}} + x_{1}) + (x_{i_{0}} + x_{1})) .

The following result is proved by Peel in [25, pp.146-147]. Peel’s notation differs from ours, in that he uses the Specht module S^{(n−r,1^{r})}, corresponding to the partition (n−r, 1^{r}) of n, in place of ∧^{r}S^{(n−1,1)}. The KΣ_{n}-modules S^{(n−r,1^{r})} and ∧^{r}S^{(n−1,1)} are isomorphic, the element Δ(x_{i_{0}}, x_{i_{1}}, …, x_{i_{r}}) in Peel’s notation corresponding to our wedge product v(i_{0}, i_{1}, …, i_{r}). We give a proof, similar to that of Peel, using our notation.

Lemma 1.4. There exists a KΣ_{n}-monomorphism

η_{r} : ∧^{r}S^{(n−1,1)} → M^{(n−r,r)}

with

η_{r}(v(i_{0}, i_{1}, …, i_{r})) = d(i_{0}, i_{1}, …, i_{r}).

Proof. It follows from equation (4) that

{v(1, i_{1}, …, i_{r}) : 1 < i_{1} < … < i_{r} ≤ n}

is a basis of ∧^{r}S^{(n−1,1)}. Using Corollary 1.3, we see that we may define a vector space isomorphism η_{r} : ∧^{r}S^{(n−1,1)} → L(n, r) with

η_{r}(v(1, i_{1}, …, i_{r})) = d(1, i_{1}, …, i_{r}).

Equations (2) and (4) show that

η_{r}(v(i_{0}, i_{1}, …, i_{r})) = d(i_{0}, i_{1}, …, i_{r})

and equations (1) and (3) show that η_{r} is a KΣ_{n}- homomorphism, as required.

Let ψ denote the restriction to S^{(n−1,1)} × S^{(n−1,1)} of the symmetric bilinear form f_{1} defined on M^{(n−1,1)} × M^{(n−1,1)}. Let v_{1}, w_{1}, …, v_{r}, w_{r} be vectors in S^{(n−1,1)}. We define a symmetric bilinear form ψ_{r} : ∧^{r}S^{(n−1,1)} × ∧^{r}S^{(n−1,1)} → K by the formula

ψ_{r}(v_{1} ∧ v_{2} ∧ … ∧ v_{r}, w_{1} ∧ w_{2} ∧ … ∧ w_{r}) = det(ψ(v_{i}, w_{j}))_{1≤i,j≤r}.

Lemma 1.5. Suppose that 1 ≤ r < n. Then

\begin{matrix}ψ_{r}(v(1, i_{1}, …, i_{r}), v(1, j_{1}, …, j_{r})) \\ = & f_{r}(d(1, i_{1}, …, i_{r}), d(1, j_{1}, …, j_{r})) \end{matrix}  (5)

for all i_{1}, j_{1}, …, i_{r}, j_{r} with 1 < i_{1} < … < i_{r} ≤ n, 1 < j_{1} < … < j_{r} ≤ n.

Proof. Let I = {i_{1}, i_{2}, …, i_{r}} and J = {j_{1}, j_{2}, …, j_{r}}. It is straightforward to check that the right hand side of equation (5) is equal to

r + 1, if I = J;
1,  if |I ∩ J| = r − 1;
0,  otherwise.

On the other hand, the left hand side of equation (5) equals

det(δ_{i_{s}j_{t}} + 1)_{1≤s,t≤r},

where δ_{ij} denotes the usual Kronecker delta. Set

A = (δ_{i_{s}j_{t}} + 1)_{1≤s,t≤r}.

If I = J, then A is the r × r matrix with zeros on the main diagonal and ones everywhere else. One can easily check that det A = (−1)^{r−1}(r − 1), which is the same as r + 1 in characteristic 2.

If |I ∩ J| = r − 1, then A is the matrix with zeros at all but one position on the main diagonal and ones everywhere else. Then we calculate that det A = 1, since K has characteristic 2. Finally, if |I ∩ J| < r − 1, then A has two identical rows (consisting entirely of ones), and thus det A = 0, as required.

Thus η_{r} identifies L(n, r) with ∧^{r}S^{(n−1,1)}. Moreover, η_{r} is an isometry between the natural forms defined on these spaces. For our next result, which shows that L(n, r) contains S^{(n−r,r)}, we quote part of the proof of [25, Theorem 4, p.148].

Lemma 1.6. Suppose that 2r ≤ n. Let a_{1}, b_{1}, …, a_{r}, b_{r} be different integers that satisfy 1 ≤ a_{1}, b_{1}, …, a_{r}, b_{r} ≤ n. Then

z(a_{1}, b_{1}, …, a_{r}, b_{r}) = ∑ d(1, c_{1}, …, c_{r}),

where the summation runs over all sequences (c_{1}, …, c_{r}) in which, for each i, either c_{i} = a_{i} or c_{i} = b_{i}.

Corollary 1.7. Suppose that 2r ≤ n. Let T_{r} be the subspace of ∧^{r}S^{(n−1,1)} spanned by all vectors of the form

(x_{a_{1}} + x_{b_{1}}) ∧ … ∧ (x_{a_{r}} + x_{b_{r}})

where a_{1}, b_{1}, …, a_{r}, b_{r} are different integers that satisfy 1 ≤ a_{1}, b_{1}, …, a_{r}, b_{r} ≤ n. Then

(i) T_{r} is a KΣ_{n}-submodule of ∧^{r}S^{(n−1,1)} and η_{r} ↓_{T_{r}} is a KΣ_{n}-isomorphism of T_{r} onto S^{(n−r,r)}. Moreover, ψ_{r}(v, w) = f_{r}(η_{r}(v), η_{r}(w)) for any v and w in ∧^{r}S^{(n−1,1)}.

(ii) If rad T_{r} denotes the radical of T_{r} with respect to ψ_{r}, rad T_{r} is a KΣ_{n}-submodule of T_{r} and

T_{r}/rad T_{r} ≅ D^{(n−r,r)}.

Proof. By Lemma 1.6, the preimage η_{r}^{−1}(S^{(n−r,r)}) is spanned by the vectors of the form

∑ v(1, c_{1}, …, c_{r}),

where the summation extends over all sequences (c_{1}, …, c_{r}) with c_{i} = a_{i} or b_{i}. However,

∑ v(1, c_{1}, …, c_{r}) = ∑(x_{c_{1}} + x_{1}) ∧ … ∧ (x_{c_{r}} + x_{1})
= (x_{a_{1}} + x_{1} + x_{b_{1}} + x_{1}) ∧ … ∧ (x_{a_{r}} + x_{1} + x_{b_{r}} + x_{1})
= (x_{a_{1}} + x_{b_{1}}) ∧ … ∧ (x_{a_{r}} + x_{b_{r}}).

This proves the first part of (i). The second part of (i) follows from Lemma 1.5. Finally, (ii) follows from (i) and Lemma 1.1.

We note that the first part of Corollary1.7 (i) is proved in [25, Theorem 4].

We wish now to bring the symplectic group into consideration. Let l be a positive integer and let E be a vector space of dimension 2l over K. Let b : E × E → K be a non-degenerate alternating bilinear form. Let {e_{1}, …, e_{l}, e_{−1}, …, e_{−l}} be a symplectic basis of E. Thus we have

b(e_{i}, e_{j}) = b(e_{−i}, e_{−j}) = 0, b(e_{i}, e_{−j}) = δ_{ij}. (1 ≤ i, j ≤ l)

The group of all isometries of b is the symplectic group Sp_{2l}(K). Sp_{2l}(K) acts on the exterior powers ∧^{r}E and it is the composition factors of this action that we wish to investigate. Suppose that r satisfies 1 ≤ r ≤ l. Elements v_{1} ∧ … ∧ v_{r} of ∧^{r}E such that v_{1}, …, v_{r} are basis vectors of a totally isotropic r-dimensional subspace of E are called isotropic r-vectors. Let F_{r}(K) denote the subspace of ∧^{r}E spanned by all the isotropic r-vectors. F_{r}(K) is clearly a KSp_{2l}(K)-submodule of ∧^{r}E. Witt’s theorem shows that Sp_{2l}(K) permutes the totally isotropic r-dimensional subspaces of E transitively. It follows that if M is a KSp_{2l}(K)- submodule of ∧^{r}E that contains an isotropic r-vector, then F_{r}(K) ≤ M. (This is true for any field K.) We shall now prove the inequality

dim F_{r}(K) ≤ \binom{2l}{r} − \binom{2l}{r−2}.

We do this by using the representation theory of the symplectic group over fields of characteristic 0.

By a theorem of MacLane [24, Theorem 2] (see also [17, Theorem 1.5 on p. 142]), there is a complete discrete valuation ring R of characteristic 0 with unique maximal ideal 2R such that R/2R is isomorphic to K. Let L be the quotient field of R. Let V be a vector space of dimension 2l over L and let β : V × V → L be a non-degenerate alternating bilinear form. Let {u_{1}, …, u_{l}, u_{−1}, …, u_{−l}} be a symplectic basis of V. Let V_{R} denote the R-lattice in V spanned by the R-linear combinations of this symplectic basis.

There are contraction maps

∂_{r} : ∧^{r}V → ∧^{r−2}V

given by

∂_{r}(v_{1} ∧ ⋯ ∧ v_{r}) = ∑_{i<j} f(v_{i}, v_{j})(−1)^{i+j−1}v_{1} ∧ ⋯ ∧ v̂_{i} ∧ ⋯ ∧ v̂_{j} ∧ ⋯ ∧ v_{r},

where v_{1}, …, v_{r} are any vectors in V. See, for example, [12, p.260]. We take ∂_{0} and ∂_{1} to be the zero maps of ∧^{0}V = L and ∧^{1}V = V.

We now define a contraction operator ∂ on the whole exterior algebra ∧V, whose restriction to ∧^{r}V equals ∂_{r}. It is not hard to see that ∂ commutes with the action of Sp_{2l}(L) on ∧V and thus it follows that ker ∂ ∩ ∧^{r}V is an LSp_{2l}(L)-submodule of ∧^{r}V. It is known that if 1 ≤ r ≤ l, ker ∂ ∩ ∧^{r}V is the irreducible LSp_{2l}(L)-module with fundamental highest weight ω_{r}, [12, 17.5]. Since ker ∂ ∩ ∧^{r}V contains, for example, the isotropic r-vector u_{1} ∧ … ∧ u_{r}, it follows from the irreducibility of ker ∂ ∩ ∧^{r}V, together with our remarks on the transitive action of Sp_{2l}(L) on isotropic r-vectors, that ker ∂ ∩∧^{r}V = F_{r}(L), where F_{r}(L) denotes the subspace of ∧^{r}V spanned by all the isotropic r-vectors. It is also true that ∂ maps ∧^{r}V onto ∧^{r−2}V in this case, and thus it follows that

dim F_{r}(L) = \binom{2l}{r} − \binom{2l}{r−2}.

Now we set F_{r}(R) = F_{r}(L)∩∧^{r}V_{R}. Then F_{r}(R) is an R-lattice in F_{r}(L) of rank dim F_{r}(L), see for example [8, ch. VII, $4, n^{∘}1].

Given a non-zero element u of V and a non-zero element λ of L, the isometry t_{λ,u} of β given by

t_{λ,u}(v) = v + λβ(v, u)u

is called a symplectic transvection. Let Sp_{2l}(R) denote the subgroup of Sp_{2l}(L) generated by all symplectic transvections as above, where λ ∈ R and u ∈ V_{R}. It is clear that V_{R} is invariant under Sp_{2l}(R) and it follows that F_{r}(R) is also invariant under Sp_{2l}(R).

Let α denote the natural homomorphism R → R/2R ≅ K. We also let α denote the epimorphism V_{R} → E given by

α( ∑_{i=−l}^{l} λ_{i}u_{i}) = ∑_{i=−l}^{l} α(λ_{i})e_{i}.

It is clear then that α(β(u, w)) = b(α(u), α(w)) for all u and w in V_{R}. There is also a group homomorphism Sp_{2l}(R) → Sp_{2l}(K), which we will denote by θ, induced by reducing coefficients modulo 2. We readily check that

θ(σ)α(v) = α(σv)  (6)

for all σ ∈ Sp_{2l}(R) and all v ∈ V_{R}. The image of the symplectic transvection t_{λ,u} under θ is the symplectic transvection t_{α(λ),α(u)}. Since we can clearly obtain all symplectic transvections of Sp_{2l}(K) in this way, and Sp_{2l}(K) is known to be generated by its transvections, it follows that θ is surjective.

We note that ∧^{r}V_{R} is an R-lattice in ∧^{r}V invariant under Sp_{2l}(R) and F_{r}(R) is an R-sublattice in ∧^{r}V_{R} also invariant under Sp_{2l}(R). Moreover, α induces an epimorphism α_{r} : ∧^{r}V_{R} → ∧^{r}E, defined by

α_{r}(v_{1} ∧ … ∧ v_{r}) = α(v_{1}) ∧ … ∧ α(v_{r})  (7)

for any v_{1}, …, v_{r} in V_{R}. Finally, for σ in Sp_{2l}(R) we have

θ(σ)α_{r}(v_{1} ∧ … ∧ v_{r}) = α_{r}σ(v_{1} ∧ … ∧ v_{r}),  (8)

a result that follows easily from equations (6) and (7). With these preliminaries described, we can proceed to the proof of our inequality.

Lemma 1.8. Let K be a field of characteristic 2 and let l be a positive integer. Let E be a vector space of dimension 2l over K and let b : E ×E → K be a non-degenerate alternating bilinear form. Given an integer r with 1 ≤ r ≤ l, let F_{r}(K) denote the subspace of ∧^{r}E spanned by the isotropic r-vectors. Then we have

dim F_{r}(K) ≤ \binom{2l}{r} − \binom{2l}{r−2}.

Proof. We identify Sp_{2l}(K) with the image of Sp_{2l}(R) under θ. The subspace α_{r}F_{r}(R) of ∧^{r}E has dimension equal to the rank of F_{r}(R), which is \binom{2l}{r} − \binom{2l}{r−2}. Equation (8) shows that it is also Sp_{2l}(K)-invariant. Since α_{r}F_{r}(R) contains the isotropic r-vector e_{1} ∧ … ∧ e_{r}, our earlier remarks show that α_{r}F_{r}(R) contains F_{r}(K) and the desired inequality follows.

Returning to Σ_{n}, we suppose that n = 2l + 1 is odd, where l is a positive integer. The Specht module S^{(2l,1)} has dimension 2l and the form ψ defined on S^{(2l,1)} × S^{(2l,1)} is alternating and non-degenerate. For notational convenience, we will identify S^{(2l,1)} with the 2l-dimensional vector space E previously studied and the form ψ with b. We thus have the inclusion Σ_{2l+1} ≤ Sp_{2l}(K).

Lemma 1.9. Let K be a field of characteristic 2 and let l be a positive integer. Let r be an integer with 1 ≤ r ≤ l and let F_{r}(K) denote the subspace of ∧^{r}S^{(2l,1)} spanned by the isotropic r-vectors. Let T_{r} be the KΣ_{2l+1} submodule of ∧^{r}S^{(2l,1)} described in Corollary 1.7. Then we have T_{r} = F_{r}(K) and thus

dim F_{r}(K) = \binom{2l}{r} − \binom{2l}{r−2}.

Proof. We know that T_{r} is spanned by r-vectors of the form (x_{a_{1}} + x_{b_{1}}) ∧ … ∧ (x_{a_{r}} + x_{b_{r}}) where a_{1}, b_{1}, …, a_{r}, b_{r} are different integers lying between 1 and 2l + 1. Now if a_{i}, b_{i}, a_{j}, b_{j} are different integers as above, x_{a_{i}} + x_{b_{i}} and x_{a_{j}} + x_{b_{j}} are orthogonal with respect to ψ. Thus, T_{r} is spanned by isotropic r-vectors and it follows that T_{r} ≤ F_{r}(K). Corollary 1.7 shows that T_{r} is isomorphic to the Specht module S^{(2l+1−r,r)}, and thus our remarks after Lemma 1.1 imply that

dim T_{r} = \binom{2l+1}{r} − \binom{2l+1}{r−1}.

But

\binom{2l}{r} − \binom{2l}{r−2} = \binom{2l+1}{r} − \binom{2l+1}{r−1}

and we deduce that

dim F_{r}(K) ≥ \binom{2l}{r} − \binom{2l}{r−2}.

This inequality and the inequality in Lemma 1.8 imply that we have

dim F_{r}(K) = \binom{2l}{r} − \binom{2l}{r−2}

and thus T_{r} = F_{r}(K), as required.

We note that Premet and Suprunenko obtained the same formula for the dimension of F_{r}(K) when K has odd prime characteristic [26, Remark, p.1337]. Their dimension formula also depends on the characteristic 0 theory, as ours does.

Let rad F_{r}(K) denote the radical of F_{r}(K) with respect to the bilinear form ψ_{r}. Then F_{r}(K)/rad F_{r}(K) is a KSp_{2l}(K)-module and we now show that it is irreducible.

Corollary 1.10. Let K be a field of characteristic 2 and let l be a positive integer. Let r be an integer with 1 ≤ r ≤ l and let F_{r}(K) denote the subspace of ∧^{r}S^{(2l,1)} spanned by the isotropic r-vectors. Then F_{r}(K)/rad F_{r}(K) is an irreducible KSp_{2l}(K)-module whose restriction to Σ_{2l+1} is isomorphic to the irreducible module D^{(2l+1−r,r)}.

Proof. The result follows from Corollary 1.7, since we have seen in Lemma 1.9 that F_{r}(K) = T_{r}.

Suppose now that K is algebraically closed and E is our vector space of dimension 2l over K. We wish to identify the irreducible KSp_{2l}(K)-module F_{r}(K)/rad F_{r}(K) and the composition factors of ∧^{r}E in this case. With the usual notation, let B = T U be a Borel subgroup of Sp_{2l}(K) containing the maximal torus T. We may choose a symplectic basis {e_{1}, …, e_{l}, e_{−1}, …, e_{−l}} of E so that these vectors are weight vectors with respect to T with corresponding weights {ε_{1}, …, ε_{l}, −ε_{1}, …, −ε_{l}} in the notation of [7, Table III]. Moreover, the l subspaces ⟨e_{1}⟩, ⟨e_{1}, e_{2}⟩, …, ⟨e_{1}, e_{2}, …, e_{l}⟩ are all B-invariant. Again with the usual notation, the fundamental dominant weights are

ω_{1} = ε_{1}, ω_{2} = ε_{1} + ε_{2}, …, ω_{l} = ε_{1} + ε_{2} + ⋯ + ε_{l}.

Let L(ω_{r}) denote the fundamental irreducible module with highest weight ω_{r}.

Theorem 1.11. Let K be an algebraically closed field of characteristic 2 and let l be a positive integer. Let E be a vector space of dimension 2l over K and let b be a non-degenerate alternating bilinear form on E×E. Let r be an integer with 1 ≤ r ≤ l and let F_{r}(K) denote the subspace of ∧^{r}E spanned by the isotropic r-vectors. Then the irreducible KSp_{2l}(K)- module F_{r}(K)/rad F_{r}(K) is isomorphic to L(ω_{r}). Thus L(ω_{r}) ↓_{Σ_{2l+1}} ≅ D^{(2l+1−r,r)}.

Proof. The one-dimensional subspace spanned by the isotropic r-vector e_{1} ∧ … ∧ e_{r} is contained in F_{r}(K) and is B-invariant. Furthermore, since

ψ_{r}(e_{1} ∧ … ∧ e_{r}, e_{−1} ∧ … ∧ e_{−r}) = 1,

e_{1} ∧ … ∧ e_{r} + F_{r}(K) is a (non-zero) maximal vector in F_{r}(K)/rad F_{r}(K) for which the associated weight is ε_{1} + ⋯ + ε_{r} = ω_{r}. The identification of F_{r}(K)/rad F_{r}(K) with L(ω_{r}) now follows from the isomorphism theorem for irreducible rational modules of semisimple algebraic groups (see, for example, [13, 31.3]).

We note that there is a subgroup Σ_{2l+2} in Sp_{2l}(K) with Σ_{2l+1} < Σ_{2l+2} ≤ Sp_{2l}(K) when l ≥ 2. Indeed, let D^{(2l+1,1)} be the 2l-dimensional irreducible quotient of the Specht module S^{(2l+1,1)}. This is a faithful module provided l ≥ 2. Then the bilinear form f_{1} on M^{(2l+1,1)} > S^{(2l+1,1)} gives rise to a non-degenerate Σ_{2l+2}-invariant alternating form on D^{(2l+1,1)}. Thus Σ_{2l+2} ≤ Sp_{2l}(K) for l ≥ 2. This agrees with the natural embedding Σ_{2l+1} < Σ_{2l+2} and our previous embedding Σ_{2l+1} < Sp_{2l}(K) since the restriction of D^{(2l+1,1)} to Σ_{2l+1} is naturally isomorphic to S^{(2l,1)}.

Corollary 1.12. Assume the hypotheses and notation of Theorem 1.11 and also that l ≥ 2. Then

L(ω_{r}) ↓_{Σ_{2l+2}} ≅ D^{(2l+2−r,r)}

for 1 ≤ r ≤ l.

Proof. It follows from Theorem 1.11 that L(ω_{r}) ↓_{Σ_{2l+2}} is irreducible. Put L(ω_{r}) ↓_{Σ_{2l+2}} ≅ D^{λ}, where λ is a 2-regular partition of 2l + 2. Then D^{λ} ↓_{Σ_{2l+1}} ≅ D^{(2l+1−r,r)}. It follows from [20, Theorem 0.6] that λ = (2l + 2 − r, r).

We turn to a consideration of the Sp_{2l}(K) composition factors of ∧^{r}E. In what follows, L(ω_{0}) will denote the trivial one-dimensional irreducible module.

Lemma 1.13. Assume the notation of Theorem 1.11. Then the composition factors of ∧^{r}E have the form L(ω_{r−2i}), where i is an integer satisfying 0 ≤ 2i ≤ r.

Proof. Following our previous notation, the weight vectors of ∧^{r}E with respect to T are the r-vectors e_{i_{1}} ∧…∧e_{i_{r}}, where the i_{j} are integers satisfying i_{1} < … < i_{r} and −l ≤ i_{j} ≤ l. The corresponding weights have the form

±ε_{j_{1}} ± … ± ε_{j_{s}},

where s = r − 2k, k is an integer satisfying 0 ≤ 2k ≤ r, and j_{1}, …, j_{s} is a strictly increasing sequence of integers, each lying between 1 and l.

Since the action of Sp_{2l}(K) on ∧^{r}E is rational, the classification of finite dimensional irreducible rational modules for semisimple algebraic groups shows that any composition factor for Sp_{2l}(K) in ∧^{r}E is determined up to isomorphism by a dominant weight. We thus seek the dominant weights among the weights ±ε_{j_{1}} ± … ± ε_{j_{s}} that occur in ∧^{r}E. Now it is known that any weight is conjugate under the Weyl group to a unique dominant weight. Given the action of the Weyl group W of Sp_{2l}(K), we easily see that ±ε_{j_{1}} ± … ± ε_{j_{s}} is W -conjugate to ε_{1} + … + ε_{s} = ω_{s}, which is a dominant weight. Thus, the only dominant weights occurring among the weights of ∧^{r}E are the ω_{s}, with s as above, and the isomorphism theorem [13, 31.3] now tells us that the composition factors of ∧^{r}E have the form L(ω_{s}), as required.

We proceed next to find the multiplicity of each composition factor L(ω_{j}) in ∧^{r}E.

Theorem 1.14. Assume the notation of Theorem 1.11. Let [∧^{r}E : L(ω_{j})] denote the multiplicity of L(ω_{j}) as a composition factor of ∧^{r}E. Then we have

[∧^{r}E : L(ω_{j})] = ∑_{i=0}^{[(r−j)/2]} \binom{2l + 2 − 2j}{r − j − 2i}′,

where [x] denotes the greatest integer not exceeding x and \binom{a}{b}′ = 0 if \binom{a}{b} is even, and \binom{a}{b}′ = 1 otherwise.

Proof. Since

E ↓_{Σ_{2l+1}}^{Sp_{2l}(K)} ≅ S^{(2l,1)},

we conclude that

∧^{r}E ↓_{Σ_{2l+1}}^{Sp_{2l}(K)} ≅ ∧^{r}S^{(2l,1)}.

Now we remarked before the proof of Lemma 1.4 that ∧^{r}S^{(n−1,1)} and S^{(n−r,1^{r})} are isomorphic as KΣ_{n}-modules. Since, by Lemma 1.13, all composition factors of ∧^{r}E are of the form L(ω_{j}), it follows from Theorem 1.11 that all composition factors of S^{(2l+1−r,1^{r})} are of the form D^{(2l+1−j,j)}, and

[∧^{r}E : L(ω_{j})] = [S^{(2l+1−r,1^{r})} : D^{(2l+1−j,j)}].

The calculation on [14, p.93] shows that

[S^{(2l+1−r,1^{r})} : D^{(2l+1−j,j)}] = ∑_{i≥0}[S^{(2l+1−r+2i,r−2i)} : D^{(2l+1−j,j)}]

and the multiplicity [S^{(2l+1−r+2i,r−2i)} : D^{(2l+1−j,j)}] is given by \binom{2l+2−2j}{r−j−2i}′ in accordance with [14, 24.15] .

While the multiplicities above depend on the parity of binomial coefficients, the multiplicity [∧E : L(ω_{j})] of L(ω_{j}) in the entire exterior algebra is more easily expressed, as it proves to be a power of 2. We begin by finding the multiplicity of the trivial module L(ω_{0}) in ∧E.

Let n be a positive integer and let

n = 2^{c_{1}} + ⋯ + 2^{c_{t}}

be the 2-adic decomposition of n, where the c_{i} are integers satisfying 0 ≤ c_{1} < … < c_{t}. Let r be an integer satisfying 1 ≤ r ≤ n and let

r = 2^{b_{1}} + ⋯ + 2^{b_{u}}

be the 2-adic decomposition of r, with 0 ≤ b_{1} < … < b_{u}. Then \binom{n}{r} is odd if and only if {b_{1}, …, b_{u}} is a subset of {c_{1}, …, c_{t}}. The case that r = 0 will be covered by taking the empty subset of {c_{1}, …, c_{t}}. Thus, since {c_{1}, …, c_{t}} has exactly 2^{t} subsets, there are 2^{t} different odd binomial coefficients \binom{n}{r}.

Consider now the exterior power ∧^{k}E. It is well known that as a KGL_{2l}(K)-module, ∧^{2l−k}E is isomorphic to the dual of ∧^{k}E. But, as an KSp_{2l}(K)-module, ∧^{k}E is isomorphic to its dual, the isomorphism being provided by an invariant non-degenerate symmetric bilinear form constructed from the underlying alternating form b in the manner described before Lemma 1.5. Thus it follows that ∧^{2l−k}E is isomorphic to ∧^{k}E as a KSp_{2l}(K)- module.

Having made these observations, we proceed to the proof of the following multiplicity theorem.

Theorem 1.15. Assume the notation of Theorem 1.11. Let

l + 1 = 2^{a_{1}} + ⋯ + 2^{a_{s}}

be the 2-adic decomposition of l + 1, where the a_{i} are integers satisfying 0 ≤ a_{1} < … < a_{s}. Then the multiplicity [∧E : L(ω_{0})] of the trivial KSp_{2l}(K)-module in the exterior algebra is 2^{s+a_{s}−1}.

Proof. It follows from Theorem 1.14 that the trivial module is a composition factor of ∧^{k}E, where 0 ≤ k ≤ l, if and only if k is even, say k = 2i. By Theorem 1.14, the multiplicity of the trivial module in ∧^{2i}E equals the number of odd binomial coefficients among

\binom{2l+2}{0}, \binom{2l+2}{2}, …, \binom{2l+2}{2i}.

Since \binom{2a}{2b} ≡ \binom{a}{b} mod 2, this multiplicity is the number of odd numbers among

\binom{l+1}{0}, \binom{l+1}{1}, …, \binom{l+1}{i}.

Suppose first that l is odd, with l = 2m + 1, where m is a non-negative integer. Then by our remarks above,

[∧E : L(ω_{0})] = 2 ∑_{i=0}^{m}[∧^{2i}E : L(ω_{0})].

Let the different integers j such that 0 ≤ j ≤ m and \binom{l+1}{j} = \binom{2m+2}{j} is odd be j_{1}, …, j_{u}. Then it follows from Lemma 1.13 that

∑_{i=0}^{m}[∧^{2i}E : L(ω_{0})] = ∑_{i=1}^{u}(m − j_{i} + 1) = u(m + 1) − ∑_{i=1}^{u} j_{i}.  (9)

We proceed to evaluate u and the sum of the j_{i}.

Let j be a positive integer with j ≤ m and \binom{2m+2}{j} odd. Let

j = 2^{b_{1}} + ⋯ + 2^{b_{t}}

be the 2-adic decomposition of j, with b_{1} < … < b_{t}. Then it is easy to see that b_{t} ≠ a_{s}, since j ≤ m, and thus t ≤ s − 1. In particular, {b_{1}, …, b_{t}} must be a subset of {a_{1}, …, a_{s−1}}, but it is easy to see that any subset of {a_{1}, …, a_{s−1}} is admissible as {b_{1}, …, b_{t}}. Taking into account the additional number j = 0 not included above, we see that there are 2^{s−1} possibilities for j and hence u = 2^{s−1}.

If we consider all those numbers j above for which the 2-adic decomposition contains exactly t summands, a straightforward count shows that the sum of these numbers is

(2^{a_{1}} + ⋯ + 2^{a_{s−1}})\binom{s−2}{t−1}.

Thus we have

∑_{i=1}^{u} j_{i} = (2^{a_{1}} + ⋯ + 2^{a_{s−1}}) ∑_{t=1}^{s−1} \binom{s−2}{t−1} = (2m + 2 − 2^{a_{s}})2^{s−2}.

We therefore obtain

∑_{i=0}^{m}[∧^{2i}E : L(ω_{0})] = 2^{s−1}(m + 1) − (2m + 2 − 2^{a_{s}})2^{s−2}
= 2^{s+a_{s}−2}

and it follows that

[∧E : L(ω_{0})] = 2 × 2^{s+a_{s}−2} = 2^{s+a_{s}−1},

as required.

To finish the proof, we assume that l = 2m is even. Then

[∧E : L(ω_{0})] = [∧^{2m}E : L(ω_{0})] + 2 ∑_{i=0}^{m−1}[∧^{2i}E : L(ω_{0})].

The multiplicity [∧^{2m}E : L(ω_{0})] is the number of odd binomial coefficients among

\binom{2m+1}{0}, \binom{2m+1}{1}, …, \binom{2m+1}{m},

which is half the number of odd binomial coefficients among \binom{2m+1}{i}, 0 ≤ i ≤ 2m+1. Since

l + 1 = 2m + 1 = 2^{a_{1}} + ⋯ + 2^{a_{s}},

we see that

[∧^{2m}E : L(ω_{0})] = 2^{s−1}.

Now let the different integers t such that 0 ≤ t ≤ m − 1 and \binom{2m+1}{t} is odd be t_{1}, …, t_{v}. Then

∑_{i=0}^{m−1}[∧^{2i}E : L(ω_{0})] = ∑_{i=1}^{v}(m − t_{i}) = vm − ∑_{i=1}^{v} t_{i}.

Let t be one of the t_{i} above. Let

t = 2^{c_{1}} + ⋯ + 2^{c_{w}}

be the 2-adic decomposition of t, with 0 ≤ c_{1} < … < c_{w}. As before, we have c_{w} ≠ a_{s}, and thus w ≤ s − 1. Therefore, {c_{1}, …, c_{w}} is again a subset of {a_{1}, …, a_{s−1}}. Furthermore, any subset of {a_{1}, …, a_{s−1}} is admissible as {c_{1}, …, c_{w}}, except if

a_{1} = 0, a_{2} = 1, …, a_{s−1} = s − 2, a_{s} = s − 1,

in which case the subset {0, 1, …, s − 2} is inadmissible. Let us omit this exceptional case 2m + 1 = 2^{s} − 1 for the time being. Then, following the earlier argument, we have v = 2^{s−1} and

∑_{i=1}^{v} t_{i} = (2m + 1 − 2^{a_{s}})2^{s−2}.

Thus, we deduce that

[∧E : L(ω_{0})] = 2^{s−1} + 2vm − 2 ∑ t_{i}
= 2^{s−1} + 2^{s}m − 2^{s−1}(2m + 1 − 2^{a_{s}})
= 2^{s+a_{s}−1},

as required.

Finally, suppose that 2m + 1 = 2^{s} − 1. Then all the binomial coefficients involved in the sum giving [∧^{2r}E : L(ω_{0})] are odd. Thus we get

[∧E : L(ω_{0})] = (m + 1) + 2 × \frac{m(m + 1)}{2} = (m + 1)^{2}
= 2^{2s−2} = 2^{s+a_{s}−1},

in accordance with our theorem.

The multiplicity [∧E : L(ω_{r})] for 1 ≤ r ≤ l is also a power of 2, for it is straightforward to see from Theorem 1.14 that this number equals the multiplicity of the trivial KSp_{2l−2r}(K)- module in the exterior algebra ∧E(2l − 2r), where E(2l − 2r) is a symplectic space of dimension 2l − 2r over K, which may be found from Theorem 1.15.

2. Investigation of further irreducible modules for Sp_{2l}(K) and Σ_{2l+2}

The previous section has shown that there are certain irreducible KSp_{2l}(K) modules that remain irreducible on restriction to both Σ_{2l+1} and Σ_{2l+2}. An obvious question to ask is whether there are additional irreducible KSp_{2l}(K) modules that remain irreducible on restriction to Σ_{2l+1}, or failing that, on restriction to Σ_{2l+2}. We have not found any further such modules that remain irreducible on restriction to Σ_{2l+1}, but the purpose of this section is to construct, using the fundamental modules for Sp_{2l}(K), additional modules that do remain irreducible on restriction to Σ_{2l+2}. It is unlikely that the modules that we consider are the only ones with the sought-for property.

Notation 2.1. Let G be a finite group and let V be a finite-dimensional KG-module. Let V_{1}, …, V_{r} be the non-isomorphic irreducible KG-modules that occur as composition factors of V and let a_{i} = [V : V_{i}] denote the multiplicity with which V_{i} occurs as a composition factor of V, 1 ≤ i ≤ r. Then we write

V ∼ a_{1}V_{1} + ⋯ + a_{r}V_{r}.

If V is a completely reducible KG-module, we write

V = a_{1}V_{1} ⊕ ⋯ ⊕ a_{r}V_{r}.

Sheth [28] described the branching rules for two-row partitions. The following result is a special case we need.

Theorem 2.2. Let n be a positive integer with n ≡ 3 mod 4 and let m be an odd positive integer with 1 ≤ m < n/2. Then, over a field of characteristic 2,

D^{(n−m,m)} ↓_{Σ_{n−1}}∼ D^{(n−m−1,m)} + 2D^{(n−m,m−1)}.

Furthermore,

D^{( \frac{n+1}{2} , \frac{n−1}{2} )} ↓_{Σ_{n−1}}∼ 2D^{( \frac{n+1}{2} , \frac{n−3}{2} )}.

Corollary 2.3. Assume the hypotheses of Theorem 2.2. Then D^{(n−m,m)} ↓_{Σ_{n−1}} is an indecomposable uniserial module, with socle and head both isomorphic to D^{(n−m,m−1)}.

Proof. The result follows from [20, Theorem 0.5] which shows that the socle of the restriction is D^{(n−m,m−1)}.

Our next result is a straightforward consequence of the Mackey subgroup theorem, and we therefore omit its proof.

Lemma 2.4. Let K be an arbitrary field and let W be a KΣ_{n}-module. Let Π be the natural permutation module for KΣ_{n} of degree n. Then we have

(W ↑^{Σ_{n+1}}) ↓_{Σ_{n}}= W ⊕ (W ⊗ Π).

We proceed to investigate the tensor products of certain irreducible KΣ_{n}-modules.

Theorem 2.5. Let K be a field of characteristic 2 and let n = 4l +2, where l is a positive integer. Let j be an integer with 1 ≤ j ≤ l − 1. Then we have

D^{(4l+1,1)} ⊗ D^{(4l−2j+2,2j)} = D^{(4l−2j+1,2j,1)} ⊕ D^{(4l−2j+1,2j+1)}

and

D^{(4l+1,1)} ⊗ D^{(2l+2,2l)} = D^{(2l+1,2l,1)}.

Proof. We assume first that 1 ≤ j ≤ l − 1. Since 4l − 2j + 2 and 2j are both even, we have

D^{(4l−2j+2,2j)} ↑^{Σ_{4l+3}}= D^{(4l−2j+2,2j,1)} ⊕ D^{(4l−2j+2,2j+1)},

by [22, Corollary D]. Furthermore,

D^{(4l−2j+2,2j,1)} ↓_{Σ_{4l+2}}= D^{(4l−2j+1,2j,1)} ⊕ D^{(4l−2j+2,2j)}

by [20, Theorem 0.6]. Since 4l + 3 ≡ 3 mod 4, Theorem 2.2 implies that

D^{(4l−2j+2,2j+1)} ↓_{Σ_{4l+2}}∼ D^{(4l−2j+1,2j+1)} + 2D^{(4l−2j+2,2j)}.

Since n is even, we have Π ∼ D^{(4l+1,1)} + 2D^{(4l+2)}, where Π is the permutation module described in Lemma 2.4. Lemma 2.4 now implies that

D^{(4l−2j+1,2j,1)} + D^{(4l−2j+1,2j+1)} + 3D^{(4l−2j+2,2j)}
∼ D^{(4l−2j+2,2j)} ⊗ (D^{(4l+1,1)} + 2D^{(4l+2)}) + D^{(4l−2j+2,2j)}.

Multiplying out, we obtain

D^{(4l+1,1)} ⊗ D^{(4l−2j+2,2j)} ∼ D^{(4l−2j+1,2j,1)} + D^{(4l−2j+1,2j+1)}.

As D^{(4l+1)} ⊗ D^{(4l−2j+2,2j)} is certainly self-dual, it must be the direct sum of the two non-isomorphic irreducible submodules above, as required.

Finally, when j = l, the only difference in the argument is that

D^{(2l+2,2l+1)} ↓_{Σ_{4l+2}}∼ 2D^{(2l+2,2l)}

and the amended result in this case is thus clear.

Remark. The assumption that n ≡ 2 mod 4 is essential to the truth of Theorem 2.5. We have been able to prove the following more involved tensor product theorems when n ≡ 0 mod 4:

D^{(n−1,1)} ⊗ D^{(n−2,2)} ∼ D^{(n−3,2,1)} + D^{(n−3,3)} + 2D^{(n−1,1)}, n ≡ 0 mod 4
D^{(n−1,1)} ⊗ D^{(n−4,4)} ∼ D^{(n−5,4,1)} + D^{(n−5,5)} + 2D^{(n−3,3)}, n ≡ 4 mod 8
D^{(n−1,1)} ⊗ D^{(n−4,4)} ∼ D^{(n−5,4,1)} + D^{(n−5,5)} + 2D^{(n−3,3)} + 2D^{(n−1,1)},
n ≡ 0 mod 8.

We proceed to give an interpretation of Theorem 2.5 in terms of the representations of Sp_{4l}(K).

Theorem 2.6. Let K be an algebraically closed field of characteristic 2. Then for the group Sp_{4l}(K) we have

L(ω_{1}) ⊗ L(ω_{0}) = L(ω_{1}),
L(ω_{1}) ⊗ L(ω_{2j}) = L(ω_{2j+1}) ⊕ L(ω_{1} + ω_{2j}), 1 ≤ j ≤ l − 1,
L(ω_{1}) ⊗ L(ω_{2l}) = L(ω_{1} + ω_{2l}).

Proof. We proceed by induction on j, the result being obvious when j = 0. We note that if λ and μ are dominant weights and L(λ) and L(μ) are the corresponding irreducible KSp_{4l}(K)-modules, L(λ)⊗L(μ) contains L(λ+μ) as a composition factor (see, for example, [13, 31.4]). Thus L(ω_{1}) ⊗ L(ω_{2j}) certainly contains L(ω_{1} + ω_{2j}) as a composition factor for 1 ≤ j ≤ l.

We know from Lemma 1.13 that, provided 1 ≤ j ≤ l, the composition factors of ∧^{2j}E consist of L(ω_{2j}), with multiplicity 1, together with various L(ω_{2j−2s}), where 1 ≤ s ≤ j. (Here we take E to be the natural vector space of dimension 4l over K on which Sp_{4l}(K) acts.) Thus the composition factors of E ⊗ ∧^{2j}E are the composition factors of L(ω_{1}) ⊗ L(ω_{2j}) and the various L(ω_{1}) ⊗ L(ω_{2j−2s}), where 1 ≤ s ≤ l. Since we have an KSp_{4l}(K)-epimorphism from E ⊗ ∧^{2j}E onto ∧^{2j+1}E given by

v ⊗ (v_{1} ∧ … ∧ v_{2j}) ↦ v ∧ v_{1} ∧ … ∧ v_{2j},

and since ∧^{2j+1}E contains L(ω_{2j+1}) provided j ≤ l−1, it follows that for 1 ≤ j ≤ l−1, there must be an index 2k such that L(ω_{1}) ⊗ L(ω_{2k}) contains L(ω_{2j+1}) as a composition factor. By induction, this index must be 2j. Thus by our earlier observation, L(ω_{1}) ⊗ L(ω_{2j}) contains both L(ω_{2j+1}) and L(ω_{1} + ω_{2j}) as composition factors, for 1 ≤ j ≤ l − 1. Now by Theorem 1.11 we have

L(ω_{1}) ↓_{Σ_{4l+2}}= D^{(4l+1,1)}, L(ω_{2j}) ↓_{Σ_{4l+2}}= D^{(4l−2j+2,2j)}

for 1 ≤ j ≤ l and by Theorem 2.5

D^{(4l+1,1)} ⊗ D^{(4l−2j+2,2j)} = D^{(4l−2j+1,2j+1)} ⊕ D^{(4l−2j+1,2j,1)}

for 1 ≤ j ≤ l−1. It therefore follows that L(ω_{1})⊗L(ω_{2j}) contains at most two composition factors for 1 ≤ j ≤ l − 1. Thus our theorem is proved for 1 ≤ j ≤ l − 1. Our theorem also follows for j = l, since we have already seen that D^{(4l+1,1)} ⊗ D^{(2l+2,2l)} = D^{(2l+1,2l,1)} is irreducible.

Corollary 2.7. Assume the notation of Theorem 2.6 above. Then

L(ω_{1} + ω_{2j}) ↓_{Σ_{4l+2}}= D^{(4l+1−2j,2j,1)}

is an irreducible KΣ_{4l+2}-module for 1 ≤ j ≤ l.

Remark. (i) While Theorem 2.5 does not apply to Σ_{4l}, and not all the statements of Theorem 2.6 are true in general for Sp_{4l−2}(K), some parts of Corollary 2.7 are true when we replace 4l + 2 by 4l. Thus, with a little more work, we can prove that

L(ω_{1} + ω_{2}) ↓_{Σ_{4l}}= D^{(4l−3,2,1)}

is an irreducible KΣ_{4l}-module. On the other hand,

L(ω_{1} + ω_{2l−1}) ↓_{Σ_{4l}}= D^{(2l,2l−1,1)} ⊕ D^{(2l+1,2l−1)},

so that not all parts of Corollary 2.7 can be translated to Σ_{4l}.

(ii) None of the modules L(ω_{1} + ω_{2j}) remains irreducible on restriction to Σ_{4l+1} by Theorem D of [19] (see also [11]).

3. Tensoring with the spin module

Let K be an algebraically closed field of characteristic 2. The irreducible module L(ω_{l}) for Sp_{2l}(K) is known as the spin module and it has dimension 2^{l}. Steinberg, [29, Theorem 11.1] has shown that, if λ is a dominant weight expressible in terms of the fundamental weights as

λ = ω_{i_{1}} + ⋯ + ω_{i_{r}}

where 1 ≤ i_{1} < i_{2} … < i_{r} ≤ l − 1, and L(λ) is the corresponding irreducible KSp_{2l}(K)- module, then L(λ)⊗L(ω_{l}) is irreducible and isomorphic to L(λ+ω_{l}). The final part of Theorem 2.6 is a very special case of this theorem. Steinberg also proved, [29, Theorem 1.3], that the irreducible modules L(λ) and L(λ+ω_{l}) remain irreducible and non-isomorphic on restriction to Sp_{2l}(F_{2}) and constitute a complete set of 2^{l} absolutely irreducible modules for Sp_{2l}(F_{2}) over K (and they may be realized over F_{2}). Since we have Σ_{2l+2} ≤ Sp_{2l}(F_{2}) for l ≥ 2, it is the restriction of these 2^{l} irreducible modules to Σ_{2l+2} that we wish to check for irreducibility.

As we have seen, the restriction of spin module L(ω_{l}) to Σ_{2l+2} is the irreducible module D^{(l+2,l)} and its restriction to Σ_{2l+1} is the irreducible module D^{(l+1,l)}. We shall refer to these modules as spin modules for the symmetric group over F_{2}. We now make use of properties of the characteristic 0 spin modules for the proper double covers of the symmetric groups to investigate these characteristic 2 spin modules in more detail.

Let Γ_{n} denote either of the two non-isomorphic proper double covers of the symmetric group Σ_{n}. Γ_{n} has a faithful irreducible complex module Δ, say, of dimension 2^{[(n−1)/2]}, known as the basic spin representation. By finding a suitable integral lattice in a basic spin module, we may reduce coefficients modulo 2 to form the module \overline{Δ}, say, for Σ_{n} in characteristic 2. It is known, [4, 5.1], that \overline{Δ} is isomorphic to the spin module described in the paragraph above. This enables us to calculate the Brauer characters of the spin modules in characteristic 2.

Let φ^{(l+2,l)} denote the Brauer character of D^{(l+2,l)} and let α be an element of odd order in Σ_{2l+2}. Let σ(α) denote the number of cycles that appear in the cycle decomposition of α. Then by [27, VII, p.203], we have

φ^{(l+2,l)}(α) = ±2^{[(σ(α)−1)/2]}.

It follows that φ^{(l+2,l)}(α) is an odd integer if and only if σ(α) = 2, in which case α is a product of two disjoint cycles of length 2r + 1 and 2l − 2r + 1, where 0 ≤ r ≤ [l/2]. Thus there are exactly [l/2] + 1 conjugacy classes of elements of odd order on which φ^{(l+2,l)} does not take even values. The restriction of φ^{(l+2,l)} to Σ_{2l+1} is the Brauer character of the irreducible module D^{(l+1,l)}, which we denote by φ^{(l+1,l)}. Our discussion above shows that the only conjugacy class of elements of odd order in Σ_{2l+1} on which φ^{(l+1,l)} does not take even values is that containing the cycles of length 2l + 1. We intend to show how this information bounds the number of non-isomorphic irreducible modules D in characteristic 2 for the symmetric group with the property that D tensored with the spin module is also irreducible. Before starting our investigation, we first note a property of Brauer characters. Let G be any finite group and let ϑ_{1}, …, ϑ_{s} be the absolutely irreducible 2-modular Brauer characters of G. Let

ϑ = a_{1}ϑ_{1} + ⋯ + a_{s}ϑ_{s},

where the a_{i} are rational integers. Suppose that ϑ(g)/2 is an algebraic integer for each element g of odd order. Then, since the ϑ_{i} are linearly independent modulo 2 (see, for example, [9, p.599]), it follows that each coefficient a_{i} is even.

Theorem 3.1. Let K be a field of characteristic 2 and let D^{λ} be an irreducible KΣ_{n}- module, where λ is a 2-regular partition of n. Let S denote the irreducible spin KΣ_{n}- module.

(a) Suppose that n is odd. Then unless D^{λ} is the trivial module (so that λ = (n)), D^{λ} ⊗ S is not irreducible. More precisely, let φ^{λ} denote the Brauer character of D^{λ} and let π denote an n-cycle in Σ_{n}. Then if φ^{λ}(π) is an even integer, each composition factor of D^{λ} ⊗ S occurs with even multiplicity. If φ^{λ}(π) is an odd integer, S occurs with odd multiplicity as a composition factor of D^{λ} ⊗ S and there are other composition factors all occuring with even multiplicity.

(b) Suppose that n is even, with n = 2m. In this case, there are at most [(m − 1)/2] + 1 2-regular partitions λ for which D^{λ} ⊗ S is irreducible.

Proof. Let φ^{λ} be the Brauer character of D^{λ} and let φ be the Brauer character of S. We note that φ^{λ} takes rational integer values, since this is true for all complex characters of Σ_{n}. Suppose first that n is odd and φ^{λ}(π) is an even integer. Then the Brauer character φ^{λ}φ of D^{λ} ⊗ S takes even integer values on all elements of odd order. By our earlier remarks, each absolutely irreducible 2-modular Brauer character of Σ_{n} occurs as a constituent of φ^{λ}φ with even multiplicity. Since the multiplicity of each composition factor of D^{λ} ⊗ S equals the multiplicity of its corresponding irreducible Brauer character as a constituent of φ^{λ}φ, we see that each composition factor of D^{λ} ⊗S occurs with even multiplicity. Suppose next that n is odd and φ^{λ}(π) is an odd integer. Then the Brauer character φ^{λ}φ + φ also takes even integer values on all elements of odd order. It follows that φ is a constituent of φ^{λ}φ with odd multiplicity and every other irreducible Brauer character is a constituent with even multiplicity. The corresponding properties of the composition factors of D^{λ} ⊗ S follow from this deduction. This proves part (a).

Suppose finally that n = 2m is even and there are t different 2-regular partitions λ_{1}, …, λ_{t} such that D^{λ_{i}} ⊗ S is irreducible. Let ϑ_{1}, …, ϑ_{t} be the Brauer characters of these irreducible tensor products. Since the irreducible Brauer characters are linearly independent modulo 2, there must exist elements g_{1}, …, g_{t} of odd order in Σ_{n} such that the t × t integer matrix

(ϑ_{i}(g_{j})), 1 ≤ i, j ≤ t

is non-singular modulo 2. Now, as each ϑ_{i} is a product of an irreducible Brauer character with φ, the g_{j} must be non-conjugate elements such that φ(g_{j}) is an odd integer. By our earlier observations, each g_{j} is a product of two disjoint odd cycles and hence t ≤ [(m − 1)/2] + 1. This completes the proof of part (b).

Concerning the construction of irreducible tensor products using the spin module, as described in part (b) of Theorem 3.1, Theorem 2.5 has already provided an example of such a non-trivial irreducible tensor product when n ≡ 2 mod 4, n ≥ 6. We take λ = (n − 1, 1) in this case. On the basis of looking at examples for n ≤ 14, we speculate that the following may be true. Suppose that n ≡ 2 mod 4 and write n = 4m + 2. Let λ be the partition (n − 2r − 1, 2r + 1). Then for 0 ≤ r ≤ m − 1, D^{λ} ⊗ S is irreducible and isomorphic to D^{d(λ)}, where d(λ) is the double of λ, as defined by Benson in [4, p.250]. Specifically, if λ = (n − 2r − 1, 2r + 1),

d(λ) = (n/2 − r, n/2 − r − 1, r + 1, r).

If this is true, then taking into account the trivial case when λ = (n), we would have m + 1 different partitions λ for which D^{λ} ⊗ S is irreducible, in accordance with the upper bound given in part (b) of Theorem 3.1. On the other hand, we speculate that there are no non-trivial examples for which D^{λ} ⊗ S is irreducible when n ≡ 0 mod 4.

We now want to prove a result on the ‘modular Littlewood-Richardson rule’ to be used later. Let us write λ^{t} for a partition conjugate to λ, i.e. the partition whose Young diagram is the transpose of that of λ. Given partitions μ = (μ_{1}, μ_{2}, …), ν = (ν_{1}, ν_{2}, …) we denote the partition (μ_{1} + ν_{1}, μ_{2} + ν_{2}, …) by μ + ν. Moreover, we denote by m(μ, ν) the partition (μ^{t} + ν^{t})^{t}. If μ, ν, λ are partitions of m, n, m + n respectively, let c_{μν}^{λ} denote the corresponding Littlewood-Richardson coefficient, see [23, (5.2)]. It follows from the Littlewood-Richardson rule [23, (9.2)] that c_{μν}^{m(μ,ν)} = 1 and c_{μν}^{λ} = 0 unless λ ⊵ m(μ, ν) where ⊳ is the dominance order on partitions, as defined in [14].

Lemma 3.2. Let F be a field of characteristic p, and let μ and ν be p-regular partitions of m and n respectively. Then

[(D^{μ} ⊗ D^{ν}) ↑_{F [Σ_{m}×Σ_{n}]}^{F Σ_{m+n}}: D^{λ}] = 0 unless λ ⊳ m(μ, ν),

and if m(μ, ν) is p-regular then

[(D^{μ} ⊗ D^{ν}) ↑_{F [Σ_{m}×Σ_{n}]}^{F Σ_{m+n}}: D^{m(μ,ν)}] = 1

Proof. By [16], (S^{μ} ⊗ S^{ν}) ↑_{F [Σ_{m}×Σ_{n}]}^{F Σ_{m+n}} has a Specht filtration with module S^{λ} appearing c_{μν}^{λ} times for any λ. Moreover D^{γ} is a composition factor of S^{λ} only if γ ⊵λ, and [S^{λ} : D^{λ}] = 1 if λ is p-regular. It follows from the remarks preceeding this theorem that

[(S^{μ} ⊗ S^{ν}) ↑_{F [Σ_{m}×Σ_{n}]}^{F Σ_{m+n}}: D^{λ}] = 0 unless λ ⊳ m(μ, ν),

and if m(μ, ν) is p-regular then

[(S^{μ} ⊗ S^{ν}) ↑_{F [Σ_{m}×Σ_{n}]}^{F Σ_{m+n}}: D^{m(μ,ν)}] = 1

Let us pick a total order ‘<’ on the pairs (μ, ν) as above such that μ′ ⊵ μ, ν′ ⊵ ν implies (μ′, ν′) ≤ (μ, ν). We prove the lemma by induction on this total order. We start induction from the minimal pair ((m), (n)). Since D^{(m)} = S^{(m)}, D^{(n)} = S^{(n)}, the result follows from the remarks above. For an arbitrary (μ, ν) the composition factors of the F [Σ_{m} × Σ_{n}]- module S^{μ} ⊗ S^{ν} are D^{μ′} ⊗ D^{ν′} with μ′ ⊵ μ, ν′ ⊵ ν, and D^{μ} ⊗ D^{ν} appears exactly once. Note that μ′ ⊵ μ, ν′ ⊵ ν and (μ′, ν′) ≠ (μ, ν) imply m(μ′, ν′) ⊳ m(μ, ν). So the result now follows from the inductive hypothesis.

If μ(i) is a partition of n_{i}, i = 1, …, a, we define m(μ(1), …, μ(a)) inductively using the formula m(μ(1), …, μ(i)) := m(m(μ(1), …, μ(i − 1)), μ(i)). The following corollary follows from Lemma 3.2 by induction.

Corollary 3.3. Let F be a field of characteristic p, and let μ(i) be a p-regular partition of n_{i}, i = 1, …, a. Then

[(D^{μ(1)} ⊗ ⋯ ⊗ D^{μ(i)}) ↑_{F [Σ_{n_{1}}×···×Σ_{na}]}^{F Σ_{n_{1}+···+na}}: D^{λ}] = 0 unless λ ⊳ m(μ(1), …, μ(a)),

and if m(μ(1), …, μ(a)) is p-regular then

[(D^{μ(1)} ⊗ ⋯ ⊗ D^{μ(i)}) ↑_{F [Σ_{n_{1}}×···×Σ_{na}]}^{F Σ_{n_{1}+···+na}}: D^{m(μ(1),...,μ(a))}] = 1.

We wish now to consider the connection between the tensor products D^{λ} ⊗ S and the modules D^{d(λ)}, where d(λ) is the double of λ, as defined by Benson, [4]. Let λ = (λ_{1}, …, λ_{r}) be a 2-regular partition of n and let Σ_{λ} = Σ_{λ_{1}} ×⋯×Σ_{λ_{r}} be the corresponding Young subgroup of Σ_{n}. Let D^{d(n)} be the spin module for Σ_{n} over F_{2}. The character formula of Schur, [27, VII, p.203], implies that

D^{d(n)} ↓_{Σ_{λ}}= 2^{k−1+[(l+1)/2]}D^{d(λ_{1})} ⊗ ⋯ ⊗ D^{d(λ_{r})},

where k is the number of parts of λ of even size and l is the number of parts of odd size. Now, provided that λ is spin regular, as defined in [4, p.250], we have

d(λ) = m(d(λ_{1}), …, d(λ_{r})).

So Corollary 3.3 implies that

D^{d(λ_{1})} ⊗ ⋯ ⊗ D^{d(λ_{r})} ↑^{Σ_{n}}

contains D^{d(λ)} with multiplicity 1, together with other D^{μ}, where μ is a 2-regular partition with μ ⊳ d(λ). It follows that the composition factors of

(D^{d(n)} ↓_{Σ_{λ}}) ↑^{Σ_{n}}

are 2^{k−1+[(l+1)/2]} copies of D^{d(λ)} together with various D^{μ}, where μ ⊳ d(λ). But we also have

(D^{d(n)} ↓_{Σ_{λ}}) ↑^{Σ_{n}} ≅ M^{λ} ⊗ D^{d(n)},

where M^{λ} is the permutation module for Σ_{n} obtained by permuting the cosets of Σ_{λ}, [9, 38.5]. By comparing these two decompositions, we can obtain some information about tensoring modules with the spin module. Before proving our main result, we briefly discuss some properties of partitions.

Let λ and μ be 2-regular partitions of n with μ ⊳ λ. Suppose that λ and μ are both spin regular. Then it is straightforward to prove that d(μ) ⊳ d(λ). However, if we assume only that λ is spin regular, we cannot deduce that μ is also spin regular. For example, taking n = 15 and λ = (9, 5, 1), λ is spin regular but there are several 2-regular partitions μ that dominate λ and are not spin regular. This seems to introduce complications into the development of the theory. In the special case that λ and μ are both partitions of n into at most two parts, with μ ⊳ λ, it is easy to see that if λ is spin regular, then μ is also spin regular. We use this fact in our final result.

Theorem 3.4. Let K be a field of characteristic 2 and let λ be a spin regular 2-row partition of n. Let d(λ) denote the double of λ and D^{d(n)} denote the spin module for Σ_{n} over K. Then the composition factors of D^{λ} ⊗ D^{d(n)} are D^{d(λ)} with multiplicity 2^{[(k+1)/2]}, where k = 0, 1 or 2 is the number of even parts of λ, and various D^{ν}, where ν ⊳ d(λ).

Proof. We shall prove by induction on the dominance order that for any spin regular partition λ into at most two parts the composition factors of D^{λ} ⊗ D^{d(n)} are D^{d(λ)} with multiplicity 2^{k−1+[(l+1)/2]}, where k is the number of even parts and l is the number of odd parts of λ, and various D^{ν}, where ν ⊳ d(λ). The result will then follow as for 2-row partitions one has k − 1 + [(l + 1)/2] = [(k + 1)/2].

The claim is clear when λ = (n). We assume that the theorem is proved for all partitions μ of n into at most two parts with μ ⊳ λ. As we noted above, any such μ is spin regular. Our earlier discussion has shown M^{λ} ⊗ D^{d(n)} contains D^{d(λ)} as a composition factor with multiplicity 2^{k−1+[(l+1)/2]}, where k is defined as above, together with various D^{ν}, where ν ⊳ d(λ). By [14, 12.1], the composition factors of M^{λ} are D^{λ}, with multiplicity 1, and various D^{μ}, where μ is a partition of n into at most two parts with μ ⊳ λ. By induction, D^{μ} ⊗ D^{d(n)} cannot contain D^{d(λ)}, since d(μ) ⊳ d(λ). It follows that D^{λ} ⊗ D^{d(n)} contains D^{d(λ)} as a composition factor with the required multiplicity.

When n is even and λ is a spin regular 2-row partition of n into parts of odd size, we see that D^{λ} ⊗ D^{d(n)} contains D^{d(λ)} as a composition factor with multiplicity 1. As we noted previously, examples suggest that D^{λ} ⊗ D^{d(n)} = D^{d(λ)} when n ≡ 2 mod 4, but that this is not true when n ≡ 0 mod 4. Thus, the decomposition matrix of Σ_{10} given in [14, p.139] shows that dim D^{(7,3)} = 48 and dim D^{(4,3,2,1)} = 768. Theorem 3.4 then implies that D^{(7,3)} ⊗ D^{(6,4)} = D^{(4,3,2,1)}. Similarly, the decomposition matrix of Σ_{14} given in [5] enables us to deduce that D^{λ} ⊗ D^{(8,6)} = D^{d(λ)} when λ is any of (13, 1), (11, 3) and (9, 5).

While we have concentrated on looking for examples of irreducible modules D^{λ} for Σ_{n} with D^{λ} ⊗ D^{d(n)} irreducible, there may be examples for which D^{λ} ⊗ D^{d(n)} is homogeneous (i.e. a multiple of an irreducible module). We describe here one such non-trivial example which may be part of a more general scheme.

Example 3.5. Let K be a field of characteristic 2 and let λ = (9, 5, 1) be a partition of 15. Let D^{(8,7)} denote the spin module for Σ_{15}. Then

D^{(9,5,1)} ⊗ D^{(8,7)} = D^{(5,4,3,2,1)} ⊕ D^{(5,4,3,2,1)},

where D^{(5,4,3,2,1)} is the unique irreducible projective module for Σ_{15}.

Proof. The partition λ is spin regular with d(λ) = (5, 4, 3, 2, 1). Our discussion before the proof of Theorem 3.4 shows that M^{λ} ⊗D^{(8,7)} contains D^{d(λ)} = D^{(5,4,3,2,1)} as a composition factor with multiplicity 2. Using Young’s rule and Benson’s decomposition matrix for Σ_{15} in [5], we find that each composition factor of M^{λ} has multiplicity at least 2, with the exception of D^{λ} = D^{(9,5,1)}, which has multiplicity 1. We also know that for any 2-regular partition μ, D^{μ} ⊗ D^{(8,7)} contains D^{(5,4,3,2,1)} with even multiplicity by Theorem 3.1(a). Thus if there is a composition factor D^{μ} of M^{λ}, with μ ≠ λ, such that D^{(5,4,3,2,1)} is a composition factor of D^{μ} ⊗ D^{(8,7)}, it follows that D^{(5,4,3,2,1)} is a composition factor of M^{λ}⊗D^{(8,7)} with multiplicity at least 4, contrary to what we already know. Thus D^{(5,4,3,2,1)} is a composition factor of D^{λ} ⊗ D^{(8,7)} with multiplicity 2. Benson’s table shows that

dim D^{λ} dim D^{(8,7)} = 2 dim D^{(5,4,3,2,1)}

and thus we deduce that D^{λ} ⊗ D^{(8,7)} has a composition series consisting of two copies of D^{(5,4,3,2,1)}. Finally, since D^{(5,4,3,2,1)} is projective, D^{λ} ⊗ D^{(8,7)} is the direct sum of two copies of D^{(5,4,3,2,1)}.

This example suggests the possibility of the following phenomenon. Suppose that n = m+m−1+⋯+1 = m(m+1)/2 is a triangular number. Then if we set λ = (2m−1, 2m− 5, 2m − 9, …), we have d(λ) = (m, m − 1, …, 1) and D^{d(λ)} is an irreducible projective module obtained from the corresponding Specht lattice in characteristic 0 by reduction modulo 2. We speculate that the decomposition

D^{λ} ⊗ D^{d(n)} = 2^{[(m−1)/4]}D^{d(λ)}

may hold, where the right hand side of the equality is the direct sum of copies of D^{d(λ)}. This decomposition holds for m ≤ 5.

References

[1] A. M. Adamovich, Analogue of space of primitive forms over a field of positive characteristic, Mosc. Univ. Math. Bull. 39 (1984), No.1, 53-56.

[2] A. M. Adamovich, The submodule lattice of Weyl modules for symplectic groups with fundamental highest weights, Mosc. Univ. Math. Bull. 41 (1986), No.2, 6-9.

[3] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76(1984), 469-514.

[4] D. Benson, Spin modules for symmetric groups, J. London Math. Soc. 38 (1988), 250-262.

[5] D. Benson, Some remarks on the decomposition numbers of the symmetric groups, Proceedings of Symposia in Pure Mathematics 47-1 (1987), 381-394. American Mathematical Society, Providence, Rhode Island, USA.

[6] C. Bessenrodt and A. Kleshchev, On Kronecker products of complex representations of the symmetric and alternating groups, Pacif. J. Math., to appear.

[7] N. Bourbaki. Groupes et Algèbres de Lie, Hermann, Paris, 1968.

[8] N. Bourbaki, Algèbre Commutative, ch.7, Hermann, Paris, 1965.

[9] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras. John Wiley, New York, 1962.

[10] K. Erdmann, Tensor products and dimensions of simple modules for symmetric groups, Manuscripta Math. 88 (1995), 357-386.

[11] B. Ford, Irreducible restrictions of representations of the symmetric groups, Bull. Lond. Math. Soc. 27(1995), 453-459.

[12] W. Fulton and J. Harris, Representation Theory. Graduate Texts in Mathematics, 129. Springer-Verlag, New York, Berlin, Heidelberg, 1991.

[13] J.E. Humphreys, Linear Algebraic Groups. Springer-Verlag, Berlin-Heidelberg-New York, 1975.

[14] G.D. James, The Representation Theory of the Symmetric Groups. Springer-Verlag, Berlin-Heidelberg-New York, 1978.

[15] G.D. James, Representations of the symmetric group over the field of order 2, J. Algebra 38 (1976), 280-308.

[16] G.D. James and M.H. Peel, Specht series for skew representations of symmetric groups, J. Algebra 56 (1979), 343-364.

[17] G. Karpilovsky, Group Representations, Vol. 4, North-Holland, Amsterdam, 1995.

[18] P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge, 1990.

[19] A. Kleshchev, On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, I, Proc. London Math. Soc.(3) 69 (1994), 515-540.

[20] A. Kleshchev, Branching rules for modular representations of symmetric groups, II, J. reine angew. Math. 459 (1995), 163- 212.

[21] A. Kleshchev, Branching rules for modular representations of symmetric groups, III: some corollaries and a problem of Mullineux, J. London Math. Soc. 54 (1996), 25- 38.

[22] A. Kleshchev, Branching rules for modular representations of symmetric groups, IV, J. Algebra 201 (1998), 547-572.

[23] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.

[24] S. MacLane, Subfields and automorphism groups of p-adic fields, Ann. of Math. 40 (1939), 423-442.

[25] M. H. Peel, Hook representations of symmetric groups, Glasgow Math. J. 12 (1971), 136-149.

[26] A. A. Premet and I. D. Suprunenko, The Weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights, Commun. Algebra 11 (1983), 1309-1342.

[27] I. Schur, Über die Darstellung der symmetrischen und alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew Math. 139 (1911), 155-250.

[28] J. Sheth, Branching rules for two-part partitions and inductive systems, Commun. Algebra (to appear).

[29] R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33-56.

[30] I. Zisser, Irreducible products of characters in A_{n}, Isr. J. Math. 84(1993), 147-151.

Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland

 E-mail address: rod.gow@ucd.ie

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

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