Spin modules for symmetric groups

Dave Benson Mathematical Institute Oxford

1 Introduction

In [2] I made a list of conjectures concerning decomposition numbers for symmetric groups, and including two “theorems to be substantiated”. My purpose here is to provide proofs for these theorems. Theorem 8 of [2] is as follows.

Theorem 1.1 Let λ = (λ_{1}, …, λ_{s}) be a partition of n into distinct parts, and let D^{λ} be the corresponding 2-modular irreducible representation (see James [4]) of the symmetric group Σ_{n}. Then D^{λ} ↓A_{n} splits as a direct sum of two non-isomorphic irreducible modules if and only if

(a) λ_{2j−1} − λ_{2j} = 1 or 2 ∀ j > 0 and

(b) λ_{2j−1} + λ_{2j} ≢ 2 (mod 4) ∀ j > 0

Otherwise D^{λ} ↓A_{n} is irreducible.

The proof of this theorem depends on the reduction modulo two of spin representations. We first recall the basic definitions and notations, and then we state Theorem 7 of [2] as our Theorem 1.2. The basic references for spin representations are Schur [12] and Morris [7, 8, 9, 10]. Let Γ_{n} be one of the two isoclinic proper double covers of Σ_{n} (it doesn’t matter which although we shall make an explicit choice later). Corresponding to a partition λ = (λ_{1}, …, λ_{s}) of n into s unequal parts, we have either one faithful irreducible representation of Γ_{n} if (n−s) is even, or two algebraically conjugate irreducible representations of Γ_{n} if (n − s) is odd. In the latter case, the characters only differ on elements of even order, and so the reductions modulo two have the same composition factors. Thus we need not distinguish them for our purposes, and in each case we shall write ⟨λ⟩ = ⟨λ_{1}, …, λ_{s}⟩ for the ordinary representation(s) of Γ_{n} corresponding to (λ_{1}, …, λ_{s}).

We now describe a certain doubling operation on partitions, in terms of which Theorem 1.2 is formulated. This is a process by which each part λ_{i} is replaced by two parts of approximately equal length to make a new partition. If λ_{i} = 2k + 1 is odd, then we replace λ_{i} by parts of lengths k + 1 and k, while if λ_{i} = 2k is even, then we replace λ_{i} by parts of lengths k +1 and k −1. We call a partition “spin regular” if the resulting parts form a partition of n into distinct parts in decreasing order. Thus for example dbl⟨11, 6, 1⟩ = (6, 5, 4, 2, 1), while ⟨11, 6, 2⟩ is not spin regular because (6, 5, 4, 2, 2) are not distinct, and ⟨11, 6, 4⟩ is not spin regular because (6, 5, 4, 2, 3, 1) are not in decreasing order.

Now if ⟨λ_{1}, …, λ_{s}⟩ is a representation of Γ_{n}, then each composition factor of \overline{⟨λ_{1}, …, λ_{s}⟩}, its reduction modulo two, has the centre of Γ_{n} in its kernel, and is hence a representation of Σ_{n}. Thus the composition factors are of the form D^{μ}. For example, we shall see in Section 5 that for the basic spin representation ⟨n⟩, we have \overline{⟨n⟩} = D^{dbl⟨n⟩}.

Theorem 1.2 Let ⟨λ⟩ = ⟨λ_{1}, …, λ_{s}⟩ be a spin regular partition of n. Then the composition factors of \overline{⟨λ_{1}, …, λ_{s}⟩} consist of D^{dbl⟨λ⟩} with multiplicity 2^{[t/2]}, where t of the λ_{i} are even, together with possibly some other composition factors, each of which is of the form D^{μ} with dbl⟨λ⟩ ⊳ μ, where ⊳ denotes strict dominance of partitions (see James [4, p. 8] for the definition of this partial order).

Finally, it should be remarked that the ideas in this paper go some way towards constructing the appropriate analogues for the Specht modules, for the spin representations. That is, the modules Δ_{λ} constructed in Section 7 contain the appropriate modules (or rather their super versions, which seem more natural and from which the desired modules may easily be reconstructed) with multiplicity one. It is now a question of finding a way of specifying the right submodule, and producing an analogue of James’ submodule theorem [4, Section 4]. We have included details of the Hermitian form on Δ_{λ} with this in mind, since this is likely to play the rôle of the orthogonal form used by James.

2 Clifford algebras and modules over Z[ζ_{8}]

Let ζ_{8} = e^{2πi/8} = \frac{1}{2} \sqrt{2}(1 + i) be a primitive eighth root of unity in C. Let K = Q[ζ_{8}] = Q[i, \sqrt{2}], so that R = Z[ζ_{8}] = Z[\frac{1}{2} \sqrt{2}(1 + i)] is the ring of integers in K. Looking at the tables in Washington [14, p. 352–3], we see that R has class number one, and is hence a principal ideal domain. Note that (2) is totally ramified in R, with (2) = ℘^{4}, ℘ = (ζ_{8} − 1).

Let X be an n-dimensional vector space over K with basis vectors e_{1}, …, e_{n}. Endow X with the quadratic form given by ⟨e_{j}, e_{k}⟩ = δ_{jk}. Let X_{0} be the linear subspace of X consisting of those elements ∑ λ_{j}e_{j} with ∑ λ_{j} = 0.

Let V be the set of vectors in X of the form ∑ λ_{j}e_{j} where

(i) Each λ_{j} ϵ ℘^{−2} (ii) An even number of the λ_{j} are in each non-zero coset of R in ℘^{−2}. Thus for example \frac{1}{2} \sqrt{2}(e_{j} − e_{k}) and \frac{1}{2} \sqrt{2}(e_{j} + ie_{k}) are in V. Note that all

inner products of elements of V are in R. We also set V_{0} = V ∩ X_{0}. Thus V and V_{0} are lattices in X and X_{0} respectively.

Let C(X) be the Z/2-graded Clifford algebra over K generated by X ⊆ C(X)^{1}, and with relations

v^{2} = −⟨v, v⟩
vw + wv = −2⟨v, w⟩ v, w ϵ X.

See Atiyah, Bott and Shapiro [1] for generalities on Z/2-graded Clifford algebras. Even products of elements of X are in C(X)^{0} while odd products are in C(X)^{1}.

Inside C(X), let C(X_{0}) be the K-subalgebra generated by X_{0}, and let C(V) and C(V_{0}) be the R-subalgebras generated by V and V_{0} respectively. Thus C(V) and C(V_{0}) are R-orders in the K-algebras C(X) and C(X_{0}) respectively.

We define elements of V

x_{jk} = \frac{1}{2} \sqrt{2}(e_{j} + ie_{k})

which satisfy

⟨x_{jk}, x_{jk}⟩ = 0
and ⟨x_{jk}, x_{lm}⟩ = 0 if {j, k} ∩ {l, m} = ∅.

Thus E, the R-span of x_{12}, x_{34}, … is an isotropic subspace of V of R-rank [n/2], and KE is a maximal isotropic subspace of X.

Let E_{0} = E ∩ V_{0}. Thus if n is even, KE_{0} is a maximal isotropic subspace of X_{0}, while if n is odd, after field extension if necessary, KE_{0} can be enlarged to a maximal isotropic subspace of one larger dimension. We shall not do this, so that later we shall have to split up some representations. The advantage is that we can keep better control of bases this way.

Since E is isotropic, the Clifford subalgebras C(E) ≤ C(V) and C(E_{0}) ≤ C(V_{0}) are isomorphic to the exterior algebras Λ(E) and Λ(E_{0}) respectively.

We make R into a C(E)-lattice by letting elements of E act as zero, and 1 ϵ C(E)^{0} act as the identity. Now define the Clifford lattice Δ for C(V) to be

Δ = R ⊗_{C(E)} C(V).

Thus the restriction of Δ to C(V_{0}) is R ⊗_{C(E_{0})} C(V_{0}).

Similarly the Clifford module Δ_{K} for C(X) is

Δ_{K} = Δ ⊗_{R} K = K ⊗_{C(KE)} C(X)

where K is a C(KE)-module in the same way as the above description of R as a C(E)-lattice. Thus Δ is a C(V)-lattice in the C(X)-module Δ_{K}. It follows from [1] that Δ is irreducible as a C(V)-lattice if n is even, and has two irreducible constituents when n is odd. We shall describe this in more detail later.

Remark : The reason for working over Z[ζ_{8}] rather than Z or even Z[i] is that Q[ζ_{8}] is a splitting field for all the Clifford algebras we are using. If we had worked over Z, we would have obtained answers depending on n modulo eight. Working over Z[i] gives answers only depending on n modulo two, but for n odd, the C(V)-lattice Δ fails to reduce. In fact its endomorphism ring is Z[ζ_{8}]. Note that over Z or Z[i] we must choose the quadratic form so that ⟨e_{j}, e_{k}⟩ = 2δ_{jk} to eliminate occurrences of \sqrt{2}.

Remember that our intention is to reduce modulo two. Since R/℘ ≅ F_{2}, we have not enlarged the residue field by adjoining eighth roots of unity. Our basis is also suitable for reducing modulo an odd prime p, and it is clear that the answers we get will depend on p modulo eight.

3 The double cover Γ_{n} of the symmetric group Σ_{n}

A double cover Γ_{n} of the symmetric group Σ_{n} sits inside C(V) as follows. Corresponding to a transposition (jk) ϵ Σ_{n}, we have the two elements

[jk] = \frac{1}{2} \sqrt{2}(e_{j} − e_{k})
[kj] = −[jk] = \frac{1}{2} \sqrt{2}(e_{k} − e_{j})

which square to −1. Thus by restriction of action, Δ is a module for Γ_{n}.

Note that negating the quadratic form ⟨ , ⟩ on X gives the other isoclinic proper double cover of Σ_{n} in which the preimages of transpositions have order two. Any representation of one may be made into a representation of the other by multiplying by i all the matrices corresponding to elements not in the subgroup of index two.

References for the representation theory of Γ_{n} are Morris [7, 8, 9, 10] and Schur [12], and for an explanation of isoclinism see Beyl and Tappe [3].

4 Description of the Clifford lattice Δ and its restriction to Γ_{n}

If n is odd, let X_{n+1} be a space of one larger dimension than X, with basis e_{1}, …, e_{n+1} and the usual quadratic form. Then C(X) ≤ C(X_{n+1}), and the Clifford module for C(X) is the restriction of the one for C(X_{n+1}). Thus we may concentrate on the case where n is even.

Let F be the R-span of x_{21}, x_{43}, …, so that F is isotropic of R-rank n/2 and X = KE ⊕ KF. If we restrict Δ_{K} from C(X) to C(KF) = Λ(KF) we obtain the regular representation of the latter. Thus Δ_{K} has a basis of the following form.

Let

u_{k} = 1 ⊗ \frac{1}{2}(e_{2k} + ie_{2k−1}) = \frac{1}{2} \sqrt{2}(1 ⊗ x_{2k,2k−1}) ϵ Δ_{K} (1 ≤ k ≤ n/2)

Let Ω = {1, …, n/2}. If I = {k_{1}, …, k_{r}} ⊆ Ω with k_{1} < ⋯ < k_{r}, let

u_{I} = u_{k_{1}} ∧ … ∧ u_{k_{r}} ϵ Δ_{K} ≅ Λ(KF).

Note that u_{∅} = 1 ⊗ 1 ϵ Δ_{K}. Then the u_{I}, I ⊆ Ω, form a basis for Δ_{K}. The action of C(X) on Δ_{K} is given by the following rules.

u_{I}e_{2j−1} = \{ \begin{matrix}−i(−1)^{|I(j)|} u_{I∪\{j\}} & if j ϵ̸ I \\ i(−1)^{|I(j)|} u_{I\\{j\}} & if j ϵ I \end{matrix}

u_{I}e_{2j} = \{ \begin{matrix}(−1)^{|I(j)|} u_{I∪\{j\}} & if j ϵ̸ I \\ (−1)^{|I(j)|} u_{I\\{j\}} & if j ϵ I , \end{matrix}

where I(j) = {i ϵ I | i ≥ j}.

The Z/2-grading on Δ_{K} is given by u_{I} ϵ (Δ_{K})^{ε}, where ε ϵ {0, 1} and |I| ≡ ε (mod 2). There is a positive definite invariant Hermitian form on Δ_{K} given

by ⟨u_{I}, u_{J}⟩ = δ_{IJ}.

It is not hard to check that the lattice Δ ⊆ Δ_{K} consists of the elements ∑ λ_{I}u_{I} where

(i) Each λ_{I} ϵ ℘^{−2}.

(ii) An even number of the λ_{I} with |I| even are in each non-identity coset of R in ℘^{−2}.

(iii) An even number of the λ_{I} with |I| odd are in each non-identity coset of R in ℘^{−2}.

Definition 4.1 We define η ϵ C(V) to be the element

η = \frac{1}{2} \sqrt{2} i^{n/2} (∑_{j=1}^{n} e_{j}) (∏_{j=1}^{n} e_{j})

Lemma 4.2 The element η commutes with all elements of C(V_{0}).

Proof : Each of ∑_{j=1}^{n} e_{j} and ∏_{j=1}^{n} e_{j} anticommutes with all elements of C(V_{0}). □

It is also easy to see that η^{2} = n/2, and that η acts as an endomorphism of Δ which is not a multiple of the identity. Thus we have

R[X]/(X^{2} − n/2) ⊆ End_{C(V_{0})}(Δ).

Since ΔC = Δ ⊗_{R} C is a direct sum of two non-isomorphic irreducible C(V_{0}) ⊗_{R} C-modules [1], we have End_{C(V_{0})}(Δ) ⊗_{R} C ≅ C ⊕ C, from which it is easy to check that the above inclusion is an equality.

Thus if n/2 is not a square in R, Δ is irreducible as a C(V_{0})-lattice, but reduces on adjoining \sqrt{n/2} to the scalars. If n/2 is a square then Δ is already reducible over R. In either case, if we let R̃ = R[\sqrt{n/2}], Δ̃ = Δ ⊗_{R} R̃ has a sublattice of finite index Δ̃^{+} ⊕ Δ̃^{−} where Δ̃^{+} = Δ̃(η + \sqrt{n/2}) and Δ̃^{−} = Δ̃(η − \sqrt{n/2}). Note that if ˜℘ ≤ R̃ lies above ℘, then R̃/˜℘ = F_{2}.

By the work of Schur [12], we know that Δ̃^{+} and Δ̃^{−} remain non-isomorphic irreducible lattices on restriction to R̃Γ_{n} ⊆ C(V_{0}) ⊗_{R} R̃. We also know that when restricted to Ã_{n} = [Γ_{n}, Γ_{n}], the double cover of A_{n}, Δ̃^{+} and Δ̃^{−} become isomorphic irreducible lattices. In this case, Δ already decomposes over RÃ_{n} as Δ^{0} ⊕ Δ^{1}, since Ã_{n} ⊆ C(V_{0})^{0}.

We now return to the case where n is odd. In this case, we have seen that Δ_{K} has a basis u_{I}, I ⊆ Ω = {1, …, (n+1)/2} with the action of C(X) as given before.

Since n is odd, the element

γ = i^{(n−1)/2} (∏_{j=1}^{n} e_{j})

is in the centre of C(V) and satisfies γ^{2} = 1. Thus Δ has a sublattice of finite index Δ^{+} ⊕ Δ^{−} where Δ^{+} = Δ(1 + γ) and Δ^{−} = Δ(1 − γ).

By [1], Δ^{+} and Δ^{−} are irreducible C(V)-lattices which are non-isomorphic, but become isomorphic when restricted to C(V_{0}). In this case, if we extend the ground ring to R̃ = R[\sqrt{n}] the splitting may be done in a way which preserves the grading, since the element

i (∑_{j=1}^{n} e_{j}) e_{n+1}

commutes with all elements of C(V_{0}) and lies in C(V_{n+1})^{0}. The modules Δ^{+} and Δ^{−} remain irreducible RΓ_{n}-lattices by Schur [12], but each splits up again into two non-isomorphic summands Δ^{+} = (Δ^{+})^{0} ⊕(Δ^{+})^{1} and Δ^{−} = (Δ^{−})^{0} ⊕(Δ^{−})^{1} on restriction to RÃ_{n} ⊆ C(V_{0})^{0}.

5 Reduction of Δ modulo ℘

Recall that R = Z[ζ_{8}], ℘ = (ζ_{8} − 1) and R/℘ ≅ F_{2}. Recall also that when n is even, R̃ = R[\sqrt{n/2}], ˜℘ is a prime ideal lying above ℘ and R̃/˜℘ ≅ F_{2}. We denote by C̄(V) the F_{2}-algebra C(V) ⊗_{R} F_{2}, C̄(V_{0}) = C(V_{0}) ⊗_{R} F_{2}, Δ̄ = Δ ⊗_{R} F_{2}, etc. When n is even, we set Δ̄^{+} = Δ̃^{+} ⊗R̃ F_{2}, Δ̄^{−} = Δ̃^{−} ⊗R̃ F_{2}, while for n odd we set Δ̄^{+} = Δ^{+} ⊗_{R} F_{2} and Δ̄^{−} = Δ^{−} ⊗_{R} F_{2}. Thus Δ̄ is a module of dimension 2^{[(n+1)/2]} for C̄(V) while Δ̄^{+} and Δ̄^{−} are modules of dimension 2^{[(n−1)/2]} for C̄(V_{0}).

Note that the image of Γ_{n} in C̄(V_{0}) identifies +1 and −1, and is hence isomorphic to Σ_{n}. Thus if M is a C(V_{0})-lattice, M̄ may be regarded as an F_{2}Σ_{n}-module.

Now recall from James [4] that for each partition λ ⊢ n, we have a Specht module S^{λ} for F_{2}Σ_{n}, and that when λ is 2-regular, S^{λ} has a unique maximal submodule. The quotient is denoted D^{λ}, and as λ runs through the 2-regular partitions D^{λ} runs through the 2-modular irreducible representations of Σ_{n} once each. Of course, this is also true for p odd, which is irrelevant here.

Theorem 5.1 As F_{2}Σ_{n}-modules, we have

Δ̄^{+} ≅ Δ̄^{−} ≅ \{ \begin{matrix}D^{(n/2+1,n/2−1)} & if n is even \\ D^{((n+1)/2,(n−1)/2)} & if n is odd \end{matrix}

an irreducible module of dimension 2^{[(n−1)/2]}.

Before we prove this theorem, we need some lemmas.

Lemma 5.2 For n even, the Specht module S^{(n/2,n/2)} has D^{(n/2+1,n/2−1)} as its unique top composition factor.

Proof : The restriction of S^{(n/2,n/2)} to Σ_{n−1} is isomorphic to S^{(n/2,n/2−1)} by the branching rule (James [4, Section 9]). Moreover it follows from James [5, Theorem 5.2 and Corollary 5.7] that the restriction of D^{(n−m,m)} to Σ_{n−1} is D^{(n−m−1,m)} for m < n/2. Since all composition factors of S^{(n/2,n/2)} are of this form, the result follows by restriction to Σ_{n−1}. □

Comment : In fact, more is true. The modules S^{(n/2,n/2)} (n even) and S^{((n+1)/2,(n−1)/2)} (n odd) have composition series in which the composition factors, from the top, are

D^{(n/2+1,n/2−1)}, D^{(n/2+3,n/2−3)}, D^{(n/2+7,n/2−7)}, …
(n even)

D^{((n−1)/2+1,(n+1)/2−1)}, D^{((n−1)/2+3,(n+1)/2−3)}, D^{((n−1)/2+7,(n+1)/2−7)}, …

(n odd)

(the j-th composition factor is D^{([n/2]+2^{j}−1,[(n+1)/2]−2^{j}+1)}). We shall not need this fact.

Lemma 5.3 We have

dimF_{2} D^{([n/2]+1,[(n−1)/2])} = 2^{[(n−1)/2]}

Proof : This follows from [5, Theorem 5.2 and Corollary 5.7]. □

Now the elements u_{∅} + ζ_{8}u_{j} ϵ Δ, for different values of j with 1 ≤ j ≤ n/2, differ by elements of ℘Δ, and hence these elements all have the same image σ in Δ̄.

Let t be the tableau

t = \{ \begin{matrix}\begin{matrix}1 & 3 & ⋯ & n − 1 \\ 2 & 4 & ⋯ & n \end{matrix} & (n even) \\ \begin{matrix}1 & 3 & ⋯ & n − 2 & n \\ 2 & 4 & ⋯ & n − 1 \end{matrix} & (n odd). \end{matrix}

Let e_{t} be the corresponding polytabloid in S^{([(n+1)/2],[n/2])}.

Lemma 5.4 The assignment e_{t} ↦ σ extends to an F_{2}Σ_{n}-module homomorphism

ϕ : S^{([(n+1)/2],[n/2])} → Δ̄

Proof : By the comment at the bottom of James and Kerber [6, p. 302], we must check that the group algebra elements described on p. 301 (7.2.1) and the appropriate Garnir elements annihilate the image of e_{t}.

(a) To check that 1 + (2j − 1, 2j) annihilates σ, note that

u_{∅}.(\frac{1}{2} \sqrt{2}(e_{2j−1} − e_{2j})) = −ζ_{8}u_{j}

and

ζ_{8}u_{j}.(\frac{1}{2} \sqrt{2}(e_{2j−1} − e_{2j})) = u_{∅}

so that

\overline{(u_{∅} + ζ_{8}u_{j})(1 + \frac{1}{2} \sqrt{2}(e_{2j−1} − e_{2j}))} = 0.

(b) Garnir relations come in two kinds, corresponding to subsets {2j − 1, 2j, 2j + 1} and {2j − 2, 2j − 1, 2j}. Now the elements of C(V_{0}) corresponding to the Garnir elements (1 − (2j, 2j + 1) − (2j + 1, 2j − 1)) and (1 − (2j, 2j − 2) − (2j − 2, 2j − 1)) are

1 − \frac{1}{2} \sqrt{2}(e_{2j} − e_{2j+1}) − \frac{1}{2} \sqrt{2}(e_{2j+1} − e_{2j−1}) = 1 + \frac{1}{2} \sqrt{2}(e_{2j−1} − e_{2j})

and

1 − \frac{1}{2} \sqrt{2}(e_{2j} − e_{2j−2}) − \frac{1}{2} \sqrt{2}(e_{2j−2} − e_{2j−1}) = 1 + \frac{1}{2} \sqrt{2}(e_{2j−1} − e_{2j}),

and so in each case, we have already verified the relation in (a)! □

Lemma 5.5 The assignment e_{t} ↦ ū_{∅} extends to an F_{2}Σ_{n}-module homomorphism

ϕ′ : S^{([(n+1)/2],[n/2])} → Δ̄/Im(ϕ).

Proof : This is proved in the same way as the last lemma. Indeed, all the necessary checks have already been made there! □

Proof of Theorem 5.1 : By Lemma 5.2, the module S^{([(n+1)/2],[n/2])} has D^{([n/2]+1,[(n−1)/2])} as its unique top composition factor. Since each of ϕ and ϕ′ is non-zero, each of Im(ϕ) and Δ̄/Im(ϕ) contains a composition factor isomorphic to D^{([n/2]+1,[(n−1)/2])}. Since dimF_{2}(Δ̄) = 2^{[(n+1)/2]}, Lemma 5.3 forces these to be the only composition factors. The theorem now follows from the fact that Δ^{+} ⊕ Δ^{−} has finite index in Δ, so that Δ̄^{+} ⊕ Δ̄^{−} has the same composition factors as Δ̄. □

6 Restriction of spin modules to the alternating group

Our goal in this section is to prove the following theorem.

Theorem 6.1 Suppose n ≢ 2 (mod 4). After extension of scalars to F_{4} if necessary, the restriction to A_{n} of the F_{2}Σ_{n}-module Δ̄^{+}(≅ Δ̄^{−}) splits as a direct sum of two non-isomorphic irreducible modules. Extension to F_{4} is necessary if and only if n ≡ 3, 4 or 5 (mod 8).

The idea of the proof of the above theorem is to show that this already happens at the level of spin modules for orthogonal groups.

Consider the natural permutation module of dimension n for F_{2}Σ_{n}. For n odd, this decomposes as a direct sum of the “all ones vector” and the subspace of codimension one on which the sum of the coördinates is zero. For n even, this vector is contained in this subspace, and the quotient of one by the other has dimension n − 2. We shall call the remaining subquotient, of dimension n − 1 if n is odd, and n − 2 if n is even, the deleted natural permutation module. In fact it is the irreducible module D^{(n−1,1)}.

Lemma 6.2 The deleted natural permutation module for Σ_{n} supports (a) a symplectic form but no orthogonal form if n ≡ 2 (mod 4)

(b) an orthogonal form of “plus” type (i.e., of Arf invariant zero) if n ≡ 0 or ± 1 (mod 8)

(c) an orthogonal form of “minus” type (i.e., of Arf invariant one) if n ≡ 3, 4 or 5 (mod 8).

Thus we have inclusions

Σ_{4m+2} ⊆ Sp_{4m}(2)
Σ_{8m−1} ⊆ SO_{8m−2}^{+}(2) A_{8m−1} ⊆ A_{8m} ⊆ O_{8m−2}^{+}(2)
Σ_{8m+1} ⊆ SO_{8m}^{+}(2)  A_{am+1} ⊆ O_{8m}^{+}(2)
Σ_{8m+3} ⊆ Σ_{8m+4} ⊆ SO_{8m+2}^{−}(2) A_{8m+3} ⊆ A_{8m+4} ⊆ O_{8m+2}^{−}(2)
Σ_{8m+5} ⊆ SO_{8m+4}^{−}(2)  A_{8m+5} ⊆ O_{8m+4}^{−}(2).

Proof : This is easy to check by examining the natural orthogonal form on the n-dimensional permutation module. □

The group O_{2m}^{±}(2) above is defined to be the subgroup of SO_{2m}^{±}(2) = P SO_{2m}^{±}(2) of index two given by the vanishing of the Dickson invariant (i.e., those elements whose fixed space has even dimension).

The group O_{2m}^{+}(2) is the adjoint Chevalley group D_{m}(2), and O_{2m}^{−}(2) is the twisted adjoint Chevalley group ^{2}D_{m}(2). Note that O_{2m+1}(2) ≅ Sp_{2m}(2) since odd dimensional quadratic forms have a null vector in characteristic two.

From the representation theory of Chevalley groups, we can see that if we take the Clifford modules Δ̃^{+} and Δ̃^{−} for Pin_{n−1}(R̃), n even, or Δ^{+} ≅ Δ^{−} for Pin_{n−1}(R), n odd (see [1] for the Pin notation), then reduction mod two gives the following modules.

The Clifford modules Δ^{+} and Δ^{−} correspond to the forked vertices of the Dynkin diagram for O_{2m}^{±}(2) (which fuse in SO_{2m}^{±}(2))

[figure: …; ^{Δ̄^{+}}; ^{Δ̄^{−}}; O_{2m}^{+}(2); …; ^{Δ̄^{+}}; ^{Δ̄^{−}}; O_{2m}^{−}(2)]

and to the double end vertex of the Dynkin diagram for SO_{2m+1}(2) = O_{2m+1}(2) ≅ Sp_{2m}(2).

[figure: …; ^{Δ̄^{+} ≅ Δ̄^{−}}; Sp_{2m}(2)]

Note that in the case of O_{2m}^{−}(2), the modules Δ̄^{+} and Δ̄^{−} need to be written over F_{4}, so that Δ̄^{−} can be obtained by applying either the field automorphism or the graph automorphism to Δ̄^{+}. The module Δ̄^{+} ⊕ Δ̄^{−} is writeable as an irreducible but not absolutely irreducible module over F_{2}. This corresponds to the inclusion O_{2m}^{−}(2) ⊆ O_{2m}^{+}(4).

The restriction of the Clifford module for Sp_{2m}(2) to O_{2m}^{±}(2) is the sum of the two Clifford modules for the latter group in each case. Thus by Lemma 6.2, the restriction of Δ̄^{+} ≅ Δ̄^{−} from Σ_{4m} to A_{4m} splits as the sum of two (necessarily non-isomorphic irreducible by Frobenius reciprocity) constituents. The decomposition happens over F_{2} if m is even and F_{4} if m is odd. The fact that these summands are non-isomorphic means that they cannot be written over F_{2}, since they are related by the field automorphism of F_{4} by the above remarks.

To complete the proof of Theorem 6.1, it remains to examine the case where n is odd. But in this case the required decomposition already happens in characteristic zero, according to Section 4.

7 Supertensors and superinduction

We now describe in modern terms the construction by Schur [12] of the spin representations ⟨λ⟩, for λ a partition of n into distinct parts. The point of view we shall adopt is that the supersymmetric sign conventions give a convenient language for describing the tensor product and induction processes which come up here. This idea also appears in some form in the work of several other people, see for example Morris [10]. We first recall the definitions.

If A and B are two Z/2-graded vector spaces, then we Z/2-grade the tensor product via

(A ˆ⊗B)^{0} = (A^{0} ⊗ B^{0}) ⊕ (A^{1} ⊗ B^{1})
(A ˆ⊗B)^{1} = (A^{0} ⊗ B^{1}) ⊕ (A^{1} ⊗ B^{0})

If A and B are Z/2-graded algebras (or superalgebras) then A ˆ⊗B is a superalgebra via

(a ˆ⊗b)(a′ ˆ⊗b′) = (−1)^{deg b deg a′}aa′ ˆ⊗bb′.

Thus at the level of algebras it is essential to distinguish between ˆ⊗ and ⊗.

For example, (C^{0} ⊕ C^{1}) ˆ⊗(C^{0} ⊕ C^{1}) = Mât_{2}(C) is the superalgebra with (Mât_{2}(C))^{0} the leading diagonal matrices and (Mât_{2}(C))^{1} the trailing diagonal matrices. This construction should not be confused with Mat_{n} of a superalgebra, which is a superalgebra defined in the obvious way. A primitive idempotent decomposition of the identity in the above algebra is 1 = \frac{1}{2}((1, 0) ˆ⊗(1, 0) + (0, 1) ˆ⊗(0, i)) + \frac{1}{2}((1, 0) ˆ⊗(1, 0) − (0, 1) ˆ⊗(0, i)).

A representation M of a superalgebra A is a Z/2-graded vector space with A-action satisfying deg(ma) = deg(m) + deg(a) (mod 2). If M and N are representations of A and B then M ˆ⊗N is a representation of A ˆ⊗B via

(m ˆ⊗n)(a ˆ⊗b) = (−1)^{deg n deg a}ma ˆ⊗nb.

These conventions are explained and exploited in the study of Clifford algebras and modules in [1].

A super division ring is a superalgebra in which every non-zero element of degree zero and every non-zero element of degree one is invertible. For example, the super division rings over C are C^{0} and C^{0} ⊕ C^{1} (the identity element of C^{1} squares to 1 ϵ C^{0}). The super division rings over R are R^{0}, C^{0}, H^{0}, R^{0} ⊕ R^{1}, C^{0} ⊕ C^{1}, H^{0} ⊕ H^{1}, C (in which \sqrt{−1} is given degree 1), and H = C ˆ⊗C.

The super analogue of Schur’s lemma says that the endomorphisms of an irreducible representation of a superalgebra form a super division ring. If we are working over the complex numbers, there are two kinds of irreducible representations of a superalgebra, according as the endomorphisms are isomorphic to C or C^{0} ⊕ C^{1}. To copy Schur’s terminology, we call these kinds of representations two-sided and associated respectively. For example, in section 4 we saw that for n even, ΔC is a two-sided representation of C(V) and an associated representation of C(V_{0}). For later reference we record that if we are working over a splitting field K or over its ring of integers R (as long as it is a PID, which we may ensure by localising if necessary) then this dichotomy still holds.

A semisimple superalgebra is one with no graded nilpotent ideals. The analogue of the Wedderburn structure theorem says that every semisimple superalgebra is a direct sum of simple superalgebras, and that the simple superalgebras are of the form Mat_{n}(D) for D a super division ring.

It follows that if we form the tensor product of irreducible representations then the splitting of the tensor product as a representation of the tensor product of the superalgebras is entirely determined by the splitting of tensor products of the corresponding endomorphism rings. For example, in the complex case, the tensor product of two associated representations is a direct sum of two isomorphic two-sided representations. All irreducibles occur as factors in these tensor products.

We now define a supergroup (this is not a special case of the usual definition of a Lie supergroup) to consist of the following data :

(i) A group G

(ii) A distinguished subgroup G^{0} of index one or two

(iii) A distinguished central element of of order two, written as −1, lying in G^{0}.

We write G^{1} for G \ G^{0}.

If G and H are supergroups, their superproduct G ˆ×H as follows. As a set, G ˆ×H is the quotient of G × H by the equivalence relation (a, b) ∼ (−a, −b). Thus we can write −1 for (−1, 1) ∼ (1, −1). Multiplication is given by

(g_{1}, h_{1})(g_{2}, h_{2}) = (−1)^{deg h_{1} deg g_{2}}(g_{1}g_{2}, h_{1}h_{2}).

The example we are interested in is as follows. We regard Γ_{n} as a graded group in the obvious way. Then the preimage of Σ_{λ_{1}} × ⋯ × Σ_{λ_{s}} ⊆ Σ_{n} in Γ_{n} is isomorphic as a group to Γ_{λ_{1}} ˆ× ⋯ ˆ×Γ_{λ_{s}}.

A Z/2-graded representation, or super representation of a supergroup G is a Z/2-graded vector space on which G^{0} acts preserving the grading, G^{1} acts reversing the grading, and the distinguished central involution acts as multiplication by −1.

If M and N are super representations of G and H respectively, then M ˆ⊗N is a super representation of G ˆ×H via the rule

(m ˆ⊗n)(g, h) = (−1)^{deg n deg g}(mg ˆ⊗nh).

The supergroup algebra R̂G of a supergroup G over a commutative ring R is the superalgebra obtained by taking the subalgebra of the group algebra RG on which −1 acts as minus the identity, and giving it the obvious grading. The dimension of the supergroup algebra is exactly half the dimension of the group algebra. A super representation of a supergroup gives rise to a representation of the supergroup algebra in the obvious way. Thus by the above discussion the complex super representations of G ˆ×H may be obtained by tensoring those of G and H and splitting those for which both were associated, using the idempotent written out above. Of course, if we wish to avoid dividing by two, as we often do, then we simply use twice the above idempotent.

Using the above constructions, it is now clear how to define the basic spin superlattices for the groups Γ_{λ_{1}} ×⋯×Γ_{λ_{s}} (over the ring R̃ obtained by adjoining to R as many square roots as seems necessary). The basic spin superlattice for Γ_{n} is taken to be Δ if n is even and the graded version of Δ^{+} (≅ Δ^{−}) discussed at the end of Section 4 if n is odd. Now for a general partition λ = (λ_{1}, …, λ_{s}) ⊢ n we take the graded tensors of these, and go down to a sublattice of half the rank wherever necessary using twice the above idempotent.

This way, we obtain a superlattice Δ_{λ} of rank 2^{[(n−s+1)/2]} for Γ_{λ_{1}} ×⋯×Γ_{λ_{s}}. It is two-sided if (n − s) is even and associated otherwise. The Hermitian form ⟨u_{I}, u_{J}⟩ = δ_{IJ} on Δ gives rise to a Hermitian form on Δ^{+} by restriction, and hence a Hermitian form on Δ_{λ}, which we shall also denote by ⟨ , ⟩.

If G is a supergroup and H is a subgroup of G with −1 ϵ H then H inherits a natural structure of supergroup with H^{0} = G^{0} ∩ H. If M is a super representation of H then M ˆ⊗_{RH} R̂G is the super representation of G obtained by

superinducing M up to G. It is written Mˆ↑^{G}.

It follows from Schur [12] that (Δ_{λ}ˆ↑Γ_{n}) ⊗R̃ K̃ consists of the irreducible spin representation ⟨λ⟩ (both of them if (n − s) is odd) with multiplicity one, together with possibly some ⟨μ⟩ with λ ⊳ μ.

It would be desirable at this stage to produce a sublattice of Δ_{λ}ˆ↑Γ_{n} corresponding to ⟨λ⟩, as pointed out in the Introduction. Instead, we offer proofs for Theorems 1.1 and 1.2, and hope that the reader is encouraged to investigate the above problem. In this context, the ideas of Morris and Yaseen [11] are probably relevant.

Proof of Theorem 1.2 : It is clear from the supertensor formulation that the reduction Δ̄_{λ} of Δ_{λ} modulo a prime ideal ˜℘ lying above (2) consists (at the level of composition factors) of 2^{[t/2]} copies of the tensor product

Δ̄_{λ_{1}} ⊗ ⋯ ⊗ Δ̄_{λ_{s}}

where t of the λ_{i} are even. By Theorem 5.1, this tensor product is isomorphic to

D^{dbl⟨λ_{1}⟩} ⊗ ⋯ ⊗ D^{dbl⟨λ_{s}⟩}

as a module for Σ_{λ_{1}} × ⋯ × Σ_{λ_{s}}. It follows from the Littlewood-Richardson rule [4, section 16] that

(D^{dbl⟨λ_{1}⟩} ⊗ ⋯ ⊗ D^{dbl⟨λ_{s}⟩}) ↑Σ^{n}

has as its composition factors S^{λ} with multiplicity 1, and possibly some S^{μ} with λ ⊳ μ. The theorem follows from this together with the above theorem of Schur, and the fact that the reduction modulo ˜℘ of a superinduced representation is the corresponding induced representation.

8 A partition identity of Schur

In this section we prove Theorem 1.1. The proof depends on a special case of a little known partition identity of Schur, and I am very grateful to G. E. Andrews for informing me of the existence of this identity. I am impressed by his knowledge of the literature!

The identity referred to above is the case k = 4, α = 1, β = 3 of Theorem V on the last page of [13], which we repeat here for the convenience of the reader.

Theorem 8.1 (Schur, my translation) Suppose k ≥ 3 is a positive integer, α a positive integer less than k/2, and denote by β the difference k − α. Then the number of partitions of an arbitrary positive integer n into distinct parts of the form kν + α or kν + β is equal to the number of partitions

n = b_{1} + b_{2} + ⋯ + b_{r} (r = 1, 2, …)

into parts of the form kν + α, kν + β or kν, where it is further stipulated that for r > 1 each difference b_{ρ} − b_{ρ+1} is at least equal to k, and if b_{ρ} and b_{ρ+1} are divisible by k then at least equal to 2k.

In the case k = 4, α = 1 and β = 3 it is not hard to check that this theorem states that the number of partitions of n into distinct odd parts is equal to the number of spin regular partitions (see the introduction for the definition) of n into parts not congruent to two modulo four. Taking doubles of these partitions gives us the following proposition.

Proposition 8.2 The number of partitions of n into distinct odd parts is equal to the number of partitions of n into distinct parts satisfying conditions (a) and (b) of Theorem 1.1.

Now by counting irreducible characters, the number of D^{λ} which split on restriction to A_{n} has to equal to the number of 2-regular conjugacy classes in Σ_{n} which split into two classes in A_{n}. These classes exactly correspond to the elements whose cycle types consist of distinct cycles of odd length. Thus the above proposition says that the number of D^{λ} which split is equal to the number of D^{λ} for which λ satisfies conditions (a) and (b) of Theorem 1.1. So to prove Theorem 1.1, it suffices to show that these D^{λ} do split.

First we treat the case where λ has just two parts, so that automatically λ = dbl⟨n⟩. In this case, by Theorem 5.1, D^{λ} ≅ Δ̄^{+} ≅ Δ̄^{−}, and so Theorem 6.1 completes the proof in this case.

Now we come to the general case. Just as in the proof of Theorem 1.2, we use the Littlewood-Richardson rule. If λ satisfies conditions (a) and (b) of Theorem 1.1, then the module

D^{dbl⟨λ_{1}⟩} ⊗ ⋯ ⊗ D^{dbl⟨λ_{s}⟩}

for Σ_{λ_{1}} × ⋯ × Σ_{λ_{s}} splits on restriction to the intersection with the alternating group, since each of the factors splits. On inducing up to Σ_{n}, we obtain just one copy of D^{λ}, together with possibly some D^{μ} with λ ⊳ μ. The result now follows from the Mackey formula

D^{dbl⟨λ_{1}⟩} ⊗ ⋯ ⊗ D^{dbl⟨λ_{s}⟩} ↑Σ^{n}↓A_{n} ≅
D^{dbl⟨λ_{1}⟩} ⊗ ⋯ ⊗ D^{dbl⟨λ_{s}⟩} ↓_{Σ_{λ_{1}} ×···×Σ_{λ_{s}} ∩A_{n}}↑A^{n} .

This completes the proof of Theorem 1.1.

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