Projective representations of symmetric groups via Sergeev duality

Jonathan Brundan and Alexander Kleshchev ^{∗}

1 Introduction

In this article, we determine the irreducible projective representations of the symmetric group S_{d} and the alternating group A_{d} over an algebraically closed field of characteristic p ≠ 2. These matters are well understood in the case p = 0, thanks to the fundamental work of Schur [24] in 1911, as well as the much more recent work of Nazarov [19, 20], Sergeev [25, 26] and others. So the focus here is primarily on the case of positive characteristic, where surprisingly little is known as yet. In particular, we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set ℛ𝒫 _{p}(d) of restricted p-strict partitions of d. Such partitions arose recently in work of Kashiwara, Miwa, Peterson and Yung [11] and Leclerc and Thibon [14] on the q-deformed Fock space of the affine Kac-Moody algebra of type A_{p−1}^{(2)}. Leclerc and Thibon proposed that ℛ𝒫 _{p}(d) should label the irreducible projective representations in some natural way, and we show here how this can be done. Note that for p = 3, 5, the labelling problem was solved in [1, 3], while if p = 2 all projective representations of S_{d} and A_{d} are linear so do not need to be considered further here.

To be more precise, recall that λ is a partition of d if λ = (λ_{1}, λ_{2}, …) is a non-increasing sequence of non-negative integers summing to d. Call λ p-strict if in addition

λ_{i} = λ_{i+1} ⇒ p|λ_{i} for each i = 1, 2, . . ..

Let 𝒫_{p}(d) denote the set of all p-strict partitions of d. Thus, the 0-strict partitions are just the partitions with no repeated non-zero parts, while a p-strict partition for p > 0 can only have repeated parts if they are divisible by p. Call λ ∈ 𝒫_{p}(d) a restricted p-strict partition if either p = 0, or p > 0 and

\{ \begin{matrix}λ_{i} − λ_{i+1} ≤ p & if p ∤ λ_{i}, \\ λ_{i} − λ_{i+1} < p & if p | λ_{i} \end{matrix}

for each i = 1, 2, . . .. Let ℛ𝒫 _{p}(d) ⊆ 𝒫_{p}(d) denote the restricted p-strict partitions of d. Also, define h_{p′}(λ) to be the number of parts of λ not divisible by p. Then, our construction leads to a labelling of the irreducible projective representations of S_{d} over an algebraically closed field of characteristic p ≠ 2 by pairs (λ, ε) where λ ∈ ℛ𝒫 _{p}(d) and ε = 0 if d − h_{p′}(λ) is even or ±1 if d − h_{p′}(λ) is odd. For A_{d}, the labelling is by pairs (λ, ε) where λ ∈ ℛ𝒫 _{p}(d) and ε = ±1 if d − h_{p′}(λ) even or 0 if d − h_{p′}(λ) is odd.

∗Authors partially supported by the NSF (grant nos DMS-9801442 and DMS-9900134).

The construction is based closely on the ideas of Sergeev and Nazarov in the characteristic 0 theory. In particular, the key step is to determine the irreducible “polynomial” representations of the supergroup Q(n) in characteristic p. These turn out to be labelled naturally according to highest weight theory by all p-strict partitions with at most n nonzero parts. ¿From this, we use Sergeev’s superalgebra analogue [25] of Schur-Weyl duality to determine the irreducible representations of a certain twisted version of the group algebra of the hyperoctahedral group. Finally, we pass from there to the symmetric group using methods of Nazarov [20] and Sergeev [26].

Acknowledgements. We would like to thank S. Ariki for bringing the work of Sergeev [25] to our attention, and S. Donkin for making [6, 18] available to us.

2 Preliminaries on superalgebras

In this section, we record a number of standard results about the representation theory of finite dimensional (associative) superalgebras. As useful general references, but sometimes with different conventions to us, we cite [17, ch.3], [15] and [10].

We will always work relative to a fixed algebraically closed field 𝕜 of characteristic p ≠ 2. By a vector superspace we mean a ℤ_{2}-graded 𝕜-vector space V = V¯0 ⊕ V¯1. Given a homogeneous vector 0 ≠ v ∈ V, we denote its degree by ∂(v) ∈ ℤ_{2}. A subsuperspace U of V means a subspace U of V such that U = (U ∩ V¯0) ⊕ (U ∩ V¯1). Define the linear map δ_{V} : V → V on homogeneous vectors by δ_{V} (v) = (−1)^{∂(v)}v. Then obviously, a subspace U ⊂ V is a subsuperspace if and only if U is stable under δ_{V}.

Given vector superspaces V and W, we view the direct sum V ⊕W and the tensor product V ⊗W as a vector superspaces with (V ⊕W)_{i} = V_{i} ⊕W_{i}, and (V ⊗W)¯0 = V¯0 ⊗W¯0 ⊕V¯1 ⊗W¯1, (V ⊗W)¯1 = V¯0⊗W¯1⊕V¯1⊗W¯0. Also, we make the vector space Hom_{𝕜}(V, W) of all linear maps from V to W into a superspace by declaring that Hom_{𝕜}(V, W)_{i} consists of the homogeneous maps of degree i for each i ∈ ℤ_{2}, that is, the maps θ : V → W with θ(V_{j}) ⊆ W_{i+j} for j ∈ ℤ_{2}. Elements of Hom_{𝕜}(V, W)¯0 will be referred to as even linear maps. The dual superspace V^{∗} is Hom_{𝕜}(V, 𝕜), where we view 𝕜 as a vector superspace concentrated in degree ¯0.

A superalgebra is a vector superspace A with the additional structure of an associative, unital 𝕜-algebra such that A_{i}A_{j} ⊆ A_{i+j} for i, j ∈ ℤ_{2}. A superalgebra homomorphism θ : A → B is an even linear map that is an algebra homomorphism in the usual sense; its kernel is a superideal, that is, an ordinary two-sided ideal that is also a subsuperspace. Most importantly, given two superalgebras A and B, we view the tensor product A ⊗ B as a superalgebra with the induced grading and multiplication defined by (a ⊗ b)(a′ ⊗ b′) = (−1)^{∂(b)∂(a′)}(aa′) ⊗ (bb′) for homogeneous elements a, a′ ∈ A, b, b′ ∈ B. We note that A ⊗ B ≅ B ⊗ A, an isomorphism being given by the supertwist map T_{A,B} : A ⊗ B → B ⊗ A, a ⊗ b ↦ (−1)^{∂(a)∂(b)}b ⊗ a for homogeneous a ∈ A, b ∈ B.

2.1. Example. Let V be a vector superspace with dim V¯0 = m, dim V¯1 = n. The tensor superalgebra is the tensor algebra T (V) regarded as a superalgebra with the induced grading.

As a quotient of T (V), we have the symmetric superalgebra, namely,

S(V) = T (V)/⟨v ⊗ w − (−1)^{∂(v)∂(w)}w ⊗ v | for all homogeneous vectors v, w ∈ V ⟩.

If we have in mind fixed bases v_{1}, …, v_{m} for V¯0 and v̄_{1}, …, v̄_{n} for V¯1, we denote the superalgebras T (V) and S(V) instead by T (m|n) and S(m|n). These are the free superalgebra and the free commutative superalgebra on m|n generators, respectively. Set S(m) := S(m|0), just the usual polynomial algebra on m generators concentrated in degree ¯0, and ⋀(n) := S(0|n), just the usual exterior algebra but with generators assigned the degree ¯1. The superalgebra ⋀(n) is called the Grassmann superalgebra. We have that

S(m) ≅ S(1) ⊗ ⋯ ⊗ S(1) (m times),
⋀(n) ≅ ⋀(1) ⊗ ⋯ ⊗ ⋀(1) (n times),
S(m|n) ≅ S(m) ⊗ ⋀(n).

2.2. Example. Another basic example that we will meet is the Clifford superalgebra, namely, the superalgebra C(n) on generators c_{1}, …, c_{n} all of degree ¯1, subject to the relations c_{i}^{2} = 1 for i = 1, …, n and c_{i}c_{j} = −c_{j}c_{i} for all i ≠ j. If, slightly more generally, one has in mind non-zero scalars λ_{1}, …, λ_{n} ∈ 𝕜^{×}, the superalgebra with generators b_{1}, …, b_{n} subject to the relations b_{i}^{2} = λ_{i}, b_{i}b_{j} = −b_{j}b_{i} is isomorphic to C(n), an obvious isomorphism sending b_{i} ↦ \sqrt{λ_{i}}c_{i}. The crucial point is that C(n_{1} + n_{2}) ≅ C(n_{1}) ⊗ C(n_{2}). Indeed, the generators c_{1} ⊗ 1, …, c_{n_{1}} ⊗ 1, 1 ⊗ c_{1}, …, 1 ⊗ c_{n_{2}} of C(n_{1}) ⊗ C(n_{2}) satisfy the same relations as the generators c_{1}, …, c_{n_{1}}, c_{n_{1}+1}, …, c_{n_{1}+n_{2}} of C(n_{1} + n_{2}). It follows at once that

C(n) ≅ C(1) ⊗ ⋯ ⊗ C(1) (n times).

Let A be a superalgebra. A left A-supermodule is a vector superspace M which is a left A-module in the usual sense, such that A_{i}M_{j} ⊆ M_{i+j} for i, j ∈ ℤ_{2}. There is of course an analogous notion of right supermodule, which we omit. A homomorphism f : M → N between two left A-supermodules means a (not necessarily homogeneous) linear map such that a(mf) = (am)f for all a ∈ A and m ∈ M. Observe we write homomorphisms between left A-supermodules on the right (and vice versa). We have now defined the category mod(A) of all left A-supermodules. It is a superadditive category in the sense of [17, ch.3,§2.7], i.e. an additive category such that each Hom_{A}(M, N) is ℤ_{2}-graded in a way that is compatible with addition and composition of morphisms. We also have the (left) parity change functor

Π : mod(A) → mod(A)

(see [17, ch.3,§1.5]). This is defined on an object M so that ΠM is the same underlying vector space but with the opposite grading, and the new left A-action is defined by a · m = (−1)^{∂(a)}am for homogeneous a ∈ A, m ∈ M. On a morphism f, Πf is the same underlying linear map as f.

A subsupermodule of an A-supermodule means an A-submodule in the usual sense that is a subsuperspace. An A-supermodule M is irreducible if it is non-zero and has no non-zero proper subsupermodules. Then M is either irreducible when viewed just as an ordinary A-module, in which case we say that M is absolutely irreducible, or else M is reducible as an A-module, in which case we call M self-associate.

2.3. Lemma. If M is a finite dimensional self-associate irreducible A-supermodule, then there exist bases v_{1}, …, v_{n} for M¯0 and v̄_{1}, …, v̄_{n} for M¯1 such that

M = span{v_{1} + v̄_{1}, …, v_{n} + v̄_{n}} ⊕ span{v_{1} − v̄_{1}, …, v_{n} − v̄_{n}}

as a direct sum of two non-isomorphic irreducible A-submodules. Moreover, the linear map J_{M} : M → M defined by v_{i} ↦ v̄_{i}, v̄_{i} ↦ v_{i} is an A-homomorphism.

Proof. We can find an irreducible A-submodule N of M that is not a subsupermodule, i.e. is not δ_{M}-stable. It is elementary to check that δ_{M}(N) is also an irreducible A-submodule of M. Hence, N ⊕ δ_{M}(N) is an A-submodule of M, even a subsupermodule since it is now δ_{M}- stable. Let u_{1}, …, u_{n} be a basis for N. Then, δ_{M}(u_{1}), …, δ_{M}(u_{n}) is a basis for δ_{M}(N), so u_{1} + δ_{M}(u_{1}), …, u_{n} + δ_{M}(u_{n}) is the required basis for M¯0 and u_{1} − δ_{M}(u_{1}), …, u_{n} − δ_{M}(u_{n}) is the required basis for M¯1.

If M is an A-supermodule, End_{A}(M) denotes the superalgebra of all A-supermodule endomorphisms of M. We stress again that we write the action of elements of End_{A}(M) on M on the opposite side to the action of A. We have the following analogue of Schur’s lemma, which is easily proved given Lemma 2.3:

2.4. Lemma (Schur’s lemma). Let M be a finite dimensional irreducible A-supermodule. Then,

End_{A}(M) = \{ \begin{matrix}span\{id_{M}\} & if M is absolutely irreducible, \\ span\{id_{M}, J_{M}\} & if M is self-associate, \end{matrix}

where J_{M} is as in Lemma 2.3.

We say that an A-supermodule M is completely reducible if it can be decomposed as a direct sum of irreducible subsupermodules. Call A a simple superalgebra if A has no nontrivial superideals, and a semisimple superalgebra if A is completely reducible viewed as a left A-supermodule. Equivalently, A is semisimple if every left A-supermodule is completely reducible. We have:

2.5. Lemma (Wedderburn’s theorem). Let A be a finite dimensional superalgebra. The following are equivalent:

(i) A is simple;

(ii) A is semisimple with only one irreducible supermodule up to isomorphism; (iii) there is a finite dimensional vector superspace V such that either A ≅ End_{𝕜}(V) or

A ≅ {θ ∈ End_{𝕜}(V) | θ ∘ J = J ∘ θ} for some involution J ∈ End_{𝕜}(V)¯1. Moreover, if A is semisimple then it is isomorphic to a direct product of simple superalgebras.

Notice in view of Lemma 2.3 that if A is semisimple as a superalgebra, then it is semisimple as an algebra. The converse is also true, and is proved e.g. in [18, (1.4c)]; it can also be deduced directly by considering the effect of the map δ_{A} on the irreducible submodules of A viewed as a left A-module. Somewhat more generally, we have:

2.6. Lemma. Let A be a finite dimensional superalgebra. Then, the Jacobson radical of A (viewed just as an ordinary algebra) can be characterized as the unique smallest superideal K of A such that A/K is a semisimple superalgebra.

Proof. Let J be the Jacobson radical of A viewed as an ordinary algebra, and let K be any superideal of A that is minimal with respect to the property that A/K is a semisimple superalgebra. We know that A/K is semisimple as an ordinary algebra by Lemma 2.3, so J ⊆ K. Conversely, we observe that J is a superideal since J is invariant under the algebra automorphism δ_{A} of A. So, A/J is a superalgebra that is semisimple as an algebra. Hence, by [18, (1.4c)], it is a semisimple superalgebra, so J = K by minimality of K.

2.7. Example. The Jacobson radical of the Grassmann superalgebra ⋀(n) coincides with the superideal generated by all degree ¯1 elements. The quotient superalgebra is isomorphic to 𝕜. It follows that ⋀(n) has a unique irreducible supermodule up to isomorphism, namely, 𝕜 itself, with all elements of ⋀(n)¯1 acting as zero.

We point out another immediate consequence of Wedderburn’s theorem and Lemma 2.6:

2.8. Corollary. Let A be a finite dimensional superalgebra, and {V_{1}, …, V_{n}} be a complete set of pairwise non-isomorphic irreducible A-supermodules such that V_{1}, …, V_{m} are absolutely irreducible and V_{m+1}, …, V_{n} are self-associate. For i = m + 1, …, n, write V_{i} = V_{i}^{+} ⊕ V_{i}^{−} as a direct sum of irreducible A-modules. Then,

{V_{1}, …, V_{m}, V_{m+1}^{±}, …, V_{n}^{±}}

is a complete set of pairwise non-isomorphic irreducible A-modules.

Given left supermodules M and N over arbitrary superalgebras A and B respectively, the (outer) tensor product M⊗N is an A⊗B-supermodule with action defined by (a⊗b)(m⊗n) = (−1)^{∂(b)∂(m)}am ⊗ bn for all homogeneous a ∈ A, b ∈ B, m ∈ M, n ∈ N. (Analogously, if M and N are right supermodules, the action of A ⊗ B on M ⊗ N is defined instead by (m ⊗ n)(a ⊗ b) = (−1)^{∂(a)∂(n)}ma ⊗ nb for all homogeneous a ∈ A, b ∈ B, m ∈ M, n ∈ N.) If f : M → M′ (resp. g : N → N′) is a homogeneous homomorphism of left A- (resp. B-) supermodules, then f ⊗ g : M ⊗ N → M′ ⊗ N′ is the A ⊗ B-supermodule homomorphism defined by (m ⊗ n)(f ⊗ g) = (−1)^{∂(n)∂(f)}mf ⊗ ng. The following lemma gives the other basic facts about outer tensor products that we need (cf. [10, (2.10)]):

2.9. Lemma. Suppose that A and B are finite dimensional superalgebras, and that M, N are irreducible supermodules over A, B respectively.

(i) If both M and N are absolutely irreducible, then M ⊗ N is an absolutely irreducible A ⊗ B-supermodule.

(ii) If one of M or N is absolutely irreducible and the other is self-associate, then M ⊗N is a self-associate irreducible A⊗B-supermodule.

(iii) If both M and N are self-associate, then M ⊗ N decomposes as a direct sum of two isomorphic, absolutely irreducible A ⊗ B-supermodules. Moreover, all irreducible A⊗B-supermodules arise as constituents of M ⊗N for some choice of M, N.

Combining Lemma 2.9 with Wedderburn’s theorem, it follows in particular that if A and B are finite dimensional semisimple superalgebras then A ⊗ B is too.

2.10. Example. Consider the Clifford superalgebra C(n) again. First, observe that C(1)

is just the simple superalgebra of 2 × 2 matrices of the form \{( \begin{matrix}a & b \\ b & a \end{matrix} ) | a, b ∈ 𝕜\}, the

generator c_{1} of C(1) corresponding to the matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). So C(1) has precisely one

irreducible supermodule U(1) which is self-associate of dimension 2, as in the second case of Lemma 2.5(iii). Hence, applying Lemma 2.9, C(2) = C(1) ⊗ C(1) has one irreducible supermodule U(2), namely the unique irreducible appearing with multiplicity two in the C(2)-supermodule U(1)⊗U(1), and U(2) is absolutely irreducible of dimension 2. Explicitly, U(2) can be described as the supermodule on basis u, ū with action defined by c_{1}u = ū, c_{1}ū = u, c_{2}u = \sqrt{−1}ū, c_{2}ū = −\sqrt{−1}u. Finally, for n > 2, C(n) = C(n − 2) ⊗ C(2), so by Lemma 2.5(i) and (ii), it has just one irreducible supermodule U(n), defined inductively by U(n) = U(n − 2) ⊗ U(2). This is absolutely irreducible if and only if U(n − 2) is absolutely irreducible, which is if and only if n is even. Observe that we have just shown that C(n) is a semisimple superalgebra with a unique irreducible supermodule. So by Lemma 2.5, C(n) is in fact a simple superalgebra, indeed, up to isomorphism, it must be the unique simple superalgebra of dimension 2^{n}. Its unique irreducible supermodule U(n) has dimension 2^{⌊(n+1)/2⌋}.

Following [25, §1.4], a ℤ_{2}-graded group is a pair (G, ∂) where G is a finite group and ∂ : G → ℤ_{2} is a group homomorphism. If (G, ∂) is a ℤ_{2}-graded group, we can regard the group algebra 𝕜G as a superalgebra, the degree of g ∈ G being ∂(g). We are interested next in counting the number of irreducible 𝕜G-supermodules in terms of conjugacy classes. Define n_{p′}(G, ¯0) to be the number of G-conjugacy classes of p′-elements (= elements of order coprime to p) of degree ¯0 and n_{p′}(G, ¯1) to be the number of G-conjugacy classes of p′-elements of degree ¯1.

2.11. Lemma. Let (G, ∂) be a ℤ_{2}-graded group. Then, there are n_{p′}(G, ¯0) pairwise nonisomorphic irreducible 𝕜G-supermodules. Of these, n_{p′}(G, ¯0) − n_{p′}(G, ¯1) are absolutely irreducible, and the remaining n_{p′}(G, ¯1) are self-associate.

Proof. We follow the proof of the analogous classical result for ordinary group algebras, see [12, §13]. For an arbitrary superalgebra A, write Z(A) = {a ∈ A | ab = ba for all b ∈ A} for its centre and S(A) = span{ab − ba | a, b ∈ A}. These are both subsuperspaces of A. Let J denote the Jacobson radical of the group algebra 𝕜G. By Lemma 2.6, J is a superideal and A := 𝕜G/J is the largest semisimple superalgebra quotient of 𝕜G. So 𝕜G and A have the same number of irreducible supermodules. Combining Lemma 2.4 and Lemma 2.5, we deduce that the number of irreducible 𝕜G-supermodules is equal to dim Z(A)¯0 and the number of self-associate irreducible 𝕜G-supermodules is equal to dim Z(A)¯1. By [12, 13.3], A = Z(A) ⊕ S(A), so dim[Z(A)]_{i} = dim[A/S(A)]_{i} for i = ¯0, ¯1. Finally, to count this dimension in either case, use formula (14) in the proof of [12, 13.8]; this tells us at once that dim[A/S(A)]_{i} = n_{p′}(G, i).

To conclude this preliminary section, we give a brief review of “Schur functors” arising from idempotents in this setting. Suppose that A is an arbitrary finite dimensional superalgebra and that e ∈ A is a homogeneous idempotent, necessarily of degree ¯0. Then, the ring eAe is a superalgebra in its own right, its identity element being the idempotent e. We have the (exact) Schur functor

R_{e} : mod(A) → mod(eAe)

given on objects by left multiplication by the idempotent e and by restriction on morphisms. Given an A-supermodule M, let O_{e}(M) (resp. O^{e}(M)) denote the largest (resp. smallest) subsupermodule N of M such that N (resp. M/N) is annihilated by e. Finally, let mod_{e}(A) denote the full subcategory of mod(A) consisting of all A-supermodules M with O_{e}(M) = 0 and O^{e}(M) = M. The following basic result is proved as in the classical case, see [9, §2]:

2.12. Lemma. The restriction of the functor R_{e} to mod_{e}(A) is an equivalence of categories between mod_{e}(A) and mod(eAe).

Suppose that {L(λ) | λ ∈ Λ} be a complete set of pairwise non-isomorphic irreducible A-supermodules, and set Λ_{e} = {λ ∈ Λ | R_{e}L(λ) ≠ 0}. Then, as an immediate consequence of Lemma 2.12, we have:

2.13. Corollary. The eAe-supermodules {R_{e}L(λ) | λ ∈ Λ_{e}} give a complete set of pairwise non-isomorphic irreducible eAe-supermodules. Moreover, for λ ∈ Λ_{e}, R_{e}L(λ) is absolutely irreducible if and only if L(λ) is absolutely irreducible.

3 The Sergeev superalgebra

Let S_{d} denote the symmetric group, acting naturally on the left on the set {1, …, d}. Denoting the basic transposition (i i+1) by s_{i}, we recall that S_{d} is generated by s_{1}, …, s_{d−1} subject to the well-known Coxeter relations.

Now let α : S_{d} × S_{d} → 𝕜^{×} be a 2-cocycle, where 𝕜 is a fixed algebraically closed field. Then, there is a corresponding twisted group algebra, namely, the 𝕜-algebra on basis {[w] | w ∈ S_{d}} with multiplication satisfying [x][y] = α(x, y)[xy] for all x, y ∈ S_{d}. Studying the projective representations of S_{d} over 𝕜 is equivalent to studying the representation theory of the twisted group algebras arising in this way, as α runs over representatives of all such 2-cocycles. The following lemma is quite standard, cf. [4] or [8, Kapitel 5, §25, Satz 12]:

3.1. Lemma. The Schur multiplier H^{2}(S_{d}, 𝕜^{×}) has exactly two elements if char 𝕜 ≠ 2 and d ≥ 4, and is trivial otherwise.

This explains in particular why all projective representations of S_{d} in characteristic 2 are linear, as remarked in the introduction. So now suppose for the remainder of the article that char 𝕜 ≠ 2. Then, Lemma 3.1 implies that S_{d} has two twisted group algebras over 𝕜 up to isomorphism (providing d ≥ 4). Of course, one of these is just the group algebra 𝕜S_{d} itself, and will not be considered further here. For the other, we may take the 𝕜-algebra S(d) on generators t_{1}, …, t_{d−1} subject to the relations

t_{i}^{2} = 1,  t_{i}t_{i+1}t_{i} = t_{i+1}t_{i}t_{i+1},  t_{i}t_{j} = −t_{j}t_{i}

for all 1 ≤ i ≤ d − 1 and all 1 ≤ j ≤ d − 1 with |i − j| > 1. In what follows, we will always view S(d) as a superalgebra, defining the grading by declaring the generators t_{1}, …, t_{d−1} to be of degree ¯1. We are interested in determining the irreducible S(d)-supermodules. Recall the definition of the set ℛ𝒫 _{p}(d) of restricted p-strict partitions of d from the introduction.

3.2. Lemma. The number of isomorphism classes of irreducible S(d)-supermodules is equal to |ℛ𝒫 _{p}(d)|.

Proof. Define Ŝ_{d} to be the double cover of S_{d} with generators ζ, ŝ_{1}, …, ŝ_{d−1} subject to the relations

ζ^{2} = ŝ_{i}^{2} = 1,  ζŝ_{i} = ŝ_{i}ζ,
ŝ_{i}ŝ_{i+1}ŝ_{i} = ŝ_{i+1}ŝ_{i}ŝ_{i+1},  ŝ_{i}ŝ_{j} = ζŝ_{j}ŝ_{i}

for all 1 ≤ i ≤ d − 1 and all 1 ≤ j ≤ d − 1 with |i − j| > 1 (see e.g. [27, p.100]). The map sending ζ ↦ 1, ŝ_{i} ↦ s_{i} determines a surjective homomorphism 𝕜Ŝ_{d} → 𝕜S_{d}, while the map defined by ζ ↦ −1, ŝ_{i} ↦ t_{i} is a surjective homomorphism 𝕜Ŝ_{d} → S(d).

Now, the elements ζ_{+} = (1−ζ)/2 and ζ_{−} = (1+ζ)/2 are orthogonal central idempotents of 𝕜Ŝ_{d} summing to the identity, so

𝕜Ŝ_{d} = ζ_{+}(𝕜Ŝ_{d}) ⊕ ζ_{−}(𝕜Ŝ_{d})

as a direct sum of two-sided ideals. Obviously, ζ_{+}(𝕜Ŝ_{d}) ≅ (𝕜Ŝ_{d})/⟨ζ−1⟩ ≅ 𝕜S_{d} and ζ_{−}(𝕜Ŝ_{d}) ≅ S(d). Making S_{d} and Ŝ_{d} into ℤ_{2}-graded groups with degree function ∂ satisfying ∂(ζ) = ¯0 and ∂(ŝ_{i}) = ∂(s_{i}) = ¯1, we deduce at once that the number of irreducible 𝕜Ŝ_{d}-supermodules is equal to the number of irreducible 𝕜S_{d}-supermodules plus the number of irreducible S(d)-supermodules. Hence, using Lemma 2.11, we deduce that the number of irreducible S(d)-supermodules is n_{p′}(Ŝ_{d}, ¯0) − n_{p′}(S_{d}, ¯0).

Finally, n_{p′}(Ŝ_{d}, ¯0) − n_{p′}(S_{d}, ¯0) can be calculated using the known labelling of the conjugacy classes of S_{d} and Ŝ_{d}, see e.g. [27, Theorem 2.1] or [24, p.172]. One deduces easily that the number of irreducible S(d)-supermodules is equal to the number of partitions λ of d with all non-zero parts of λ being odd and not divisible by p. In turn, to see that this number equals |ℛ𝒫 _{p}(d)|, we appeal to the partition identity

∑_{d≥0} |ℛ𝒫 _{p}(d)|t^{d} = ∏_{i odd, p∤i} \frac{1}{1 − t^{i}} ,

from [14, (40)], which is a special case of [2, Theorem 2].

Next, let C(d) be the Clifford superalgebra on odd generators c_{1}, …, c_{d} as in Example 2.2, so c_{i}^{2} = 1 for each i. There is a unique right action of S_{d} on C(d) by superalgebra automorphisms so that c_{i} · w = c_{w^{−1}i} for all i = 1, …, d and w ∈ S_{d}. The Sergeev superalgebra is the vector superspace

W (d) = 𝕜S_{d} ⊗ C(d)

(here, 𝕜S_{d} is viewed as a superspace concentrated in degree ¯0) with multiplication defined on generators by the rule

(x ⊗ c)(y ⊗ d) = xy ⊗ (c · y)d

for x, y ∈ S_{d}, c, d ∈ C(d). As observed by Sergeev in [26], a check of relations shows:

3.3. Lemma. There is an injective superalgebra homomorphism ω : S(d) → W (d) defined on generators by

ω(t_{i}) = \frac{1}{\sqrt{−2}}s_{i} ⊗ (c_{i} − c_{i+1})

for each i = 1, …, d − 1. Moreover, ω(t_{i})(1 ⊗ c_{j}) = −(1 ⊗ c_{j})ω(t_{i}) for each i = 1, …, d − 1 and j = 1, …, d.

Henceforth, we identify S(d) with a subsuperalgebra of W (d) via the embedding ω from the lemma, and also identify C(d) with the subsuperalgebra 1 ⊗ C(d) of W (d). Then, Lemma 3.3 shows that multiplication defines a superalgebra isomorphism

C(d) ⊗ S(d) ⟶^{∼} W (d), c ⊗ s ↦ cs,

the tensor product of superalgebras on the left hand side being defined according to the usual rule of signs.

So we can define an exact functor

F : mod(S(d)) → mod(W (d))

on an object M by F M = U(d) ⊗ M, and on a morphism f : M → M′ by F f = id_{U(d)} ⊗f. Thus, the action of a homogeneous s ∈ S(d) ⊂ W (d) on m ⊗ u ∈ U(d) ⊗ M is by s(u ⊗ m) = (−1)^{∂(s)∂(u)}u ⊗ (sm), the action of c ∈ C(d) ⊂ W (d) is by c(u ⊗ m) = (cu) ⊗ m, and (u ⊗ m)(id_{U(d)} ⊗f) = u ⊗ (mf). We also have an exact functor

G : mod(W (d)) → mod(S(d)).

This is defined on an object N by GN = Hom_{C(d)}(U(d), N), the action of a homogeneous s ∈ S(d) on f ∈ Hom_{C(d)}(U(d), N) being determined by u(sf) = (−1)^{∂(u)∂(s)}s(uf) for all homogeneous u ∈ U(d). On a morphism g : N → N′, Gg : Hom_{C(d)}(U(d), N) → Hom_{C(d)}(U(d), N′) is defined by u(f(Gg)) = (uf)g for u ∈ U(d) and f ∈ Hom_{C(d)}(U(d), N).

Recall the parity change functor Π defined in the previous section.

3.4. Theorem. The functors F and G form an adjoint pair, that is, there is a natural (even) isomorphism

Hom_{W (d)}(F M, N) ≅ Hom_{S(d)}(M, GN)

for each S(d)-supermodule M and W (d)-supermodule N. Moreover:

(a) if d is even, then F ∘ G ≅ Id and G ∘ F ≅ Id;

(b) if d is odd, then F ∘ G ≅ Id ⊕Π and G ∘ F ≅ Id ⊕Π.

Proof. For adjointness, there are natural maps

Hom_{W (d)}(U(d) ⊗ M, N) → Hom_{S(d)}(M, Hom_{C(d)}(U(d), N)),  f ↦ f̂;
Hom_{S(d)}(M, Hom_{C(d)}(U(d), N)) → Hom_{W (d)}(U(d) ⊗ M, N),  g ↦ g̃.

Here, f̂ is defined by u(mf̂) = (u ⊗ m)f and g̃ is defined by (u ⊗ m)g̃ = u(mg). Now check that ˜ f̂ = f and ˆ g̃ = g.

Now we prove (b), the argument for (a) being similar (and considerably easier!). Let E = End_{C(d)}(U(d)) for short, a vector superspace on basis I = id_{U(d)}, J = J_{U(d)} as in Lemma 2.4. On any category of left supermodules, the functor Id ⊕Π is naturally isomorphic to the tensor functor E⊗?, which sends a module M to E ⊗_{𝕜}M and a morphism f to id_{E} ⊗f (written on the right). We will actually show that G ∘ F ≅ E⊗? and that F ∘ G ≅ E⊗?.

We first show that G ∘ F ≅ E⊗?. Define a natural transformation η : E⊗? → G ∘ F by defining the map

η_{M} : E ⊗ M → Hom_{C(d)}(U(d), U(d) ⊗ M)

for an S(d)-supermodule M by the formula uη_{M}(f ⊗ m) = uf ⊗ m for each u ∈ U(d), m ∈ M, f ∈ E. To see that η is actually a natural isomorphism, it suffices to consider the special case M = 𝕜 when it is obvious.

Now we show that F ∘ G ≅ E⊗?. Define a natural transformation η : F ∘ G → E⊗? by letting

η_{N} : U(d) ⊗ Hom_{C(d)}(U(d), N) → E ⊗ N

for each W (d)-supermodule N be the map

u ⊗ f ↦ I ⊗ uf + (−1)^{∂(u)}J ⊗ uJf

for homogeneous u ∈ U(d) and f ∈ Hom_{C(d)}(U(d), N). To see that η is actually a natural isomorphism, it suffices (since C(d) is a simple superalgebra) to consider the special case N = U(d). We can pick a homogeneous basis u_{1}, …, u_{n}, ū_{1}, …, ū_{n} for U(d) so that u_{i}J = ū_{i}, ū_{i}J = u_{i} as in Lemma 2.3. Then, the map η_{U(d)} maps u_{i} ⊗ I ↦ I ⊗ u_{i} + J ⊗ ū_{i}, ū_{i} ⊗ I ↦ I ⊗ ū_{i} − J ⊗ u_{i}, u_{i} ⊗ J ↦ I ⊗ ū_{i} + J ⊗ u_{i} and ū_{i} ⊗ J ↦ I ⊗ u_{i} − J ⊗ ū_{i}. It is obvious from this that it is a bijection.

3.5. Corollary. (a) Suppose d is even. The functors F and G induce mutually inverse bijections between isomorphism classes of irreducible (resp. absolutely irreducible) S(d)- supermodules and irreducible (resp. absolutely irreducible) W (d)-supermodules.

(b) Suppose d is odd. The functor F induces a bijection between isomorphism classes of absolutely irreducible S(d)-supermodules and self-associate irreducible W (d)-supermodules. The functor G induces a bijection between isomorphism classes of absolutely irreducible W (d)-supermodules and self-associate irreducible S(d)-supermodules.

Proof. (a) This is obvious since F and G are mutually inverse equivalences of categories.

(b) Let D be an irreducible S(d)-supermodule. By Lemma 2.9, F D is a self-associate irreducible W (d)-supermodule in case D is absolutely irreducible, and decomposes as a direct sum of two isomorphic absolutely irreducible W (d)-supermodules in case D is self-associate.

It is now straightforward to complete the proof of the corollary using the properties of F and G from Theorem 3.4.

For the remainder of the article, we in fact work with the Sergeev superalgebra W (d) instead of with S(d), this being justified by Theorem 3.4 and its corollary. To conclude the section, we develop notation for products of arbitrary elements in W (d).

First, let W_{d} denote the hyperoctahedral group, that is, the semidirect product of S_{d} and ℤ_{2}^{d}. To be more precise, denote elements of the Abelian group ℤ_{2}^{d} by d-tuples ε = (ε_{1}, …, ε_{d}) with each ε_{i} ∈ ℤ_{2}. There is a right action of S_{d} on ℤ_{2}^{d} given by ε · w = (ε_{w1}, ε_{w2}, …, ε_{wd}) for w ∈ S_{d}, ε ∈ ℤ_{2}^{d}. Then, elements of W_{d} are pairs (w, ε) with w ∈ S_{d}, ε ∈ ℤ_{2}^{d}, the product of two such elements being defined by

(x, ε)(y, δ) = (xy, ε · y + δ).

Henceforth, we will identify w ∈ S_{d} (resp. ε ∈ ℤ_{2}^{d}) with the element (w, 0) ∈ W_{d} (resp. (1, ε) ∈ W_{d}). It will also be convenient to extend the action of S_{d} on ℤ_{2}^{d} to an action of all of W_{d} on ℤ_{2}^{d}, so that ε · (w, δ) = ε · w + δ for ε ∈ ℤ_{2}^{d}, (w, δ) ∈ W_{d}.

For ε ∈ ℤ_{2}^{d}, let

c^{ε} = c_{1}^{ε_{1}} … c_{d}^{ε_{d}} ∈ C(d)

Then, the {c^{ε} | ε ∈ ℤ_{2}^{d}} form a basis for the Clifford superalgebra C(d). The product of two such basis elements is given explicitly by the rule

c^{ε}c^{δ} = α(ε; δ)c^{ε+δ}  where  α(ε; δ) = ∏_{1≤s<t≤d}(−1)^{δ_{s}ε_{t}}

for ε, δ ∈ ℤ_{2}^{d}. It is worth remarking for later calculations that α(ε + ε′; δ) = α(ε; δ)α(ε′; δ) and α(ε; δ + δ′) = α(ε; δ)α(ε; δ′).

We obtain a basis {w ⊗ c^{ε} | w ∈ S_{d}, ε ∈ ℤ_{2}^{d}} for the Sergeev superalgebra W (d) = 𝕜S_{d} ⊗ C(d). The right action of w ∈ S_{d} on the basis element c^{ε} of C(d) is given by the formula

c^{ε} · w = α(ε; w)c^{ε·w}  where  α(ε; w) = ∏_{{1≤s<t≤d \atop w^{−1}s>w^{−1}t}}(−1)^{ε_{s}ε_{t}}.

Hence, the product of two basis elements of W (d) given by the formula

(x ⊗ c^{ε})(y ⊗ c^{δ}) = α(x, ε; y, δ)xy ⊗ c^{ε·y+δ}  where  α(x, ε; y, δ) = α(ε; y)α(ε · y; δ).

It follows that the resulting function α : W_{d} × W_{d} → {±1}, ((x, ε), (y, δ)) ↦ α(x, ε; y, δ) is a 2-cocycle on W_{d}. So W (d) is a twisted group algebra of the hyperoctahedral group W_{d} over 𝕜. In particular, the twisted group algebra analogue of Maschke’s theorem gives:

3.6. Lemma. If p = 0 or p > d, then W (d) is a semisimple (super)algebra.

We finally record a technical property about the cocycle α just constructed.

3.7. Lemma. For all ε, δ ∈ ℤ_{2}^{d} and g = (w, γ) ∈ W_{d},

α(ε + δ; w) = α(ε; g)α(δ; g)α(ε + δ; δ)α(ε · g + δ · g; δ · g).

Proof. Expand both sides of the equation (c^{ε+δ}c^{δ}) · w = (c^{ε+δ} · w)(c^{δ} · w) to show that α(ε + δ; w) = α(ε; w)α(δ; w)α(ε + δ; δ)α(ε · w + δ · w; δ · w). Now expand the definition of α(ε; g)α(δ; g)α(ε · g + δ · g; δ · g) to see that it equals

α(ε; w)α(ε · w; γ)α(δ; w)α(δ · w; γ)α(ε · w + δ · w; δ · w + γ)
= α(ε; w)α(δ; w)α(ε · w + δ · w; δ · w)α(ε · w; γ)α(δ · w; γ)α(ε · w + δ · w; γ)
= α(ε; w)α(δ; w)α(ε · w + δ · w; δ · w),

and the result follows.

4 The Schur superalgebra

We introduce some further notation. Suppose that 0 ≠ i, j ∈ ℤ. Define ∂_{i} = ¯0 if i > 0 or ¯1 if i < 0; define ∂_{i,j} = ∂_{i} + ∂_{j} ∈ ℤ_{2}. More generally, given d-tuples i = (i_{1}, …, i_{d}) and j = (j_{1}, …, j_{d}) of non-zero integers, let

∂_{i} = ∂_{i_{1}} + ⋯ + ∂_{i_{d}} ∈ ℤ_{2},  ∂_{i,j} = ∂_{i} + ∂_{j} ∈ ℤ_{2},
ε_{i} = (∂_{i_{1}}, ∂_{i_{2}}, …, ∂_{i_{d}}) ∈ ℤ_{2}^{d},  ε_{i,j} = ε_{i} + ε_{j} ∈ ℤ_{2}^{d}.

Let ℤ_{2}^{d} act on the left on {±1, …, ±d} so that for ε = (ε_{1}, …, ε_{d}) ∈ ℤ_{2}^{d} and s = 1, …, d, ε(±s) = (−1)^{ε_{s}}(±s). Extend the natural action of S_{d} on {1, …, d} to an action on {±1, …, ±d} so that w(−s) = −(ws) for s = 1, …, d. These two actions combine to give a well-defined left action of the hyperoctahedral group W_{d} on the set {±1, …, ±d}.

Now let I(n, d) denote the set of all functions i : {±1, …, ±d} → {±1, …, ±n} such that i(−s) = −i(s) for s = 1, …, d. We often denote the value i(s) of the function i ∈ I(n, d) at s ∈ {±1, …, ±d} by i_{s}. Then, the element i ∈ I(n, d) can be thought of simply as the d-tuple (i_{1}, …, i_{d}): the original function i can be recovered uniquely from knowledge of this d-tuple since i(−s) = −i(s). The group W_{d} acts on the right on I(n, d) by composition of functions, so (i · g)(s) = i(gs) for i ∈ I(n, d), g ∈ W_{d} and s ∈ {±1, …, ±d}. Write i ∼ j if i, j ∈ I(n, d) lie in the same W_{d}-orbit. Also let W_{d} act diagonally on the right on the set I(n, d) × I(n, d) of double indexes, and write (i, j) ∼ (k, l) if the double indexes (i, j) and (k, l) lie in the same orbit.

Let V denote the vector superspace with basis v_{±1}, …, v_{±n}, where ∂(v_{i}) = ∂_{i}. Then, the tensor product V^{⊗d} is also a vector superspace with the induced grading. A basis is given by the monomials v_{i} = v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{d}} for all i ∈ I(n, d), and ∂(v_{i}) = ∂_{i}. We make V^{⊗d} into a right W (d)-supermodule by setting

v_{i}(w ⊗ c^{δ}) = α(ε_{i}; w, δ)v_{i·(w,δ)}

for all i ∈ I(n, d), (w, δ) ∈ W_{d}. The fact that this is well-defined follows from the fact that α is a 2-cocycle. To be more explicit, the action of the generator s_{i} of S_{d} ⊂ W (d) is as the linear map id ⊗ ⋯ ⊗ id ⊗T_{V,V} ⊗ id ⊗ ⋯ ⊗ id where the supertwist map T_{V,V} is in the ith position, and the generator c_{j} of C(d) ⊂ W (d) acts on the right (with our usual convention regarding signs) as the linear map id ⊗ ⋯⊗id ⊗J_{V} ⊗id ⊗ ⋯⊗id where the map J_{V} : v_{i} ↦ v_{−i} is in the jth tensor.

Now define the Schur superalgebra of type Q

Q̇(n, d) := End_{W (d)}(V^{⊗d}).

So, Q̇(n, d) acts on V^{⊗d} on the left. Note Q̇(n, d) is naturally a subsuperalgebra of the Schur superalgebra Ṡ(n|n, d) = End_{𝕜S_{d}}(V^{⊗d}) of type GL, which was studied in [18, 6]. We observe right away by Lemma 3.6 that:

4.1. Lemma. If p = 0 or p > d, then Q̇(n, d) is a semisimple (super)algebra.

The initial goal is to describe an explicit basis for Q̇(n, d).

4.2. Lemma. For (i, j) ∈ I(n, d) × I(n, d), the following properties are equivalent: (i) ∂_{i_{s},j_{s}}∂_{i_{t},j_{t}} = ¯0 whenever |i_{s}| = |i_{t}| and |j_{s}| = |j_{t}| for some 1 ≤ s < t ≤ d; (ii) α(ε_{i,j}; w) = 1 for all (w, δ) ∈ Stab_{W_{d}}(i, j).

Proof. Using the fact that Stab_{S_{d}}(i, j) is generated by transpositions and that α is a 2cocycle, property (ii) is equivalent to the condition that α(ε_{i,j}; w) = 1 for all (w, δ) ∈ Stab_{W_{d}}(i, j) with w a transposition. This weaker statement is precisely the condition (i), by the definition of α.

Call the double index (i, j) ∈ I(n, d) × I(n, d) strict if it satisfies the properties in the lemma, and let I^{2}(n, d) denote the set of all strict double indexes. Observe using Lemma 4.2(i) that I^{2}(n, d) is W_{d}-stable. Given (i, j) ∼ (k, l) ∈ I^{2}(n, d), choose (w, δ) ∈ W_{d} such that (i, j) · (w, δ) = (k, l) and define the sign σ(i, j; k, l) to be α(ε_{i,j}; w). In view of Lemma 4.2(ii), this definition of σ(i, j; k, l) is independent of the choice of (w, δ).

Given i, j ∈ {±1, …, ±d}, let ė_{i,j} ∈ End_{𝕜}(V) denote the linear map with ė_{i,j}v_{k} = δ_{j,k}v_{i} for all k. Given i, j ∈ I(n, d), let

ė_{i,j} = ė_{i_{1},j_{1}} ⊗ ė_{i_{2},j_{2}} ⊗ ⋯ ⊗ ė_{i_{d},j_{d}} ∈ End_{𝕜}(V)^{⊗d}.

Now there is an isomorphism between the superalgebras End_{𝕜}(V)^{⊗d} and End_{𝕜}(V^{⊗d}) under which our element ė_{i,j} ∈ End_{𝕜}(V)^{⊗d} corresponds to the linear map V^{⊗d} → V^{⊗d} with

ė_{i,j}v_{k} = δ_{j,k}α(ε_{i,j}; ε_{j})v_{i}.  (4.3)

We will henceforth identify End_{𝕜}(V)^{⊗d} and End_{𝕜}(V^{⊗d}) in this way. Given strict (i, j) ∈ I^{2}(n, d), define the linear map ξ̇_{i,j} ∈ End_{𝕜}(V^{⊗d}) by

ξ̇_{i,j} = ∑_{(k,l)∼(i,j)} σ(i, j; k, l)ė_{k,l}.  (4.4)

Obviously, if (i, j) ∼ (k, l) ∈ I^{2}(n, d), then ξ̇_{i,j} = σ(i, j; k, l)ξ̇_{k,l}. Now choose some set Ω(n, d) of orbit representatives for the action of W_{d} on I^{2}(n, d). Then:

4.5. Theorem. The elements {ξ̇_{i,j} | (i, j) ∈ Ω(n, d)} give a basis for Q̇(n, d). Moreover, given (i, j), (k, l) ∈ I^{2}(n, d),

ξ̇_{i,j}ξ̇_{k,l} = ∑_{(s,t)∈Ω(n,d)} a_{i,j,k,l,s,t}ξ̇_{s,t}

where

a_{i,j,k,l,s,t} = ∑_{{h∈I(n,d) with \atop (s,h)∼(i,j),(h,t)∼(k,l)}} σ(i, j; s, h)σ(k, l; h, t)α(ε_{s,h}; ε_{h,t}).

Proof. Obviously, the given elements are linearly independent. To show that they span End_{W (d)}(V^{⊗d}), let

θ = ∑_{i,j∈I(n,d)} a_{i,j}ė_{i,j}

be an arbitrary element of End_{𝕜}(V^{⊗d}). Take w ∈ S_{d}, δ ∈ ℤ_{2}^{d} and set g = (w, δ) ∈ W_{d}. For j ∈ I(n, d), we have that (θv_{j})(w ⊗ c^{δ}) = θ(v_{j}(w ⊗ c^{δ})) if and only if

∑_{i∈I(n,d)} a_{i,j}α(ε_{i,j}; ε_{j})α(ε_{i}; g)v_{i·g} = ∑_{i∈I(n,d)} a_{i·g,j·g}α(ε_{i·g,j·g}; ε_{j·g})α(ε_{j}; g)v_{i·g}

Simplifying using Lemma 3.7, we see that θ ∈ End_{W (d)}(V^{⊗d}) if and only if

a_{i·g,j·g} = α(ε_{i,j}; w)a_{i,j}

for all i, j ∈ I(n, d) and g = (w, δ) ∈ W_{d}. So by Lemma 4.2(ii), we must have that a_{i,j} = 0 unless (i, j) is strict, and for strict (h, k) ∼ (i, j), we have that a_{h,k} = σ(i, j; h, k)a_{i,j}. This shows that θ ∈ Q̇(n, d) if and only if θ = ∑_{(i,j)∈Ω(n,d)} a_{i,j}ξ̇_{i,j}, completing the proof of the first part of the theorem.

Now we show how to deduce the product rule. To calculate a_{i,j,k,l,s,t} in the product expansion, we need by (4.4) to determine the coefficient of ė_{s,t} in

ξ̇_{i,j}ξ̇_{k,l} = ∑_{(i′,j′)∼(i,j)} ∑_{(k′,l′)∼(k,l)} σ(i, j; i′, j′)σ(k, l; k′, l′)ė_{i′,j′}ė_{k′,l′}.

We have that ė_{i′,j′}ė_{k′,l′} = δ_{j′,k′}α(ε_{i′,j′}; ε_{k′,l′})ė_{i′,l′}. Using this the ė_{s,t}-coefficient of ξ̇_{i,j}ξ̇_{k,l} is therefore precisely as in the theorem (with h = j′ = k′).

5 The coordinate ring

Now we proceed to give an entirely different construction of the Schur superalgebra in the spirit of Green’s monograph [7]. We begin by reviewing some basic facts about cosuperalgebras and bisuperalgebras, following [18].

A cosuperalgebra is a vector superspace A with the additional structure of a 𝕜-coalgebra, such that both the comultiplication Δ : A → A ⊗ A and the counit ϵ : A → 𝕜 are even linear maps. Given two cosuperalgebras A and B, A⊗B is a cosuperalgebra with comultiplication id_{A} ⊗T_{A,B} ⊗ id_{B} ∘(Δ_{A} ⊗ Δ_{B}). A cosuperalgebra homomorphism θ : A → B means an even linear map that is a coalgebra homomorphism in the usual sense. Cosuperideals and subcosuperalgebras are also the obvious graded version of the usual notions.

Given a cosuperalgebra A, a right A-cosupermodule is a vector superspace M together with an even linear map η : M → M ⊗A, called the structure map of M, which makes M into a right A-comodule in the usual sense. A homomorphism between two A-cosupermodules means an A-comodule homomorphism in the usual sense; note we write homomorphisms between right A-cosupermodules on the left (and vice versa). We let comod(A) denote the (superadditive) category of all right A-cosupermodules.

A bisuperalgebra is a vector superspace A that is both a superalgebra and a cosuperalgebra, such that the comultiplication Δ : A → A ⊗ A (recall how A ⊗ A is viewed as a superalgebra!) and counit ϵ : A → 𝕜 are superalgebra homomorphisms. If A is a bisuperalgebra, we have a natural notion of (inner) tensor product of two right A-cosupermodules M and N, namely, the vector superspace M ⊗N with structure map defined by the composition

M ⊗ N ⟶^{η_{M} ⊗η_{N}} M ⊗ A ⊗ N ⊗ A ⟶^{id ⊗T_{A,N} ⊗id} M ⊗ N ⊗ A ⊗ A ⟶^{id ⊗ id ⊗μ} M ⊗ N ⊗ A,

where η_{M} : M → M ⊗A and η_{N} : N → N ⊗A are the structure maps of M, N, respectively, and μ : A ⊗ A → A denotes the multiplication in A (one needs to know here that μ is a cosuperalgebra homomorphism, see e.g. [6, §2.2]).

Let A be a finite dimensional cosuperalgebra. We make the dual superspace A^{∗} into a superalgebra by defining the product f_{1}f_{2} of homogeneous f_{1}, f_{2} ∈ A^{∗} by (f_{1}f_{2})(a) = (f_{1} ¯⊗f_{2})Δ(a), interpreting the right hand side according to the usual rule of signs. Given a right A-cosupermodule M with structure map η : M → M ⊗ A, we can view M as a left A^{∗}-supermodule, with action defined by fm = (id_{M} ¯⊗f)η(m) for f ∈ A^{∗}, m ∈ M. Now suppose that θ : M → N is a homogeneous morphism of right A-cosupermodules and define θ̃: M → N by mθ̃:= (−1)^{∂(m)∂(θ)}θm for homogeneous m ∈ M. Then, viewing M and N as left A^{∗}-supermodules as just explained, the map θ̃ is a morphism of left A^{∗}-supermodules. One obtains in this way an isomorphism between the categories comod(A) and mod(A^{∗}).

Finally in this review of definitions, we mention a standard general result about direct sums of cosuperalgebras. Suppose A is a (possibly infinite dimensional) cosuperalgebra and that A = ⨁_{i∈I} A_{i} as a direct sum of subcosuperalgebras. Then, as in [7, p.20] we have:

5.1. Lemma. With the preceeding notation, let M be a right A-cosupermodule with structure map η : M → M ⊗ A. Then, M = ⨁_{i∈I} M_{i} where M_{i} is the unique maximal subcosupermodule of M with η(M_{i}) ⊆ M_{i} ⊗ A_{i}.

As a corollary, one obtains that the category of right A-cosupermodules is equivalent to the product of the categories of right A_{i}-cosupermodules for all i ∈ I.

Now we begin the alternative construction of the Schur superalgebra. Start with the free superalgebra F (n) on non-commuting generators {f_{i,j} | i, j = ±1, …, ±n}, where ∂(f_{i,j}) = ∂_{i,j}. Then, F (n) is naturally ℤ-graded by degree as

F (n) = ⨁_{d≥0} F (n, d).

Given a double index (i, j) ∈ I(n, d) × I(n, d), define f_{i,j} = f_{i_{1},j_{1}}f_{i_{2},j_{2}} … f_{i_{d},j_{d}}. The elements {f_{i,j} | (i, j) ∈ I(n, d) × I(n, d)} form a basis for F (n, d). One checks that the unique superalgebra maps ϵ : F (n) → 𝕜 and Δ : F (n) → F (n) ⊗ F (n) defined on generators by

ϵ(f_{i,j}) = δ_{i,j},
Δ(f_{i,k}) = ∑_{j∈{±1,...,±n}}(−1)^{∂_{i,j}∂_{j,k}}f_{i,j} ⊗ f_{j,k}

make F (n) into a bisuperalgebra. We point out that for (i, k) ∈ I(n, d) × I(n, d),

Δ(f_{i,k}) = ∑_{j∈I(n,d)}(−1)^{∂_{i,j}∂_{j,k}}α(ε_{j,k}; ε_{i,j})f_{i,j} ⊗ f_{j,k}.

Hence, each F (n, d) is a finite dimensional subcosuperalgebra of F (n). Make the vector superspace V from the previous section into a right F (n)-cosupermodule with structure map V → V ⊗ F (n) defined by

v_{j} ↦ ∑_{i∈{±1,...,±n}}(−1)^{∂_{i}∂_{i,j}}v_{i} ⊗ f_{i,j}.

Then, for each d ≥ 1, V^{⊗d} is also automatically a right F (n)-cosupermodule with structure map V^{⊗d} → V^{⊗d} ⊗ F (n) given explicitly by the formula

v_{j} ↦ ∑_{i∈I(n,d)}(−1)^{∂_{i}∂_{i,j}}α(ε_{i,j}; ε_{i})v_{i} ⊗ f_{i,j}.

In particular, V^{⊗d} can be viewed as a right F (n, d)-cosupermodule.

Let E(n, d) = F (n, d)^{∗} be the dual superalgebra. Let e_{i,j} denote the element of E(n, d) with

e_{i,j}(f_{i,j}) = α(ε_{i,j}; ε_{i,j}), e_{i,j}(f_{k,l}) = 0 for (k, l) ≠ (i, j).

Then, the {e_{i,j} | i, j ∈ I(n, d)} give a basis for E(n, d).

The right F (n, d)-cosupermodule V^{⊗d} is a left E(n, d)-supermodule in the way described above. Let ρ_{d} : E(n, d) → End_{𝕜}(V^{⊗d}) be the resulting representation.

5.2. Lemma. The representation ρ_{d} is an isomorphism between E(n, d) and End_{𝕜}(V^{⊗d}). Moreover, ρ_{d}(e_{i,j}) = ė_{i,j} for all i, j ∈ I(n, d).

Proof. It suffices to check that e_{i,j}v_{k} = ė_{i,j}v_{k} for all i, j, k ∈ I(n, d). By the definition of the action of E(n, d), we have that

e_{i,j}v_{k} = (id ¯⊗e_{i,j}) ( ∑_{h∈I(n,d)}(−1)^{∂_{h}∂_{h,k}}α(ε_{h,k}; ε_{h})v_{h} ⊗ f_{h,k})
= δ_{j,k}α(ε_{i,j}; ε_{i})α(ε_{i,j}; ε_{i,j})v_{i} = δ_{j,k}α(ε_{i,j}; ε_{j})v_{i} = ė_{i,j}v_{k}.

This completes the proof.

Now consider the superideal ℐ (n) of F (n) generated by the elements

{f_{i,j} − f_{−i,−j}, f_{i,j}f_{k,l} − (−1)^{∂_{i,j}∂_{k,l}}f_{k,l}f_{i,j} | i, j, k, l = ±1, …, ±n}.

A short calculation reveals that this is actually a bisuperideal, so the quotient

B(n) := F (n)/ℐ (n)

is a bisuperalgebra quotient of F (n). Let b_{i,j} = f_{i,j} + ℐ (n). Then, B(n) is just the free commutative superalgebra on the degree ¯0 generators b_{i,j} = b_{−i,−j} and degree ¯1 generators b_{i,−j} = b_{−i,j}, for all 1 ≤ i, j ≤ n. The superideal ℐ (n) is homogeneous, so graded as ℐ (n) = ⨁_{d≥0} ℐ (n, d). So B(n) is also ℤ-graded by degree as B(n) = ⨁_{d≥0} B(n, d), with B(n, d) ≅ F (n, d)/ℐ (n, d). Moreover, B(n, d) is spanned by all monomials b_{i,j} = b_{i_{1},j_{1}} … b_{i_{d},j_{d}} for i, j ∈ I(n, d). The monomial b_{i,j} is non-zero if and only if (i, j) is strict, and for strict (i, j) ∼ (k, l), we have that

b_{i,j} = σ(i, j; k, l)b_{k,l}.

It follows that B(n, d) has basis {b_{i,j} |(i, j) ∈ Ω(n, d)}, where Ω(n, d) is the choice of W_{d}-orbit representatives in I^{2}(n, d) made in the previous section.

Now, let Q(n, d) denote the dual superalgebra B(n, d)^{∗}. Since B(n, d) = F (n, d)/ℐ (n, d), Q(n, d) is naturally identified with the annihilator ℐ (n, d)^{∘} ⊆ E(n, d). For (i, j) ∈ I^{2}(n, d), let ξ_{i,j} ∈ Q(n, d) ⊆ E(n, d) denote the unique function with

ξ_{i,j}(b_{i,j}) = α(ε_{i,j}; ε_{i,j}),  and  ξ_{i,j}(b_{k,l}) = 0 for (k, l) ≁ (i, j).

The {ξ_{i,j} | (i, j) ∈ Ω(n, d)} give a basis for Q(n, d).

We can regard the F (n, d)-cosupermodule V^{⊗d} instead as a B(n, d)-cosupermodule by restriction. Dualizing, we obtain a natural representation Q(n, d)→ End_{𝕜}(V^{⊗d}), which is nothing more than the restriction of the representation ρ_{d} : E(n, d) ⟶^{∼} End_{𝕜}(V^{⊗d}) defined earlier to the subsuperalgebra Q(n, d) ⊆ E(n, d). Then:

5.3. Theorem. The representation ρ_{d} gives an isomorphism between Q(n, d) and the Schur superalgebra Q̇(n, d). Moreover, ρ_{d}(ξ_{i,j}) = ξ̇_{i,j} for all (i, j) ∈ I^{2}(n, d).

Proof. Pick (i, j) ∈ I^{2}(n, d). Since Q(n, d) ⊆ E(n, d), we can write

ξ_{i,j} = ∑_{k,l∈I(n,d)} a_{k,l}e_{k,l}

for coefficients a_{k,l} ∈ 𝕜. To calculate the coefficient a_{k,l}, evaluate both sides at the element f_{k,l} ∈ F (n, d) to see that a_{k,l}α(ε_{k,l}; ε_{k,l}) = ξ_{i,j}(f_{k,l}) = ξ_{i,j}(b_{k,l}). So by the definition of ξ_{i,j}, a_{k,l} is zero unless (k, l) ∼ (i, j), in which case, a_{k,l} = α(ε_{k,l}; ε_{k,l})σ(i, j; k, l)ξ_{i,j}(b_{i,j}) = σ(i, j; k, l). This shows that

ξ_{i,j} = ∑_{(k,l)∼(i,j)} σ(i, j; k, l)e_{k,l}.

Now the theorem follows at once from Lemma 5.2, Theorem 4.5 and the definition (4.4).

We will henceforth identify Q(n, d), which we defined as the dual of the cosuperalgebra B(n, d), with Q̇(n, d), which we defined as the commutant of W (d) on tensor space V^{⊗d}. So the dual basis element ξ_{i,j} ∈ Q(n, d) is identified with the linear transformation ξ̇_{i,j} ∈ Q̇(n, d).

6 Weights and idempotents

Let Λ(n, d) denote the set of all tuples λ = (λ_{1}, …, λ_{n}) of non-negative integers with λ_{1} + ⋯ + λ_{n} = d. We partially order Λ(n, d) by the usual dominance order, so λ ≥ μ if and only if ∑_{s=1}^{t} λ_{s} ≥ ∑_{s=1}^{t} μ_{s} for each t = 1, …, n. For i ∈ I(n, d), define its weight wt(i) to be the composition λ = (λ_{1}, …, λ_{n}) ∈ Λ(n, d) where λ_{s} = |{t | 1 ≤ t ≤ d, |i_{t}| = s}|. Conversely, given λ ∈ Λ(n, d), let i_{λ} denote the element (1, …, 1, 2, …, 2, 3, …) ∈ I(n, d) where there are λ_{1} ones, λ_{2} twos, etc., so that wt(i_{λ}) = λ. Define

ξ_{λ} := ξ_{i_{λ},i_{λ}} ∈ Q(n, d).

We call the elements {ξ_{λ} | λ ∈ Λ(n, d)} weight idempotents, motivated by the following lemma:

6.1. Lemma. For (i, j) ∈ I^{2}(n, d),

ξ_{λ}ξ_{i,j} = \{ \begin{matrix}ξ_{i,j} & if wt(i) = λ, \\ 0 & otherwise. \end{matrix} ξ_{i,j}ξ_{λ} = \{ \begin{matrix}ξ_{i,j} & if wt(j) = λ, \\ 0 & otherwise. \end{matrix}

In particular, {ξ_{λ} | λ ∈ Λ(n, d)} is a set of mutually orthogonal idempotents whose sum is the identity element of Q(n, d).

Proof. It is elementary to check that the matrix units {e_{h,h} | h ∈ I(n, d)} in E(n, d) are a set of mutually orthogonal idempotents whose sum is the identity, with e_{h,h}e_{i,j} = δ_{h,i}e_{i,j} and e_{i,j}e_{h,h} = δ_{h,j}e_{i,j} for all h, i, j ∈ I(n, d). Now according to (4.4), ξ_{λ} = ∑_{h} e_{h,h} summing over all h ∈ I(n, d) with wt(h) = λ, as an element of E(n, d). The lemma follows easily from these remarks.

Let ω denote the weight (1^{d}), which is an element of Λ(n, d) providing n ≥ d. Assuming this, the weight idempotent ξ_{ω} is a well-defined element of Q(n, d), and ξ_{ω}Q(n, d)ξ_{ω} is naturally a superalgebra in its own right, its identity element being the idempotent ξ_{ω}. We have the following double centralizer property:

6.2. Theorem. Assume that n ≥ d.

(i) The map ϕ : Q(n, d)ξ_{ω} → V^{⊗d}, ξ_{i,i_{ω}} ↦ v_{i} for i ∈ I(n, d) is an even isomorphism of Q(n, d)-supermodules. In particular, V^{⊗d} is a projective Q(n, d)-supermodule.

(ii) The map ψ : W (d) → ξ_{ω}Q(n, d)ξ_{ω}, x ⊗ c^{δ} ↦ ξ_{i_{ω}·(x,δ),i_{ω}} for all (x, δ) ∈ W_{d}, is a superalgebra isomorphism.

(iii) End_{Q(n,d)}(V^{⊗d}) ≅ W (d).

Proof. For (i), we first claim that ξ_{i,i_{ω}}v_{i_{ω}} = v_{i}. Well, ξ_{i,i_{ω}} = ∑_{(k,l)∼(i,i_{ω})} e_{k,l}, and e_{k,l}v_{i_{ω}} = δ_{l,i_{ω}}v_{k}. Now observe that (k, i_{ω}) ∼ (i, i_{ω}) if and only if k = i, since Stab_{W_{d}}(i_{ω}) = 1. It now follows easily that ξ_{i,i_{ω}}v_{i_{ω}} = v_{i} as claimed. So in particular, ξ_{ω}v_{i_{ω}} = v_{i_{ω}}, so there is a well-defined Q(n, d)-module homomorphism Q(n, d)ξ_{ω} → V^{⊗d} such that ξ_{ω} ↦ v_{i_{ω}}. By the claim, this is precisely the map ϕ. Finally, observe that Q(n, d)ξ_{ω} has as basis the elements {ξ_{i,i_{ω}} | i ∈ I(n, d)}, so that ϕ is an isomorphism.

For (ii) and (iii), ξ_{ω} is an idempotent, so the superalgebras End_{Q(n,d)}(Q(n, d)ξ_{ω}) and ξ_{ω}Q(n, d)ξ_{ω} are naturally isomorphic. There is a homomorphism W (d) → End_{Q(n,d)}(V^{⊗d}) defined by the representation of W (d) on V^{⊗d}. Combining these with (i), we obtain a superalgebra homomorphism ψ : W (d) → ξ_{ω}Q(n, d)ξ_{ω}. By definition, it maps the element x⊗c^{δ} ∈ W (d) to the unique element ξ of ξ_{ω}Q(n, d)ξ_{ω} with ξϕ = v_{i_{ω}}(x⊗c^{δ}). But v_{i_{ω}}(x⊗c^{δ}) = v_{i_{ω}·(x,δ)}, so ψ(x ⊗ c^{δ}) = ξ_{i_{ω}·(x,δ),i_{ω}} as in the lemma. It remains to observe that the elements {ξ_{i_{ω}·(x,δ),i_{ω}} | (x, δ) ∈ W_{d}} give a basis for ξ_{ω}Q(n, d)ξ_{ω}, so that ψ is an isomorphism.

Using Theorem 6.2(ii), Corollary 2.13, Lemma 3.2 and Corollary 3.5, we deduce:

6.3. Lemma. For n ≥ d, the number of irreducible Q(n, d)-supermodules not annihilated by ξ_{ω} is equal to |ℛ𝒫 _{p}(d)|.

There is one other situation where Schur functors arising from weight idempotents will be useful. Suppose now that m ≥ n. We embed Λ(n, d) into Λ(m, d) as the set of all weights of the form (λ_{1}, …, λ_{n}, 0, …, 0), and I(n, d) into I(m, d) as the set of all i ∈ I(m, d) with i_{s} ∈ {±1, …, ±n} for each s = 1, …, d. To avoid confusion with the corresponding elements of Q(n, d), we denote the elements ξ_{λ}, ξ_{i,j} ∈ Q(m, d) for λ ∈ Λ(m, d), (i, j) ∈ I^{2}(m, d) instead by ξ̂_{λ}, ξ̂_{i,j} respectively. Let e ∈ Q(m, d) denote the idempotent

e = ∑_{λ∈Λ(n,d)⊆Λ(m,d)} ξ̂_{λ}.  (6.4)

If i, j ∈ I(n, d) ⊆ I(m, d), the element ξ̂_{i,j} ∈ Q(m, d) lies in eQ(m, d)e.

6.5. Lemma. The map ι : Q(n, d) → eQ(m, d)e, ξ_{i,j} ↦ ξ̂_{i,j} for all (i, j) ∈ I^{2}(n, d), is a superalgebra isomorphism.

Proof. Consider the ℤ-graded superideal 𝒥 (m) = ⨁_{d≥0} 𝒥 (m, d) of B(m) generated by the elements

{b_{i,j} | i or j equals ± (n + 1), ±(n + 2), …, ±m}.

One checks easily that Δ(𝒥 (m)) ⊆ 𝒥 (m)⊗B(m)+B(m)⊗𝒥 (m), so that the comultiplication Δ on B(m) induces a well-defined comultiplication on B(m)/𝒥 (m) (though 𝒥 (m) is not a cosuperideal). Evidently, B(m)/𝒥 (m) ≅ B(n) as superalgebras, the induced comultiplication on B(m)/𝒥 (m) corresponding to the usual comultiplication on B(n) under the isomorphism. Dualizing, we obtain a multiplicative even isomorphism between Q(n, d) and 𝒥 (m)^{∘} ⊆ eQ(m, d)e, being precisely the map ι. Finally, observe that eQ(m, d)e = 𝒥 (m)^{∘} to complete the proof.

Next, we introduce a subsuperalgebra of Q(n, d) which plays the role of Cartan subalgebra. Let 𝒥_{0}(n) = ⨁_{d≥0} 𝒥_{0}(n, d) denote the ℤ-graded superideal of B(n) generated by the elements

{b_{i,j} | i, j = ±1, …, ±n, |i| ≠ |j|}.

It is elementary to check that 𝒥_{0}(n) is a bisuperideal of B(n), so we can form the bisuperalgebra quotient B_{0}(n) := B(n)/𝒥_{0}(n). For i = 1, …, n, let x_{i} denote the image of b_{i,i} = b_{−i,−i} in B_{0}(n), and x̄_{i} denote the image of b_{i,−i} = b_{−i,i}. Then B_{0}(n) is precisely the free commutative superalgebra on the generators x_{1}, …, x_{n}, x̄_{1}, …, x̄_{n}. Comultiplication Δ : B_{0}(n) → B_{0}(n) ⊗ B_{0}(n) is given explicitly on these generators by

Δ(x_{i}) = x_{i} ⊗ x_{i} − x̄_{i} ⊗ x̄_{i}, Δ(x̄_{i}) = x_{i} ⊗ x̄_{i} + x̄_{i} ⊗ x_{i}.

As usual, B_{0}(n) is ℤ-graded by degree as ⨁_{d≥0} B_{0}(n, d), with B_{0}(n, d) ≅ B(n, d)/𝒥_{0}(n, d) being a subsupercoalgebra of B_{0}(n) for each d ≥ 0. The dual superalgebra Q_{0}(n, d) = B_{0}(n, d)^{∗} can be identified with the annihilator 𝒥_{0}(n, d)^{∘} ⊆ Q(n, d), giving us a subsuperalgebra of Q(n, d).

Consider the special case Q_{0}(1, d) for d ≥ 1 in more detail (obviously, Q_{0}(1, 0) = 𝕜). Writing x = x_{1}, x̄ = x̄_{1}, the elements {x^{d}, x^{d−1}x̄} give a basis for B_{0}(1, d), with comultiplication Δ : B_{0}(1, d) → B_{0}(1, d) ⊗ B_{0}(1, d) satisfying

Δ(x^{d}) = x^{d} ⊗ x^{d} − dx^{d−1}x̄ ⊗ x^{d−1}x̄, Δ(x^{d−1}x̄) = x^{d−1}x̄ ⊗ x^{d} + x^{d} ⊗ x^{d−1}x̄.

As a basis for Q_{0}(1, d), take the dual basis {y_{d}, ȳ_{d}} to the basis {x^{d}, x^{d−1}x̄} of B_{0}(1, d). The superalgebra multiplication, dual to the comultiplication in B_{0}(1, d), is then given by y_{d}y_{d} = y_{d}, y_{d}ȳ_{d} = ȳ_{d} = ȳ_{d}y_{d}, ȳ_{d}ȳ_{d} = dy_{d}. Hence, for d ≥ 1,

Q_{0}(1, d) ≅ \{ \begin{matrix}C(1) & if p ∤ d, \\ ⋀(1) & if p|d, \end{matrix}

recalling Example 2.2.

Now in general, the subsuperalgebra Q_{0}(n, d) ⊆ Q(n, d) contains each weight idempotent ξ_{λ} for λ ∈ Λ(n, d) in its center. So,

Q_{0}(n, d) ≅ ∏_{λ∈Λ(n,d)} ξ_{λ}Q_{0}(n, d).  (6.6)

Moreover, one can see that

ξ_{λ}Q_{0}(n, d) ≅ Q_{0}(1, λ_{1}) ⊗ ⋯ ⊗ Q_{0}(1, λ_{n}) ≅ C(h_{p′}(λ)) ⊗ ⋀(h_{p}(λ))  (6.7)

where h_{p}(λ) denotes the number of non-zero parts of λ that are divisible by p, and h_{p′}(λ) denotes the number of parts of λ that are coprime to p. We deduce immediately using Lemma 2.9, Example 2.7 and Example 2.10 that ξ_{λ}Q_{0}(n, d) has a unique irreducible supermodule up to isomorphism, of dimension 2^{⌊(h_{p′}(λ)+1)/2⌋}. We pick one such and denote by U(λ). Note U(λ) is absolutely irreducible if and only if h_{p′}(λ) is even. Finally, regarding U(λ) as an Q_{0}(n, d)-supermodule by inflation, we have shown:

6.8. Lemma. The supermodules {U(λ) | λ ∈ Λ(n, d)} give a complete set of pairwise nonisomorphic irreducible Q_{0}(n, d)-supermodules. The dimension of U(λ) is 2^{⌊(h_{p′}(λ)+1)/2⌋}, and U(λ) is absolutely irreducible if and only if h_{p′}(λ) is even.

Recalling Lemma 5.1, we have thus determined the irreducible B_{0}(n)-cosupermodules, namely, the B_{0}(n)-cosupermodules {U(λ) | λ ∈ Λ(n)}, where Λ(n) := ⋃_{d≥0} Λ(n, d). Now let M be an arbitrary finite dimensional B(n)-cosupermodule with structure map η : M → M ⊗B(n). By Lemma 5.1, M decomposes as M = ⨁_{d≥0} M_{d} where M_{d} is the largest subcosupermodule with η(M_{d}) ⊆ M_{d} ⊗ B(n, d). Each M_{d} is naturally a B(n, d)-cosupermodule, hence a Q(n, d)-supermodule. Then, for λ ∈ Λ(n, d), we define the λ-weight space of M to be the space M_{λ} := ξ_{λ}M_{d}. Recalling (6.6), M_{λ} is a Q_{0}(n, d)-subsupermodule of M_{d}. Equivalently, M_{λ} is a B_{0}(n)-subcosupermodule of M, viewing M as a B_{0}(n)-cosupermodule by restriction, and

M = ⨁_{λ∈Λ(n)} M_{λ}.

Let X(n) denote the free polynomial algebra ℤ[x_{1}, …, x_{n}] and for λ ∈ Λ(n), set x^{λ} = x_{1}^{λ_{1}}x_{2}^{λ_{2}} … x_{n}^{λ_{n}}. Define the formal character

ch M = ∑_{λ∈Λ(n)} dim M_{λ}x^{λ} ∈ X(n).

Note that for finite dimensional B(n)-cosupermodules M, N, we have that ch(M ⊕ N) = ch M +ch N and ch(M ⊗N) = ch M. ch N. In other words, the map ch : Grot(B(n)) → X(n) is a ring homomorphism from the Grothendieck ring of the category of finite dimensional right B(n)-cosupermodules to X(n).

7 The big cell

Let 𝒥_{♭}(n) = ⨁_{d≥0} 𝒥_{♭}(n, d) and 𝒥_{♯}(n) = ⨁_{d≥0} 𝒥_{♯}(n, d) denote the ℤ-graded superideals of B(n) generated by the elements

{b_{i,j} | i, j = ±1, …, ±n, |i| < |j|}, {b_{i,j} | i, j = ±1, …, ±n, |i| > |j|}

respectively. One easily checks that these are cosuperideals. Hence, we can form the bisuperalgebras quotients

B_{♭}(n) := B(n)/𝒥_{♭}(n), B_{♯}(n) := B(n)/𝒥_{♯}(n).

Both B_{♭}(n) and B_{♯}(n) are ℤ-graded with degree d component, denoted B_{♭}(n, d) and B_{♯}(n, d) respectively, being cosuperalgebra quotients of B(n, d). The corresponding dual superalgebras to these, namely Q_{♭}(n, d) = 𝒥_{♭}(n, d)^{∘} and Q_{♯}(n, d) = 𝒥_{♯}(n, d)^{∘}, are therefore subsuperalgebras of Q(n, d), called the negative Borel and positive Borel subsuperalgebras respectively. They are spanned by the elements

{ξ_{i,j} | (i, j) ∈ I^{2}(n, d), |i| ≥ |j|} and {ξ_{i,j} | (i, j) ∈ I^{2}(n, d), |i| ≤ |j|}

respectively, where |i| ≥ |j| means that |i_{k}| ≥ |j_{k}| for each k = 1, …, d. Let π_{♭} : B(n) → B_{♭}(n) and π_{♯} : B(n) → B_{♯}(n) denote the natural quotient maps and set b_{i,j}^{♭} = π_{♭}(b_{i,j}), b_{i,j}^{♯} = π_{♯}(b_{i,j}) for i, j ∈ I(n, d). In particular, b_{i,j}^{♭} = 0 unless |i| ≥ |j| and b_{i,j}^{♯} = 0 unless |i| ≤ |j|. Let

π : B(n) → B_{♭}(n) ⊗ B_{♯}(n)

be the map (π_{♭} ⊗ π_{♯}) ∘ Δ. We wish to prove that this map π is injective, this being an analogue of the existence of the big cell in reductive algebraic groups, crucial for highest weight theory. It is possible to give a quick proof in the setting of algebraic supergroups. Since we wish to avoid introducing this language, we content ourselves with an elementary direct proof, though it is rather lengthy:

7.1. Theorem. π is injective.

Proof. We proceed in a number of steps. Observe right away that it is enough to prove that π is injective on each B(n, d) separately. So, fix d ≥ 1 and consider the restriction π : B(n, d) → B_{♭}(n, d) ⊗ B_{♯}(n, d). Let

Y = {(i, k, l, j) ∈ I(n, d) × I(n, d) × I(n, d) × I(n, d) | |i| ≥ |k|, |l| ≤ |j|}.

Write (i, k, l, j) ≈ (i′, k′, l′, j′) if both (i, k) ∼ (i′, k′) and (l, j) ∼ (l′, j′). Also call (i, k, l, j) strict if both (i, k) and (l, j) are strict in the sense of Lemma 4.2. Then:

7.2. If Z is a choice of representatives for the ≈-equivalence classes of strict (i, k, l, j) ∈ Y, then {b_{i,k}^{♭} ⊗ b_{l,j}^{♯} | (i, k, l, j) ∈ Z} is a basis for B_{♭}(n, d) ⊗ B_{♯}(n, d).

Now define m(i, j), for any i, j ∈ I(n, d), to be the unique element m ∈ I(n, d) with

m_{s} = \{ \begin{matrix}i_{s} & if |i_{s}| < |j_{s}| \\ j_{s} & if |i_{s}| ≥ |j_{s}| \end{matrix}

for all s = 1, …, d. Observe that m(i · g, j · g) = m(i, j) · g for all g ∈ W_{d}. We claim:

7.3. Suppose i, j ∈ I(n, d) and g ∈ W_{d} are such that m(i, j) = m(i, j · g) = m(i · g, j · g). Then, (i, j) ∼ (i, j · g).

We prove (7.3) by induction on d. Let m = m(i, j). If d = 1, then the assumption that m · g = m forces g = 1, and the lemma follows trivially. Now suppose that d > 1 and that we have proved (7.3) for all smaller d. Write {±1, …, ±d} = I ⊔ J where

I = {±s | 1 ≤ s ≤ d, |i_{s}| ≥ |j_{s}|},
J = {±s | 1 ≤ s ≤ d, |i_{s}| < |j_{s}|}.

Suppose first that g stabilizes I. Then, we can write g = xy where x fixes J pointwise and y fixes I pointwise. The assumption that m = m·g implies that both m = m·x and m = m·y. For s ∈ J, m_{s} = i_{s} and m_{ys} = i_{ys}, so since m_{s} = m_{ys}, we see that i_{s} = i_{ys}. Hence i · y = i, and a similar argument gives that j · x = j. So, (i, j · g) = (i · y, j · y) ∼ (i, j) as required.

Now suppose that g does not stabilize I. Then, we can pick s ∈ I such that gs ∈ J. Let t = gs ∈ J and define x to be the unique element of W_{d} with xs = t, xt = s and fixing all other elements of {±1, …, ±d} \ {±s, ±t}. Set g′ = xg, j′ = j · x, so j′ · g′ = jg. Using that m · g = m, we have that j_{s} = m_{s} = m_{t} = i_{t}. So, |j_{t}| > |i_{t}| = |m_{t}| = |m_{s}|. Using m = m(i, j · g), we must therefore have that m_{s} = i_{s} = i_{t} = m_{t}. This shows that i · x = i and m · x = m. Now,

m(i, j) = m(i · x, j · x) = m(i, j′),
m(i, j · g) = m(i, j′ · g′),
m(i · g, j · g) = m(i · g′, j′ · g′).

So by our assumption, m(i, j′) = m(i, j′ · g′) = m(i · g′, j′ · g′). Now, g′s = s, so we deduce by induction that (i, j′) ∼ (i, j′ · g′). Hence, (i, j) ∼ (i · x, j · x) = (i, j′) ∼ (i, j′ · g′) = (i, j · g) as required to complete the proof of (7.3).

Now we apply (7.3) to show:

7.4. Let i, j, i′, j′ ∈ I(n, d) and m = m(i, j), m′ = m(i′, j′). If (i, m, m, j) ≈ (i′, m′, m′, j′) then (i, j) ∼ (i′, j′).

Indeed, take g, h ∈ W_{d} such that (i, m) = (i′ · g, m′ · g) and (m, j) = (m′ · gh, j′ · gh). Set k = j′ · g. Now,

m = m(i, j) = m(i, j′ · gh) = m(i, k · h),
m′ · g = m(i′ · g, j′ · g) = m(i, k),
m′ · gh = m(i′ · gh, j′ · gh) = m(i · h, k · h).

So, observing that m = m′ · g = m′ · gh, we have that m(i, k) = m(i, k · h) = m(i · h, k · h). Hence by (7.3), (i, k) ∼ (i, k · h). So (i′, j′) ∼ (i′ · g, j′ · g) = (i, k) ∼ (i, k · h) = (i, j).

Next we claim:

7.5. Let i, j ∈ I(n, d) and m = m(i, j). If (i, j) is strict, then (i, m, m, j) is strict.

To prove this, take (i, j) strict and suppose that (i, m) is not strict. Then, there exist 1 ≤ s < t ≤ d with |i_{s}| = |i_{t}|, |m_{s}| = |m_{t}| and ∂_{i_{s},m_{s}}∂_{i_{t},m_{t}} = ¯1. So, i_{s} ≠ m_{s}, i_{t} ≠ m_{t}, hence by the definition of m, m_{s} = j_{s}, m_{t} = j_{t}. But this contradicts the fact that (i, j) is strict. Hence, (i, m) is strict, and a similar argument shows that (m, j) is strict.

Recall that Ω(n, d) is some set of representatives of the ∼-equivalence classes of strict (i, j) ∈ I(n, d) × I(n, d). In view of (7.4) and (7.5), all {(i, m, m, j) | (i, j) ∈ Ω(n, d), m = m(i, j)} are strict and lie in different ≈-equivalence classes. So they are linearly independent by (7.2), and we have now proved:

7.6. The elements {b_{i,m}^{♭} ⊗ b_{m,j}^{♯} | (i, j) ∈ Ω(n, d), m = m(i, j)} are linearly independent.

Now we can prove the theorem. Call (i, k, l, j) ∈ Y special if there exists g ∈ W_{d} such that

i_{gs} = k_{gs} = l_{s} whenever |l_{s}| < |j_{s}|,
l_{s} = j_{s} = k_{gs} whenever |l_{s}| = |j_{s}|

for all s = 1, …, d. We point out that if m = m(i, j), then (i, m, m, j) is special. Now, if (i, k, l, j) ≈ (i′, k′, l′, j′) and (i, k, l, j) is special, then (i′, k′, l′, j′) is too. So the property of being special is a property of ≈-equivalence classes. Choose a total order ≻ on the set of all special ≈-equivalence classes such that the following hold for all special (i, k, l, j), (i′, k′, l′, j′) ∈ Y:

(1) if wt(k′) > wt(k) (in the dominance order) then (i′, k′, l′, j′) ≻ (i, k, l, j);

(2) if wt(k) = wt(k′) and |{s | 1 ≤ s ≤ d, i_{s} = k_{s}}| > |{s | 1 ≤ s ≤ d, i′_{s} = k′_{s}}| then (i′, k′, l′, j′) ≻ (i, k, l, j).

We need one more claim:

7.7. Let i, j ∈ I(n, d) and m = m(i, j). Then,

π(b_{i,j}) = ±b_{i,m}^{♭} ⊗ b_{m,j}^{♯} + A + B

where A is a linear combination of terms of the form b_{i,k}^{♭} ⊗ b_{k,j}^{♯} with (i, k, k, j) special and (i, k, k, j) ≻ (i, m, m, j), and B is a linear combination of terms of the form b_{i,k}^{♭} ⊗ b_{k,j}^{♯} with (i, k, k, j) not special.

To prove (7.7), we have from the definition of π that

π(b_{i,j}) = ±b_{i,m}^{♭} ⊗ b_{m,j}^{♯} ± b_{i,−m}^{♭} ⊗ b_{−m,j}^{♯} + (a linear combination of b_{i,k}^{♭} ⊗ b_{k,j}^{♯} with |k| < |m|)

where m = min(|i|, |j|). So, writing m = m(i, j),

π(b_{i,j}) = ∑_{δ∈ℤ_{2}^{d}} ±b_{i,m·δ}^{♭} ⊗ b_{m·δ,j}^{♯} + (a linear combination of b_{i,k}^{♭} ⊗ b_{k,j}^{♯} with wt(k) > wt(m).)

Therefore, we just need to show that for all (¯0, ¯0, …, ¯0) ≠ δ ∈ ℤ_{2}^{d} such that (i, m · δ, m · δ, j) is special, we have that |{s | 1 ≤ s ≤ d, i_{s} = m_{s}}| > |{s | 1 ≤ s ≤ d, i_{s} = m_{δs}}|. Take δ ∈ ℤ_{2}^{d} such that (i, m · δ, m · δ, j) is special. Then certainly we have that m_{δs} = j_{s} whenever |m_{s}| = |j_{s}|, when m_{s} = j_{s} by definition of m. So for s with |m_{s}| = |j_{s}|, we have that m_{δs} = m_{s}, whence δ_{s} = ¯0. Instead, take t with |m_{t}| < |j_{t}|. Then, m_{t} = i_{t} so m_{t} = i_{δt} if and only if δ_{t} = ¯0. These observations establish that

|{s | 1 ≤ s ≤ d, i_{s} = m_{s}}| ≥ |{s | 1 ≤ s ≤ d, i_{s} = m_{δs}}|

with equality if and only if δ = (¯0, ¯0, …, ¯0). This completes the proof of (7.7).

Now the theorem follows easily from (7.6), (7.7) and a unitriangular argument involving the order ≻.

7.8. Corollary. The natural multiplication map μ : Q_{♭}(n, d) ⊗ Q_{♯}(n, d) → Q(n, d) is surjective.

8 Highest weight theory

Now we can classify the irreducible Q(n, d)-supermodules using highest weight theory. Recall that Q_{♯}(n, d) denotes the positive Borel subsuperalgebra of Q(n, d). We begin by determining the irreducible Q_{♯}(n, d)-supermodules.

The superideal 𝒥_{♯}(n) from §7 is contained in the superideal 𝒥_{0}(n) from §6. It follows that Q_{0}(n, d) ⊆ Q_{♯}(n, d). On the other hand, let Q_{+}(n, d) denote the subsuperspace of Q_{♯}(n, d) spanned by the elements

{ξ_{i,j} | (i, j) ∈ I^{2}(n, d), |i| ≤ |j|, |i_{s}| < |j_{s}| for some s}.

It follows from Lemma 6.1 that Q_{+}(n, d) is a superideal of Q_{♯}(n, d). Moreover, Q_{♯}(n, d) = Q_{0}(n, d) ⊕ Q_{+}(n, d) as a vector superspace, and Q_{♯}(n, d)/Q_{+}(n, d) ≅ Q_{0}(n, d). Analogously, Q_{−}(n, d) denotes the superideal spanned by the elements {ξ_{i,j}|(i, j) ∈ I^{2}(n, d), |i| ≥ |j|, |i_{s}| > |j_{s}| for some s}, and Q_{♭}(n, d) = Q_{0}(n, d) ⊕ Q_{−}(n, d).

If M is any Q_{0}(n, d)-supermodule, we can view it as a Q_{♯}(n, d)-supermodule by inflation along the quotient map Q_{♯}(n, d) ↠ Q_{0}(n, d). In particular, we obtain irreducible Q_{♯}(n, d)- modules denoted {U(λ) | λ ∈ Λ(n, d)}, namely, the inflations of the irreducible Q_{0}(n, d)- supermodules constructed in Lemma 6.8.

Now suppose that M is a non-zero Q_{♯}(n, d)-supermodule and λ ∈ Λ(n, d). By Lemma 6.1, for ξ ∈ Q_{+}(n, d), ξM_{λ} ⊆ ⨁_{μ>λ} M_{μ}. It follows at once that for any weight λ maximal in the dominance order such that M_{λ} ≠ 0 (such a weight certainly exists as there are finitely many weights!), the weight space M_{λ} is annihilated by Q_{+}(n, d). So M_{λ} is a Q_{♯}(n, d)-subsupermodule of M and the action of Q_{♯}(n, d) on M_{λ} factors through the quotient Q_{0}(n, d). In particular, if M is an irreducible Q_{♯}(n, d)-supermodule, M ≅ U(λ).

Given an arbitrary weight λ, we call a Q(n, d)-supermodule M a highest weight module of highest weight λ if the following conditions hold:

(1) M_{λ} is a Q_{♯}(n, d)-subsupermodule of M isomorphic to U(λ);

(2) M is generated as an Q(n, d)-supermodule by M_{λ}. For λ ∈ Λ(n, d), define

V (λ) := Q(n, d) ⊗_{Q_{♯}(n,d)} U(λ).  (8.1)

Call the weight λ an admissible weight if V (λ) ≠ 0.

8.2. Lemma. For admissible λ, V (λ) is a highest weight module of highest weight λ. Moreover, V (λ)_{μ} = 0 unless μ ≤ λ.

Proof. Recalling Corollary 7.8, we certainly have that

V (λ) = Q_{♭}(n, d) ⊗ U(λ) = Q_{−}(n, d) ⊗ U(λ) ⊕ Q_{0}(n, d) ⊗ U(λ).

All weights of Q_{−}(n, d) ⊗ U(λ) are strictly lower than λ in the dominance order. So the λ-weight space of V (λ) is equal to 1⊗U(λ), a homomorphic image of U(λ). The assumption that λ is admissible is equivalent to 1 ⊗ U(λ) being non-zero, in which case it is isomorphic to U(λ) as U(λ) is irreducible.

The admissible V (λ) have the following universal property:

8.3. Lemma. Suppose that M is a highest weight module of highest weight λ. Then, λ is admissible and M is a homomorphic image of V (λ). In particular, M_{μ} = 0 unless μ ≤ λ.

Proof. There is a natural isomorphism

Hom_{Q_{♯}(n,d)}(U(λ), M ↓) ⟶^{∼} Hom_{Q(n,d)}(V (λ), M).

Choose an isomorphism θ : U(λ) → M_{λ} ⊆ M of Q_{♯}(n, d)-supermodules and let θ ↑: V (λ) → M be the corresponding Q(n, d)-supermodule homomorphism. This is non-zero, hence λ is admissible, and is surjective as M is generated by M_{λ}. This shows that M is a quotient of V (λ), and the final statement about weights follows from Lemma 8.2.

For admissible λ, define L(λ) to be the head of V (λ), i.e. L(λ) is the largest completely reducible quotient supermodule of V (λ). We remark that if p = 0 or p > d, then Q(n, d) is semisimple by Lemma 4.1, so that L(λ) = V (λ) in these cases.

8.4. Lemma. The set {L(λ) | for all admissible λ ∈ Λ(n, d)} is a complete set of pairwise non-isomorphic irreducible Q(n, d)-supermodules. Moreover, L(λ) is absolutely irreducible if and only if h_{p′}(λ) is even.

Proof. Let λ be admissible. We first claim that V (λ) has a unique maximal subsupermodule, so that L(λ) is irreducible. For let M, N be two maximal subsupermodules of V (λ). Since V (λ)_{λ} is irreducible over Q_{0}(n, d) and generates V (λ) over Q(n, d), we must have that M_{λ} = N_{λ} = 0, so (M + N)_{λ} = 0. This shows that M + N is a proper subsupermodule of V (λ). Hence, M = M + N = N by maximality, as required.

Evidently, for admissible λ ≠ μ, L(λ) and L(μ) are not isomorphic, as they have different highest weights. Now suppose that L is an arbitrary irreducible Q(n, d)-supermodule. Choose λ maximal in the dominance order such that L_{λ} ≠ 0. Then, by irreducibility, L must be a highest weight module of highest weight λ, so a quotient of V (λ) by Lemma 8.3. Hence, L ≅ L(λ).

It remains to prove the statement about absolute irreducibility. First observe by adjointness that Hom_{Q(n,d)}(V (λ), L(λ)) ≅ Hom_{Q_{♯}(n,d)}(U(λ), L(λ) ↓) ≅ End_{Q_{0}(n,d)}(U(λ)). Now there is a natural embedding Hom_{Q(n,d)}(L(λ), L(λ)) ↪ Hom_{Q(n,d)}(V (λ), L(λ)). To see that it is an isomorphism, observe that any Q(n, d)-homomorphism V (λ) → L(λ) annihilates the unique maximal submodule of V (λ), hence induces a well-defined homomorphism L(λ) → L(λ). We have shown that End_{Q(n,d)}(L(λ)) ≅ End_{Q_{0}(n,d)}(U(λ)). Now the final part of the lemma follows from Lemma 6.8 and Schur’s lemma.

9 Classification of admissible weights

We now proceed to give a combinatorial description of the admissible weights, to complete the classification of the irreducible Q(n, d)-supermodules. We make some definitions. Let Λ^{+}(n, d) denote the set of all λ ∈ Λ(n, d) such that λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n}, i.e. λ is a partition of d with at most n non-zero parts. Let Λ_{p}^{+}(n, d) denote the set of all λ ∈ Λ^{+}(n, d) such that

λ_{i} = λ_{i+1} ⇒ p|λ_{i} for each i = 1, 2, …, n − 1.

Call λ ∈ Λ_{p}^{+}(n, d) restricted if either p = 0 or p > 0 and

\{ \begin{matrix}λ_{i} − λ_{i+1} ≤ p & if p ∤ λ_{i}, \\ λ_{i} − λ_{i+1} < p & if p | λ_{i} \end{matrix}

for each i = 1, 2, …, n − 1. Let Λ_{p}^{+}(n, d)_{res} denote the set of all restricted λ ∈ Λ_{p}^{+}(n, d).

We first construct another subsuperalgebra of Q(n, d). Let 𝒦 (n) = ⨁_{d≥0} 𝒦 (n, d) denote the ℤ-graded superideal of B(n) generated by the elements

{b_{i,j} | i = 1, …, n, j = −1, …, −n}.

It is a bisuperideal, so we can form the bisuperalgebra quotient

A(n) = B(n)/𝒦 (n),

this being ℤ-graded as A(n) = ⨁_{d≥0} A(n, d) where A(n, d) ≅ B(n, d)/𝒦 (n, d). For i, j = 1, …, n, set c_{i,j} = b_{i,j} + 𝒦 (n). Observing that each c_{i,j} has degree ¯0, A(n) = A(n)¯0 is precisely the free polynomial algebra on the generators {c_{i,j} | 1 ≤ i, j ≤ n}. So the dual superalgebra S(n, d) = A(n, d)^{∗} is just the usual classical Schur algebra as in [7] concentrated in degree ¯0. We identify S(n, d) with the subsuperalgebra 𝒦 (n, d)^{∘} ⊆ Q(n, d)¯0 ⊆ Q(n, d).

Now we treat the case n = 2, copying an argument due to Penkov [21, §7] in our setting.

9.1. Lemma. Suppose that n = 2 and that λ ∈ Λ(2, d) is an admissible weight. Then, either λ_{1} > λ_{2}, or λ_{1} = λ_{2} = c for some c ≥ 0 with p | c.

Proof. The restriction of L(λ) to the ordinary Schur algebra S(2, d) ⊆ Q(2, d) gives us an S(2, d)-module with maximal weight λ. We deduce from the classical theory that λ_{1} ≥ λ_{2}. To complete the proof, suppose for a contradiction that λ_{1} = λ_{2} = c but that p ∤ c. So d = 2c. Now, there are no μ ∈ Λ^{+}(2, 2c) with μ < λ. Since we also know that dim L(λ)_{λ} = dim U(λ) = 2, we deduce by the classical representation theory of S(2, 2c) that L(λ) ↓ S(2, 2c) splits as a direct sum of two irreducible S(2, 2c)-modules both of highest weight λ. But such S(2, 2c)-modules are one dimensional (being just a tensor power of the determinant module). This shows that L(λ) = L(λ)_{λ}, of dimension exactly two. Hence, L(λ)_{ν} = 0 for all ν ≠ λ.

Define the following elements of I(2, 2c):

i = (1, …, 1, −2; 2, …, 2, 2),  j = (1, …, 1, 2; 2, …, 2, 2),
k = (1, …, 1, 1; 2, …, 2, −1),  l = (1, …, 1, 1; 2, …, 2, 1),
s = (1, …, 1, −1; 2, …, 2, 2),  t = (1, …, 1, 1; 2, …, 2, −2),
u = (1, …, 1, −2; 2, …, 2, 1),  i_{λ} = (1, …, 1, 1; 2, …, 2, 2)

where the symbol ; is between the cth and (c + 1)th entries. Now an explicit calculation using the product rule Theorem 4.5 shows that

ξ_{i_{λ},j}ξ_{i,i_{λ}} = ξ_{s,i_{λ}} + ξ_{u,i_{λ}}  and  ξ_{i_{λ},k}ξ_{l,i_{λ}} = ξ_{t,i_{λ}} + ξ_{u,i_{λ}}.

Hence,

ξ_{i_{λ},j}ξ_{i,i_{λ}} − ξ_{i_{λ},k}ξ_{l,i_{λ}} = ξ_{s,i_{λ}} − ξ_{t,i_{λ}}.

Using the previous paragraph and a weight argument, both terms on the left hand side of this equation act as zero on L(λ)_{λ}. Hence, the term ξ_{s,i_{λ}} − ξ_{t,i_{λ}} ∈ ξ_{λ}Q_{0}(n, d) on the right hand side acts as zero on L(λ)_{λ} ≅ U(λ). But ξ_{λ}Q_{0}(n, d) ≅ C(2) according to (6.7), so as U(2) is a faithful C(2)-supermodule, the non-zero element ξ_{s,i_{λ}} − ξ_{t,i_{λ}} of ξ_{λ}Q_{0}(n, d) cannot act as zero on U(λ), a contradiction.

Now observe that for λ ∈ Λ(n, d), λ lies in Λ_{p}^{+}(n, d) if and only if for each i = 1, …, n−1 (λ_{i}, λ_{i+1}) lies in Λ_{p}^{+}(2, λ_{i}+λ_{i+1}). So by an argument involving restriction to various quotients of B(n) isomorphic to B(2), we have the following corollary of Lemma 9.1:

9.2. Corollary. If λ ∈ Λ(n, d) is admissible, then λ ∈ Λ_{p}^{+}(n, d).

It remains to prove that every λ ∈ Λ_{p}^{+}(n, d) is admissible, i.e. that there does exist some highest weight module of highest weight λ for each λ ∈ Λ_{p}^{+}(n, d). We first give a construction of some highest weight modules in the case p > 0 using a Frobenius twist argument. Recall from earlier in the section that A(n) denotes the free polynomial algebra on generators {c_{i,j} | 1 ≤ i, j ≤ n}, viewed as a bialgebra as in the classical polynomial representation theory of GL(n) [7]. In particular, we can view A(n) is a bisuperalgebra concentrated in degree ¯0.

9.3. Lemma. If p > 0, the unique algebra map σ : A(n) → B(n), such that c_{i,j} ↦ b_{i,j}^{p} for all 1 ≤ i, j ≤ n, is a bisuperalgebra embedding.

Proof. This is a routine check of relations, similar to that carried out in [6, §2.3].

In view of the lemma, there is a natural restriction functor

Fr : mod(A(n)) → mod(B(n)).

On objects, Fr is defined by sending an A(n)-cosupermodule M with structure map η : M → M ⊗A(n) to the B(n)-cosupermodule equal to M as a superspace with structure map (id ⊗σ) ∘ η; we call Fr M the Frobenius twist of M. On morphisms, Fr sends a morphism to the same linear map but regarded instead as a B(n)-cosupermodule map. We note that if M is a polynomial A(n)-cosupermodule of degree d, then Fr M is a B(n, pd)-cosupermodule. Also, let Fr : X(n) → X(n) be the linear map determined by Fr(x^{λ}) = x^{pλ} for each λ ∈ Λ(n), where pλ denotes (pλ_{1}, …, pλ_{n}). Then, the formula

ch(Fr M) = Fr(ch M)

describes the effect of the functor Fr at the level of characters.

9.4. Lemma. Suppose that λ ∈ Λ(n, d_{1}) is an admissible weight, and that μ ∈ Λ^{+}(n, d_{2}) is arbitrary. Then, λ + pμ ∈ Λ(n, d_{1} + pd_{2}) is an admissible weight. Moreover, all non-zero weights of L(λ + pμ) are of the form λ′ + pμ′ for λ′ ≤ λ and μ′ ≤ μ.

Proof. If p = 0, there is nothing to prove. Otherwise, by the classical theory, there exists an irreducible A(n)-comodule L′(μ) of highest weight μ. Regard L′(μ) instead as an A(n)- cosupermodule concentrated in degree ¯0 (say) and consider the B(n)-cosupermodule

M = L(λ) ⊗ Fr L′(μ).

It is a B(n, d_{1} + pd_{2})-cosupermodule, hence a Q(n, d_{1} + pd_{2})-supermodule. Its non-zero weights are of the form λ′ + pμ′ for λ ≤ λ and μ′ ≤ μ, and the weight λ + pμ definitely appears as a weight of M. Hence, there exists a highest weight module of highest weight λ + pμ, so λ + pμ is admissible. The statement about weights follows because L(λ + pμ) must then be a subquotient of M.

Now we are in a position to complete the classification of admissible weights by a counting argument. Recall the definition of the idempotent ξ_{ω} from §6.

9.5. Theorem. (i) λ ∈ Λ(n, d) is admissible if and only if λ ∈ Λ_{p}^{+}(n, d).

(ii) Assuming that n ≥ d and λ ∈ Λ_{p}^{+}(n, d), we have that ξ_{ω}L(λ) ≠ 0 if and only if λ ∈ Λ_{p}^{+}(n, d)_{res}.

Proof. Recalling Corollary 9.2, we just need to show for (i) that if λ ∈ Λ_{p}^{+}(n, d), then λ is admissible. We consider first the case n ≥ d, and proceed by induction on d = 0, 1, …, n. The result is trivially true in case d = 0. For n ≥ d > 0, take λ ∈ Λ_{p}^{+}(n, d). Suppose first that λ ∉ Λ_{p}^{+}(n, d)_{res}. Then, we can write λ = λ_{1} + pλ_{2} where λ_{1} ∈ Λ_{p}^{+}(n, d_{1}) and λ_{2} ∈ Λ^{+}(n, d_{2}) for some d_{1}, d_{2} with d = d_{1} + pd_{2} and d_{2} ≠ 0. By induction, λ_{1} is admissible, so we deduce from Lemma 9.4 that λ is admissible, and moreover that ξ_{ω}L(λ) = 0. But by Lemma 6.3, there are exactly |ℛ𝒫 _{p}(d)| = |Λ_{p}^{+}(n, d)_{res}| non-isomorphic irreducible Q(n, d)-supermodules not annihilated by ξ_{ω}. In view of Corollary 9.2, this means that all λ ∈ Λ_{p}^{+}(n, d)_{res} must both be admissible and satisfy ξ_{ω}L(λ) ≠ 0, else we end up with too few such modules.

Now suppose that n < d and choose m ≥ d. Let e ∈ Q(m, d) be the idempotent defined in (6.4), and also recall the embedding Λ(n, d) ↪ Λ(m, d) there. Take λ ∈ Λ_{p}^{+}(n, d). Then, viewing λ as an element of Λ_{p}^{+}(m, d), we have already shown in the previous paragraph that λ is admissible for Q(m, d), so that there exists an irreducible Q(m, d)-supermodule L(λ) of highest weight λ. Clearly, eL(λ)_{λ} ≠ 0 as λ ∈ Λ(n, d). Taking into account Lemma 6.5 and Corollary 2.13, eL(λ) is an irreducible Q(n, d)-supermodule of highest weight λ.

10 Decomposition numbers

In Theorem 9.5(i) and Lemma 8.4, we have classified the irreducible Q(n, d)-supermodules; they are precisely the supermodules {L(λ) | λ ∈ Λ_{p}^{+}(n, d)}. Applying Lemma 5.1, we have equivalently determined the irreducible B(n)-cosupermodules. Let Λ_{p}^{+}(n) = ⋃_{d≥0} Λ_{p}^{+}(n, d) denote the set of all p-strict partitions with at most n non-zero parts. Then, we have shown:

10.1. Theorem. The B(n)-cosupermodules {L(λ) | λ ∈ Λ_{p}^{+}(n)} give a complete set of pairwise non-isomorphic irreducible B(n)-cosupermodules. Moreover, L(λ) is absolutely irreducible if and only if h_{p′}(λ) is even.

Next we turn our attention to constructing the irreducible representations of the Sergeev superalgebra W (d). Let n ≥ d, and identify Λ_{p}^{+}(n, d) with the set 𝒫_{p}(d) of all p-strict partitions of d. Then, Λ_{p}^{+}(n, d)_{res} is identified with ℛ𝒫 _{p}(d) ⊆ 𝒫_{p}(d). Also let ξ_{ω} ∈ Q(n, d) be the idempotent from §6. For λ ∈ ℛ𝒫 _{p}(d), define the W (d)-supermodule

M(λ) := ξ_{ω}L(λ).

We should note that this definition is independent of the particular choice of n ≥ d up to natural isomorphism (this is proved in a standard way, see e.g. [5, §3.5]). The following result is immediate from Theorem 9.5(ii) and Corollary 2.13:

10.2. Theorem. The modules {M(λ) | λ ∈ ℛ𝒫 _{p}(d)} give a complete set of pairwise nonisomorphic irreducible W (d)-supermodules. Moreover, M(λ) is absolutely irreducible if and only if h_{p′}(λ) is even.

In order to obtain a labelling for all irreducible W (d)-modules, not just supermodules, we know by Lemma 2.3 that if M(λ) is self-associate, it decomposes as M(λ, +) ⊕ M(λ, −) for two non-isomorphic irreducible W (d)-modules M(λ, +), M(λ, −). By Corollary 2.8, the modules

{M(λ) | λ ∈ ℛ𝒫 _{p}(d), h_{p′}(λ) even} ∪ {M(λ, +), M(λ, −) | λ ∈ ℛ𝒫 _{p}(d), h_{p′}(λ) odd}

then give a complete set of pairwise non-isomorphic irreducible W (d)-modules.

To pass to the projective representations of the symmetric group, we use the functors F and G from §3 together with Corollary 3.5. Suppose first that d is even. For λ ∈ ℛ𝒫 _{p}(d), set D(λ) = GM(λ), an irreducible S(d)-supermodule which is absolutely irreducible if and only if M(λ) is absolutely irreducible, which is if and only if h_{p′}(λ) is even. In the case that d is odd, take λ ∈ ℛ𝒫 _{p}(d). If h_{p′}(λ) is even, we set D(λ) = GM(λ) as before, giving us a self-associate irreducible S(d)-supermodule. If h_{p′}(λ) is odd, there is an absolutely irreducible S(d)-supermodule D(λ), unique up to isomorphism, such that M(λ) ≅ F D(λ). Then, recalling Corollary 3.5, we have:

10.3. Theorem. The modules {D(λ) | λ ∈ ℛ𝒫 _{p}(d)} give a complete set of pairwise nonisomorphic irreducible S(d)-supermodules. Moreover, D(λ) is absolutely irreducible if and only if d − h_{p′}(λ) is even.

If λ ∈ ℛ𝒫 _{p}(d) and d − h_{p′}(λ) is odd, we can decompose D(λ) ≅ D(λ, +) ⊕ D(λ, −) as a direct sum of two non-isomorphic irreducible S(d)-modules, and by Corollary 2.8 the modules

{D(λ) | λ ∈ ℛ𝒫 _{p}(d), d − h_{p′}(λ) even} ∪ {D(λ, +), D(λ, −) | λ ∈ ℛ𝒫 _{p}(d), d − h_{p′}(λ) odd}

then give a complete set of pairwise non-isomorphic irreducible S(d)-modules. We have thus determined the irreducible projective representations of S_{d} over 𝕜.

The next theorem explains how to obtain the irreducible projective representations of the alternating group A_{d} from these. Let A(d) = S(d)¯0. Providing d > 7, this is up to isomorphism the only twisted group algebra of A_{d} over 𝕜, other than the group algebra 𝕜A_{d} itself. The following theorem is proved by arguments analogous to the Clifford theory for groups with normal subgroups of index two.

10.4. Theorem. Let λ ∈ ℛ𝒫 _{p}(d). If d − h_{p′}(λ) is even, D(λ) ↓_{A(d)}≅ E(λ, +) ⊕ E(λ, −) for two non-isomorphic irreducible A(d)-modules E(λ, +), E(λ, −). If d − h_{p′}(λ) is odd, D(λ) ↓_{A(d)}≅ E(λ) ⊕ E(λ) for a single irreducible A(d)-module E(λ). The modules

{E(λ) | λ ∈ ℛ𝒫 _{p}(d), d − h_{p′}(λ) odd} ∪ {E(λ, +), E(λ, −) | λ ∈ ℛ𝒫 _{p}(d), d − h_{p′}(λ) even}

then give a complete set of pairwise non-isomorphic irreducible A(d)-modules.

10.5. Remark. We have assumed up to now that 𝕜 is algebraically closed. In fact, the construction of the irreducible (super)modules of Q(n, d), W (d), S(d) and A(d) that we have described can be carried out in precisely the same way over any field 𝕜 of characteristic different from 2 providing only that 𝕜 contains square roots of all ±1, …, ±d. In fact, any such field is a splitting field for each of the algebras Q(n, d), W (d), S(d) and A(d). This is proved by reducing using a Schur functor argument to the case of Q(n, d), where as explained in the proof of Lemma 8.4,

End_{Q(n,d)}(L(λ)) ≅ End_{Q_{0}(n,d)}(U(λ)).

If 𝕜 contains square roots of all ±1, …, ±d, then 𝕜 is a splitting field for each of the Clifford superalgebras C(1), …, C(d), hence for Q_{0}(n, d). So the right hand side is then one or two dimensional according to whether L(λ) is absolutely irreducible or self-associate, as required to prove that 𝕜 is a splitting field.

We conclude with some discussion of decomposition numbers. It is immediate from highest weight theory that the character map ch : Grot(B(n)) → X(n) described at the end of §6 is an embedding of the Grothendieck ring of the category of B(n)-cosupermodules into X(n). Set L_{λ} = ch L(λ), for λ ∈ Λ_{p}^{+}(n). Then, the elements

{L_{λ} | λ ∈ Λ_{p}^{+}(n)}

of X(n) form a ℤ-basis for the image of ch. For λ ∈ Λ^{+}(n), Schur’s P -function P_{λ} is defined by:

P_{λ} = ∑_{w∈S_{n}/S_{λ}} w \{x^{λ} \frac{∏_{λ_{i}>λ_{j}}(x_{i} + x_{j})}{∏_{λ_{i}>λ_{j}}(x_{i} − x_{j})}\} ,  (10.6)

where S_{λ} denotes the stablizier of x^{λ} in S_{n} and S_{n}/S_{λ} is some choice of left coset representatives. This is the definition from [16, III(2.2)] (with t there equal to −1, compare [16, III.8]). For λ ∈ Λ_{p}^{+}(n), let

E_{λ} = 2^{⌊(h_{p′}(λ)+1)/2⌋}P_{λ}.

The E_{λ} arise naturally as certain Euler characteristics, in an analogous way to the construction in the work of Penkov and Serganova in characteristic 0, see [22, Prop.1] and [23].

(Fuller details in the positive characteristic case will appear elsewhere.) In particular, E_{λ} is an alternating sum of characters of B(n)-cosupermodules. Since E_{λ} and L_{λ} have the same leading term 2^{⌊(h_{p′}(λ)+1)/2⌋}x^{λ} plus a linear combination of lower terms lower with respect to the dominance order, it follows easily that

{E_{λ} | λ ∈ Λ_{p}^{+}(n)}

also forms a ℤ-basis for the image of ch. So we can write

E_{λ} = ∑_{μ∈Λ_{p}^{+}(n)} d_{λ,μ}L_{μ}

for uniquely determined d_{λ,μ} ∈ ℤ with d_{λ,λ} = 1 and d_{λ,μ} = 0 if μ ≰ λ. We will call the matrix D = (d_{λ,μ})_{λ,μ∈Λ_{p}^{+}(n,d)} the decomposition matrix of Q(n, d) in characteristic p.

Now suppose that (𝕜, R, K) is a p-modular system with K sufficiently large (specifically, containing square roots of ±1, …, ±d). So, R is a complete discrete valuation ring, K is its field of fractions of characteristic 0 and our fixed algebraically closed field 𝕜 of characteristic p is its residue field. The bisuperalgebra B(n) can be defined in exactly the same as in §5 but over the ground ring R, giving us an R-free R-bisuperalgebra B(n)_{R} such that B(n) ≅ B(n)_{R} ⊗_{R} 𝕜. Set Q(n, d)_{R} = Hom_{R}(B(n, d)_{R}, R) to obtain an R-form of the Schur superalgebra Q(n, d). So, Q(n, d)_{R} is R-free as an R-module and Q(n, d) ≅ Q(n, d)_{R} ⊗_{R} 𝕜; we will from now on identify the two. Also, set Q(n, d)_{K} = Q(n, d)_{R} ⊗_{R} K, the analogous Schur superalgebra over the ground field K. Similarly, we can define an R-form Q_{0}(n, d)_{R} of Q_{0}(n, d), and set Q_{0}(n, d)_{K} = Q_{0}(n, d)_{R} ⊗_{R} K. We will view Q(n, d)_{R} and Q_{0}(n, d)_{R} as R-subsuperalgebras of Q(n, d)_{K}.

For λ ∈ Λ_{0}^{+}(n, d), let V (λ)_{K} denote the irreducible Q(n, d)_{K}-supermodule of highest weight λ, constructed as in (8.1). By Sergeev’s character formula [25, Theorem 4],

ch V (λ)_{K} = 2^{⌊(h(λ)+1)/2⌋}P_{λ}

where h(λ) is the number of non-zero parts of λ. Denote the highest weight space of V (λ)_{K} by U(λ)_{K}; this is precisely the Q_{0}(n, d)_{K}-supermodule defined as in §6. Now, the construction of U(λ)_{K} can be carried out over R instead, because R contains square roots of each ±λ_{i}, giving us a finitely generated R-free Q_{0}(n, d)_{R}-subsupermodule U(λ)_{R} of U(λ)_{K} such that U(λ)_{K} ≅ U(λ)_{R} ⊗_{R} K. Let V (λ)_{R} denote the Q(n, d)_{R}-subsupermodule of V (λ)_{K} generated by U(λ)_{R}. Then, V (λ)_{R} is a finitely generated R-free R-module such that V (λ)_{K} ≅ V (λ)_{R} ⊗_{R} K. Now set \overline{V} (λ) := V (λ)_{R} ⊗_{R} 𝕜. This gives us a Q(n, d)-supermodule such that

ch \overline{V} (λ) = ch V (λ)_{K} = 2^{⌊(h(λ)+1)/2⌋−⌊(h_{p′}(λ)+1)/2⌋}E_{λ}.

In particular, we deduce:

10.7. Theorem. For λ ∈ Λ_{0}^{+}(n) and μ ∈ Λ_{p}^{+}(n), the decomposition number d_{λ,μ} defined above is a non-negative integer.

One can hope that in fact d_{λ,μ} ≥ 0 for all λ, μ ∈ Λ_{p}^{+}(n).

Finally, we relate the decomposition matrix D of Q(n, d) for n ≥ d to the decomposition matrices of the superalgebras W (d) and S(d). Using the subscript K to indicate that we are working over the ground field K instead of our usual 𝕜, we have irreducible W (d)_{K}- (resp. S(d)_{K}-) supermodules labelled by strict partitions λ ∈ 𝒫_{0}(d), which we denote by M(λ)_{K} and D(λ)_{K} respectively. By a straightforward extension of Brauer’s theory, we can reduce these modulo p to obtain W (d)- (resp. S(d)-) supermodules \overline{M}(λ) and \overline{D}(λ). These are not uniquely determined up to isomorphism, but at least the multiplicities of composition factors are unique. So we obtain well-defined decomposition matrices D^{S} = (d_{λ,μ}^{S}) and D^{W} = (d_{λ,μ}^{W}) of S(d) and W (d) respectively, for λ ∈ 𝒫_{0}(d), μ ∈ ℛ𝒫 _{p}(d), determined by the equations

[\overline{M}(λ)] = ∑_{μ∈ℛ𝒫 _{p}(d)} d_{λ,μ}^{W}[M(μ)], [\overline{D}(λ)] = ∑_{μ∈ℛ𝒫 _{p}(d)} d_{λ,μ}^{S}[D(μ)]

written in the Grothendieck groups of mod(W (d)) and mod(S(d)) respectively. The final theorem relates these decomposition numbers to those of the Schur superalgebra Q(n, d):

10.8. Theorem. Let D = (d_{λ,μ})_{λ,μ∈Λ_{p}^{+}(n,d)} be the decomposition matrix of Q(d, d) in characteristic p, as defined above. Then, for any λ ∈ 𝒫_{0}(d) and μ ∈ ℛ𝒫 _{p}(d),

d_{λ,μ}^{W} = 2^{⌊(h(λ)+1)/2⌋−⌊(h_{p′}(λ)+1)/2⌋}d_{λ,μ}.

Moreover, if d is even,

d_{λ,μ}^{S} = d_{λ,μ}^{W},

while if d is odd,

d_{λ,μ}^{S} = \{ \begin{matrix}d_{λ,μ}^{W} & if h(λ) − h_{p′}(μ) is even, \\ 2d_{λ,μ}^{W} & if h(λ) is even and h_{p′}(μ) is odd, \\ \frac{1}{2}d_{λ,μ}^{W} & if h(λ) is odd and h_{p′}(μ) is even. \end{matrix}

Proof. The Schur functor coming from the idempotent ξ_{ω} can be defined over the ground ring R, using an R-integral version of Theorem 6.2. Using that Schur functors commute with base change, one sees that [ξ_{ω}\overline{V} (λ)] = [\overline{M}(λ)] (equality written in the Grothendieck group). In particular, it follows from this by exactness of Schur functors that d_{λ,μ}^{W} = d_{λ,μ}.

Similarly, the functors F from §3 can be defined over the ground ring R, and F evidently commutes with base change. So in the case that d is even, [F \overline{D}(λ)] = [\overline{M}(λ)] and F D(μ) = M(μ) by Theorem 3.4 over K or 𝕜 respectively, hence d_{λ,μ}^{S} = d_{λ,μ}^{W}.

Finally, suppose that d is odd. Applying Theorem 3.4 and Lemma 2.9 over K or 𝕜 respectively, we have that

[F \overline{D}(λ)] = \{ \begin{matrix}[\overline{M}(λ)] & if h(λ) is odd, \\ 2[\overline{M}(λ)] & if h(λ) is even; \end{matrix}
[F D(μ)] = \{ \begin{matrix}[M(μ)] & if h_{p′}(μ) is odd, \\ 2[M(μ)] & if h_{p′}(λ) is even. \end{matrix}

The theorem follows from these equations together with exactness of F.

Thus our results show that the decomposition matrices for projective representations of the symmetric group S_{d} can be deduced from knowledge of the decomposition matrix of the Schur superalgebra Q(d, d). In [14], a precise conjecture is made relating decomposition matrices for projective representations of S_{d} to the specialization at q = 1 of certain polynomials d_{λ,μ}(q) arising as coefficients of the canonical basis of the identity component of the Fock space of U_{q}(A_{p−1}^{(2)}). Indeed, it appears that for λ ∈ 𝒫_{p}(d), μ ∈ ℛ𝒫 _{p}(d), the integer d_{λ,μ}(1) as defined in [14] should equal the decomposition number d_{λ,μ} of Q(d, d) (as defined above) providing d < p^{2}. This statement is essentially a reformulation of the conjecture made by Leclerc and Thibon in [14]. It would be interesting to extend the Leclerc-Thibon construction of the canonical basis of the identity component of the Fock space of U_{q}(A_{p−1}^{(2)}) to the entire Fock space, as was done in [13] for the case of U_{q}(A_{p−1}^{(1)}), to obtain a conjectural algorithm for computing d_{λ,μ} for all μ ∈ 𝒫_{p}(d).

References

[1] G. E. Andrews, C. Bessenrodt and J. B. Olsson, Partition identities and labels for some modular characters, Trans. Amer. Math. Soc. 344 (1994), 597–615.

[2] G. E. Andrews and J. B. Olsson, Partition identities with an application to group representation theory, J. reine angew. Math. 413 (1991), 198–212.

[3] C. Bessenrodt, A. O. Morris, and J. B. Olsson, Decomposition matrices for spin characters of symmetric groups at characteristic 3, J. Algebra 164 (1994), 146–172.

[4] N. Blackburn, The extension theory of the symmetric and alternating groups, Math. Z. 117 (1970), 191–206.

[5] J. Brundan, R. Dipper and A. Kleshchev, Quantum linear groups and representations of GL_{n}(𝔽_{q}), preprint, University of Oregon, 1999.

[6] S. Donkin, Symmetric and exterior powers, linear source modules and representations of supergroups, preprint, Queen Mary and Westfield College, 1999.

[7] J. A. Green, Polynomial representations of GL_{n}, Lecture Notes in Math., vol. 830, Springer-Verlag, 1980.

[8] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.

[9] J. C. Jantzen and G. M. Seitz, On the representation theory of the symmetric groups, Proc. London Math. Soc. 65 (1992), 475–504.

[10] T. Józefiak, Semisimple superalgebras, in: Algebra – some current trends (Varna, 1986), pp. 96–113, Lecture Notes in Math. 1352, Springer, Berlin-New York, 1988.

[11] M. Kashiwara, T. Miwa, J.-U. Peterson and C. Yung, Perfect crystals and q-deformed Fock spaces, Selecta Math. (N.S.) 2 (1996), 415–499.

[12] P. Landrock, Finite Group Algebras and their Modules, Cambridge University Press, Cambridge, 1983.

[13] B. Leclerc and J.-Y. Thibon, Canonical bases of q-deformed Fock spaces, Internat. Math. Res. Notices 9 (1996), 447–456.

[14] B. Leclerc and J.-Y. Thibon, q-Deformed Fock spaces and modular representations of spin symmetric groups, J. Phys. A 30 (1997), 6163–6176.

[15] D.A. Leites, Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1980), 1–64.

[16] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, second edition, OUP, 1995.

[17] Yu I. Manin, Gauge field theory and complex geometry, Grundlehren der mathematischen Wissenschaften 289, second edition, Springer, 1997.

[18] N. J. Muir, Polynomial representations of the general linear Lie superalgebra, Ph.D. thesis, University of London, 1991.

[19] M. Nazarov, Young’s orthogonal form of irreducible projective representations of the symmetric group, J. London Math. Soc. 42 (1990), 437–451.

[20] M. Nazarov, Young symmetrizers for projective representations of the symmetric group, Advances in Math. 127 (1997), 190–257.

[21] I. Penkov, Characters of typical irreducible finite dimemsional 𝔮(n)-modules, Func. Anal. Appl. 20 (1986), 30–37.

[22] I. Penkov and V. Serganova, Cohomology of G/P for classical complex Lie supergroups G and characters of some atypical G-modules, Ann. Inst. Fourier 39 (1989), 845–873.

[23] I. Penkov and V. Serganova, Characters of irreducible G-modules and cohomology of G/P for the Lie supergroup G = Q(N), J. Math. Sci. 84 (1997), no. 5, 1382–1412.

[24] I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155– 250.

[25] A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras GL(n, m) and Q(n), Math. USSR Sbornik 51 (1985), 419–427.

[26] A. N. Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434.

[27] J. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Advances in Math. 74 (1989), 87–134.

brundan@darkwing.uoregon.edu, klesh@math.uoregon.edu

Department of Mathematics, University of Oregon, Eugene, Oregon 97403, U.S.A.