Modular Littlewood-Richardson Coefficients

Jonathan Brundan and Alexander Kleshchev

Introduction

Let 𝔽 be an algebraically closed field of characteristic p ≥ 0. We are interested in polynomial representations of the general linear group GL(n) := GL_{n}(𝔽) and modular representations of the symmetric group Σ_{r} over 𝔽. If p = 0 then the classical Littlewood-Richardson coefficients describe the decomposition of all of the following modules:

(M1) the restriction of an irreducible GL(n)-module to a Levi subgroup of GL(n); (M2) the tensor product of irreducible GL(n)-modules; (M3) the restriction of an irreducible Σ_{r}-module to a Young subgroup of Σ_{r}; (M4) the Σ_{r}-module induced from an irreducible module for a Young subgroup. However in positive characteristic, very little is known about the structure of these mod-

ules. The goal of the article is to describe three quite different sources of connections between them, hence obtaining connections between various ‘modular Littlewood-Richardson coefficients’.

In the modular case, we also have tilting modules (for general linear groups) and Young modules (for symmetric groups). So we also consider:

(M1′) the restriction of an indecomposable tilting module to a Levi subgroup of GL(n); (M2′) the tensor product of indecomposable tilting modules; (M3′) the restriction of a Young module to a Young subgroup of Σ_{r}; (M4′) the Σ_{r}-module induced from a Young module for a Young subgroup. Our main results, in sections 2, 3 and 4 respectively, obtain the following connections

between these modules:

First, we construct a polynomial induction functor from a Levi subgroup of GL(n) to GL(n), which plays the same role in the GL(n)-setting as ordinary induction does for symmetric groups. Our main result about this induction functor (Theorem 2.7) shows that, when applied to an outer tensor product of modules for the Levi subgroup, it gives an inner tensor product of GL(n)-modules in a precise way. This gives a functorial connection between branching rules and tensor products via ‘Frobenius reciprocity’. For example, it allows us to give a direct relationship between the spaces of high weight vectors and the socles of the modules (M1) and (M2) (see Theorem 2.8 and Theorem 2.20). As a special case, taking the Levi subgroup to be a maximal torus of GL(n), we recover a character formula due to Donkin [D3] (see Corollary 2.11).

Next, we use dual pairs to explain the connection between composition multiplicities in (M1) to the tilting module decomposition multiplicities in (M2′), and similarly between the composition multiplicities in (M2) and the tilting decomposition multiplicities in (M1′). The main result here is Theorem 3.7. The proof depends on a modular version of Howe duality (as in [D3] and [AR]) allowing us to construct a dual pair associated to a Levi subgroup of GL(n). As a special case, taking the Levi subgroup to be a maximal torus of GL(n) again, we recover two character formulae (see Corollary 3.8 and Corollary 3.9), the first of which is a result of Mathieu-Papadopoulo [MP].

Finally, we obtain the analogues of these results for the symmetric groups. To do this, we first explain how to apply Schur functors to tensor products of GL(n)-modules (resp. restrictions of GL(n)-modules to Levi subgroups), to obtain induced modules from Young subgroups (resp. restrictions to Young subgroups) in the symmetric group setting. Using this, we then translate our earlier connections in the GL(n)-setting to analogous connections for symmetric groups. See Theorems 4.10, 4.11, 4.16, 4.17 and 4.19.

To give the reader a flavour of the questions to be considered in this paper we formulate some special cases of our results. If G and H are groups, M is a G-module and N is an H-module, we denote by M ⊠ N the outer tensor product (which is a G × H-module). If L is another G-module then M ⊗ L is the inner tensor product (which is a G-module with the diagonal action). If L is irreducible we denote by [M : L] its multiplicity in M, and if L is indecomposable its multiplicity as a direct summand of M will be denoted (M : L). For a subgroup G_{1} < G we write M ↓_{G_{1}} (or just M ↓ if the subgroup is clear from the context) for the restriction of M to G_{1}. Finally, S^{i}(M) and ⋀^{i}(M) denote the ith symmetric and exterior powers of M respectively.

The compositions (resp. partitions) with at most n non-zero parts are identified with ‘polynomial’ weights (resp. ‘polynomial’ dominant weights) for GL(n) and we denote the set of all such by Λ(n) (resp. Λ^{+}(n)). The partitions of r in Λ^{+}(n) are denoted by Λ^{+}(n, r), and we write |λ| = r to indicate that λ is a partition of r. The partitions λ with at most n non-zero parts and such that the first part λ_{1} is at most m are denoted Λ^{+}(n × m). The transpose λ^{t} of a partition λ is the partition whose Young diagram is the transpose of the Young diagram λ (with respect to the main diagonal). If λ ∈ Λ^{+}(n×m) then λ^{t} ∈ Λ^{+}(m×n). For λ ∈ Λ^{+}(n), we write L_{n}(λ), Δ_{n}(λ), ∇_{n}(λ), and T_{n}(λ) for the irreducible, standard (or Weyl), costandard and the indecomposable tilting modules over GL(n) with highest weight λ, respectively. We always consider GL(m)×GL(n) as a subgroup of GL(m+n) (embedded diagonally).

Theorem A. Let μ, ν ∈ Λ^{+}(m+n) be partitions such that μ has at most m non-zero parts, and ν has at most n non-zero parts. Set μ̄ = (μ_{1}, …, μ_{m}) ∈ Λ^{+}(m) and ν̄ = (ν_{1}, …, ν_{n}) ∈ Λ^{+}(n). Let M be any polynomial GL(m + n)-module. Then,

(i) Hom_{GL(m+n)}(M, ∇_{m+n}(μ) ⊗ ∇_{m+n}(ν)) ≅ Hom_{GL(m)×GL(n)}(M ↓, ∇_{m}(μ̄) ⊠ ∇_{n}(ν̄)). If in addition |μ| ≤ m and |ν| ≤ n, then

(ii) Hom_{GL(m+n)}(M, L_{m+n}(μ) ⊗ L_{m+n}(ν)) ≅ Hom_{GL(m)×GL(n)}(M ↓, L_{m}(μ̄) ⊠ L_{n}(ν̄)).

If we take M = Δ_{n}(λ) for some λ ∈ Λ^{+}(n), the numbers appearing in Theorem A(i) are – by standard properties of ∇-filtrations – the usual Littlewood-Richardson coefficients. If instead we take M = L_{n}(λ), we obtain our first candidate for modular Littlewood-Richardson coefficients. The other candidates are the numbers appearing in (i) and (ii) of the next theorem. We note that the GL(m) × GL(n)-module T_{m}(μ) ⊠ T_{n}(ν) appearing in Theorem B is an indecomposable tilting module.

Theorem B. Let λ ∈ Λ^{+}((m + n) × k), μ ∈ Λ^{+}(m × k) and ν ∈ Λ^{+}(n × k). Then, (i) (T_{m+n}(λ) ↓_{GL(m)×GL(n)}: T_{m}(μ) ⊠ T_{n}(ν)) = [L_{k}(μ^{t}) ⊗ L_{k}(ν^{t}) : L_{k}(λ^{t})]. (ii) [L_{m+n}(λ) ↓_{GL(m)×GL(n)}: L_{m}(μ) ⊠ L_{n}(ν)] = (T_{k}(μ^{t}) ⊗ T_{k}(ν^{t}) : T_{k}(λ^{t})).

Now we state the character formulae mentioned earlier.

Theorem C. Let λ ∈ Λ^{+}(n × m), μ = (μ_{1}, …, μ_{n}) ∈ Λ(n × m). Let V and W be the natural GL(n)- and GL(m)-modules respectively. Then,

(i) (Donkin) dim L_{n}(λ)_{μ} = (S^{μ_{1}}(V) ⊗ ⋯ ⊗ S^{μ_{n}}(V) : Q_{n}(λ)), where Q_{n}(λ) is the the injective hull of L_{n}(λ) in the category of polynomial GL(n)-modules.

(ii) (Mathieu-Papadopoulo) dim L_{n}(λ)_{μ} = (⋀^{μ_{1}}(W) ⊗ ⋯ ⊗ ⋀^{μ_{n}}(W) : T_{m}(λ^{t})).

(iii) dim T_{n}(λ)_{μ} = (⋀^{μ_{1}}(W) ⊗ ⋯ ⊗ ⋀^{μ_{n}}(W) : L_{m}(λ^{t})).

Now, let Σ_{n} be the symmetric group on n letters. The irreducible 𝔽Σ_{n} module corresponding to a p-regular partition λ of n is denoted by D^{λ} as in [J1]. Let Y^{μ} be the Young module corresponding to (an arbitrary) partition μ of n (see [J2]).

Theorem D. Let λ, μ and ν be partitions of m + n, m, and n, respectively. Then, (i) (Y^{λ} ↓_{Σ_{m}×Σ_{n}}: Y^{μ} ⊠ Y^{ν}) = [L_{m+n}(μ^{t}) ⊗ L_{m+n}(ν^{t}) : L_{m+n}(λ^{t})]; (ii) ((Y^{μ} ⊠ Y^{ν}) ↑^{Σ_{m+n}}: Y^{λ}) = [L_{m+n}(λ) ↓_{GL(m)×GL(n)}: L_{m}(μ^{t}) ⊠ L_{n}(ν^{t})].

Moreover, if λ, μ and ν are all p-regular, then (iii) [D^{λ} ↓_{Σ_{m}×Σ_{n}}: D^{μ} ⊠ D^{ν}] = (T_{m+n}(μ^{t}) ⊗ T_{m+n}(ν^{t}) : T_{m+n}(λ^{t})); (iv) [(D^{μ} ⊠ D^{ν}) ↑^{Σ_{m+n}}: D^{λ}] = (T_{m+n}(λ^{t}) ↓_{GL(m)×GL(n)}: T_{m}(μ^{t}) ⊠ T_{n}(ν^{t})).

The main results of the article will be applied in [BK], where we use our earlier work [K2, K3, K4, B1, B2] on modular branching rules from GL(n) to GL(n − 1) to obtain new results about versions of ‘translation functors’ in general linear and symmetric groups. In particular, we will compute certain modular Littlewood-Richardson coefficients exactly.

Acknowledgement. The material on dual pairs in section 3 of the paper was explained to us by O. Mathieu. In particular, he communicated to us Lemma 3.1 and Theorem D(iii). We are very grateful to Prof. Mathieu for the illuminating discussion.

1 Preliminaries

We introduce some notation for rational and polynomial representations of GL(n) and its Levi subgroups. We begin at the level of rational representations, following Jantzen [J]. A GL(n)-module always means a rational 𝔽GL(n)-module as in [J], and we denote the category of all GL(n)-modules by GL(n) -Rat. More generally, given any algebraic group G over 𝔽, a G-module means a rational 𝔽G-module and G -Rat denotes the category of all such G-modules.

Let T (n) denote the fixed maximal torus of GL(n) consisting of all diagonal invertible matrices, and let B^{+}(n) denote the positive Borel subgroup consisting of all upper triangular invertible matrices. The character group of T (n) is the free abelian group with generators ε_{1}, …, ε_{n}, where ε_{i} denotes the standard character defined by ε_{i}(diag(t_{1}, …, t_{n})) = t_{i} for all t_{1}, …, t_{n} ∈ 𝔽^{×}. The root system of GL(n) (relative to T (n)) is the set {ε_{i} − ε_{j} | 1 ≤ i, j ≤ n, i ≠ j}, and the root ε_{i} − ε_{j} is positive if i < j.

We let X(n) denote all n-tuples λ = (λ_{1}, …, λ_{n}) of integers, and call the elements of X(n) weights. We identify X(n) with the character group of T (n), by letting (λ_{1}, …, λ_{n}) ∈ X(n) correspond to the character ∑_{i} λ_{i}ε_{i}. We have the usual dominance order on X(n), defined by λ > μ if (λ − μ) is a sum of positive roots. A weight (λ_{1}, …, λ_{n}) ∈ X(n) is dominant (relative to B^{+}(n)) precisely when λ_{1} ≥ ⋯ ≥ λ_{n}, and we let X^{+}(n) denote all such dominant weights. Say λ ∈ X^{+}(n) is p-restricted if λ_{i} − λ_{i+1} < p for i = 1, …, n − 1.

For λ ∈ X^{+}(n), we have the GL(n)-modules L_{n}(λ), Δ_{n}(λ) (denoted V (λ) in [J]) and ∇_{n}(λ) (denoted H^{0}(λ) in [J]) which are the irreducible, standard and costandard modules of highest weight λ respectively. We also denote the natural GL(n)-module Δ_{n}(ε_{1}) by V, with standard basis e_{1}, …, e_{n}. Let f_{1}, …, f_{n} denote the dual basis for V^{∗} ≅ Δ_{n}(−ε_{n}).

Let Λ(n) ⊂ X(n) denote all n-tuples (α_{1}, …, α_{n}) satisfying α_{i} ≥ 0 for i = 1, …, n, and Λ^{+}(n) := Λ(n) ∩ X^{+}(n). Let Λ(n, r) ⊂ Λ(n) denote all n-tuples (α_{1}, …, α_{n}) satisfying |α| := α_{1} + ⋯ + α_{n} = r, and Λ^{+}(n, r) := Λ(n, r) ∩ X^{+}(n). We call elements of Λ(n, r) compositions of r (with at most n non-zero parts), and elements of Λ^{+}(n, r) partitions of r (with at most n non-zero parts).

Fix a ≥ 1 and ν = (n_{1}, …, n_{a}) ∈ Λ(a, n) (a composition of n with at most a nonzero parts). Let GL(ν) = GL(n_{1}) × ⋯ × GL(n_{a}) denote the standard Levi subgroup of GL(n) consisting of all invertible block diagonal matrices with block sizes n_{1}, …, n_{a}. Of course, if ν = (n) then GL(ν) = GL(n) while, at the other extreme, if ν = (1, …, 1) then GL(ν) = T (n).

The torus T (n) is also a maximal torus of GL(ν). Given a weight λ ∈ X(n) which is dominant with respect to the Borel subgroup B^{+}(ν) := GL(ν)∩B^{+}(n) of GL(ν), we denote the corresponding irreducible, standard and costandard GL(ν)-modules of highest weight λ by L_{ν}(λ), Δ_{ν}(λ) and ∇_{ν}(λ) respectively. Let

X(ν) := X(n_{1}) × ⋯ × X(n_{a}).

The dominance order on X(ν) is defined as the product of the orders on X(n_{1}), …, X(n_{a}). The bijective map X(n) → X(ν) defined by λ ↦ (λ^{(1)}, …, λ^{(a)}), where for j = 1, …, a, λ^{(j)} := (λ_{n_{1}+···+n_{j−1}+1}, …, λ_{n_{1}+···+n_{j}}), induces a bijection between the B^{+}(ν)-dominant weights in X(n) and the set

X^{+}(ν) := X^{+}(n_{1}) × ⋯ × X^{+}(n_{a}).

Moreover, for λ ∈ X^{+}(ν), the GL(ν)-modules L_{ν}(λ), Δ_{ν}(λ) and ∇_{ν}(λ) correspond under this bijection to the GL(n_{1}) × ⋯ × GL(n_{a})-modules L_{n_{1}}(λ^{(1)}) ⊠ ⋯ ⊠ L_{n_{a}}(λ^{(a)}), ∇_{n_{1}}(λ^{(1)}) ⊠ ⋯ ⊠ ∇_{n_{a}}(λ^{(a)}) and Δ_{n_{1}}(λ^{(1)}) ⊠ ⋯ ⊠ Δ_{n_{a}}(λ^{(a)}) respectively, where ⊠ denotes outer tensor product.

Following [V], we say that the GL(ν)-module M has a ∇-filtration (resp. a Δfiltration) if it has an ascending filtration 0 = M_{0} < M_{1} < … with ⋃_{i≥0} M_{i} = M such that each factor M_{i}/M_{i−1} is isomorphic to a (possibly infinite) direct sum of copies of ∇_{ν}(λ_{i}) (resp Δ_{ν}(λ_{i})) for some λ_{i} ∈ X^{+}(ν).

We say a GL(ν)-module is tilting if it has both a Δ-filtration and a ∇-filtration. Donkin [D3] has shown that, for each λ ∈ X^{+}(ν), there a unique (finite dimensional) indecomposable tilting module T_{ν}(λ) of highest weight λ, satisfying [T_{ν}(λ) : L_{ν}(λ)] = 1 and [T_{ν}(λ) : L_{ν}(μ)] = 0 for μ ≰ λ.

For a left exact additive functor F from an abelian category with enough injectives to another category, R^{i}F denotes its ith (right) derived functor. The category GL(ν) -Rat contain enough injectives, and Ext_{GL(ν)}^{i}(M, −) denotes R^{i} Hom_{GL(ν)}(M, −). We recall the cohomological criterion for ∇-filtrations (see [J, II.4.16(b)] and the proof of [V, Theorem 3.2.7]):

1.1. A module M ∈ GL(ν) -Rat has a ∇-filtration if and only if Ext_{GL(ν)}^{1}(Δ(λ), M) = 0 for all λ ∈ X^{+}(ν).

Note that the last few definitions, and (1.1), include GL(n) as a special case, taking ν = (n).

Now, there are two commuting left actions of GL(n) on its coordinate ring 𝔽[GL(n)], the left regular and right regular actions, which we define for g, g′ ∈ GL(n), f ∈ 𝔽[GL(n)] by (gf)(g′) = f(g^{−1}g′) and (gf)(g′) = f(g′g) respectively. Regarding GL(ν) as a closed subgroup of GL(n), we have the usual restriction functor res_{GL(ν)}^{GL(n)} : GL(n) -Rat → GL(ν) -Rat.

For M ∈ GL(ν) -Mod, we define the induced module ind_{GL(ν)}^{GL(n)} M to be the set of GL(ν)- fixed points (M ⊗ 𝔽[GL(n)])^{GL(ν)} where the GL(n)-action on the induced module comes from the right regular action of GL(n) on 𝔽[GL(n)] and the trivial action on M, and the action of GL(ν) on M ⊗ 𝔽[GL(n)] under which we are taking fixed points comes from the given action on M and the left regular action on 𝔽[GL(n)].

This gives a functor ind_{GL(ν)}^{GL(n)} : GL(ν) -Rat → GL(n) -Rat which (precisely as in [J, I.3.4])

is right adjoint to the exact functor res_{GL(ν)}^{GL(n)}, so sends injectives to injectives. Note that this is not quite the same definition as in [J, I.3.3] – the roles of the left and right regular actions are swapped there. This is important for us in relating our results to Schur algebras in the most natural way. Our induction functor is naturally isomorphic to the one in [J, I.3.3] (for example, because both definitions give functors which are right adjoint to res_{GL(ν)}^{GL(n)}).

Now, by [R], the quotient GL(n)/GL(ν) is an affine variety, so [J, I.5.13(a)] implies:

1.2. The functor ind_{GL(ν)}^{GL(n)} is exact.

Hence [J, I.4.6] gives us generalized Frobenius reciprocity:

1.3. For M ∈ GL(n) -Rat, N ∈ GL(ν) -Rat and i ≥ 0,

Ext_{GL(n)}^{i}(M, ind_{GL(ν)}^{GL(n)} N) ≅ Ext_{GL(ν)}^{i}(res_{GL(ν)}^{GL(n)} M, N)

We recall the Donkin-Mathieu theorem [W, D1, M]:

1.4. (i) If M, N ∈ GL(ν) -Rat are modules with ∇-filtrations (resp. tilting modules), then M ⊗ N has a ∇-filtration (resp. is tilting);

(ii) If M ∈ GL(n) -Rat has a ∇-filtration (resp. is tilting), then res_{GL(ν)}^{GL(n)} M has a ∇- filtration (resp. is tilting).

In particular, (1.4)(ii) combined with (1.3) and (1.1) shows:

1.5. The functor ind_{GL(ν)}^{GL(n)} sends modules with ∇-filtrations to modules with ∇-filtrations.

We also note:

1.6. For λ ∈ X^{+}(ν), T_{ν}(λ) ≅ T_{n_{1}}(λ^{(1)}) ⊠ ⋯ ⊠ T_{n_{a}}(λ^{(a)}).

To prove this, note that the outer tensor product is certainly a tilting module and it has the correct highest weight. So it suffices to show that the outer tensor product of two indecomposable tilting modules is indecomposable. This follows from the following general fact for which we were unable to find a reference:

1.7. Lemma. Let A and B be two 𝔽-algebras and let M and N be finite dimensional indecomposables for A and B respectively. Then, M ⊠ N is an indecomposable A ⊗ B-module.

Proof. Note that as M and N are indecomposable, the identity is a primitive idempotent in each of the finite dimensional algebras End_{A}(M) and End_{B}(N). Consequently (by the commutant correspondence described in section 3) End_{A}(M) and End_{B}(N) each possess a unique irreducible module, of dimension 1. So by [CR, (10.38)(iii)] and the fact that 𝔽 is algebraically closed, End_{A}(M) ⊗ End_{B}(N) possesses a unique irreducible module, also of dimension 1. Hence, as End_{A⊗B}(M ⊠ N) ≅ End_{A}(M) ⊗ End_{B}(N) by [CR, (10.37)], the identity is a primitive idempotent of End_{A⊗B}(M ⊠ N), so M ⊠ N is indecomposable.

Now we specialize to polynomial representations of GL(n), following Green [G] and the appendix to Donkin’s monograph [D5]. We note however that Green and Donkin consider only finite dimensional modules whereas we include infinite dimensional (locally finite) modules. All the results from [G, D5] that we need are valid in this slightly more general setting (without significant alterations to the proofs).

The coordinate ring 𝔽[GL(n)] of GL(n) has the structure of a Hopf algebra, with comultiplication, counit and antipode coming from multiplication, unit and inversion in GL(n). Given a coalgebra A over 𝔽, Comod-A denotes the category of all right A-comodules (a right A-comodule means an 𝔽-module M with structure map μ : M → M ⊗ A as in [S]). Every GL(n)-module is naturally a right 𝔽[GL(n)]-comodule, and this gives an equivalence of categories between GL(n) -Rat and Comod-𝔽[GL(n)].

Let A(n) denote the subalgebra of 𝔽[GL(n)] generated by the functions {c_{ij}|1 ≤ i, j ≤ n}, where c_{ij} picks out the ij-entry of a matrix g ∈ GL(n). Then, A(n) is a sub-bialgebra of 𝔽[GL(n)], isomorphic to the free polynomial algebra 𝔽[c_{ij} |1 ≤ i, j ≤ n], and 𝔽[GL(n)] is the localization of A(n) at the determinant function. For r ≥ 0, the subspace A(n, r) < A(n) consisting of all homogeneous polynomials of degree r in the c_{ij} is a sub-coalgebra of A(n).

Let M be a left GL(n)-module with structure map μ : M → M ⊗ 𝔽[GL(n)]. We say M is a polynomial module (resp. a polynomial module of degree r) if the image of μ lies in M ⊗A(n) (resp. M ⊗A(n, r)). We denote the category of all left polynomial GL(n)-modules (resp. polynomial modules of degree r) by M_{𝔽}(n) (resp. M_{𝔽}(n, r)). The category M_{𝔽}(n) (resp. M_{𝔽}(n, r)) is equivalent to Comod-A(n) (resp. Comod-A(n, r)). Moreover, by [G, 2.2c], any polynomial module is a direct sum of polynomial modules of various degrees, and M_{𝔽}(n) ≅ ⨁_{r≥0} M_{𝔽}(n, r).

The dual space S(n, r) := A(n, r)^{∗} of the coalgebra A(n, r) inherits a natural algebra structure from the counit and comultiplication in A(n, r); S(n, r) is the Schur algebra of [G]. Given any algebra S over 𝔽, S -Mod denotes the category of all (possibly infinite dimensional) left S-modules. By [G, §1] there is an equivalence of categories between S(n, r) -Mod and Comod-A(n, r), hence between M_{𝔽}(n, r) and S(n, r) -Mod. This can be described more directly using the surjective homomorphism e : 𝔽GL(n) → S(n, r) of [G, 2.4], defined for g ∈ GL(n) by letting e(g) ∈ S(n, r) = A(n, r)^{∗} be the unique element satisfying e(g)(a) = a(g) for all a ∈ A(n, r). We can regard any S(n, r)-module as an 𝔽GL(n)-module via this surjection e, and Green shows that this gives the equivalence of categories between M_{𝔽}(n, r) and S(n, r) -Mod.

A rational GL(n)-module M is polynomial (resp. polynomial of degree r) if and only if all weights of M are elements of Λ(n) (resp. Λ(n, r)). Hence, the modules L_{n}(λ), Δ_{n}(λ), ∇_{n}(λ) and T_{n}(λ) lie in M_{𝔽}(n) (resp. M_{𝔽}(n, r)) if and only if λ ∈ Λ^{+}(n) (resp. Λ^{+}(n, r)). Noting that M_{𝔽}(n) has enough injectives (which look like direct sums of injectives for various S(n, r)), we let Q_{n}(λ) denote the injective hull of L_{n}(λ) in the category M_{𝔽}(n), for λ ∈ Λ^{+}(n).

Now, the category M_{𝔽}(n) is a full subcategory of GL(n) -Rat. So, given M, N ∈ M_{𝔽}(n), we can compute Ext^{i}(M, N) either by computing Ext_{M_{𝔽}(n)}^{i}(M, N) in the category M_{𝔽}(n), or by regarding M and N as elements of GL(n) -Rat and computing Ext_{GL(n)}^{i}(M, N) in the category GL(n) -Rat. By [D2, Theorem 2.1f], these two calculations of Ext^{i}(M, N) give the same answer:

1.8. Given modules M, N ∈ M_{𝔽}(n), Ext_{M_{𝔽}(n)}^{i}(M, N) ≅ Ext_{GL(n)}^{i}(M, N).

Define the polynomial truncation functor Pol : GL(n) -Rat → M_{𝔽}(n) by letting Pol M equal the largest polynomial submodule of M for M ∈ GL(n) -Rat, and by restriction on morphisms. Note Pol is a special case of Donkin’s truncation functor O_{π} from [D2, 2.1] (take π := Λ^{+}(n), a saturated subset of X^{+}(n)). We record the following basic property of Pol [D2, 2.1b]:

1.9. For a module M ∈ GL(n) -Rat with a ∇-filtration, (R^{i} Pol)M = 0 for i > 0.

Note that Pol is obviously right adjoint to the inclusion functor M_{𝔽}(n) → GL(n) -Rat (which is exact). So, Pol is left exact and sends injectives to injectives.

1.10. Lemma. Take M ∈ M_{𝔽}(n) and N ∈ GL(n) -Rat, and suppose that N has a ∇- filtration. Then, Ext_{GL(n)}^{i}(M, N) ≅ Ext_{GL(n)}^{i}(M, Pol N).

Proof. By the adjoint functor property, there is an isomorphism of functors between Hom_{GL(n) -Rat}(M, −) and Hom_{M_{𝔽}(n)}(M, −)∘Pol. Moreover, Pol sends injectives to injectives and Hom_{M_{𝔽}(n)}(M, −) is left exact, so for each M ∈ M_{𝔽}(n), there is a Grothendieck spectral sequence with

E_{2}^{i,j} = Ext_{M_{𝔽}(n)}^{i}(M, (R^{j} Pol)N) ⇒ Ext_{GL(n)}^{i+j}(M, N).

If N has a ∇-filtration, then (R^{j} Pol)N = 0 for j > 0 by (1.9), so this degenerates to give the isomorphism

Ext_{M_{𝔽}(n)}^{i}(M, Pol N) ≅ Ext_{GL(n)}^{i}(M, N).

Now the lemma follows by (1.8).

Combining (1.1) and Lemma 1.10, we deduce:

1.11. The functor Pol sends modules with ∇-filtrations to modules with ∇-filtrations.

We finally extend the notion of polynomial representations from GL(n) to the standard Levi subgroup GL(ν) < GL(n), for fixed ν ∈ Λ(a, n). The coordinate ring of GL(ν) is isomorphic to 𝔽[GL(n_{1})] ⊗ ⋯ ⊗ 𝔽[GL(n_{a})], which is the localization of A(ν) := A(n_{1}) ⊗ ⋯ ⊗ A(n_{a}) at determinant. Now, A(ν) can be regarded as a free polynomial algebra in indeterminates c_{ij}, where 1 ≤ i, j ≤ n and both i and j lie in the same one of the following intervals: [1, n_{1}], [n_{1} + 1, n_{1} + n_{2}], …, [n_{1} + ⋯ + n_{a−1} + 1, n]. If we let A(ν, r) denote the sub-coalgebra of A(ν) consisting of all homogeneous polynomials of degree r, then we have the coalgebra isomorphism

A(ν, r) ≅ ⨁_{ρ∈Λ(a,r)} A(n_{1}, ρ_{1}) ⊗ ⋯ ⊗ A(n_{a}, ρ_{a}).

Define the category M_{𝔽}(ν) (resp. M_{𝔽}(ν, r)) of polynomial GL(ν)-modules (resp. of degree r) to be all rational GL(ν)-modules M for which the image of the structure map μ : M → M ⊗𝔽[GL(ν)] lies in M ⊗A(ν) (resp. M ⊗A(ν, r)). Copying Green’s construction, the category M_{𝔽}(ν, r) is equivalent to the category S(ν, r) -Mod of left modules for the algebra S(ν, r) := A(ν, r)^{∗}. Obviously,

(1.12)  S(ν, r) ≅ ⨁_{ρ∈Λ(a,r)} S(n_{1}, ρ_{1}) ⊗ ⋯ ⊗ S(n_{a}, ρ_{a}).

Let J(ν) be the ideal of A(n) generated by all c_{ij} with i and j lying in distinct intervals out of [1, n_{1}], [n_{1}+1, n_{1}+n_{2}], …, [n_{1}+⋯+n_{a−1}+1, n]. It is graded by degree as ⨁_{r≥0} J(ν, r).

1.13. Lemma. S(ν, r) is isomorphic as an algebra to the annihilator J(ν, r)^{∘} of J(ν, r) in S(n, r).

Proof. The inclusion GL(ν) ↪ GL(n) induces a surjection 𝔽[GL(n)] → 𝔽[GL(ν)] of the coordinate rings. Since 𝔽[GL(n)] and 𝔽[GL(ν)] are the localizations of A(n) and A(ν) respectively at determinant, this restricts to a surjection A(n) → A(ν). So we can regard A(ν) as the bialgebra quotient A(n)/J(ν), and so A(ν, r) ≅ A(n, r)/J(ν, r). Hence, S(ν, r) = A(ν, r)^{∗} is isomorphic as an algebra to the annihilator J(ν, r)^{∘}.

1.14. Lemma. If e : 𝔽GL(n) → S(n, r) is the natural surjection, e(𝔽GL(ν)) ≅ S(ν, r).

Proof. By definition of the map e, the annihilator e(𝔽GL(ν))^{∘} < A(n, r) of e(𝔽GL(ν)) < S(n, r) is {f ∈ A(n, r) | f(g) = 0 for all g ∈ GL(ν)}, which is precisely J(ν, r). Taking annihilators, we deduce e(𝔽GL(n)) = J(ν, r)^{∘} ≅ S(ν, r), by Lemma 1.13.

Because of this lemma, we call S(ν, r) a standard Levi subalgebra of S(n, r) (see [D4, Section 2] for the quantum analogue). We can also describe an explicit basis for S(ν, r) in terms of Green’s basis for S(n, r). Let I(n, r) denote the set of all functions {1, …, r} → {1, …, n}. We regard a function i ∈ I(n, r) as an r-tuple i = (i_{1}, …, i_{r}) where i_{k} := i(k). Define a right action of Σ_{r} on I(n, r) by letting iπ be the composition of functions i ∘ π, for i ∈ I(n, r), π ∈ Σ_{r}. The set

I^{2}(n, r) := {(i, j) ∈ I(n, r) × I(n, r) | j_{1} ≤ ⋯ ≤ j_{r} and i_{k} ≤ i_{k+1} whenever j_{k} = j_{k+1}}

is a set of orbit representatives for the diagonal action of Σ_{r} on I(n, r) × I(n, r). Then {c_{i,j} | (i, j) ∈ I^{2}(n, r)} is a basis for A(n, r). We let {ξ_{i,j} | (i, j) ∈ I^{2}(n, r)} denote the dual basis for S(n, r). With this notation, Lemma 1.13 also implies:

1.15. The set of all ξ_{i,j} with (i, j) ∈ I^{2}(n, r) such that i_{k} and j_{k} belong to the same one of the intervals [1, n_{1}], [n_{1} + 1, n_{1} + n_{2}], …, [n_{1} + ⋯ + n_{a−1} + 1, n], k = 1, 2 …, r, is a basis of S(ν, r).

Let Λ^{+}(ν, r) denote the set of all λ = (λ^{(1)}, …, λ^{(a)}) ∈ X^{+}(ν) with 0 ≤ λ_{i}^{(j)} for all i, j and |λ^{(1)}| + ⋯ + |λ^{(a)}| = r. Then, by (1.12), the modules L_{ν}(λ), Δ_{ν}(λ), ∇_{ν}(λ) and T_{ν}(λ) for λ ∈ X^{+}(ν) lie in M_{𝔽}(ν, r) if and only if λ ∈ Λ^{+}(ν, r).

2 Polynomial induction functors

In this section, we introduce a ‘polynomial induction’ functor from Levi subgroups, and use it to obtain our first functorial connection between tensor products and branching rules.

Fix ν = (n_{1}, …, n_{a}) ∈ Λ(a, n) for some a, and let GL(ν) < GL(n) be the standard Levi subgroup. By definition, the algebra A(ν) (resp. A(ν, r)) is the image of A(n) (resp. A(n, r)) under the quotient map 𝔽[GL(n)] → 𝔽[GL(ν)]. Hence, the restriction of a polynomial module (resp. a polynomial module of degree r) from GL(n) to GL(ν) is again a polynomial module (resp. a polynomial module of degree r). So we have the exact restriction functor

R_{ν}^{n} : M_{𝔽}(n) → M_{𝔽}(ν),

which sends M_{𝔽}(n, r) into M_{𝔽}(ν, r).

Now, the left and right regular actions of GL(n) on 𝔽[GL(n)] stabilize A(n) and A(n, r). So, for M ∈ M_{𝔽}(ν), we can define GL(n)-modules (M ⊗ A(n))^{GL(ν)} and (M ⊗ A(n, r))^{GL(ν)}, in the same way as the induced module ind_{GL(ν)}^{GL(n)} = (M ⊗ 𝔽[GL(n)])^{GL(ν)} was defined in section 1. Note that both (M ⊗ A(n))^{GL(ν)} and (M ⊗ A(n, r))^{GL(ν)} are submodules of ind_{GL(ν)}^{GL(n)}. Define the functor

I_{ν}^{n} : M_{𝔽}(ν) → M_{𝔽}(n)

by letting I_{ν}^{n}M := (M ⊗ A(n))^{GL(ν)}, with the obvious definition on morphisms.

We regard S(n, r) as a left GL(n)-module by letting g.s be the unique element of S(n, r) such that (g.s)(a) = s(g^{−1}.a) (left regular action), for s ∈ S(n, r), g ∈ GL(n). Similarly, we regard S(n, r) as a right GL(n)-module by defining (s.g)(a) = s(g.a) (right regular action), for s ∈ S(n, r), g ∈ GL(n). Note that g.s = e(g)s and s.g = se(g), where e : 𝔽GL(n) → S(n, r) is the natural surjection. Now, Hom_{GL(ν)}(S(n, r), M) is naturally a left GL(n)-module, with action (gf)(s) = f(sg) for g ∈ GL(n), s ∈ S(n, r) and f ∈ Hom_{GL(ν)}(S(n, r), M).

2.1. Lemma. For M ∈ M_{𝔽}(ν) and r ≥ 0, the left GL(n)-modules (M ⊗ A(n, r))^{GL(ν)} and Hom_{GL(ν)}(S(n, r), M) are naturally isomorphic.

Proof. Identify M ⊗ A(n, r) with Hom_{𝔽}(S(n, r), M) by letting a generator m ⊗ a ∈ M ⊗ A(n, r) correspond to the homomorphism s ↦ s(a)m (s ∈ S(n, r)).

Take M ∈ M_{𝔽}(ν, r). By the isomorphism A(n) ≅ ⨁_{s≥0} A(n, s) and Lemma 2.1,

I_{ν}^{n}M ≅ ⨁_{s≥0} Hom_{GL(ν)}(S(n, s), M).

Now S(n, s) is by definition a polynomial GL(ν)-module of degree s. But M is of degree r and there are no GL(ν)-homomorphisms between polynomial modules of different degrees. Hence,

I_{ν}^{n}M ≅ Hom_{GL(ν)}(S(n, r), M) ≅ (M ⊗ A(n, r))^{GL(ν)}.

So I_{ν}^{n} sends M_{𝔽}(ν, r) into M_{𝔽}(n, r). Moreover, it sends finite dimensional modules to finite dimensional modules (unlike ind_{GL(ν)}^{GL(n)}).

2.2. Lemma. (i) The functor I_{ν}^{n} is right adjoint to R_{ν}^{n}. Hence, I_{ν}^{n} is left exact and sends injectives in M_{𝔽}(ν) to injectives in M_{𝔽}(n).

(ii) The functors I_{ν}^{n} : M_{𝔽}(ν) → M_{𝔽}(n) and Pol ∘ ind_{GL(ν)}^{GL(n)} : M_{𝔽}(ν) → M_{𝔽}(n) are isomorphic.

(iii) Let M ∈ M_{𝔽}(n) and N ∈ M_{𝔽}(ν). If N has a ∇-filtration, then

Ext_{GL(ν)}^{i}(R_{ν}^{n}M, N) ≅ Ext_{GL(n)}^{i}(M, I_{ν}^{n}N)

for i ≥ 0.

(iv) I_{ν}^{n} sends modules with ∇-filtrations to modules with ∇-filtrations.

Proof. For (i), it suffices to show that Hom_{GL(ν)}(R_{ν}^{n}M, N) ≅ Hom_{GL(n)}(M, I_{ν}^{n}N) for M ∈ M_{𝔽}(n, r) and N ∈ M_{𝔽}(ν, r). The functor I_{ν}^{n} : M_{𝔽}(ν, r) → M_{𝔽}(n, r) is isomorphic to the functor Hom_{S(ν,r)}(S(n, r), −) by Lemma 2.1, and it is a well-known fact about finite dimensional algebras that this is right adjoint to restriction between S(n, r) -Mod and S(ν, r) -Mod. To prove (ii), we simply note that the functor Pol ∘ ind_{GL(ν)}^{GL(n)} is also right ad-

joint to R_{ν}^{n}, which follows as ind_{GL(ν)}^{GL(n)} is right adjoint to res_{GL(ν)}^{GL(n)} and Pol is right adjoint to the inclusion functor M_{𝔽}(n) → GL(n) -Mod. Now (iii) follows easily from (ii), (1.5), Lemma 1.10 and (1.3), while (iv) follows from (ii), (1.5) and (1.11).

Now fix m ≤ n, and let ν = (m, 1, …, 1), a composition of n. Then, GL(m) is a normal subgroup of the Levi subgroup GL(ν) ≤ GL(n), and we have an exact inflation functor Infl_{m}^{ν} : GL(m) -Rat → GL(ν) -Rat which sends each M_{𝔽}(m, r) into M_{𝔽}(ν, r). Let H(m) ≤ GL(ν) be the (n−m)-dimensional torus such that GL(ν) = GL(m)×H(m). Then, the fixed point functor M ↦ M^{H(m)} is right adjoint to Infl_{ν}^{m} (see eg [J, I.6.4]). Now define functors I_{m}^{n} : M_{𝔽}(m) → M_{𝔽}(n) and R_{m}^{n} : M_{𝔽}(n) → M_{𝔽}(m) by the compositions

I_{m}^{n} := I_{ν}^{n} ∘ Infl_{m}^{ν}, R_{m}^{n} := (−)^{H(m)} ∘ R_{ν}^{n}.

We note that I_{m}^{n} is right adjoint to R_{m}^{n}, hence left exact. The exact functor R_{m}^{n} just takes the 0-weight space of a GL(n)-module with respect to the torus H(m) so, as special cases of [J, II.2.11] and [J, II.5.21], we know:

2.3. Fix λ ∈ Λ^{+}(n) with λ_{m+1} = ⋯ = λ_{n} = 0 (so λ has at most m non-zero rows). Let λ̄ = (λ_{1}, …, λ_{m}) ∈ Λ^{+}(m). Then, R_{m}^{n}∇_{n}(λ) ≅ ∇_{m}(λ̄) and R_{m}^{n}L_{n}(λ) ≅ L_{m}(λ̄).

On the other hand, if λ ∈ Λ^{+}(n) has λ_{m+1} > 0, then all weights μ of ∇_{n}(λ) also satisfy μ_{m+1} > 0. Hence:

2.4. For λ ∈ Λ^{+}(n) with λ_{m+1} > 0, we have R_{m}^{n}∇_{n}(λ) = R_{m}^{n}L_{n}(λ) = 0.

Moreover by Lemma 2.2(iv), I_{m}^{n} sends modules with ∇-filtrations to modules with ∇- filtrations. Hence, for μ ∈ Λ^{+}(m), I_{m}^{n}∇_{m}(μ) has a ∇-filtration, and on calculating the factors occuring in a ∇-filtration of I_{m}^{n}∇_{m}(μ) using [J, II.4.16(a)], together with the isomorphism Hom_{GL(n)}(Δ_{n}(λ), I_{m}^{n}∇_{m}(μ)) ≅ Hom_{GL(m)}(R_{m}^{n}Δ_{n}(λ), ∇_{m}(μ)) and (2.3),(2.4), one deduces:

2.5. Let λ and λ̄ be as in (2.3). Then, I_{m}^{n}∇_{m}(λ̄) ≅ ∇_{n}(λ).

Now, let λ and λ̄ be as in (2.3). Since I_{m}^{n} sends injectives to injectives, I_{m}^{n}Q_{m}(λ̄) is injective. Moreover, for μ ∈ Λ^{+}(n), Hom_{GL(n)}(L_{n}(μ), I_{m}^{n}Q_{m}(λ̄)) ≅ Hom_{GL(m)}(R_{m}^{n}L_{n}(μ), Q_{m}(λ̄)). If μ_{m+1} = 0 this is zero by (2.3) unless μ̄ = λ̄, when it is one dimensional. If μ_{m+1} > 0, then R_{m}^{n}L_{n}(μ) = 0 by (2.4), so again the Hom space is zero. This shows that I_{m}^{n}Q_{m}(λ̄) is injective with simple socle L_{n}(λ). Hence:

2.6. Let λ and λ̄ be as in (2.3). Then, I_{m}^{n}Q_{m}(λ̄) = Q_{n}(λ).

It is not in general true that I_{m}^{n}L_{m}(λ̄) ≅ L_{n}(λ) (but see (2.19)). For instance, consider I_{1}^{n}L_{1}(aε_{1}) which equals ∇_{n}(aε_{1}) by (2.5), which is not necessarily irreducible. This example also shows that Lemma 2.2(iv) is false if we replace ∇-filtrations with Δ-filtrations.

Our main reason for introducing the functors I_{m}^{n} and I_{ν}^{n} is the following result describing polynomial induction applied to an outer tensor product:

2.7. Theorem. Let ν ∈ Λ(a, n). Take any modules M_{1}, …, M_{a} with M_{i} ∈ M_{𝔽}(n_{i}), so that M_{1} ⊠ ⋯ ⊠ M_{a} ∈ M_{𝔽}(ν). Then

I_{ν}^{n}(M_{1} ⊠ ⋯ ⊠ M_{a}) ≅ (I_{n_{1}}^{n}M_{1}) ⊗ ⋯ ⊗ (I_{n_{a}}^{n}M_{a}).

Proof. Let M = M_{1} ⊠ ⋯ ⊠ M_{a}. We can identify A(n) with the symmetric algebra S(V^{∗} ⊗ V), letting the generator c_{ij} ∈ A(n, 1) correspond to f_{i} ⊗ e_{j} ∈ V^{∗} ⊗ V. Having done this, it is routine to check that the left regular (resp. right regular) action of GL(n) on A(n) corresponds to the action of GL(n) on S(V^{∗} ⊗ V) coming from the action on V^{∗} (resp. V). By the definition of I_{ν}^{n} and these remarks,

I_{ν}^{n}M ≅ [M ⊗ S(V^{∗} ⊗ V)]^{GL(ν)}

where the GL(n)-action on I_{ν}^{n}M comes just from its action on V and the GL(ν)-action under which we are taking fixed points comes from the action on M and V^{∗}. As a GL(ν)- module, V^{∗} ≅ V_{1}^{∗} ⊕ ⋯ ⊕ V_{a}^{∗}, where V_{i} is the natural module for the ith factor GL(n_{i}) of GL(ν) (and GL(n_{j}) acts trivially on V_{i} for j ≠ i). So, as a GL(ν) × GL(n)-module,

M ⊗ S(V^{∗} ⊗ V) ≅ M ⊗ S((V_{1}^{∗} ⊗ V) ⊕ ⋯ ⊕ (V_{a}^{∗} ⊗ V))
≅ M ⊗ S(V_{1}^{∗} ⊗ V) ⊗ ⋯ ⊗ S(V_{a}^{∗} ⊗ V)
≅ [M_{1} ⊗ S(V_{1}^{∗} ⊗ V)] ⊗ ⋯ ⊗ [M_{a} ⊗ S(V_{a}^{∗} ⊗ V)] ,

where the GL(n_{i})-action comes from action on M_{i} and V_{i}^{∗}, and the GL(n)-action comes from the action on V only. Taking fixed points, we deduce

I_{ν}^{n}(M_{1} ⊠ ⋯ ⊠ M_{a}) ≅ [M_{1} ⊗ S(V_{1}^{∗} ⊗ V)]^{GL(n_{1})} ⊗ ⋯ ⊗ [M_{a} ⊗ S(V_{a}^{∗} ⊗ V)]^{GL(n_{a})} .

So it remains to show that if m ≤ n and N is a polynomial GL(m)-module, then I_{m}^{n}N ≅ [N ⊗ S(W^{∗} ⊗ V)]^{GL(m)} where W denotes the natural GL(m)-module. But by definition, I_{m}^{n}N ≅ [Infl_{m}^{n} N ⊗ S(V^{∗} ⊗ V)]^{GL(m)×H(m)} . Noting that S(V^{∗} ⊗ V)^{H(m)} ≅ S(W^{∗} ⊗ V) as a GL(m) × GL(n)-module, the result follows on taking H(m)-fixed points.

2.8. Theorem. Let ν ∈ Λ(a, n) and μ^{(1)}, …, μ^{(a)} ∈ Λ^{+}(n) be partitions such that μ^{(i)} has at most n_{i} non-zero rows for each i. Let μ̄^{(i)} = (μ_{1}^{(i)}, …, μ_{n_{i}}^{(i)}) ∈ Λ^{+}(n_{i}). For any M ∈ M_{𝔽}(n),

Hom_{GL(n)}(M, ∇_{n}(μ^{(1)}) ⊗ ⋯ ⊗ ∇_{n}(μ^{(a)})) ≅ Hom_{GL(ν)}(R_{ν}^{n}M, ∇_{n_{1}}(μ̄^{(1)}) ⊠ ⋯ ⊠ ∇_{n_{a}}(μ̄^{(a)})).

Proof. This is just the fact that I_{ν}^{n} is right adjoint to R_{ν}^{n}, together with Theorem 2.7 and (2.5).

We record an elementary lemma:

2.9. Lemma. Let M be a GL(n − 1)-module all of whose weights are of the form μ with μ_{1} + ⋯ + μ_{n−1} = s and N be a GL(n)-module all of whose weights are of the form ν with n_{1} + ⋯ + n_{n} = r. Then, Hom_{GL(n−1)}(M, N) ≅ Hom_{GL(n−1,1)}(M ⊠ Δ_{1}((r − s)ε_{1}), N).

Proof. The image of M under any GL(n − 1)-homomorphism lies in the subspace of N consisting of the sum of all weight spaces N_{γ} with γ_{1} + ⋯ + γ_{n−1} = s, hence γ_{n} = (r − s). Now the conclusion is obvious.

We now state some corollaries of Theorem 2.8. The first will be used in [BK].

2.10. Corollary. Fix λ, μ ∈ Λ^{+}(n) with μ_{n} = 0. Then,

Hom_{GL(n)}(L_{n}(λ), ∇_{n}(μ) ⊗ S^{ℓ}(V)) ≅ Hom_{GL(n−1)}(Δ_{n−1}(μ̄), L_{n}(λ) ↓_{GL(n−1)})

where ℓ = |λ| − |μ|.

Proof. By Theorem 2.8 and contravariant duality, the left hand side is isomorphic to Hom_{GL(n−1,1)}(Δ_{n−1}(μ̄) ⊠ Δ_{1}(ℓε_{1}), L_{n}(λ) ↓_{GL(n−1,1)}). Now apply Lemma 2.9.

The second corollary is a result of Donkin [D3, Lemma 3.4(i)]:

2.11. Corollary (Donkin). Let λ ∈ Λ^{+}(n) and μ ∈ Λ(n). Then,

dim L_{n}(λ)_{μ} = (S^{μ_{1}}(V) ⊗ ⋯ ⊗ S^{μ_{n}}(V) : Q_{n}(λ)),

the multiplicity of Q_{n}(λ) as summand of the injective module S^{μ_{1}}(V) ⊗ ⋯ ⊗ S^{μ_{n}}(V).

Proof. We first need to observe that S^{μ_{1}}(V) ⊗ ⋯ ⊗ S^{μ_{n}}(V) is indeed injective in M_{𝔽}(n), which is proved in [D3, Lemma 3.4(i)] (it is a summand of A(n)). Now, taking GL(ν) = T (n), dim L(λ)_{μ} = dim Hom_{GL(n)}(L_{n}(λ), S^{μ_{1}}(V) ⊗ ⋯ ⊗ S^{μ_{a}}(V)) by Theorem 2.8. The result follows immediately as Q_{n}(λ) has simple socle L_{n}(λ).

2.12. Corollary. For r ≤ n, V^{⊗r} ≅ ⨁_{λ∈Λ^{+}(n,r)} Q_{n}(λ)^{d_{λ}} where d_{λ} := dim L_{n}(λ)_{ω} and ω = (1, …, 1, 0, …, 0) ∈ Λ(n, r). In particular, V^{⊗r} has p-restricted socle.

Proof. The first statement is immediate from Corollary 2.11. To deduce the second statement, note that Q_{n}(λ) has simple socle L_{n}(λ), while by [G, 6.4b], d_{λ} is non-zero if and only if λ is p-restricted.

In the remainder of the section, we will study the functors R_{m}^{n} and I_{m}^{n} in more detail, by relating them to Green’s functor d_{n,m} from [G, 6.5]. For later use, we first recall some standard properties of Schur functors, then specialize to the case in hand. The basic references here are [G, Chapter 6] and [JS], but as we work with the right adjoint ‘induction’ functor whereas [G, JS] work with the left adjoint, we have given some of the proofs.

Suppose that S is a finite dimensional algebra over a field 𝔽, and e ∈ S is an idempotent. The Schur functor R_{e} is the functor R_{e} : S -Mod → eSe -Mod defined on modules by M ↦ eM, and by restriction on morphisms. Equivalently, R_{e}(M) is the module Hom_{S}(Se, M), regarded as an eSe-module by the action (sθ)(s′) = θ(s′s) for s ∈ eSe, s′ ∈ Se, θ ∈ Hom_{S}(Se, M). By [G, 6.2a], the functor R_{e} is exact. We recall [G, 6.2g]:

2.13. Let {L(λ) | λ ∈ X} be a full set of non-isomorphic irreducible modules in S -Mod, indexed by a set X. Let X′ = {λ ∈ X | R_{e}L(λ) ≠ 0}. Then, {R_{e}L(λ) | λ ∈ X′} is a full set of non-isomorphic irreducible modules in eSe -Mod.

Given N ∈ eSe -Mod, we let I_{e}(N) := Hom_{eSe}(eS, N), regarded as an element of S -Mod by defining the S-action to be (sθ)(s′) = θ(s′s), for s ∈ S, s′ ∈ eS, θ ∈ I_{e}(N). With the obvious definition on morphisms, this gives a functor I_{e} : eSe -Mod → S -Mod which is right adjoint to R_{e}. In other words:

2.14. For M ∈ S -Mod, N ∈ eSe -Mod, Hom_{S}(M, I_{e}N) ≅ Hom_{eSe}(R_{e}M, N).

Hence, I_{e} is left exact and sends injective eSe-modules to injective S-modules. Note R_{e}(I_{e}(N)) = e Hom_{eSe}(eS, N) ≅ Hom_{eSe}(eSe, N) ≅ N, so I_{e} is a ‘right inverse’ to R_{e}:

2.15. For N ∈ eSe -Mod, R_{e}(I_{e}(N)) ≅ N.

For a module M ∈ S -Mod, we let M_{(e)} denote the largest submodule of M annihilated by e, and M^{(e)} denote the smallest submodule such that M/M^{(e)} is annihilated by e. We say a module M is e-restricted if M^{(e)} = M and M_{(e)} = 0. Observe that M_{(e)} = 0 if and only if the socle of M is e-restricted, while M^{(e)} = M if and only if the head of M is e-restricted. Moreover, [JS, Lemma 2.1] proves that:

2.16. For M ∈ S -Mod, M^{(e)} = SeM.

Now, take N ∈ eSe -Mod and let N̂ := I_{e}(R_{e}(N)). Since N ≅ Hom_{S}(S, N) and N̂ = Hom_{eSe}(eS, eN), there is a natural restriction map res : N → N̂.

2.17. Lemma. (i) For N ∈ S -Mod and N̂ := I_{e}(R_{e}(N)), the kernel of the natural restriction map res : N → N̂ equals N_{(e)}, while the image of res contains N̂^{(e)}.

(ii) Suppose N ∈ S -Mod has e-restricted socle and M ∈ S -Mod has e-restricted head. Then, Hom_{eSe}(R_{e}M, R_{e}N) ≅ Hom_{S}(M, N).

(iii) If SeS = S, the functors R_{e} and I_{e} induce an equivalence of categories between S -Mod and eSe -Mod.

Proof. (i) The image under res of n ∈ N is the homomorphism res(n) ∈ Hom_{eSe}(eS, eN) where res(n)(es) = esn for all es ∈ eS. So, res(n) = 0 if and only if eSn = 0, which is if and only if n ∈ N_{(e)}. Hence, ker res = N_{(e)}. For the second statement, it suffices by (2.16) to show that eN̂ ⊂ res(N). But, eN̂ = e Hom_{eSe}(eS, eN) = res(eN) ⊂ res(N) as required. (This is an analogue of [JS, 2.11(ii)].)

(ii) By (2.14), Hom_{eSe}(R_{e}M, R_{e}N) ≅ Hom_{S}(M, N̂) where N̂ = I_{e}(R_{e}(N)). By (i) and the fact that N_{(e)} = 0, N̂ is an extension of N by a module annihilated by e. But M has e-restricted head, so any homomorphism from M to N̂ must have image wholly contained in N, and Hom_{S}(M, N̂) ≅ Hom_{S}(M, N).

(iii) For N ∈ eSe -Mod, (2.15) shows that R_{e}(I_{e}(N)) ≅ N. The assumption SeS = S is equivalent by (2.16) to S having e-restricted head as a left S-module, or all irreducible S-modules being e-restricted. Hence, for all M ∈ S -Mod, M_{(e)} = 0 and M^{(e)} = M. Given this, (i) implies that M ≅ I_{e}(R_{e}(M)) for M ∈ S -Mod. Finally, one checks that these isomorphisms are functorial.

Now we specialize to the functors I_{m}^{n} and R_{m}^{n} defined earlier, for n ≥ m ≥ 1. On restricting I_{m}^{n} and R_{m}^{n} to the subcategories M_{𝔽}(m, r) and M_{𝔽}(n, r) respectively, we obtain functors I_{m,r}^{n,r} : M_{𝔽}(m, r) → M_{𝔽}(n, r) and R_{m,r}^{n,r} : M_{𝔽}(n, r) → M_{𝔽}(m, r). As R_{m,r}^{n,r} amounts to picking certain weight spaces, it can be identified with the Schur functor R_{e_{n,m}}, where e_{n,m} ∈ S(n, r) is the idempotent defined as in [G, 6.5] (R_{e_{n,m}} is the functor denoted d_{n,m} by Green). As I_{m}^{n} is right adjoint to R_{m}^{n}, I_{m,r}^{n,r} is right adjoint to R_{m,r}^{n,r}. Now, I_{e_{n,m}} is also right adjoint to R_{m,r}^{n,r} = R_{e_{n,m}}, so I_{m,r}^{n,r} can be identified with I_{e_{n,m}}. Now, (2.15) and the isomorphism M_{𝔽}(m) ≅ ⨁_{r≥0} M_{𝔽}(m, r) imply:

2.18. For M ∈ M_{𝔽}(m), R_{m}^{n}(I_{m}^{n}(M)) ≅ M.

If r ≤ m ≤ n, then R_{m,r}^{n,r}L_{n}(λ) is non-zero for all λ ∈ Λ^{+}(n, r) by (2.3). Hence, in this case all S(n, r)-modules have e_{n,m}-restricted head, so S(n, r) = S(n, r)e_{n,m}S(n, r) by (2.16). Now Lemma 2.17(iii) shows that I_{m,r}^{n,r} : M_{𝔽}(m, r) → M_{𝔽}(n, r) is an equivalence of categories for r ≤ m ≤ n, which is a result of Green [G, 6.5g]. Hence:

2.19. Let λ and λ̄ be as in (2.3). If |λ| ≤ m ≤ n, then I_{m}^{n}L_{m}(λ̄) ≅ L_{n}(λ).

Now we can prove an analogue of Theorem 2.8 for irreducible modules:

2.20. Theorem. With the notation of Theorem 2.8, assume in addition |μ^{(i)}| ≤ n_{i} for each i. Then,

Hom_{GL(n)}(M, L_{n}(μ^{(1)}) ⊗ ⋯ ⊗ L_{n}(μ^{(a)})) ≅ Hom_{GL(ν)}(R_{ν}^{n}M, L_{n_{1}}(μ̄^{(1)}) ⊠ ⋯ ⊠ L_{n_{a}}(μ̄^{(a)})).

Proof. Use the fact that I_{ν}^{n} is right adjoint to R_{ν}^{n}, together with Theorem 2.7 and (2.19).

2.21. Remarks. (I) Of particular interest are the cases when M = Δ_{n}(λ) or L_{n}(λ) for some λ ∈ Λ^{+}(n). In the latter case, the theorem then gives a connection between the socle of a tensor product and the head of a restriction to a Levi subgroup.

(II) We end with an example to show that Theorem 2.20 is in general false if we remove the assumption |μ^{(i)}| ≤ n_{i}. Let p = 2. By Steinberg’s tensor product theorem, L_{3}(ε_{1} +ε_{2})⊗ L_{3}(2ε_{1}) ≅ L_{3}(3ε_{1} + ε_{2}). So,

Hom_{GL(3)}(L_{3}(2ε_{1} + ε_{2} + ε_{3}), L_{3}(ε_{1} + ε_{2}) ⊗ L_{3}(2ε_{1})) = 0.

But L_{3}(2ε_{1} + ε_{2} + ε_{3}) is just the natural GL(3)-module tensored with determinant, so its restriction to the Levi subgroup GL(2, 1) contains L_{2}(ε_{1} + ε_{2}) ⊠ L_{1}(2ε_{1}) as a summand. So,

Hom_{GL(2,1)}(L_{3}(2ε_{1} + ε_{2} + ε_{3}) ↓, L_{2}(ε_{1} + ε_{2}) ⊠ L_{1}(2ε_{1})) ≠ 0.

3 Dual pairs and tilting modules

Next, we use dual pairs to explain the connection between tensor products and restrictions of irreducible GL(n)-modules to restrictions and tensor products of indecomposable tilting modules. We begin in a general setting and then specialize to Schur algebras.

Let M be a finite dimensional vector space over 𝔽. Let A and C be subalgebras of End_{𝔽}(M). We say that (A, C) is a dual pair on M if

(D1) C = End_{A}(M);

(D2) A = End_{C}(M).

Assume now that only (D1) holds. Let

M = ⨁_{i∈I} T_{i}

be a decomposition of M into a direct sum of indecomposable A-modules T_{i}, over some index set I. Let e_{i} ∈ C be the idempotent such that e_{i}M = T_{i}, and let P_{i} := Hom_{A}(M, T_{i}), regarded as a right C-module by defining the product θc to be the composition of functions θ ∘ c, for θ ∈ Hom_{A}(M, T_{i}) and c ∈ C. Observe that P_{i} is isomorphic to the right ideal e_{i}C of C. Fitting’s theorem (see eg [L, 1.4]) says that ∑_{i∈I} e_{i} is a decomposition of the identity in C into primitive idempotents, and T_{i} ≅ T_{j} as left A-modules if and only if P_{i} ≅ P_{j} as right C-modules.

Define an equivalence relation ∼ on I by i ∼ j if T_{i} ≅ T_{j}, or equivalently if P_{i} ≅ P_{j}. Let X denote the set of ∼-equivalence classes in I, and for λ ∈ X, let T_{A}(λ) := T_{i} and P_{C}(λ) := P_{i}, for some arbitrary i ∈ λ. Let L_{C}(λ) := hd P_{C}(λ) be the head of P_{C}(λ). Then, {P_{C}(λ) | λ ∈ X} is a complete set of non-isomorphic PIMs in the category Mod-C of right C-modules, and {L_{C}(λ) | λ ∈ X} is a complete set of non-isomorphic irreducible right C-modules. Moreover, dim L_{C}(λ) is equal to the cardinality |λ| of the equivalence class λ, and so

M ≅ ⨁_{λ∈X} T_{A}(λ)^{dim L_{C}(λ)}

as an A-module.

Now, let B < A be a subalgebra of A, and let D > C be its commutant End_{B}(M). Performing the same construction for B and D, we can write

M = ⨁_{j∈J} T_{j}

as indecomposable B-modules, over some index set J (disjoint from I). We obtain a corresponding decomposition ∑_{j∈J} f_{j} of the identity of D into primitive idempotents in D. Let Y denote the equivalence classes of J under i ∼ j if T_{i} ≅ T_{j}, and for μ ∈ Y, let T_{B}(μ) denote T_{j} and P_{D}(μ) denote f_{j}D for some j ∈ μ. Finally, set L_{D}(μ) := hd P_{D}(μ); we obtain all irreducible right D-modules in this way. Moreover,

M ≅ ⨁_{μ∈Y} T_{B}(μ)^{dim L_{D}(μ)}

as a B-module.

We denote the restriction functors A -Mod → B -Mod and Mod-D → Mod-C by ↓_{B} and ↓_{C} respectively. We also use the induction functor ↑^{D}: Mod-C → Mod-D, defined as usual for finite dimensional algebras by N ↑^{D}:= N ⊗_{C} D, for N ∈ Mod-C. Restriction is exact, and induction is left adjoint to restriction, hence is right exact and sends projectives to projectives.

The following result is presumably known.

3.1. Lemma. For λ ∈ X, μ ∈ Y, we have

(T_{A}(λ) ↓_{B}: T_{B}(μ)) = (P_{C}(λ) ↑^{D}: P_{D}(μ)) = [L_{D}(μ) ↓_{C}: L_{C}(λ)].

Proof. Pick an idempotent e ∈ C such that T_{A}(λ) ≅ eM, so P_{C}(λ) ≅ eC. Write e = f_{1} + ⋯ + f_{n} where f_{1}, …, f_{n} are primitive idempotents in D. Then,

T_{A}(λ) ↓_{B}= ⨁_{i=1}^{n} f_{i}M

where f_{i}M are indecomposable B-modules, and f_{i}M ≅ T_{B}(μ) if and only if f_{i}D ≅ P_{D}(μ), for μ ∈ Y. Hence, (T_{A}(λ) ↓_{B}: T_{B}(μ)) = (eD : P_{D}(μ)).

Next, observe that the natural map eC ⊗_{C} D → eD is clearly onto. It is injective, because if x := ∑ ec_{i} ⊗ d_{i} lies in the kernel, for c_{i} ∈ C, d_{i} ∈ D, then y := ∑_{i} ec_{i}d_{i} = 0, and x = ∑_{i} ec_{i} ⊗ d_{i} = ∑_{i} e ⊗ ec_{i}d_{i} = e ⊗ y = 0. Hence, eD ≅ P_{C}(λ) ↑^{D}, and we have proved the first equality (T_{A}(λ) ↓_{B}: T_{B}(μ)) = (P_{C}(λ) ↑^{D}: P_{D}(μ)). Finally, observe that

(P_{C}(λ) ↑^{D}: P_{D}(μ)) = dim Hom_{D}(P_{C}(λ) ↑^{D}, L_{D}(μ))
= dim Hom_{C}(P_{C}(λ), L_{D}(μ) ↓_{C})
= [L_{D}(μ) ↓_{C}: L_{C}(λ)],

which completes the proof.

Fix m, n ≥ 1. Let Λ(n × m) be the set of all weights α ∈ Λ(n) with α_{i} ≤ m for all i and define Λ^{+}(n × m) := Λ(n × m) ∩ X^{+}(n). For λ ∈ Λ^{+}(n × m), we let λ^{t} denote the transpose partition, which can be regarded as an element of Λ^{+}(m × n). By definition, λ^{t} = (μ_{1}, …, μ_{m}) where μ_{i} is equal to the number of λ_{j} (1 ≤ j ≤ n) with λ_{j} ≥ i.

We now briefly recall a special case of Donkin’s construction of generalized Schur algebras from [D2]. Observe that Λ^{+}(n × m) is a saturated set of dominant weights of X^{+}(n) in the sense of [D2]. We say a GL(n)-module belongs to Λ^{+}(n × m) if all composition factors are of the form L_{n}(λ) for λ ∈ Λ^{+}(n × m). Regarding 𝔽[GL(n)] as a left GL(n)-module via the right regular action, we let A(n × m) be the largest submodule of 𝔽[GL(n)] (or equivalently, of A(n)) that belongs to Λ^{+}(n × m). By [D2, 1.2], A(n × m) is a subcoalgebra of 𝔽[GL(n)] and by [D2, 2.2c]:

3.2. dim A(n × m) = ∑_{λ∈Λ^{+}(n×m)}(dim Δ_{n}(λ))^{2}.

Hence, S(n×m) := A(n×m)^{∗} is naturally a finite dimensional algebra. In fact, the results in [D2] establish that S(n × m) is a quasihereditary algebra with weight poset Λ^{+}(n × m), in the sense of [D5, Appendix].

Now the arguments of Green [G, §1, 2.4] (see also [D2, §3] for the generalization to hyperalgebras over ℤ) show that there is a natural surjection e : 𝔽GL(n) → S(n × m), defined (as in section 1) by e(g)(a) := a(g) for all g ∈ GL(n) and a ∈ A(n × m), such that every GL(n)-module belonging to Λ^{+}(n × m) factors through e to give a well-defined S(n×m)-module. Moreover, this gives an equivalence of categories between S(n × m) -Mod and the category of all GL(n)-modules belonging to Λ^{+}(n × m).

Fix m, n ≥ 1. Given any GL(m)-module M, we let M̃ denote the right 𝔽GL(m)- module obtained from the left 𝔽GL(m)-module M by twisting the action of GL(m) with the antiautomorphism τ corresponding to matrix transposition.

Our results depend centrally on the following theorem of Donkin [D4] and Adamovich and Rybnikov [AR]. Actually, Donkin does not prove his version in quite this generality, while in [AR], a slightly different module M is used. Therefore, we have included a sketch of the proof, following the arguments in [D4] and [AR] closely.

3.3. Theorem (Donkin, Adamovich-Rybnikov). Let V and W denote the natural module for GL(n) and GL(m), respectively. Regard M := ⋀(V ⊗ W) as a GL(n) × GL(m)- module, by letting GL(n) act trivially on W and GL(m) act trivially on V. Let A (resp. C) denote the image of 𝔽GL(n) (resp. 𝔽GL(m)) in End_{𝔽}(M). Then,

(i) the set of indecomposable summands of M as a GL(n)-module is

{T_{n}(λ) | λ ∈ Λ^{+}(n × m)};

(ii) C ≅ S(m × n);

(iii) C = End_{A}(M);

(iv) for each λ ∈ Λ^{+}(n × m), there is an idempotent e_{λ} ∈ C such that e_{λ}M ≅ T_{n}(λ) and e_{λ}C is the projective cover in Mod-C of L̃_{m}(λ^{t}).

Proof. (i) We first decompose M = ⋀(V ⊗ W) as a GL(n) × T (m)-module to deduce

M ≅ ⋀((V ⊗ 𝔽_{ε_{1}}) ⊕ ⋯ ⊕ (V ⊗ 𝔽_{ε_{m}})) ≅ ⋀(V ⊗ 𝔽_{ε_{1}}) ⊗ ⋯ ⊗ ⋀(V ⊗ 𝔽_{ε_{m}})

where 𝔽_{ε_{i}} denotes the 1-dimensional T (m)-module of weight ε_{i}. Consequently, for α ∈ X(m), the α-weight space M_{α} of M with respect to T (m) is zero unless α ∈ Λ(m × n), in which case

M_{α} ≅ ⋀^{α_{1}}V ⊗ ⋯ ⊗ ⋀^{α_{m}}V. (3.4)

The highest weight of this GL(n)-module M_{α} is the transpose of the dominant conjugate of α, so lies in Λ^{+}(n × m, r). In particular, this shows that all weights of M with respect to T (n) lie in Λ(n×m). By (1.4)(i), M is tilting as a GL(n)-module, so is a direct sum of tilting modules T_{n}(λ) for certain λ (necessarily) in Λ^{+}(n×m). Moreover, for every λ ∈ Λ^{+}(n×m), T_{n}(λ) definitely appears as a summand of M, since λ is the highest weight of the summand M_{λ^{t}}.

(ii) The argument of (i) applies equally well to the action of GL(m) on M. In particular, M belongs to Λ^{+}(m × n) as a GL(m)-module, so the action of 𝔽GL(m) factors through the quotient e : 𝔽GL(m) → S(m × n) to give a well-defined S(m × n)-module. Hence, C is equal to the image of S(m × n) in End_{𝔽}(M). But we have also shown that the set of indecomposable summands of M as an S(m × n)-module is {T_{m}(λ) | λ ∈ Λ^{+}(m × n)}. Hence, M is a full tilting module for the quasihereditary algebra S(m × n), so faithful by the argument of [AR, Proposition 4.4]. This shows that C ≅ S(m × n).

(iii) As M is a (contravariantly self-dual) tilting GL(n)-module, [J, II.4.13] implies that

dim End_{A}(M) = ∑_{λ∈Λ^{+}(n×m)}(dim Hom_{A}(M, ∇_{n}(λ)))^{2}.

We know that C ≤ End_{A}(M). By (ii) and (3.2), dim C = ∑_{λ∈Λ^{+}(m×n)}(dim Δ_{m}(λ))^{2}. So it now suffices to show by dimension that dim Hom_{A}(M, ∇_{n}(λ)) = dim Δ_{m}(λ^{t}) for any λ ∈ Λ^{+}(n × m).

Let X_{m}(λ) := Hom_{A}(M, ∇_{n}(λ)), regarded as a GL(m)-module with action (gθ)(m) = θ(τ(g)m) for g ∈ GL(m), θ ∈ Hom_{A}(M, ∇_{n}(λ)) and m ∈ M. By 3.4 and the observation that τ is the identity on restriction to T (m),

dim X_{m}(λ)_{α} = dim Hom_{GL(n)}(M_{α}, ∇_{n}(λ)) = dim Hom_{GL(n)}(⋀^{α_{1}}V ⊗ ⋯ ⊗ ⋀^{α_{m}}V, ∇_{n}(λ)) for α ∈ Λ(m × n). But by the Littlewood-Richardson rule (or a calculation involving symmetric functions), the right hand dimension is equal to the number of standard λ^{t}- tableaux of weight α, which is precisely dim Δ_{m}(λ^{t})_{α} (see [BKS, section 2] for these wellknown definitions). Hence, the character of the GL(m)-module X_{m}(λ) is equal to the character of Δ_{m}(λ^{t}). In particular, their dimensions agree as required to prove that C = End_{A}(M).

For use in (iv), we claim now that X_{m}(λ) ≅ Δ_{m}(λ^{t}). We know already that the characters agree, so it suffices to show that X_{m}(λ) is a standard module for GL(m), or equivalently that it is a standard module for the quasihereditary algebra C^{op}, which we have just shown is the Ringel dual of A (see [D4, Appendix A4]). But by definition, the C^{op}-module X_{m}(λ) is Hom_{A}(M, ∇_{n}(λ)) which is certainly a standard module according to [D4, Appendix A4].

(iv) Given λ ∈ Λ^{+}(n × m), we can find an idempotent e_{λ} ∈ C such that e_{λ}M ≅ T_{n}(λ) by (i) and (iii). This means that e_{λ}C is a projective indecomposable right C-module, so the projective cover of the irreducible right C-module L̃_{m}(μ) for some μ ∈ Λ^{+}(m × n). To show that μ equals λ^{t} as required, it suffices to show that the GL(m)-module Y_{m}(λ) := Hom_{GL(n)}(M, T_{n}(λ)) (with action defined as in (iii)) contains L_{m}(λ^{t}) in its head.

As T_{n}(λ) has a ∇-filtration and λ is its highest weight, we can find a submodule K < T_{n}(λ) with a ∇-filtration such that

0 → K → T_{n}(λ) → ∇_{n}(λ) → 0

is exact. Applying Hom_{GL(n)}(M, −) to this, we obtain the long exact sequence

⋯ → Hom_{GL(n)}(M, T_{n}(λ)) → Hom_{GL(n)}(M, ∇_{n}(λ)) → Ext_{GL(n)}^{1}(M, K) → ….

But M has a Δ-filtration and K has a ∇-filtration, so Ext_{GL(n)}^{1}(M, K) = 0 by [J, II.4.13]. This shows that X_{m}(λ) is a quotient of Y_{m}(λ). But we showed in (iii) that X_{m}(λ) ≅ Δ_{m}(λ^{t}). Hence, Y_{m}(λ) contains L_{m}(λ^{t}) in its head, as required.

We remark that the situation in the theorem is symmetric. So, with notation as in the theorem, the theorem shows in fact that (A, C) is a dual pair on M.

Now fix in addition a ≥ 1. Applying the theorem a times to the outer tensor product, the following corollary is immediate because of [CR, 10.37] and (1.6):

3.5. Corollary. For i = 1, …, a, let n_{i} (resp. m_{i}) be a positive integer and let V_{i} (resp. W_{i}) denote the natural module for GL(n_{i}) (resp. GL(m_{i})). Regard

M := ⋀(V_{1} ⊗ W_{1}) ⊠ ⋯ ⊠ ⋀(V_{a} ⊗ W_{a})

as a module for (GL(n_{1}) × ⋯ × GL(n_{a})) × (GL(m_{1}) × ⋯ × GL(m_{a})) in the obvious way. Let A_{1} ⊗ ⋯ ⊗ A_{a} (resp. C_{1} ⊗ ⋯ ⊗ C_{a}) denote the image of the group algebra 𝔽GL(n_{1}) ⊗ ⋯ ⊗ 𝔽GL(n_{a}) (resp. 𝔽GL(m_{1}) ⊗ ⋯ ⊗ 𝔽GL(m_{a})) in End_{𝔽}(M). Then,

(i) the set of indecomposable summands of M as a GL(n_{1}) × ⋯ × GL(n_{a})-module is

{T_{n_{1}}(λ^{(1)}) ⊠ ⋯ ⊠ T_{n_{a}}(λ^{(a)}) | λ^{(i)} ∈ Λ^{+}(n_{i} × m_{i}), i = 1, …, a};

(ii) C_{i} ≅ S(m_{i} × n_{i}) for i = 1, …, a; (iii) C_{1} ⊗ ⋯ ⊗ C_{a} = End_{A_{1}⊗···⊗A_{a}}(M); (iv) for λ^{(i)} ∈ Λ^{+}(n_{i} × m_{i}), i = 1, …, a, there is an idempotent

e := e_{λ^{(1)}} ⊗ ⋯ ⊗ e_{λ^{(a)}} ∈ C_{1} ⊗ ⋯ ⊗ C_{a}

such that eM ≅ T_{n_{1}}(λ^{(1)}) ⊠ ⋯ ⊠ T_{n_{a}}(λ^{(a)}) and e(C_{1} ⊗ ⋯ ⊗ C_{a}) is the projective cover in Mod-(C_{1} ⊗ ⋯ ⊗ C_{a}) of L̃_{m}((λ^{(1)})^{t}) ⊠ ⋯ ⊠ L̃_{m}((λ^{(a)})^{t}).

Again, the situation in this corollary is symmetric, so, with the notation of the corollary, it implies that (A_{1} ⊗ ⋯ ⊗ A_{a}, C_{1} ⊗ ⋯ ⊗ C_{a}) is a dual pair on M.

Now we are ready for our application. Let GL(n) × GL(m) act on M := ⋀(V ⊗ W) as in Theorem 3.3, and let A (resp. C) be the image of 𝔽GL(n) (resp. 𝔽GL(m)) in End_{𝔽}(M). So, by Theorem 3.3, (A, C) is a dual pair on M.

Next we choose ν = (n_{1}, …, n_{a}) ∈ Λ(a, n) for some a, and let B < A be the image of 𝔽GL(ν) < 𝔽GL(n) in End_{𝔽}(M). Let V_{i} be the natural GL(n_{i})-module, so V ≅ V_{1} ⊕ ⋯ ⊕ V_{a} as a GL(ν)-module. Observe that as a GL(ν)-module,

M ≅ ⋀((V_{1} ⊗ W) ⊕ ⋯ ⊕ (V_{a} ⊗ W)) ≅ ⋀(V_{1} ⊗ W) ⊠ ⋯ ⊠ ⋀(V_{a} ⊗ W). (3.6)

Consequently, GL(ν) = GL(n_{1}) × ⋯ × GL(n_{a}) acts on M in the same way as GL(n_{1}) × ⋯ × GL(n_{a}) in Corollary 3.5.

Let GL(m) × ⋯ × GL(m) (a times) act on M via its natural action on W in the outer tensor product of 3.6. This action obviously commutes with the action of GL(ν), and is precisely the action of GL(m_{1}) × ⋯ × GL(m_{a}) in Corollary 3.5 with m_{1} = ⋯ = m_{a} = m. We also note that the original action of GL(m) on M is the restriction of the GL(m) × ⋯ × GL(m)-action just defined if we embed GL(m) into GL(m) × ⋯ × GL(m) diagonally. We let D > C denote the image of the group algebra 𝔽GL(m) ⊗ ⋯ ⊗ 𝔽GL(m) in End_{𝔽}(M). Now, Corollary 3.5 immediately implies that (B, D) is a dual pair on M.

So we are now in the situation of the beginning of the section. We apply Lemma 3.1 twice to obtain our main result:

3.7. Theorem. Fix a, n, m ≥ 1, and choose ν ∈ Λ(a, n). Let GL(ν) denote the standard Levi subgroup of GL(n). Choose λ ∈ Λ^{+}(n × m) and μ^{(i)} ∈ Λ^{+}(n_{i} × m) for i = 1, …, a. Then,

(i) (T_{n}(λ) ↓_{GL(ν)}: T_{n_{1}}(μ^{(1)})⊠⋯⊠T_{n_{a}}(μ^{(a)})) = [L_{m}((μ^{(1)})^{t})⊗⋯⊗L_{m}((μ^{(a)})^{t}) : L_{m}(λ^{t})]. (ii) [L_{n}(λ) ↓_{GL(ν)}: L_{n_{1}}(μ^{(1)})⊠⋯⊠L_{n_{a}}(μ^{(a)}))] = (T_{m}((μ^{(1)})^{t})⊗⋯⊗T_{m}((μ^{(a)})^{t}) : T_{m}(λ^{t})).

Proof. For (i), we apply Lemma 3.1 to the pairs (A, C) and (B, D), using Theorem 3.3(i) (resp. Corollary 3.5(i)) to identify the indecomposable summands of M as an A- (resp. B-) module and Theorem 3.3(iv) (resp. Corollary 3.5(iv)) to identify the corresponding irreducible C- (resp. D-) modules arising from the commutant construction. The only subtlety is in showing that the restriction from D to C of the D-module corresponding to the outer tensor product

L̃_{m}((μ^{(1)})^{t}) ⊠ ⋯ ⊠ L̃_{m}((μ^{(a)})^{t})

(which is a right 𝔽GL(m) ⊗ ⋯ ⊗ 𝔽GL(m)-module) is the C-module corresponding to the right 𝔽GL(m)-module

L̃_{m}((μ^{(1)})^{t}) ⊗ ⋯ ⊗ L̃_{m}((μ^{(a)})^{t}).

This follows because the embedding of C into D is induced from the diagonal embedding of GL(m) into GL(m) × ⋯ × GL(m). Given this (i) follows easily. The argument for (ii) is identical, but swapping the roles of A and D and the roles of B and C.

As special cases of Theorem 3.7(i) and (ii) (taking GL(ν) = T (n)), we recover the following character formulae, the first of which is a result of Mathieu and Papadopoulo [MP]:

3.8. Corollary (Mathieu-Papadopoulo). For λ ∈ Λ^{+}(n × m) and μ ∈ Λ(n × m),

dim L_{n}(λ)_{μ} = (⋀^{μ_{1}}(W) ⊗ ⋯ ⊗ ⋀^{μ_{n}}(W) : T_{m}(λ^{t}))

where W denotes the natural GL(m)-module.

3.9. Corollary. For λ ∈ Λ^{+}(n × m) and μ ∈ Λ(n × m),

dim T_{n}(λ)_{μ} = [⋀^{μ_{1}}(W) ⊗ ⋯ ⊗ ⋀^{μ_{n}}(W) : L_{m}(λ^{t})]

where W denotes the natural GL(m)-module.

The final corollary clarifies the connection between the arguments in [BKS] and [MP]:

3.10. Corollary. For λ ∈ Λ^{+}(n×m) and μ ∈ Λ^{+}(n×(m−1)) with |λ| ≥ |μ| put ℓ = |λ|−|μ|. Then

(T_{n}(μ) ⊗ ⋀^{ℓ}(V) : T_{n}(λ)) = [L_{m}(λ^{t}) ↓_{GL(m−1)}: L_{m−1}(μ^{t})].

Proof. By Theorem 3.7(ii), the left hand side is equal to

[L_{m}(λ^{t}) ↓_{GL(m−1,1)}: L_{m−1}(μ^{t}) ⊠ L_{1}(ℓε_{1})].

Now a similar argument to Lemma 2.9 gives the conclusion.

4 Symmetric groups and Schur functors

We now obtain our final general link between modular Littlewood-Richardson coefficients, by explaining the relationship with representations of symmetric groups.

Throughout the section, we fix integers n, r and compositions ν = (n_{1}, …, n_{a}) ∈ Λ(a, n) and ρ = (r_{1}, …, r_{a}) ∈ Λ(a, r) for some a such that n ≥ r and n_{i} ≥ r_{i} for i = 1, …, a.

We denote the symmetric group on {1, …, r} by Σ_{r}. Note that we regard the natural action of Σ_{r} on {1, …, r} as a left action and always work with left modules for the group algebra 𝔽Σ_{r}. First, we recall some basic facts about the representation theory of the symmetric group Σ_{r}, following [J1].

Given a composition λ = (λ_{1}, λ_{2}, …) we write λ ⊨ r if λ_{1} + λ_{2} + ⋯ = r. If λ ⊨ r is actually a partition we say that λ is a partition of r and write λ ⊢ r. For every λ ⊢ r, there is an associated left 𝔽Σ_{r}-module S^{λ}, defined as in [J1], called a Specht module.

Say λ ⊢ r is p-regular if at most (p−1) of the non-zero parts λ_{i} of λ are equal. According to [J1]:

4.1. If λ is p-regular, then S^{λ} has simple head denoted D^{λ}, and {D^{λ} | λ ⊢ r, λ p-regular} is a complete set of non-isomorphic irreducible 𝔽Σ_{r}-modules.

We mention some other important 𝔽Σ_{r}-modules. For any λ ⊨ r, let M^{λ} denote the corresponding permutation module, which is the module induced from the trivial module for the Young subgroup Σ_{λ} := Σ_{λ_{1}} × Σ_{λ_{2}} × … of Σ_{r}. Now, according to [J2], for λ ⊢ r, M^{λ} contains a unique submodule isomorphic to S^{λ}, and there is a unique indecomposable summand Y^{λ} of M^{λ} containing this submodule S^{λ}. These modules Y^{λ} are the Young modules. Moreover, any permutation module M^{μ} splits as a direct sum of Young modules.

Next, we recall Green’s construction of the Schur functor R_{ξ} : M_{𝔽}(n, r) → 𝔽Σ_{r} -Mod from [G, §6]. We write S := S(n, r) for short. Fix any weight ω ∈ Λ(n, r) of the form ω = ε_{h_{1}} + ⋯ + ε_{h_{r}}, h_{1} < h_{2} < ⋯ < h_{r}. Let h = h^{ω} = (h_{1}, …, h_{r}) ∈ I(n, r). Set ξ = ξ^{ω} := ξ_{h,h} ∈ S (see section 1). By [G, §3], ξ is an idempotent and for any M ∈ M_{𝔽}(n, r), ξM is precisely the ω-weight space M_{ω} of M. Moreover, as in [G, 6.1d]:

4.2. The set {ξ_{hπ,h} | π ∈ Σ_{r}} is a basis for the algebra ξSξ. The linear map 𝔽Σ_{r} → ξSξ, defined by π ↦ ξ_{hπ,h} for π ∈ Σ_{r}, is an algebra isomorphism.

Now define the Schur functor R_{ξ} : S -Mod → 𝔽Σ_{r} -Mod as in section two, using (4.2) to identify 𝔽Σ_{r} with ξSξ. Explicitly, given M ∈ S -Mod, we regard the ξSξ-module R_{ξ}M as an 𝔽Σ_{r}-module by letting π ∈ Σ_{r} act in the same way as ξ_{hπ,h} ∈ ξSξ.

In this section we are also going to work with a well-known equivalent, but for us more convenient, description of this functor, which we now explain. Let Perm_{r} denote the subgroup of GL(n) consisting of all permutation matrices g ∈ GL(n) with g_{i,i} = 1 whenever i ∉ {h_{1} …, h_{r}}. Obviously, Perm_{r} is isomorphic to Σ_{r}. More precisely:

4.3. For w ∈ Perm_{r}, define w̄ ∈ Σ_{r} to be the unique element such that the (h_{w̄i}, h_{i})-entry of the permutation matrix w is 1 for all i = 1, …, r. This gives an isomorphism Perm_{r} → Σ_{r}.

Given any M ∈ M_{𝔽}(n, r), the ω-weight space M_{ω} of M is stable under the action of Perm_{r}. We now define a functor

F_{n,r} = F_{n,r}^{ω} : M_{𝔽}(n, r) → 𝔽Perm_{r} -Mod

on objects by letting F_{n,r}M := M_{ω}, and by restriction on morphisms. (If we use another ω of the form ε_{h_{1}} + ⋯ + ε_{h_{r}} we get an isomorphic functor.)

4.4. Lemma. Identify Perm_{r} and Σ_{r} using the map w ↦ w̄ from (4.3). Then, F_{n,r}M and R_{ξ}M are isomorphic as 𝔽Σ_{r}-modules.

Proof. As vector spaces, F_{n,r}M and R_{ξ}M both equal M_{ω}. So it suffices to show that for any w ∈ Perm_{r}, the action of w on M_{ω} is the same as the action of w̄ ∈ Σ_{r}. Recall by definition that w̄ acts on M as ξ_{hw̄,h}. The action of w on M factors through the quotient e : 𝔽GL(n) → S, so w acts on M_{ω} in the same way as the element e(w) ∈ S. Since M_{ω} = ξM and ξ is idempotent, it therefore suffices to show that ξe(w)ξ = ξ_{hw̄,h}. To prove this, since both lie in ξSξ which has basis ξ_{hπ,h} for all π ∈ Σ_{r}, we just need to check that

(ξe(w)ξ)(c_{hπ,h}) = ξ_{hw̄,h}(c_{hπ,h})

for all π ∈ Σ_{r}. The right hand side is clearly zero unless π = w̄, when it is 1. On the other hand note that (ξe(w)ξ)(c_{hπ,h}) = (e(w))(c_{hπ,h}) = c_{hπ,h}(w) = w_{h_{π1},h_{1}} … w_{h_{πr} ,h_{r}}. This is zero unless π = w̄, when it is 1, by definition of w̄.

Let sgn denote the 1-dimensional sign representation for 𝔽Σ_{r}. The effect of F_{n,r} on the various modules in M_{𝔽}(n, r) is given by the following:

4.5. Lemma. Fix λ ∈ Λ^{+}(n, r).

(i) F_{n,r}∇_{n}(λ) ≅ S^{λ};

(ii) F_{n,r}Δ_{n}(λ) ≅ (S^{λ})^{∗} ≅ S^{λ^{t}} ⊗ sgn;

(iii) F_{n,r}L_{n}(λ) is zero unless λ is p-restricted, in which case F_{n,r}L_{n}(λ) ≅ D^{λ^{t}} ⊗ sgn . (iv) for any μ ∈ Λ(n, r), F_{n,r}(⋀^{μ_{1}}(V) ⊗ ⋯ ⊗ ⋀^{μ_{n}}(V)) ≅ M^{μ} ⊗ sgn;

(v) F_{n,r}T_{n}(λ) = Y^{λ^{t}} ⊗ sgn.

Proof. (i), (ii) and (iii) are well-known, and are proved in [G, §6]. Also, (v) follows easily from (ii) and (iv), since by definition Y^{λ^{t}} ⊗ sgn is the unique indecomposable summand of M^{λ^{t}} ⊗ sgn containing a submodule isomorphic to S^{λ^{t}} ⊗ sgn, see [D3, (3.6)(ii)]. Finally, (iv) is proved in [D3, (3.5)(ii)] (or follows easily from Theorem 4.13).

Recall that we have fixed ν = (n_{1}, …, n_{a}) ∈ Λ(a, n) and ρ = (r_{1}, …, r_{a}) ∈ Λ(a, r) with r_{i} ≤ n_{i} for i = 1, …, a. Let GL(ν) be the corresponding Levi subgroup of GL(n), and Σ_{ρ} be the corresponding Young subgroup Σ_{r_{1}} × ⋯ × Σ_{r_{a}} of Σ_{r}. First, we consider the effect of Schur functors on restrictions from GL(n) to GL(ν).

Choose the weight ω above to be the specific weight

(4.6) ω = ( \underbrace{\overbrace{1, …, 1}^{r_{1} entries}, 0, …, 0}_{n_{1} entries}, \underbrace{\overbrace{1, …, 1}^{r_{2} entries}, 0, …, 0}_{n_{2} entries}, …, \underbrace{\overbrace{1, …, 1}^{r_{a} entries}, 0, …, 0}_{n_{a} entries} ).

Let h = h^{ω} and ξ = ξ^{ω} Then the idempotent ξ ∈ S lies in the Levi subalgebra L := S(ν, r) < S (for example by (1.15)).

Let Perm_{ρ} := Perm_{r}∩GL(ν). Then Perm_{ρ} is isomorphic to the Young subgroup Σ_{ρ} < Σ_{r} (under the map from (4.3)).

4.7. Lemma. ξLξ ≅ 𝔽Σ_{ρ}.

Proof. Note that ξLξ = ξSξ ∩ L since ξ ∈ L. Now the result follows from (4.2) and Lemma 1.13.

Consequently, we obtain a Schur functor R̄_{ξ} : L -Mod → 𝔽Σ_{ρ} -Mod. Precisely as before, we can identify this Schur functor with the functor

F_{ν,ρ} : M_{𝔽}(ν, r) → 𝔽Perm_{ρ} -Mod

given on M ∈ M_{𝔽}(ν, r) by letting F_{ν,ρ}M denote the weight space M_{ω}, noting that this is stable under the action of Perm_{ρ} < GL(ν). It is now obvious that:

4.8. For any M ∈ M_{𝔽}(n, r), F_{ν,ρ}(R_{ν}^{n}M) ≅ (F_{n,r}M) ↓_{𝔽Perm_{ρ}}^{𝔽Perm_{r}} .

The action of the functor F_{ν,ρ} on outer tensor products is also easily understood:

4.9. Lemma. Given a GL(ν)-module that is an outer tensor product M_{1} ⊠ ⋯ ⊠ M_{a} for M_{i} ∈ M_{𝔽}(n_{i}, r_{i}),

F_{ν,ρ}(M_{1} ⊠ ⋯ ⊠ M_{a}) ≅ (F_{n_{1},r_{1}}M_{1}) ⊠ ⋯ ⊠ (F_{n_{a},r_{a}}M_{a}).

Proof. This follows because the ω-weight space of M_{1}⊠⋯⊠M_{a} is (M_{1})_{ω^{(1)}}⊠⋯⊠(M_{a})_{ω^{(a)}} where ω^{(i)} ∈ Λ(n_{i}, r_{i}) is the weight (1, …, 1, 0, …, 0).

We now give applications of this Schur functor F_{ν,ρ}. First, we have the following results on composition and tilting module multiplicities:

4.10. Theorem. Fix λ ⊢ r and μ^{(i)} ⊢ r_{i} for i = 1, …, a. Regard λ^{t} as an element of Λ^{+}(n, r) and each (μ^{(i)})^{t} as elements of Λ^{+}(n_{i}, r_{i}). Then,

(Y^{λ} ↓_{Σ_{ρ}}: Y^{μ^{(1)}} ⊠ ⋯ ⊠ Y^{μ^{(a)}}) = (T_{n}(λ^{t}) ↓_{GL(ν)}: T_{n_{1}}((μ^{(1)})^{t}) ⊠ ⋯ ⊠ T_{n_{a}}((μ^{(a)})^{t})).

Moreover, if in addition all the partitions are p-regular, then

[D^{λ} ↓_{Σ_{ρ}}: D^{μ^{(1)}} ⊠ ⋯ ⊠ D^{μ^{(a)}}] = [L_{n}(λ^{t}) ↓_{GL(ν)}: L_{n_{1}}((μ^{(1)})^{t}) ⊠ ⋯ ⊠ L_{n_{a}}((μ^{(a)})^{t})].

Proof. We prove this for the irreducible modules; precisely the same argument (together with Lemma 1.7) gives the statement about tilting modules. We now identify Perm_{r} with Σ_{r}. Observe that

F_{ν,ρ}(R_{ν}^{n}L_{n}(λ^{t})) ≅ (F_{n,r}L_{n}(λ^{t})) ↓_{Σ_{ρ}} ≅ (D^{λ} ⊗ sgn) ↓_{Σ_{ρ}}

by (4.8) and Lemma 4.5(iii). By Lemma 4.5(iii) and Lemma 4.9,

F_{ν,ρ}(L_{n_{1}}((μ^{(1)})^{t}) ⊠ ⋯ ⊠ L_{n_{a}}((μ^{(a)})^{t})) ≅ (D^{μ^{(1)}} ⊠ ⋯ ⊠ D^{μ^{(a)}}) ⊗ sgn,

which is a non-zero irreducible for 𝔽Σ_{ρ}. Since the Schur functor F_{ν,ρ} is exact, it preserves composition multiplicities, so the result follows on tensoring with sgn.

Special cases of the next more general result on homomorphisms have appeared in [JS] and [K1].

4.11. Theorem. Let M be a GL(n)-quotient of V_{n}^{⊗r}, and for i = 1, …, a, let M_{i} be a module in M_{𝔽}(n_{i}, r_{i}) with p-restricted socle. Then,

Hom_{GL(ν)}(M ↓_{GL(ν)}, M_{1} ⊠⋯⊠M_{a}) ≅ Hom_{𝔽Σ_{ρ}}((F_{n,r}M) ↓_{Σ_{ρ}}, (F_{n_{1},r_{1}}M_{1})⊠⋯⊠(F_{n_{a},r_{a}}M_{a})).

Proof. By assumption, M ∈ M_{𝔽}(n, r). As a GL(ν)-module, M splits as a direct some

⨁_{ρ′∈Λ(a,r)} M(ρ′)

where M(ρ′) denotes the summand of M that is homogeneous of polynomial degree ρ′_{i} for GL(n_{i}) for each i = 1, …, a. The GL(ν)-module M_{1} ⊠ ⋯ ⊠ M_{a} is of polynomial degree r_{i} for each GL(n_{i}), and there are no GL(ν)-homomorphisms between modules of different degrees. Consequently,

Hom_{GL(ν)}(M ↓_{GL(ν)}, M_{1} ⊠ ⋯ ⊠ M_{a}) ≅ Hom_{GL(ν)}(M(ρ), M_{1} ⊠ ⋯ ⊠ M_{a}).

Also note that by weights and (4.8) we have

(F_{n,r}M) ↓_{Σ_{ρ}}= F_{ν,ρ}(R_{ν}^{n}M) = F_{ν,ρ}M(ρ),

while by Lemma 4.9,

(F_{n_{1},r_{1}}M_{1}) ⊠ ⋯ ⊠ (F_{n_{a},r_{a}}M_{a}) = F_{ν,ρ}(M_{1} ⊠ ⋯ ⊠ M_{a}).

So, if L = S(ν, r), we need to prove that

Hom_{L}(M(ρ), M_{1} ⊠ ⋯ ⊠ M_{a}) ≅ Hom_{𝔽Σ_{ρ}}(F_{ν,ρ}M(ρ), F_{ν,ρ}(M_{1} ⊠ ⋯ ⊠ M_{a})).

By Lemma 2.17(ii), the proposition follows once we have shown that M(ρ) has ξ-restricted head and M_{1} ⊠ ⋯ ⊠ M_{a} has ξ-restricted socle. By Lemma 4.5(iii) and Lemma 4.9, an Lmodule M_{1} ⊠ ⋯ ⊠ M_{a} has a ξ-restricted socle if and only if each M_{i} has a p-restricted socle, which we have by assumption. To see that M(ρ) has p-restricted head, we first decompose V_{n}^{⊗r} as a GL(ν)-module:

V_{n}^{⊗r} ≅ ⨁_{ρ′∈Λ(r,a)} a(ρ′)V_{n_{1}}^{ρ′_{1}} ⊠ ⋯ ⊠ V_{n_{a}}^{ρ′_{a}}

for some multiplicities a(ρ′). Now, M(ρ) is a submodule of the factor of this direct sum with ρ′ = ρ, which has p-restricted head again by Corollary 2.12 (and contravariant duality).

4.12. Remark. By Corollary 2.12, we can take the module M in Theorem 4.11 to be Δ_{n}(λ) or L_{n}(λ) for p-restricted λ ∈ Λ(n, r), and each M_{i} to be ∇_{n_{i}}(μ^{(i)}) or L_{n_{i}}(μ^{(i)}) for p-restricted μ^{(i)} ∈ Λ(n_{i}, r_{i}).

Now we turn our attention to tensor products. Our main result describing the Schur functor F_{n,r} applied to a tensor product of GL(n)-modules is as follows:

4.13. Theorem. Given modules M_{i} ∈ M_{𝔽}(n, r_{i}) for i = 1, …, a,

F_{n,r}(M_{1} ⊗ ⋯ ⊗ M_{a}) ≅ (F_{n,r_{1}}M_{1} ⊠ ⋯ ⊠ F_{n,r_{a}}M_{a}) ↑_{𝔽Σ_{ρ}}^{𝔽Σ_{r}} .

Proof. For notational simplicity, we prove this in the special case that a = 2; the general case then follows quite easily by induction on a, using transitivity of induction and the tensor identity.

So, assume that a = 2. Take M ∈ M_{𝔽}(n, r_{1}) and N ∈ M_{𝔽}(n, r_{2}). We will show that

F_{n,r}(M ⊗ N) ≅ (F_{n,r_{1}}M ⊠ F_{n,r_{2}}N) ↑Perm_{r} Perm_{ρ} .

Let J = {1, …, r_{1}, n_{1} + 1, …, n_{1} + r_{2}} = {i | ω_{i} ≠ 0} (see (4.6)). Given a subset I = {i_{1}, …, i_{k}} ⊆ J, we write ω_{I} for the weight ε_{i_{1}} +⋯+ε_{i_{k}} ∈ Λ(n) . For a module M ∈ M_{𝔽}(n), denote the ω_{I}-weight space of M by M_{I}. Then, as vector spaces,

F_{n,r}(M ⊗ N) = ⨁_{I⊆J,|I|=r_{1}} M_{I} ⊗ N_{J\I}. (4.14)

Taking K = {1, 2, …, r_{1}} we see that W := M_{K} ⊠N_{J\K} is a 𝔽Perm_{ρ}-submodule of F_{n,r}(M ⊗ N), isomorphic to F_{n,r_{1}}M ⊠ F_{n,r_{2}}N. Now it suffices to prove that F_{n,r}(M ⊗ N), as a vector space, equals

⨁_{t∈T} w_{t}W (4.15)

where {w_{t} | t ∈ T } is a complete system of left coset representatives of Perm_{ρ} in Perm_{r}.

For w ∈ Perm_{r} and I = {i_{1}, …, i_{r}} ⊂ J we write gI for {gi_{1}, …, gi_{r}}. Note that gL_{(I)} = L_{(gI)}. For every I ⊂ J with |I| = r_{1} pick w_{I} ∈ Perm_{r} such that g_{I}K = I. Then

{w_{I} | I ⊂ [1, r], |I| = r_{1}}

is a complete system of left coset representatives of Perm_{ρ} in Perm_{r}. Since

w_{I}(M_{(K)} ⊠ N_{(J\K)}) = M_{(w_{I}K)} ⊠ N_{(w_{I}(J\K))} = M_{(I)} ⊠ N_{(J\I)},

(4.15) follows from (4.14).

Again, this gives direct connections between modular Littlewood-Richardson coefficients for general linear groups and the corresponding coefficients for symmetric groups. The proofs of these are essentially identical to the corresponding proofs for branching rules in Theorem 4.10 and Theorem 4.11, so we omit the details. First we have the tensor product analogue of Theorem 4.10:

4.16. Theorem. Fix partitions λ ⊢ r and μ^{(i)} ⊢ r_{i} for i = 1, …, a. Regard λ^{t} as an element of Λ^{+}(n, r) and each (μ^{(i)})^{t} as elements of Λ^{+}(n, r_{i}). Then,

(Y^{μ^{(1)}} ⊠ ⋯ ⊠ Y^{μ^{(a)}} ↑^{Σ_{r}}: Y^{λ}) = (T_{n}((μ^{(1)})^{t}) ⊗ ⋯ ⊗ T_{n}((μ^{(a)})^{t}) : T_{n}(λ^{t})).

Moreover, if all the partitions are p-regular, then

[D^{μ^{(1)}} ⊠ ⋯ ⊠ D^{μ^{(a)}} ↑^{Σ_{r}}: D^{λ}] = [L_{n}((μ^{(1)})^{t}) ⊗ ⋯ ⊗ L_{n}((μ^{(a)})^{t}) : L_{n}(λ^{t})].

The analogue of Theorem 4.11 for tensor products is as follows:

4.17. Theorem. Let M be a GL(n)-module with p-restricted head, and for i = 1, …, a, let M_{i} be a GL(n)-submodule of V_{n}^{⊗r_{i}}. Then,

Hom_{GL(n)}(M, M_{1} ⊗ ⋯ ⊗ M_{a}) ≅ Hom_{𝔽Σ_{r}}(F_{n,r}M, ((F_{n,r_{1}}M_{1}) ⊠ ⋯ ⊠ (F_{n,r_{a}}M_{a})) ↑^{Σ_{r}}).

As in Remark 4.12, we can take the module M in Theorem 4.17 to be Δ_{n}(λ) or L_{n}(λ) for p-restricted λ ∈ Λ^{+}(n, r), while M_{i} can be ∇_{n}(μ^{(i)}) or L_{n}(μ^{(i)}) for p-restricted μ^{(i)} ∈ Λ^{+}(n, r_{i}). In particular, this gives the following special cases of Theorem 4.17, which will be used in [BK]:

4.18. Corollary. Fix p-regular partitions μ ⊢ r and λ ⊢ (r + 1), and take n > r. Regard μ^{t} and λ^{t} as elements of Λ^{+}(n). Then,

Hom_{Σ_{r+1}}(D^{λ}, D^{μ} ↑^{Σ_{r+1}}) ≅ Hom_{GL(n)}(L_{n}(λ^{t}), L_{n}(μ^{t}) ⊗ V_{n}),
Hom_{Σ_{r+1}}(S^{λ}, D^{μ} ↑^{Σ_{r+1}}) ≅ Hom_{GL(n)}(Δ_{n}(λ^{t}), L_{n}(μ^{t}) ⊗ V_{n}),
Hom_{Σ_{r+1}}(S^{μ} ↑^{Σ_{r+1}}, D^{λ}) ≅ Hom_{GL(n)}(L_{n}(λ^{t}), ∇_{n}(μ^{t}) ⊗ V_{n}),
Hom_{Σ_{r+1}}(S^{μ} ↑^{Σ_{r+1}}, (S^{λ})^{∗}) ≅ Hom_{GL(n)}(Δ_{n}(λ^{t}), ∇_{n}(μ^{t}) ⊗ V_{n}).

Finally, observe that on combining Theorem 4.10 and Theorem 4.16 with the main result Theorem 3.7 from section 3, one obtains (under our usual assumptions n ≥ r, n_{i} ≥ r_{i}) the final connections between modular Littlewood-Richardson coefficients in the symmetric group setting:

4.19. Theorem. Fix partitions λ ⊢ r and μ^{(i)} ⊢ r_{i} for i = 1, …, a. Regard λ^{t} as an element of Λ^{+}(n, r) and each (μ^{(i)})^{t} as elements of Λ^{+}(n_{i}, r_{i}) or Λ^{+}(n, r_{i}). Then,

(i) (Y^{λ} ↓_{Σ_{ρ}}: Y^{μ^{(1)}} ⊠ ⋯ ⊠ Y^{μ^{(a)}}] = (L_{n}((μ^{(1)})^{t}) ⊗ ⋯ ⊗ L_{n}((μ^{(a)})^{t}) : L_{n}(λ^{t})); (ii) ((Y^{μ^{(1)}} ⊠ ⋯ ⊠ Y^{μ^{(a)}}) ↑^{Σ_{r}}: Y^{λ}) = [L_{n}(λ^{t}) ↓_{GL(ν)}: L_{n_{1}}((μ^{(1)})^{t}) ⊠ ⋯ ⊠ L_{n_{a}}((μ^{(a)})^{t})].

Moreover, if all the partitions are p-regular, then (iii) [D^{λ} ↓_{Σ_{ρ}}: D^{μ^{(1)}} ⊠ ⋯ ⊠ D^{μ^{(a)}}] = (T_{n}((μ^{(1)})^{t}) ⊗ ⋯ ⊗ T_{n}((μ^{(a)})^{t}) : T_{n}(λ^{t})); (iv) [(D^{μ^{(1)}} ⊠ ⋯ ⊠ D^{μ^{(a)}}) ↑^{Σ_{r}}: D^{λ}] = (T_{n}(λ^{t}) ↓_{GL(ν)}: T_{n_{1}}((μ^{(1)})^{t}) ⊠ ⋯ ⊠ T_{n_{a}}((μ^{(a)})^{t})).

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

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