EXTENSIONS OF MODULES OVER SCHUR ALGEBRAS, SYMMETRIC GROUPS AND HECKE

ALGEBRAS

STEPHEN R. DOTY, KARIN ERDMANN, AND DANIEL K. NAKANO

Abstract. We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.

1. Introduction

1.1. We begin by explaining our motivation. Let E be the vector space of n-dimensional column vectors over a field k. We assume that k is infinite just for this introduction. The general linear group GL_{n} over k acts naturally on E by matrix multiplication, and this action extends to an action on the tensor power E^{⊗d}, by acting the same in each tensor position. This action commutes with the action of the symmetric group Σ_{d}, acting by place-permutation. But much more is true. Let

kGL_{n} ⟶^{ρ} End(E^{⊗d}) ⟵^{π} kΣ_{d}

be the corresponding representations, extended by linearity to each group algebra. Then the image of each representation is the complete centralizer of the other, i.e.

im ρ = End_{Σ_{d}}(E^{⊗d}), im π = End_{GL_{n}}(E^{⊗d}).

This double-centralizer theorem, which goes back to Schur [S2] in zero characteristic and DeConcini and Procesi [DP] in general, has had a strong impact on the deep interactions between the representation theories of general linear and symmetric groups, and is in some sense our starting point.

Date: June 2000.

The third author was partially supported by NSF grant DMS-9800960.

A main goal of this paper is to exploit the above setup to relate the cohomology of the two groups.

We do this by means of the Schur algebra S(n, d), which can be defined as the image of the map ρ above. The modules for this finitedimensional algebra are precisely the GL_{n}-modules whose coefficients depend on the matrix coordinate functions of GL_{n} as homogeneous polynomials of total degree d. Many interesting modules for GL_{n} (e.g. Weyl modules, simple modules) are homogeneous or are closely related to homogeneous modules.

When n is sufficiently large (n ≥ d) the module E^{⊗d} is a projective summand of the algebra A = S(n, d), i.e. there is an idempotent e ∈ A such that Ae ≅ E^{⊗d} [Gr1, (6.4f)]. It follows immediately from the double centralizer property that eAe is isomorphic with kΣ_{d}. Thus one has an exact covariant functor F (sometimes called the “Schur functor”) going from A-mod to eAe-mod, which is given by M ↦ eM. Ever since the publication of Green’s monograph [Gr1], this functor has been exploited to relate the modular representation theory of GL_{n} to that of Σ_{d}; its use in characteristic zero goes all the way back to Schur’s thesis [S1].

Everything we have said thus far admits of quantization, and indeed we are also interested in studying the relation between the cohomology of quantum general linear groups and Hecke algebras in Type A. Here the Schur algebra must be replaced by its quantization, the q-Schur algebra introduced by Dipper and James [DJ1, DJ2].

Thus we arrive at our general context: we have an idempotent e ≠ 1 in an arbitrary finite-dimensional algebra A, and we want to relate extensions in A-mod to extensions in eAe-mod. We do this by means of the two natural adjoints G_{Hom}, G_{⊗} to the Schur functor F arising from the identifications

eM ≅ Hom_{A}(Ae, M) ≅ eA ⊗_{A} M

and the associated Grothendieck spectral sequences which are implicit in the interactions of these functors. We then obtain several applications, in several cases obtaining new proofs of results obtained earlier by more ad hoc methods. In a sense, it seems that these standard homological methods have been overlooked in the literature concerned with the relation between symmetric and general linear groups, so one of our purposes is to point out the applicability of these tools to such questions.

An earlier preprint version of this paper has been circulated for some time, and now there are several papers which rely on techniques and reductions from this one to obtain new results about symmetric group cohomology. In [KN], extensions for the Schur algebras are compared with corresponding extensions for the symmetric group in a certain explicit range of degrees. The stability results obtained there provide a proof of a conjecture made by Burichenko, Kleshchev and Martin [BKM] on the cohomology of dual Specht modules. Other applications are given in [DN]. We should add that the use of quasi-hereditary algebras to compute group cohomology also appears in recent work of Bendel, Pillen and the second author [BNP]. In that paper, it is shown that that there are spectral sequences which start with modules in a bounded category of G-modules where G is a reductive algebraic group G, and converge to the extensions over the finite Chevalley group G(𝔽_{q}).

1.2. Summary. The paper is organized as follows. In the next section we consider the general situation of an arbitrary finite-dimensional algebra A and idempotent e ∈ A. We construct the Grothendieck spectral sequences mentioned above. The resulting five-term exact sequences give relationships between Ext^{1}-groups in the categories of A-modules and eAe-modules. In particular, to describe Ext^{1} in the category of eAe-modules, one needs to compute images under the adjoints G_{Hom} and G_{⊗} of the Schur functor F.

In the third section we investigate properties of these adjoint functors. We give a description of the image G(N) for an eAe-module N where G = G_{Hom} or G_{⊗}, and a more concrete description in the case when N is simple. We also give a characterization of eAe-modules which are acyclic for the functor G, in the case when A has finite global dimension.

In the fourth section we assume that the category A-mod is a highest weight category. In this situation, we obtain more precise information about the relation between Ext^{1} and Ext^{2} for A and eAe. A short proof of results in [F], [Mar2] then falls out as a corollary. We also show that one can give an interpretation of extensions between simple eAemodules in terms of the socle of a quotient of a corresponding injective indecomposable A-module by the image of a simple eAe-module under G. This shows that the structure of injective A-modules carries vital information about the extension theory for eAe. Consequences of a well-known vanishing condition for quasi-hereditary algebras (see [Jan, II 2.14]) and for symmetric groups (studied in [KS]) are discussed. It is also shown how the spectral sequence may be used to relate extensions for general linear groups of different ranks.

The fifth section looks specifically at the situation when A is the Schur algebra S_{q}(d, d) and eAe is the Hecke algebra H_{q}(d). We show that composition factor multiplicities of the higher derived functors of G_{Hom} in this context can be obtain by calculating extensions between Young and simple modules. When the characteristic is odd (resp. q is an odd root of unity), the image of G_{Hom} on the sign representation of eAe is the determinant representation. This fact in conjunction with the previous results, is used to make a direct connection between Ext^{1} for H_{q}(d) and Ext^{1} for quantum GL_{n}(k). Therefore, computing cohomology for symmetric groups in odd characterstics is equivalent to computing cohomology for GL_{n}(k) for small primes (i.e. p ≤ n). The Lusztig conjecture, which would yield information on extensions for GL_{n}(k) for p greater than the Coxeter number of the root system would give no information in this case. Guralnick and Kleshchev conjecture that if p ≥ 3 and V is an irreducible faithful kΣ_{d}-module then dim_{k} H^{1}(Σ_{d}, V) ≤ 1. Our results show that one can reformulate this problem to calculations solely in terms of GL_{n}(k)-cohomology. For symmetric groups, we give several results explicitly describing G(L), L a simple module.

Finally, in the last section examples are given to demonstrate the back and forth interplay between symmetric group and general linear group extensions.

1.3. Acknowledgements. The first author thanks the Department of Mathematics and Statistics at Utah St. University and the third author thanks the Mathematical Institute at Oxford University for hospitality and support. The second and third authors also acknowledge the support of the workshop on representations of algebraic groups at the Newton Institute in Cambridge during the completion of the paper.

We are grateful to Robert Guralnick and Leonard Scott for sharing their insights on the known cohomological results for symmetric and algebraic groups, and to Stephen Donkin and Alexander Kleshchev for suggesting improvements to an earlier version of this manuscript.

2. Spectral sequences

In this section we construct the spectral sequences and derive some immediate applications. The results in this section apply to an arbitrary finite-dimensional algebra.

2.1. Schur functor and its adjoints. Let A be a finite-dimensional associative algebra (with 1) over a field k. Denote by A-mod the category of finite-dimensional left A-modules with A-module homomorphisms. Let e be an idempotent of A. Following [Gr1, 6.2] we consider the exact covariant functor F from A-mod to eAe-mod defined on objects M ∈ A-mod by

F(M) = eM ≅ eA ⊗_{A} M ≅ Hom_{A}(Ae, M) (2.1.1)

and defined on morphisms in the obvious way. Thanks to the identifications in (2.1.1) above, it is clear that F admits two natural adjoint functors, both arising from the standard adjointness between tensor and Hom. These adjoint functors, which we shall denote by G_{Hom} and G_{⊗}, are defined on objects N ∈ eAe-mod by

G_{Hom}(N) = Hom_{eAe}(eA, N), G_{⊗}(N) = Ae ⊗_{eAe} N

and on morphisms in the obvious way. Note that the functor G_{⊗} was considered already in [Gr1]. The higher right (resp. left) derived functors of G_{Hom} (resp. G_{⊗}) will be denoted by R^{•}G_{Hom} (resp. L_{•}G_{⊗}). These derived functors can be identified as

R^{j}G_{Hom}(N) = Ext_{eAe}^{j}(eA, N), L_{j}G_{⊗}(N) = Tor_{j}^{eAe}(Ae, N)

for j ≥ 0.

For all M ∈ A-mod, N ∈ eAe-mod we have natural isomorphisms

(2.1.2) Hom_{eAe}(eM, N) ≅ Hom_{A}(M, Hom_{eAe}(eA, N))
(2.1.3) Hom_{eAe}(N, eM) ≅ Hom_{A}(Ae ⊗_{eAe} N, M)

i.e., G_{Hom} (resp., G_{⊗}) is right (resp., left) adjoint to F. The following lemma says that G_{Hom} and G_{⊗} are both right “inverse” to the functor F.

Lemma . For any object N ∈ eAe-mod we have isomorphisms:

F ∘ G_{Hom}(N) ≅ N, (2.1.4)

F ∘ G_{⊗}(N) ≅ N. (2.1.5)

Proof. We have F ∘ G_{Hom}(N) = Hom_{A}(Ae, Hom_{eAe}(eA, N)) by (2.1.1). But by standard adjointness between tensor and Hom we have the isomorphism

Hom_{A}(Ae, Hom_{eAe}(eA, N)) ≅ Hom_{eAe}(eA ⊗_{A} Ae, N)

and since eA ⊗_{A} Ae ≅ eAe this reduces to Hom_{eAe}(eAe, N), which is isomorphic with N.

Moreover, F ∘ G_{⊗}(N) = e(Ae ⊗_{eAe} N) ≅ eAe ⊗_{eAe} N ≅ N.

2.2. The spectral sequences. Fix an object M ∈ A-mod, and consider the functors from A-mod to k-mod given by

H_{M}(−) = Hom_{A}(M, −), H′_{M}(−) = Hom_{A}(−, M).

We have the following result.

Theorem . For M ∈ A-mod, N ∈ eAe-mod there are two first-quadrant Grothendieck spectral sequences, with E_{2}-pages respectively given by

(2.2.1) E_{2}^{i,j} = Ext_{A}^{i}(M, Ext_{eAe}^{j}(eA, N)) ⟹ Ext_{eAe}^{i+j}(eM, N)

and

(2.2.2) E_{2}^{i,j} = Ext_{A}^{i}(Tor_{j}^{eAe}(Ae, N), M) ⟹ Ext_{eAe}^{i+j}(N, eM).

Proof. The functors H_{M} and G_{Hom} are covariant, and left exact. We have to show that G_{Hom} takes injectives to H_{M}-acyclics. It is clearly enough to show that G_{Hom} takes injectives to injectives, since any injective A-module will be H_{M}-acyclic. Let I be injective in eAe-mod. Then G_{Hom}(I) is an injective A-module if and only if the functor Hom_{A}(−, G_{Hom}(I)) is exact. But by adjointness (2.1.2) this functor is isomorphic with the functor

Hom_{eAe}(e(−), I) = Hom_{eAe}(−, I) ∘ F

i.e. it is the composite of two exact functors, hence exact. This proves that G_{Hom} takes injectives to injectives, as desired.

Now by applying the adjointness again we have

H_{M} ∘ G_{Hom}(N) = Hom_{A}(M, G_{Hom}(N)) ≅ Hom_{eAe}(eM, N)

and so the first result follows from the usual Grothendieck spectral sequence (see [Jan, Prop. I 4.1]).

For the proof of the second result, let P be a projective eAe-module. We want to show that Hom_{A}(G_{⊗}(P), −) is exact. But by adjointness, this functor is isomorphic to the functor

Hom_{eAe}(P, −) ∘ F,

which is the composite of two exact functors, hence exact.

We remark that another way to see that G_{⊗} takes projectives to projectives is by using the following argument. Observe that Ae ⊗_{eAe} eAe ≅ Ae, so G_{⊗} takes free objects to projectives, and hence takes projectives to projectives.

Note that the differentials d_{r} in the spectral sequences have bidegree (r, 1 − r). It follows that if A has finite global dimension then the spectral sequences stop, i.e. E_{s} = E_{∞} for s sufficiently large.

Corollary . For M ∈ A-mod, N ∈ eAe-mod we have the following five-term exact sequences:

(2.2.3) 0 → Ext_{A}^{1}(M, G_{Hom}(N)) → Ext_{eAe}^{1}(eM, N)
→ Hom_{A}(M, R^{1}G_{Hom}(N)) → Ext_{A}^{2}(M, G_{Hom}(N))
→ Ext_{eAe}^{2}(eM, N)
(2.2.4) 0 → Ext_{A}^{1}(G_{⊗}(N), M) → Ext_{eAe}^{1}(N, eM)
→ Hom_{A}(L^{1}G_{⊗}(N), M) → Ext_{A}^{2}(G_{⊗}(N), M)
→ Ext_{eAe}^{2}(N, eM).

Proof. This follows from the theorem by applying [B, Ch. X, §2, exerc. 15c].

2.3. Regular and singular factors of modules. We recall the following standard facts from the representation theory of algebras. Suppose that L is a simple A-module. Then eL is either 0 or is a simple eAe-module. Moreover, all simple eAe-modules have the form eL, for some simple A-module (see [Gr1, (6.2g)]). We say that a simple Amodule L is regular if eL ≠ 0, and singular if eL = 0. By induction on the composition length we have for any A-module M

(2.3.1) eM = 0 ⇔ all composition factors of M are singular.

Observe that we have the equivalences

(2.3.2) AeM = M ⇔ M/R = AeM/R = Ae(M/R)
⇔ M/R has only regular simple factors,

where R = rad_{A}M.

A surjective homomorphism P → M/R with P projective lifts to a surjective homomorphism P → M.

Now every regular simple A-module is a homomorphic image of the projective A-module Ae, so any direct sum of regular simple A-modules is a homomorphic image of a direct sum of copies of Ae. Thus we have

(2.3.3) AeM = M ⇔ M is a homomorphic image of a

direct sum of copies of Ae.

We also have

(2.3.4) M/AeM is the largest quotient of M which has only singular composition factors.

To prove the last claim, suppose that S is an A-submodule of M such that M/S has only singular composition factors. We must show that S ⊃ AeM. The short exact sequence

0 → S → M → M/S → 0

induces the exact sequence

0 → eS → eM → e(M/S) → 0.

Since e(M/S) = 0, we conclude that eS = eM. Hence AeS = AeM and since S ⊃ AeS the claim is established.

We define S(M) for an A-module M to be the set of all m ∈ M such that eAm = 0. One can easily check that S(M) is an A-submodule of M. If N is any A-submodule of M all the composition factors of which are singular, then eN = 0 by (2.3.1), so eAN = 0 and thus N ⊂ S(M). Moreover, it is clear that eS(M) = 0, so S(M) itself is such a submodule. It follows that:

(2.3.5) S(M) is the largest A-submodule of M all the composition factors of which are singular.

The following observation will be used later.

Lemma . Suppose that L is a regular simple A-module. Then G_{Hom}(eL) has simple socle, isomorphic with L, and G_{⊗}(eL) has a unique simple quotient, again isomorphic with L.

Proof. By adjointness (2.1.2), we have an isomorphism

Hom_{A}(L′, G_{Hom}(eL)) ≅ Hom_{eAe}(eL′, eL)

and by [Gr1, (6.2g)] this is isomorphic with k if L′ ≅ L, and 0 otherwise. This proves the first claim. The other claim follows from (2.1.3) by an entirely similar argument.

2.4. Relating extensions. The spectral sequences and the Schur functor can be used to give a precise relationship between Ext^{1}-groups for A-mod and eAe-mod.

Theorem (A). Let M ∈ A-mod and N ∈ eAe-mod.

(i) eR^{1}G_{Hom}(N) = 0, i.e. the A-module Ext_{eAe}^{1}(eA, N) has only singular composition factors.

(ii) Suppose that M = AeM. Then

Ext_{A}^{1}(M, G_{Hom}(N)) ≅ Ext_{eAe}^{1}(eM, N),

i.e. adjointness between F, G_{Hom} holds at the Ext^{1} level. Furthermore, there exists an injective map

Ext_{A}^{2}(M, G_{Hom}(N)) ↪ Ext_{eAe}^{2}(eM, N).

Proof. (i) Let I be the injective hull of N in eAe-mod. There exists an exact sequence of the form

0 → N → I → C → 0.

Since Ext_{eAe}^{1}(eA, I) = 0, the long exact sequence induced from the above short exact sequence yields the exact sequence

0 → G_{Hom}(N) → G_{Hom}(I) → G_{Hom}(C) → R^{1}G_{Hom}(N) → 0.

Applying the (exact) functor F to this sequence and applying Lemma 2.1 we get an exact sequence isomorphic to

0 → N → I → C → eR^{1}G_{Hom}(N) → 0.

which forces eR^{1}G_{Hom}(N)) = 0. This proves the first claim in (i). The second statement in (i) now follows by (2.3.1).

(ii) The result follows immediately from the five term exact sequence, once we have established the equality Hom_{A}(M, R^{1}G_{Hom}(N)) = 0. But, by (2.3.4), the equality M = AeM tells us that no quotient of M can have any singular composition factors. Combined with the result from part (i), this shows that Hom_{A}(M, R^{1}G_{Hom}(N)) = 0.

Theorem (B). Let M ∈ A-mod and N ∈ eAe-mod.

(i) L_{1}G_{⊗}(N), i.e. the A-module Tor_{1}^{eAe}(eA, N) has only singular composition factors. (ii) Suppose that S(M) = 0. Then

Ext_{A}^{1}(G_{⊗}(N), M) ≅ Ext_{eAe}^{1}(N, eM),

i.e. adjointness between G_{⊗}, F holds at the Ext^{1} level. Furthermore, there exists an injective map

Ext_{A}^{2}(G_{⊗}(N), M) ↪ Ext_{eAe}^{2}(N, eM).

Proof. The argument is similar to the proof of Theorem 2.4A, except that one uses the projective cover instead of the injective hull.

10 STEPHEN R. DOTY, KARIN ERDMANN, AND DANIEL K. NAKANO

2.5. Extensions of simple eAe-modules. The next result describes how extensions of simple eAe-modules arise as homomorphic images in the quotient of an injective hull.

Theorem (A). Let L be a simple object in A-mod, N any object in eAe-mod, and I the injective hull of N in eAe-mod.

(i) G_{Hom}(I) is the injective hull of G_{Hom}(N) in A-mod; (ii) Ext_{eAe}^{1}(eL, N) ≅ Hom_{A}(L, G_{Hom}(I)/G_{Hom}(N)); in particular,

dim_{k} Ext_{eAe}^{1}(eL, N) = [soc_{A}(G_{Hom}(I)/G_{Hom}(N)) : L] · dim_{k} End_{A}(L).

Proof. (i) Since G_{Hom} takes injectives to injectives, G_{Hom}(I) is injective. Moreover, by the adjointness (see 2.1) we have

Hom_{A}(L, G_{Hom}(N)) ≅ Hom_{eAe}(eL, N)
= Hom_{eAe}(eL, I)
≅ Hom_{A}(L, G_{Hom}(I)).

Thus the socles of G_{Hom}(N) and G_{Hom}(I) agree, and the result follows.

(ii) Apply the long exact sequence in cohomology to the short exact sequence 0 → G_{Hom}(N) → G_{Hom}(I) → G_{Hom}(I)/G_{Hom}(N) → 0 to get the exact sequence

0 → Hom_{A}(L, G_{Hom}(N)) →^{f} Hom_{A}(L, G_{Hom}(I))
→ Hom_{A}(L, G_{Hom}(I)/G_{Hom}(N)) → Ext_{A}^{1}(L, G_{Hom}(N)) → 0.

As we have seen from above f must be an isomorphism, thus

Ext_{A}^{1}(L, G_{Hom}(N)) ≅ Hom_{A}(L, G_{Hom}(I)/G_{Hom}(N)).

The result now follows by Theorem 2.4 in case L is regular. In case L is singular, Ext_{eAe}^{1}(eL, N) is zero. Moreover, by the five term exact sequence it follows that Ext_{A}^{1}(L, G_{Hom}(N)) = 0, and so the result holds vacuously in this case.

Theorem (B). Let L be a simple object in A-mod, N any object in eAe-mod, and P the projective cover of N in eAe-mod.

(i) G_{⊗}(P) is the projective cover of G_{⊗}(N) in A-mod; (ii) If K is the kernel of the surjection G_{⊗}(P) → G_{⊗}(N), then Ext_{eAe}^{1}(N, eL) ≅ Hom_{A}(K, L); in particular,

dim_{k} Ext_{eAe}^{1}(N, eL) = [hd_{A} K : L] · dim_{k} End_{A}(L).

Proof. The argument is similar to the proof of Theorem 2.5(A).

2.6. Composition factors. The next results gives information on the composition factor multiplicities of the derived functors of G_{Hom} and G_{⊗}.

Proposition (A). Let L be a simple A-module, P (L) its projective cover. Let N be an object in eAe-mod. The composition factor multiplicity of L in the A-module R^{j}G_{Hom}(N) is given by

(2.6.1) [R^{j}G_{Hom}(N) : L] = dim_{k} Ext_{eAe}^{j}(eP (L), N), j ≥ 0.

In particular, we have

(2.6.2) [G_{Hom}(D) : L] = [hd_{eAe}(eP (L)) : D] · dim_{k} End_{A}(L)

for any simple object D in eAe-mod.

Proof. The functor H_{P (L)} = Hom_{A}(P (L), −) is exact, so taking M = P (L), the spectral sequence of (2.2.1) collapses, yielding isomorphisms

Hom_{A}(P (L), Ext_{eAe}^{j}(eA, N)) ≅ Ext_{eAe}^{j}(eP (L), N)

for all j ≥ 0. For any A-module M′, the composition factor multiplicity [M′ : L] of L in M′ is given by dim_{k} Hom_{A}(P (L), M′). Taking M′ = Ext_{eAe}^{j}(eA, N) and combining with the isomorphism above yields the first claim.

In particular, the case j = 0 (with N replaced by D) gives us the equalities

[Hom_{eAe}(eA, D) : L] = dim_{k} Hom_{eAe}(eP (L), D)
= [hd_{eAe}(eP (L)) : D] · dim_{k} End_{A}(L)

and this establishes the second claim.

Proposition (B). Let L be a simple A-module, I(L) its injective envelope. Let N be an object in eAe-mod. The composition factor multiplicity of L in L_{j}G_{⊗}(N) is given by

(2.6.3) [L_{j}G_{⊗}(N) : L] = dim_{k} Ext_{eAe}^{j}(N, eI(L)), j ≥ 0.

In particular,

(2.6.4)  [G_{⊗}(D) : L] = [soc_{eAe} eI(L) : D] · dim_{k} End_{A}(L)

for any simple object D in eAe-mod.

Proof. The proof is entirely similar, using the other spectral sequence from Theorem 2.2. We leave the details to the reader.

2.7. Extensions of singular simples. The spectral sequences can be used to compute extensions of singular simple A-modules, as follows.

Theorem (A). Let N be an eAe-module and let L_{S}, L_{R} simple objects in A-mod with L_{S} singular and L_{R} regular. Then:

(i) Ext_{A}^{1}(L_{S}, G_{Hom}(N)) = 0; (ii) Ext_{A}^{2}(L_{S}, G_{Hom}(N)) ≅ Hom_{A}(L_{S}, R^{1}G_{Hom}(N)); (iii) Ext_{A}^{1}(L_{S}, L_{R}) ≅ Hom_{A}(L_{S}, G_{Hom}(eL_{R})/L_{R}); in particular,

dim_{k} Ext_{A}^{1}(L_{S}, L_{R}) = [soc_{A} G_{Hom}(eL_{R})/L_{R} : L_{S}] · dim_{k} End_{A}(L_{S}).

Proof. The first two statements are immediate from the five term exact sequence (2.2.3), with M = L_{S}.

For (iii), consider the short exact sequence 0 → L_{R} → G_{Hom}(eL_{R}) → C → 0 with C = G_{Hom}(eL_{R})/L_{R}. Applying the long exact sequence in cohomology, one gets the exact sequence

0 → Hom_{A}(L_{S}, L_{R}) → Hom_{A}(L_{S}, G_{Hom}(eL_{R})) → Hom_{A}(L_{S}, C)
→ Ext_{A}^{1}(L_{S}, L_{R}) → Ext_{A}^{1}(L_{S}, G_{Hom}(eL_{R})) → …

By Lemma 2.3 the socle of G_{Hom}(eL_{R}) is L_{R}, thus Hom_{A}(L_{S}, G_{Hom}(eL_{R})) = 0. On the other hand, Ext_{A}^{1}(L_{S}, G_{Hom}(eL_{R})) = 0 by part (i), so part (iii) holds.

Theorem (B). Let N be an eAe-module and let L_{S}, L_{R} simple objects in A-mod with L_{S} singular and L_{R} regular. Then:

(i) Ext_{A}^{1}(G_{⊗}(N), L_{S}) = 0; (ii) Ext_{A}^{2}(G_{⊗}(N), L_{S}) ≅ Hom_{A}(L_{1}G_{⊗}(N), L_{S}); (iii) If K is the kernel of the surjection Ae ⊗_{eAe} eL_{R} → L_{R}, then Ext_{A}^{1}(L_{R}, L_{S}) ≅ Hom_{A}(K, L_{S}); in particular,

dim_{k} Ext_{A}^{1}(L_{R}, L_{S}) = [hd_{A} K : L_{S}] · dim_{k} End_{A}(L_{S}).

Proof. The argument is similar to the proof of Theorem 2.7(A) above.

In our later applications, we will often assume that k is algebraically closed, in which case End_{A}(L) ≅ k for all simple A-modules L. This fact will be used without further comment.

3. Properties of the functors G_{⊗} and G_{Hom}

3.1. Let M ∈ A-mod and P (M) be its projective cover. Denote by Ω^{1}(M) the kernel of a surjection P (M) → M. Recursively, let Ω^{m}(M) be the kernel of an essential map P (Ω^{m−1}(M)) → Ω^{m−1}(M). On the other hand, let I(M) be the injective hull of M and Ω^{−1}(M) be the cokernel of an injection M → I(M), and define Ω^{−m}(M) to be the cokernel of Ω^{−m+1}(M) → I(Ω^{−m+1}(M)).

Let μ be the natural transformation G_{⊗} ∘ F = Ae ⊗_{eAe} e(−) → Ae(−) given by multiplication. Consider also the natural transformation Ae(−) → id_{A-mod} given by inclusion, and let μ′ be its composite with μ. Consider A-modules Y such that μ′_{Y} is an isomorphism; we claim that this is the case for all modules of the form Y = G_{⊗}(Z). To see this, it is enough to check that μ′_{Y} is surjective, since F∘G_{⊗}(Y) ≅ Y. A general element of such a Y is of the form

∑_{i} a_{i}e ⊗ z_{i}

for a_{i} ∈ A and z_{i} ∈ Z. Then e ⊗ z_{i} ∈ Y = Ae ⊗_{eAe} Z and ∑_{i} a_{i}e ⊗ e(e ⊗ z_{i}) ∈ Ae ⊗_{eAe} eY is mapped by μ′_{Y} onto the above element. This proves the claim.

Let X be the full subcategory of A-mod with objects all X such such μ′_{X} and μ′_{Ω^{1}(X)} are isomorphisms. Then we have the following.

Proposition (A). The following are equivalent for an eAe-module N:

(i) L_{1}G_{⊗}(N) = 0. (ii) G_{⊗}(N) belongs to X. (iii) N ≅ F(X) for some X ∈ X.

Proof. (ii) ⇒ (iii): Set X = G_{⊗}(N) and assume this is in X. Then N ≅ F ∘ G_{⊗}(N) = F(X).

(iii) ⇒ (ii): Suppose N ≅ F(X) for some X ∈ X. Then we have G_{⊗}(N) ≅ G_{⊗} ∘ F(X). But X ∈ X, so μ′_{X} is an isomorphism from G_{⊗} ∘ F(X) to X.

(i) ⇒ (ii) Let L_{1}G_{⊗}(N) = 0 and set X = G_{⊗}(N). Then by the above remark, μ′_{X} is an isomorphism. Let P be a projective cover of N and 0 → Ω^{1}(N) → P → N → 0 exact. Since L_{1}G_{⊗}(N) = 0 this gives an exact sequence

0 → G_{⊗}(Ω^{1}(N)) → G_{⊗}(P) → G_{⊗}(N) = X → 0

Then G_{⊗}(P) is projective, hence the kernel is isomorphic to Ω^{1}(X) ⊕ Q with Q projective. By the above remark, μ′_{G_{⊗}(Ω^{1}(N))} is an isomorphism and hence μ′_{Ω^{1}(X)} is an isomorphism as well. This shows that X belongs to X.

(iii) ⇒ (i) Take some X ∈ X, and let N = F(X) = eX. We must show that L_{1}G_{⊗}(eX) = 0. Let P be a projective cover of X and

0 → Ω^{1}(X) ⟶^{ϕ} P ⟶^{η} X → 0

be exact. We get a commutative diagram with exact rows

L_{1}G_{⊗}(eX) ⟶ G_{⊗}(eΩ^{1}(X)) ⟶^{1⊗ϕ} G_{⊗}(eP) ⟶ G_{⊗}(eX) ⟶ 0
μ′_{Ω^{1}(X)}↓  ^{μ′_{P}} ↓  ^{μ′_{X}} ↓
0  ⟶ Ω^{1}(X) ⟶^{ϕ} P ⟶ X ⟶ 0

We assume that μ′_{Ω^{1}(X)} and μ′_{X} are isomorphisms. Then by the Snake Lemma, μ′_{P} is an isomorphism and then eP is a projective eAe-module, hence L_{1}G_{⊗}(eP) = 0. So we have L_{1}G_{⊗}(eX) = Ker(1⊗ϕ). Since μ′_{Ω^{1}(X)} is injective, so is ϕ ∘ μ′_{Ω^{1}(X)}, which is equal to μ′_{P} (1 ⊗ ϕ), and hence 1 ⊗ ϕ is also injective, as required.

Let σ′ be the natural transformation X → Hom_{eAe}(eA, eX) for X an A-module defined in the following way. Each x ∈ X is sent to the map f_{x} : eA → eX where f_{x}(ea) = eaex. Observe that S(−) is sent of zero under this transformation, and S(−) = ker(σ′). Therefore σ′ = σ ∘ η where η : (−) → (−)/S(−) and σ : (−)/S(−) → Hom_{eAe}(eA, e(−)). Let Y be the full subcategory in A-mod of objects Y such that σ′_{Y} and σ′_{Ω^{−1}(Y)} are isomorphisms. Observe that if Y = G_{Hom}(Z) then S(Y) = 0 by using adjointness, and σ′_{Y} is an isomorphism. The following result (dual to Proposition 3.1(A)) provides us with a characterization of F(Y) for Y ∈ Y.

Proposition (B). The following are equivalent for an eAe-module N:

(i) R^{1}G_{Hom}(N) = 0. (ii) G_{Hom}(N) belongs to Y. (iii) N ≅ F(Y) for some Y ∈ Y.

Proof. The proof is completely analogous to the one given in Proposition 3.1(A).

The preceding two propositions can be used to prove for which modules there is adjointness for the Ext^{1}-functor.

Corollary . Let M ∈ A-mod and N ∈ eAe-mod.

(a) If N ≅ F(X) for some X ∈ X then

Ext_{A}^{1}(G_{⊗}(N), M) ≅ Ext_{eAe}^{1}(N, F(M)).

(b) If N ≅ F(Y) for some Y ∈ Y then

Ext_{A}^{1}(M, G_{Hom}(N)) ≅ Ext_{eAe}^{1}(F(M), N).

Proof. (a) Since X ∈ X we have L_{1}G_{⊗}F(X) = 0 by the Proposition 3.1(A), Consider the 5-term exact sequence obtained from the spectral sequence in Theorem 2.2. The third term is Hom_{A}(L_{1}G_{⊗}(N), M), which is zero, and so the claim follows.

(b) This statement follows by using a similar argument to part (a). This involves using the other spectral sequence (2.2.1) and the corresponding five term exact sequence.

3.2. The main result of this section gives a description of G_{⊗}(eL) for a regular simple A-module L. This is a special case of the following.

Theorem . Suppose M is an A-module such that M = AeM. Then there is an exact sequence

0 → Ω^{1}(M)/AeΩ^{1}(M) → G_{⊗}(eM) → M → 0

and

G_{⊗}(eM) ≅ P (M)/AeΩ^{1}(M).

Moreover, for any simple A-module L we have

Hom_{A}(M, L) ≅ Hom_{A}(G_{⊗}(eM), L)

and this is zero for L singular.

Proof. We begin with the exact sequence 0 → Ω^{1}(M) ⟶^{ι} P (M) → M → 0, apply G_{⊗} ∘ F, and obtain a commutative diagram with exact rows

G_{⊗}(eΩ^{1}(M)) ⟶^{1⊗ι} G_{⊗}(eP (M)) ⟶^{γ} G_{⊗}(eM) ⟶ 0
μ_{1}↓  ^{μ}↓  ^{μ_{2}}↓
0 ⟶ Ω^{1}(M) ⟶^{ι} P (M) ⟶ M ⟶ 0.

The vertical maps are given by multiplication and ι is inclusion. By hypothesis, all top composition factors of M, and hence of P (M), are regular. Consequently we have P (M) = AeP (M) and μ, μ_{2} are surjective. We claim that μ is an isomorphism. Then by the Snake Lemma we obtain

ker(μ_{2}) ≅ coker(μ_{1}) = Ω^{1}(M)/AeΩ^{1}(M),

and this provides the exact sequence.

It remains to show that μ is an isomorphism. To be precise, let μ be the natural transformation Ae ⊗_{eAe} e(−) → Ae(−), given by multiplication. We must argue that if P is projective over A and P = AeP then μ_{P} is an isomorphism. Without loss of generality, we may assume that P is indecomposable. Then P is isomorphic to a direct summand of Ae since P = AeP. So it suffices to show that μ_{Ae} is an isomorphism. But μ_{Ae} is the same as the canonical map Ae⊗_{eAe}eAe → Ae (= AeAe), an isomorphism of right eAe-modules and in particular a vector space isomorphism, as claimed.

Now we have a surjection γμ^{−1} : P (M) → G_{⊗}(eM). We claim that this has kernel AeΩ^{1}(M). By comparing composition lengths we see that it is enough to show one inclusion. Let x ∈ AeΩ^{1}(M), then it is straightforward to show that γμ^{−1}(ιx) = 0, which proves the claim and the second statement in the theorem.

Finally, let L be simple. Applying Hom_{A}(−, L) to the exact sequence, we obtain the exact sequence

0 → Hom_{A}(M, L) → Hom_{A}(G_{⊗}(eM), L) → Hom_{A}(Ω^{1}(M)/AeΩ^{1}(M), L).

If L = AeL then the last term is zero since e(Ω^{1}(M)/AeΩ^{1}(M)) = 0. On the other hand, if eL = 0 then by the adjointness (2.1.3), Hom_{A}(G_{⊗}(eM), L) = 0.

Corollary . Let L be a simple A-module with L = AeL. Then G_{⊗}(eL) is the largest quotient of the projective cover of L whose radical has only singular composition factors.

Proof. By (2.3.4), the largest quotient of P (L) whose radical has only singular composition factors is P (L)/AeΩ^{1}(L) (since Ω^{1}(L) = rad P (L)). The preceding theorem shows that G_{⊗}(eL) ≅ P (L)/AeΩ^{1}(L).

3.3. The following theorem is a result dual to Theorem 3.2. A complete proof is included because the proof actually relies on using Theorem 3.2.

Theorem . Suppose M is an A-module such that S(M) = 0. Then there is an exact sequence

0 → M → G_{Hom}(eM) → S(Ω^{−1}(M)) → 0

and

I(M)/G_{Hom}(eM) ≅ Ω^{−1}(M)/S(Ω^{−1}(M)).

Moreover, for any simple A-module L we have

Hom_{A}(L, M) ≅ Hom_{A}(L, G_{Hom}(eM))

and this is zero for L singular.

Proof. Consider the exact sequence 0 → M ⟶^{ι}I(M) → Ω^{−1}(M) → 0. By applying G_{Hom} ∘ F, one obtains a commutative diagram with exact rows

0 ⟶ G_{Hom}(eM) ⟶^{γ} G_{Hom}(eI(M)) ⟶ G_{Hom}(eΩ^{−1}(M))
μ_{1}↑  ^{μ}↑  ^{μ_{2}}↑
0 ⟶  M  ⟶^{ι}  I(M)  ⟶  Ω^{−1}(M)  ⟶ 0.

By assumption S(M) = 0, thus S(I(M)) = 0. Consequently, μ, μ_{1} are injective. We claim that μ is an isomorphism. If μ is an isomorphism then by the Snake Lemma

coker(μ_{1}) ≅ ker(μ_{2}) = S(Ω^{−1}(M)),

and this provides the exact sequence.

It remains to show that μ is an isomorphism. Without a loss of generality, it suffices to prove that μ is an isomorphism for I(L) for L simple with S(I(L)) = 0. This is because I(M) is a direct sum of such indecomposable injective modules. Let eS be a simple eAe-module. One can apply the five term exact sequence from the spectral sequence involving G_{⊗} to get

0 → Ext_{A}^{1}(G_{⊗}(eS), I(L)) → Ext_{eAe}^{1}(eS, eI(L)) → Hom_{A}(L_{1}G_{⊗}(eS), I(L)) → ...

Since S(I(L)) = 0 and L_{1}G_{⊗}(eS) has only singular composition factor, it follows that Hom_{A}(L_{1}G_{⊗}(eS), I(L)) = 0. On the other hand, I(L) is injective so Ext_{A}^{1}(G_{⊗}(eS), I(L)) = 0. Therefore, Ext_{eAe}^{1}(eS, eI(L)) = 0 for all simple eAe-modules eS, thus eI(L) is injective. Now G_{Hom} takes injectives to injectives so G_{Hom}(eI(L)) is injective.

Let S be a simple A-module. Then

Hom_{A}(S, G_{Hom}(eI(L))) ≅ Hom_{eAe}(eS, eI(L))
≅ Hom_{A}(G_{⊗}(eS), I(L))
≅ \begin{cases}k & S ≅ L \\ 0 & otherwise. \end{cases}

The last isomorphism uses Corollary 3.2. and the fact that S(I(L)) =

0. Therefore, I(L) ≅ G_{Hom}(eI(L)) and μ is an isomorphism.

Next observe that one can use the exact sequence to obtain the following isomorphisms.

Ω^{−1}(M)/S(Ω^{−1}(M)) ≅ (I(M)/M)/(G_{Hom}(eM)/M) ≅ I(M)/G_{Hom}(eM).

Finally, let L be simple. Applying Hom_{A}(L, −) to the exact sequence, we obtain the exact sequence

0 → Hom_{A}(L, M) → Hom_{A}(L, G_{Hom}(eM)) → Hom_{A}(L, S(M)).

If L is regular then the last term is zero. On the other hand, if L is singular (i.e. eL = 0) then, by the adjointness (2.1.3), Hom_{A}(L, G_{Hom}(eM)) = 0.

Corollary . Let L be a simple A-module with S(L) = 0. Then G_{Hom}(eL) is the largest submodule of the injective hull of L whose socle quotient has only singular composition factors.

Proof. The largest submodule of I(L)/L which has singular composition factors is S(I(L)/L). Now G_{Hom}(eL) ↪ I(L) and by Theorem 3.3 G_{Hom}(eL)/L ≅ S(Ω^{−1}(L)) ≅ S(I(L)/L).

3.4. Given reasonable assumptions on our algebras A and eAe, we can provide a necessary and sufficient condition for a module being acyclic for G_{⊗} or G_{Hom}.

Theorem . Let A have finite global dimension and suppose that eAe is a self-injective algebra.

(a) N ∈ eAe-mod is G_{⊗}-acyclic if and only if N is projective.

(b) N ∈ eAe-mod is G_{Hom}-acyclic if and only if N is injective.

Proof. (a) If N is projective, then N is G_{⊗}-acyclic. For the converse, suppose that L_{n}G_{⊗}(N) = 0 for n > 0. Let Λ be the set of isomorphism classes of simple A-modules, and let L(λ) be a representative of the class of λ ∈ Λ. Set M equal to the direct sum of the L(λ) as λ varies over Λ. With these choices, the spectral sequence in Section 2.4 collapses, yielding the isomorphism

Ext_{A}^{m}(G_{⊗}(N), M) ≅ Ext_{eAe}^{m}(N, F(M)))

for m ≥ 0. Since A has finite global dimension, Ext_{A}^{m}(G_{⊗}(N), M) = 0 for sufficiently large values of m, thus Ext_{eAe}^{m}(N, F(M)) = 0 for sufficiently large values of m. Now,

Ext_{eAe}^{m}(N, F(M)) ≅ \underline{Hom}_{eAe}(Ω^{m}(N), F(M)),

where \underline{Hom} is the quotient modulo maps which factor through projectives. But F(M) is the direct sum of all simple eAe-modules, so if this space is zero the Ω^{m}(N) is projective, and hence N is projective as well. The previous statement uses the fact that eAe is self-injective.

(b) This follows by an analogous (dual) argument to part (a).

We remark that for arbitrary eAe the proof shows that N is G_{Hom}- acyclic if and only if the injective dimension of N is less than or equal to the global dimension of A.

This theorem applies when A is a Schur algebra or a q-Schur algebra and e is the usual idempotent defining the usual Schur functor, i.e. in these cases it is known that A has finite global dimension and that eAe is self-injective. The theorem cannot be applied to the infinitesimal Schur algebras, because in general the infinitesimal Schur algebras will not have finite global dimension [DNP]; an interesting problem would be to determine an acyclicity condition for these algebras.

4. Applications to Highest Weight Categories

4.1. We will now examine the situation where mod(A) is a highest weight category, or equivalently when the algebra A is quasi-hereditary. (These notions were introduced in [CPS].) Let Λ be a poset of weights indexing the simple modules and {L(λ)}_{λ∈Λ} be the complete set of non-isomorphic simple A-modules. For each λ ∈ Λ, let I(λ) denote the injective hull of L(λ). The two intermediate families of modules will be denoted by {Δ(λ)}_{λ∈Λ} and {∇(λ)}_{λ∈Λ}. These families of modules satisfy the properties that for λ ∈ Λ; L(λ) = soc_{A}∇(λ) = Δ(λ)/rad_{A}Δ(λ). Moreover, if L(μ) occurs as a composition factor of rad_{A}Δ(λ), or of ∇(λ)/L(λ), then μ < λ.

In the category of eAe-modules, for each λ ∈ Λ, let D_{λ} = eL(λ). Set

Λ_{reg} = {λ ∈ Λ : eL(λ) ≠ 0}
Λ_{sing} = {λ ∈ Λ : eL(λ) = 0}.

The set Λ_{reg} gives a one-to-one parametrization of the simple eAemodules. Moreover, for each λ ∈ Λ, set Y^{λ} = eI(λ), S^{λ} = e∇(λ) and S_{λ} = eΔ(λ). A concrete interpretation of these modules is given below for the Schur algebras and the q-Schur algebras.

(4.1.1) Schur algebras. Consider the Schur algebra A = S(n, d) over an infinite field k of characteristic p > 0 when d ≤ n. The simple S(n, d)-modules are in bijective correspondence with set of partitions of d with at most n parts, denoted by Λ = Λ^{+}(n, d). One can also identify Λ as the set of dominant polynomial weights of G = GL_{n}(k) of degree d. For λ ∈ Λ, ∇(λ) is H^{0}(λ) = ind_{B}^{G}λ where B is the lower triangular Borel subgroup of G. Also, Δ(λ) is V (λ), the Weyl module for G; here Δ(λ) ≅ (∇(λ))^{τ} where τ denotes the usual contravariant duality.

Let e = ζ_{(1,2,...,d)(1,2,...,d)} be the idempotent in S(n, d) described in [Mar1, p. 94] or [Gr1, (6.1)]. Then eS(n, d)e identifies with kΣ_{d}. The simple modules for kΣ_{d} are in one to one correspondence with Λ_{reg} = Λ^{(p)}(d), the set of p-restricted partitions of d. Recall that a partition (λ_{1}, λ_{2}, …, λ_{d}) is p-restricted (column p-regular in [Gr1]) if and only if 0 ≤ λ_{i} − λ_{i+1} ≤ p − 1 for 1 ≤ i ≤ d − 1 and 0 ≤ λ_{d} ≤ p − 1. In this case the Young modules are {Y^{λ}}_{λ∈Λ}, the Specht modules are {S^{λ}}_{λ∈Λ}, and the dual Specht modules are {S_{λ}}_{λ∈Λ}.

(4.1.2) q-Schur algebras. The setup is similar to the one given above. Let A = S_{q}(n, d) be the q-Schur algebra over an algebraically closed field k and assume that q is a primitive lth root of unity. Let d ≤ n, and let e be the appropriate idempotent in S_{q}(n, d) (such that eS_{q}(n, d)e ≅ H_{q}(d), the Hecke algebra of Type A). The simple S_{q}(n, d)-modules are indexed by partitions of d. For each λ ∈ Λ, Δ(λ) = V_{q}(λ) (the q-Weyl module) and ∇(λ) = H_{q}^{0}(λ) the quantum induced module. The simple modules for H_{q}(d) are indexed by the set of l-restricted partitions of d, denoted by Λ_{reg} = Λ^{(l)}(d). For each Λ_{reg}, the q-Young modules are {Y^{λ}}, the q-Specht modules are {S^{λ}}_{λ∈Λ}, and the duals of the q-Specht modules are {S_{λ}}_{λ∈Λ}.

It is well-known that both the Schur algebras and the q-Schur algebras are in fact quasi-hereditary. The result in the classical case was first given in [Don1, (2.2)]. Other proofs have been given by Parshall [P] and Green [Gr2]. The proof in the quantum case appeared in [PW] with a restriction on q. A proof for general q is given by Donkin [Don2].

4.2. Ext^{1} for eAe-modules. The general results in Section 2 can be applied to obtain information about extensions between the {D_{λ}}, {S^{λ}}, and {S_{λ}}. Our first result concerns Ext^{1} between the simple eAe-modules.

Theorem . Let λ, μ ∈ Λ_{reg}.

(i) Ext_{eAe}^{1}(D_{λ}, D_{μ}) ≅ Ext_{A}^{1}(L(λ), G_{Hom}(D_{μ})). (ii) If G_{Hom}(D_{μ}) = L(μ) then we have an isomorphism

Ext_{eAe}^{1}(D_{λ}, D_{μ}) ≅ Ext_{A}^{1}(L(λ), L(μ)).

(iii) There is an exact sequence of the form

0 → G_{Hom}(D_{μ}) → I(μ) → I(μ)/G_{Hom}(D_{μ}) → 0,

and one has an isomorphism

Ext_{eAe}^{1}(D_{λ}, D_{μ}) ≅ Hom_{A}(L(λ), I(μ)/G_{Hom}(D_{μ})).

In particular,

dim_{k} Ext_{eAe}^{1}(D_{λ}, D_{μ}) = [soc_{A} I(μ)/G_{Hom}(D_{μ}) : L(λ)].

Proof. (i) Set M = L(λ) and N = D_{μ}. Then F(M) = D_{λ}. The result now follows by Theorem 2.4(A)(ii).

(ii) This is clear from (i).

(iii) Let L = L(λ), N = D_{μ} and let I^{μ} be the injective hull of D_{μ}. According to Theorem 2.5(A)(i), G_{Hom}(I^{μ}) = I(μ). Since G_{Hom} is left exact, G_{Hom}(D_{μ}) injects in I(μ) and this gives the exact sequence. The rest of the statements in (iii) now follow by Theorem 2.5(A)(ii).

4.3. Recall that the Ext^{1}-quiver of a finite-dimensional algebra B has vertices the simple modules, and the number of arrows from S to S′ is equal to the dimension of Ext_{B}^{1}(S, S′). The next two results show that a portion of the Ext^{1}-quiver for mod(A), consisting of the vertices corresponding to simple modules with weights in Λ_{reg}, appears as a subquiver in the Ext^{1}-quiver for mod(eAe).

Theorem . Let μ ∈ Λ_{reg} and M ∈ A-mod such that AeM = M. Then there is an injective homomorphism

Ext_{A}^{1}(M, L(μ)) ↪ Ext_{eAe}^{1}(F(M), D_{μ}).

In particular, if λ, μ ∈ Λ_{reg} then there exist injective homomorphisms (i) Ext_{A}^{1}(L(λ), L(μ)) ↪ Ext_{eAe}^{1}(D_{λ}, D_{μ}); (ii) Ext_{A}^{1}(Δ(λ), L(μ)) ↪ Ext_{eAe}^{1}(S_{λ}, D_{μ}).

Proof. By Corollary 3.3, there exists a short exact sequence of the form 0 → L(μ) → G_{Hom}(D_{μ}) → C → 0, where C contains only composition factors with singular weights. The short exact sequence induces the long exact sequence:

0 → Hom_{A}(M, L(μ)) → Hom_{A}(M, G_{Hom}(D_{μ})) → Hom_{A}(M, C)
→ Ext_{A}^{1}(M, L(μ)) → Ext_{A}^{1}(M, G_{Hom}(D_{μ})) → ⋯

Since M = AeM, M/ rad_{A} M is a direct sum of simples L(λ) with λ ∈ Λ_{reg}, so Hom_{A}(M, C) = 0. Consequently, Ext_{A}^{1}(M, L(μ)) injects into Ext_{A}^{1}(M, G_{Hom}(D_{μ})). The theorem now follows by Theorem 2.4(A)(ii).

As a consequence of the preceding theorem we recover (as a corollary) the results proved by Fettes [F] and Martin [Mar2].

Corollary . Let λ, μ ∈ Λ_{reg}.

(i) If Ext_{eAe}^{1}(D_{λ}, D_{μ}) = 0 then Ext_{A}^{1}(L(λ), L(μ)) = 0. (ii) If Ext_{eAe}^{1}(S_{λ}, D_{μ}) = 0 then Ext_{A}^{1}(Δ(λ), L(μ)) = 0.

4.4. A vanishing result. Let (Λ, ≤) denote the weight poset and ordering for the quasi-hereditary algebra A. The following vanishing condition holds. Let λ, μ ∈ Λ.

(4.4.1) If μ ≯ λ then Ext_{A}^{1}(Δ(λ), L(μ)) = 0.

For algebraic groups this is proved in [Jan, II 2.14]; in general it follows from [CPS, 3.1(c)(ii)]. Similarly one shows that

(4.4.2) Ifμ ≯ λ then Ext_{A}^{1}(L(μ), ∇(λ)) = 0.

We can formulate a vanishing condition by using the spectral sequence in Theorem 2.4.

Proposition . Suppose μ ≯ λ and assume λ, μ ∈ Λ_{reg}.

(i) If G_{Hom}(D_{μ}) = L(μ) then Ext_{eAe}^{1}(S_{λ}, D_{μ}) = 0. (ii) If G_{⊗}(D_{μ}) = L(μ) then Ext_{eAe}^{1}(D_{μ}, S^{λ}) = 0.

Proof. (i) Let M = Δ(λ) and N = D_{μ}. Then M = AeM if λ is regular, and the claim follows from Theorem 2.4(A)(ii) and (4.4.1).

(ii) Take M = ∇(λ) and N = D_{μ}. Then S(M) = 0 since the socle of M is simple regular. Apply Theorem 2.4(B)(ii) and (4.4.2).

Lemma . Suppose μ ≯ λ, where λ, μ ∈ Λ_{reg}. Assume that all δ > λ are regular. Then

Ext_{eAe}^{1}(S_{λ}, D_{μ}) = 0 and Ext_{eAe}^{1}(D_{μ}, S^{λ}) = 0.

Proof. We prove the first part, the other is dual. We have an exact sequence

0 → L(μ) → G_{Hom}(D_{μ}) =: X → Y → 0

and Y has only singular composition factors. By (4.4.1) and the hypothesis, we get from the long exact sequence

0 = Ext_{A}^{1}(Δ(λ), L(μ)) → Ext_{A}^{1}(Δ(λ), X) → Ext_{A}^{1}(Δ(λ), Y)

Let L(δ) be a composition factor of Y, this is singular and hence by the hypothesis δ ≯ λ. Therefore, by using (4.4.1) we deduce that Ext_{A}^{1}(Δ(λ), Y) = 0. So we obtain 0 = Ext_{A}^{1}(Δ(λ), G_{Hom}(D_{μ})) ≅ Ext_{eAe}^{1}(S_{λ}, D_{μ}), as stated.

4.5. Assume that the set of weights Λ_{sing} is saturated, in other words it is an ideal in the poset Λ. There are many important situations where this occurs. For example, let A = S(n′, d) and n ≤ n′. Then there is an idempotent e of A such that the algebra eAe is isomorphic to S(n, d), and Λ_{reg} = Λ^{+}(n, d) (see [Gr1, 6.5], or [E2, 3.9]). This corresponds to comparing representations of GL_{n′}(k) and the restrictions to a certain Levi subgroup isomorphic to GL_{n}(k) × k^{∗} × ⋯ × k^{∗}.

If λ ∈ Λ_{reg}, write λ̃ for λ viewed as a weight of eAe. In general, if Λ_{sing} is saturated then mod(eAe) is a highest weight category with weight poset (Λ_{reg}, ≤). Furthermore eΔ(λ) ≅ Δ(λ̃) and e∇(λ) ≅ ∇(λ̃) (see [E2]).

Theorem . If Λ_{sing} is saturated then for any λ ∈ Λ_{reg} one has an isomorphism G_{⊗}(eΔ(λ)) ≅ Δ(λ). Moreover for λ, μ ∈ Λ_{reg} we have

Ext_{A}^{1}(Δ(λ), Δ(μ)) ≅ Ext_{eAe}^{1}(Δ(λ̃), Δ(μ̃)).

Proof. We have Δ(λ) = AeΔ(λ), so the first part will follow from Theorem 3.2 if we show that Ω^{1}(Δ(λ)) = AeΩ^{1}(Δ(λ)). Since A is quasi-hereditary, the module Ω^{1}(Δ(λ)) has a Δ-filtration, with quotients isomorphic to Δ(γ) with γ > λ. Then γ must belong to Λ_{reg}, which implies that all top composition factors of Ω^{1}(Δ(λ)) are regular, and hence AeΩ^{1}(Δ(λ)) = Ω^{1}(Δ(λ)).

For the second part we wish to apply Corollary 3.1. Provided we show that Δ(λ) ∈ X then we can take N = eΔ(λ)(= Δ(λ̃)) and M = Δ(μ).

We have just seen that G_{⊗}(eΔ(λ)) ≅ Δ(λ), so μ′_{Δ(λ)} is an isomorphism. Moreover Ω^{1}(Δ(λ)) is filtered by Δ(γ) with γ ∈ Λ_{reg}. For these, also G_{⊗}(eΔ(γ)) ≅ Δ(γ) and therefore μ′_{Δ(γ)} is an isomorphism. The category of modules Y for which μ′_{Y} is an isomorphism is closed under extensions (which is seen by an application of the Snake Lemma), and hence μ′_{Ω^{1}(Δ(λ))} is an isomorphism. This shows that Δ(λ) belongs to X.

5. Symmetric groups and Hecke algebras

Throughout this section, let A be the q-Schur algebra S_{q}(n, d) where n ≥ d. Then eAe ≅ H_{q}(d) where q is a primitive lth root of unity. Many of the results will be stated for the Hecke algebra. One can recover analogous results for the ordinary Schur algebra and the symmetric group by setting q = 1. Let σ = D_{(1^{d})} and δ = L(1^{d}). Then σ is the one-dimensional module affording the sign character for the Hecke algebra, and δ is the one-dimensional A-module corresponding to the determinant character of quantum GL_{n}(k). Note that eδ ≅ σ, whence F(δ) ≅ σ. Note that for l = 2, the module σ is isomorphic to the trivial module k. For this reason, the case l = 2 is somewhat special and should be treated separately from the other cases.

For each λ ∈ Λ, let M^{λ} be the permutation module for H_{q}(d) [Mar1, 4.6, 6.6]. This module M^{λ} for the Hecke algebra can be realized by inducing the “trivial module” of an appropriate Young subalgebra.

5.1. Composition factors of R^{•}G_{Hom}(N). The following result provides an interpretation of composition factor multiplicities of L(λ) in R^{j}G_{Hom}(N) as the dimensions of the extension groups between Y^{λ} and N for λ ∈ Λ. Note in the case of symmetric groups and Hecke algebras the Young modules Y^{λ} are self-dual [Mar1, Lemma 4.6.2(ii)]. Moreover, since Y^{λ} has simple socle D_{λ} for all λ ∈ Λ_{reg}, part (ii) shows again that G_{Hom}(D_{μ}) can have no regular composition factors except possibly L(μ); compare with Corollary 3.2.

Proposition . Let λ, μ ∈ Λ, n ≥ d, and N ∈ mod(H_{q}(d)).

(i) For j ≥ 0, [R^{j}G_{Hom}(N) : L(λ)] = dim_{k} Ext_{H_{q}(d)}^{n}(Y^{λ}, N). (ii) In particular, [G_{Hom}(D_{μ}) : L(λ)] = [soc_{H_{q}(d)} Y^{λ} : D_{μ}]. (iii) For j > 0, R^{j}G_{Hom}(N) only has composition factors L(λ) where λ ∈ Λ_{sing}.

Proof. (i) Let P (λ) be the projective cover of L(λ) in S_{q}(n, d)-mod. Since Y^{λ} is self-dual, it follows that eP (λ) = Y^{λ}. Hence, by Proposition 2.6(A), for j ≥ 0,

[R^{j}G_{Hom}(N) : L(λ)] = dim_{k}Ext_{H_{q}(d)}^{j}(eP (λ), N)
= dim_{k}Ext_{H_{q}(d)}^{j}(Y^{λ}, N)

For (ii), apply Theorem 2.6(A) and self-duality.

(iii) For λ ∈ Λ_{reg}, Y^{λ} is the indecomposable injective corresponding to λ. Moreover, Y^{λ} is self dual, so Y^{λ} is projective for λ ∈ Λ_{reg}. The result now follows by part (i).

5.2. Comparison of cohomology. For S_{q}(d, d) let l ≥ 3. Under these assumptions we now establish a direct connection between Ext^{1} for Hecke algebras H_{q}(d) and extensions for quantum GL_{d}.

Theorem . Let q+1 ≠ 0 and V ∈ S_{q}(d, d)-mod such that V^{τ} = AeV^{τ}.

(i) G_{Hom}(σ) = δ; (ii) Ext_{H_{q}(d)}^{1}(k, eV ⊗ σ) ≅ Ext_{H_{q}(d)}^{1}(σ, eV) ≅ Ext_{S_{q}(d,d)}^{1}(δ, V) (iii) There exists an injective homomorphism

Ext_{S_{q}(d,d)}^{2}(δ, V) ↪ Ext_{H_{q}(d)}^{2}(σ, eV).

Proof. (i) Let M^{λ} = ind_{(H_{q}(d))_{λ}}^{H_{q}(d)}k where (H_{q}(d))_{λ} is the Young subalgebra. Since M^{λ} contains Y^{λ}, we have by Frobenius reciprocity that

Hom_{H_{q}(d)}(σ, Y^{λ}) ⊂ Hom_{H_{q}(d)}(σ, M^{λ}) ≅ Hom_{(H_{q}(d))_{λ}}(σ, k).

Since q + 1 ≠ 0, it follows that Hom_{(H_{q}(d))_{λ}}(σ, k) ≅ k for λ = (1^{d}) and zero otherwise. If it is zero then of course also Hom_{H_{q}(d)}(σ, Y^{λ}) = 0. In the other case one sees directly that Hom_{H_{q}(d)}(σ, Y^{λ}) = k since λ = (1^{d}) and so σ is the Specht module S^{λ}. Therefore, by Proposition 5.1(ii), G_{Hom}(σ) = δ.

(ii) (iii) Set N = σ and M = V^{τ}. The results follow immediately by combining part (i) with Theorem 2.4(ii), and applying duality twice.

As a corollary of the preceding theorem one can easily recover some results of [BKM]. This uses the labelling of simple modules as in [Jam]. Namely, if λ is regular and if we denote by λ′ the partition conjugate to λ then the Specht module S^{λ′} has a unique simple quotient, denoted by D^{λ′}. We have D^{λ′} ≅ D_{λ} ⊗ σ.

Corollary (A). Let q + 1 ≠ 0 and λ ∈ Λ_{reg}. Then

(i) Ext_{H_{q}(d)}^{1}(k, D^{λ′}) ≅ Ext_{S_{q}(d,d)}^{1}(δ, L(λ)); (ii) Ext_{H_{q}(d)}^{1}(k, S^{λ} ⊗ σ) ≅ Ext_{H_{q}(d)}^{1}(σ, S^{λ}) ≅ Ext_{S_{q}(d,d)}^{1}(δ, ∇(λ)) = 0.

Proof. Part (i) and the first equality in part (ii) follows immediately from part (ii) of the Theorem. Since δ = ∇(1^{d}), the last equality in part (ii) follows from (4.4.1).

The next result gives some information about the second socle and radical layer of I(1^{d}), the injective hull of δ. It implies that the only simple modules that can appear are ones with regular highest weight.

Corollary (B). Let q + 1 ≠ 0 and λ ∈ Λ_{sing}.

(i) Ext_{S_{q}(d,d)}^{1}(L(λ), δ) = 0; (ii) Ext_{S_{q}(d,d)}^{2}(L(λ), δ) ≅ Hom_{S_{q}(d,d)}(L(λ), R^{1}G_{Hom}(σ)).

Proof. This follows by combining part (i) of Theorem 5.1 with Theorem 2.7.

5.3. Self-extensions between simple modules. We will now show that self extensions of simple modules do not exist, given suitable hypotheses on images under G_{Hom}.

Proposition . Let λ ∈ Λ_{reg} and suppose that G_{Hom}(D_{λ}) ≅ L(λ). Then Ext_{eAe}^{1}(D_{λ}, D_{λ}) = 0.

Proof. According to Theorem 4.2(i), we have

Ext_{eAe}^{1}(D_{λ}, D_{λ}) ≅ Ext_{A}^{1}(L(λ), G_{Hom}(D_{λ}))
≅ Ext_{A}^{1}(L(λ), L(λ)).

The algebra A is quasi-hereditary and therefore Ext_{A}^{1}(L(λ), L(λ)) = 0. This follows from (4.6.1) and the fact that rad_{A}Δ(λ) does not have L(λ) as a composition factor (see 4.1). In the classical case, this also follows by [Jan, II 2.12], since Ext_{GL_{d}(k)}^{1}(−, −) = Ext_{A}^{1}(−, −).

Kleshchev and Martin have conjectured that Ext_{kΣ_{d}}^{1}(D_{λ}, D_{λ}) = 0 for all λ, provided that p ≥ 3. The result above shows that this might be verifiable if one has information about G_{Hom}(D_{λ}). Corollary 5.2(A) demonstates how G_{Hom}(D_{μ}) can depend on the prime being larger than two. Recently, Kleshchev and the third author have verified that G_{Hom}(D_{λ}) ≅ L(λ) for D_{λ} a non-trivial completely splittable kΣ_{d}-module [KN, Thm. 3.4].

5.4. Guralnick conjecture. We consider only the classical case (A = S(d, d)) in this section. Guralnick has conjectured that, for a finite group L and a faithful irreducible representation V, dim_{k} H^{1}(L, V) ≤ 2. The question of bounding H^{1} has been found to be related to looking at the generating sets in L. It has been shown (by using the classification of finite simple groups) that H^{1}(L, V) ≤ dim_{k} V. Better bounds were found for symmetric and alternating groups in [GK]. Let G = GL_{d}(k), B a Borel subgroup of G containing a maximal torus T, and U the unipotent radical of B. We will now show that for symmetric groups one can also obtain information about symmetric group cohomology from information about the cohomology for general linear groups.

Theorem . Assume p ≥ 3 and let λ ∈ Λ_{reg}.

(a) H^{1}(Σ_{d}, D^{λ′}) ≅ Ext_{GL_{d}(k)}^{1}(L(λ), δ);

(b) If L(λ) ≠ δ and k[G/T] is the coordinate algebra of G/T then

dim_{k} Ext_{kΣ_{d}}^{1}(k, D^{λ′}) = [soc_{G} (k[G/T] ⊗ δ)/δ : L(λ)] − dim_{k} D^{λ′}.

Proof. (a) This follows by Corollary 5.2(A).

(b) First observe that by the tensor identity k[G/T]⊗δ ≅ ind_{T}^{G}δ. The module δ is an injective T -module, so ind_{T}^{G}δ is an injective G-module. Furthermore, by Frobenius reciprocity, δ appears in the socle of ind_{T}^{G}δ as a G-module. Consider the short exact sequence

0 → δ → ind_{T}^{G}δ → ind_{T}^{G}δ/δ → 0.

One can use the injectivity of ind_{T}^{G}δ to see that the first part of the long exact sequence in cohomology takes on the following form

0 → Hom_{GL_{d}(k)}(L(λ), δ) → Hom_{GL_{d}(k)}(L(λ), ind_{T}^{G}δ)
→ Hom_{GL_{d}(k)}(L(λ), ind_{T}^{G}δ/δ) → Ext_{GL_{d}(k)}^{1}(L(λ), δ) → 0.

Since L(λ) ≠ δ, this sequence above shows that

dim_{k} Ext_{GL_{d}(k)}^{1}(L(λ), δ) = dim_{k} Hom_{GL_{d}(k)}(L(λ), ind_{T}^{G}δ/δ)
− dim_{k} Hom_{GL_{d}(k)}(L(λ), ind_{T}^{G}δ)
= [soc_{G} ind_{T}^{G}δ/δ : L(λ)] − dim_{k} Hom_{T} (L(λ), δ)
= [soc_{G} ind_{T}^{G}δ/δ : L(λ)] − dim_{k} D^{λ′}.

The result now follows by applying part (a).

5.5. Images of some simple modules under the functor G_{⊗}. From our previous discussion, the modules G_{⊗}(eL(μ)) for μ regular play an important role in the cohomology theory. We have the following general result for the case of Schur algebras. Let A = S(d, d). By [Gr1, 6.2g] the set of non-zero eL(μ) are a full set of simple modules for the symmetric group Σ_{d}. In particular there is a unique restricted weight λ such that eL(λ) ≅ k.

Theorem . Let A = S(d, d) and let λ be the unique weight in Λ^{+}(d) such that F(L(λ)) = eL(λ) is isomorphic to k, the trivial kΣ_{d}−module. Then λ = ((p−1)^{a}, b) with 0 ≤ b < p−1. Then G_{⊗}(eL(λ)) is isomorphic with ∇(d), the d-th symmetric power of the natural module E.

Proof. Set λ = ((p − 1)^{a}, b) with 0 ≤ b < p − 1. ¿From the definition of the symmetric power it is clear that eS^{d}(E) ≅ k, since the unique monomial in S^{d}(E) of weight (1^{d}) is stabilized by the action of the symmetric group. The submodule lattice of S^{d}(E) is completely understood (see [Dot1]), and L(λ) is the unique simple quotient of S^{d}(E). Moreover, L(λ) is the unique regular composition factor of S^{d}(E). (In fact, L(λ) is isomorphic with the truncated symmetric power; see [Dot2], [DW].) Therefore, by Corollary 3.2, the essential map P (λ) → S^{d}(E) = ∇(d) factors through Ae ⊗_{eAe} eL(λ) and, in particular, eL(λ) = k.

Now it suffices to show that both modules have the same composition factor multiplicities. It is clear that the symmetric power is multiplicity-free (all the weight spaces are one-dimensional), so as a first step we claim:

(5.5.1) Ae ⊗_{eAe} eL(λ) is multiplicity-free. To see this, let γ be a singular weight. Then, by Proposition 2.6(B), the multiplicity of L(γ) as a composition factor in this module is equal to the dimension of Hom_{kΣ_{d}}(k, Y^{γ}). Moreover, Hom_{kΣ_{d}}(k, Y^{γ}) embeds in Hom_{kΣ_{d}}(k, M^{γ}) since Y^{γ} is a submodule of M^{γ}, and by Frobenius reciprocity, this is Hom_{kΣ_{γ}}(k, k), which is 1-dimensional.

(5.5.2) Suppose L(γ) does not occur as a composition factor in ∇(d) = S^{d}(E). It suffices to show that Hom_{kΣ_{d}}(k, Y^{γ}) = 0. According to Krop’s tensor product theorem ([Kr], [Dot2, Thm. 4.3]), every composition factor of S^{d}(E) can be expressed as a twisted tensor product of truncated symmetric powers. Let γ = ∑_{i} γ(i)p^{i} be the p-adic expansion of γ, with γ(i) p-restricted (see (4.1.1)). Since L(γ) is not a composition factor of S^{d}(E), for some j the partition γ(j) is not of the form ((p − 1)^{u}, v). Now we use the Grabmeier correspondence for Young modules [Gb], [Mar1, 4.6]. The Young module Y^{γ} has a Young vertex Σ_{ρ} which is a Young subgroup corresponding to the partition ρ with all parts powers of p and where the number of times p^{i} occurs is exactly the degree of γ(i).

Let N be the normalizer of Σ_{ρ} in Σ_{d}. The factor group N/Σ_{ρ} is isomorphic to a Young subgroup Σ_{β}, where β_{i} is the degree of λ(i) for each i. It follows that the Grabmeier correspondent of Y^{γ} is the N-module on which Σ_{ρ} acts trivially, and which, as a module for the quotient Σ_{β}, is isomorphic to the tensor product ⨂_{i} Y^{γ(i)} (see [Mar1, 4.6.1]). If f(Y^{γ}) is this correspondent, then Y^{γ} is a direct summand of the induced module f(Y^{γ}) ⊗_{kN} kΣ_{d}.

Now Y^{γ(i)} ≅ P (k) (as a module for Σ_{β_{i}}) if and only if γ(i) has all except possibly one part equal to p−1. Now by our hypothesis, Y^{γ(j)} is not P (k), and it follows that the Grabmeier correspondent has a simple top which is not the trivial module. Hence by Frobenius reciprocity

Hom_{kΣ_{d}}(f(Y^{γ}) ⊗ kΣ_{d}, k) ≅ Hom_{kN}(f(Y^{γ}), k) = 0

Now, Hom_{kΣ_{d}}(Y^{γ}, k) is a direct summand, so it must be zero. Since the Young modules are self-dual, it follows that Hom_{kΣ_{d}}(k, Y^{γ}) = 0, which completes the proof.

5.6. The following result on representations of symmetric groups can be found in [Jam, Cor. 13.17]. Unless p = 2 and λ is row 2-singular,

Hom_{kΣ_{d}}(S^{λ}, M^{μ}) ≠ 0 ⇒ λ ≥ μ.

We will apply this by using a simple quotient of a Specht module. Recall that, for λ regular, D_{λ} = D^{λ′} ⊗ σ, where D^{λ′} is the unique simple quotient of S^{λ′}. Let m(λ′) be the partition such that D_{λ} = D^{m(λ′)}, a quotient of S^{m(λ′)}. The next result generalizes Theorem 5.2 for symmetric groups (when λ = (1^{d}) and m(λ′) = (1^{d})).

Proposition . Let A = S(d, d) and suppose p > 2. Assume λ is regular such that all μ with m(λ′) ≥ μ are regular. Then

G_{⊗}(eL(λ)) ≅ L(λ).

Proof. Suppose not, then by Corollary 3.2 there is some γ which is not restricted such that L(γ) occurs as a composition factor of Ae ⊗_{eAe} eL(λ). Therefore,

0 ≠ Hom_{A}(Ae⊗_{eAe}eL(λ), I(γ)) = Hom_{kΣ_{d}}(D_{λ}, Y^{γ}) = Hom_{kΣ_{d}}(D^{m(λ′)}, Y^{γ}),

and since D^{m(λ′)} is a quotient of the Specht module S^{m(λ′)} this is contained in Hom_{kΣ_{d}}(S^{m(λ′)}, M^{γ}). By [Jam], m(λ′) ≥ γ, and by the hypothesis γ is restricted, which is a contradiction.

5.7. Hook partitions. Let A = S(d, d). The m-th symmetric power S^{m}(E) of the natural GL_{d}(k)-module E is isomorphic to ∇(m), and the a-th exterior power of E, for a ≤ d, is isomorphic to ∇(1^{a}) (see e.g. [Gr1, 4.4]). For 1 ≤ a ≤ d − 1 there is an exact sequence of A-modules

0 → ∇(d − a, 1^{a}) → ∇(1^{a}) ⊗ ∇(d − a) → ∇(d − a + 1, 1^{a−1}) → 0. (5.7.1)

To obtain (5.7.1), one uses the canonical long exact sequence

0 → Λ^{d}(E) → Λ^{d−1}(E) ⊗ S^{d−1}(E) → ⋯ Λ^{a}(E) ⊗ S^{d−a}(E) → ⋯,

see for instance [B, Ch. 10, §9.3]. By the general result in [Mat] (completing earlier work of Wang Jian-pan and Donkin), each term has a good filtration. The category of modules with good filtration is closed under cokernels of monomorphisms. By induction starting at the left we get that all images and kernels of the sequence have a good filtration. By the Littlewood-Richardson rule, one finds the ∇-quotients for the terms of the sequence (≤ 2) and thus one gets (5.7.1).

Applying the Schur functor gives an exact sequence of kΣ_{d}-modules

0 → S^{(d−a,1^{a})} → e(∇(1^{a}) ⊗ ∇(d − a)) → S^{(d−a+1,1^{a−1})} → 0. (5.7.2)

If p does not divide d then the end terms are in different blocks (since the partitions have different p-cores, by the ’Nakayama conjecture’). So the sequence splits. So suppose now that p divides d. We also assume that p > 2. We claim that then the middle term of (5.7.1) is indecomposable, and has a simple socle L(d − a, 1^{a}) if d − a < p. Moreover, in (5.7.2) the middle is self-dual and both sequences are nonsplit. This is well-known, but we give a proof for the convenience of the reader.

The middle term in (5.7.2) is isomorphic to ind_{Σ_{a}×Σ_{d−a}}^{Σ_{d}}(σ ⊗k), hence it is self-dual. It was proved by Peel (see [Jam]) that in this case the Specht modules for hook partitions have two distinct composition factors, except for the trivial and the alternating Specht module. Furthermore, two distinct ones which both occur in one sequence (5.7.2) have one common composition factor in common. This implies that the middle term in (5.7.2) must be indecomposable and the sequence is non-split.

Hence (5.7.1) is non-split as well. Moreover, if the middle term in (5.7.1) were decomposable then it would be the direct sum of the end terms, since it has a ∇-filtration with only two quotients, but then the sequence would split. So it is indecomposable and moreover, if d−a < p then the middle is self-dual and its socle must be simple. Part (a) of the next result is a consequence of [Don1] and [FK].

Theorem . Assume p > 2. Suppose d is divisible by p and d − a < p. Then

(a) ∇(d − a + 1, 1^{a−1}) has a simple top, isomorphic to L(d − a, 1^{a}).

(b) G_{⊗}(eL(d − a, 1^{a})) ≅ L(d − a, 1^{a}) unless d = p and a = 1.

Proof. (a) Since d − a < p the symmetric power ∇(d − a) is simple. It follows that ∇(1^{a}) ⊗ ∇(d − a) is self-dual. This fact in conjunction with the aforementioned observations implies (a).

(b) If λ = (d−a, 1^{a}) then from the above we have D_{λ} is the top factor of the Specht module S^{(d−a+1,1^{a−1})}. So m(λ′) = (d − a + 1, 1^{a−1}). Now apply Proposition 5.6. (Its hypotheses are satisfied because d − a < p.)

6. Examples

In the first two examples we will show how information about the cohomology of GL_{n}(k) can be used to compute symmetric group cohomology. The last two examples proceed in the opposite direction, using information about symmetric groups to determine information about general linear group cohomology.

6.1. [kΣ_{2}, p = 2]. Let p = 2 and A = S(2, 2). There are two simple Amodules up to isomorphism: L(1^{2}) and L(2). One can easily compute the structure of the induced modules ∇(1^{2}) and ∇(2): ∇(1^{2}) = L(1^{2}) is simple and ∇(2) has two composition factors, namely L(2) and L(1^{2}). One knows that each injective indecomposable has a ∇ filtration and that the number of times a given ∇(λ) appears as a section in the filtration of I(μ) equals the composition factor multiplicity of L(μ) in ∇(λ). This information determines the composition factors and structure diagrams of the injective indecomposables, shown below.

I(1^{2}) : L(1^{2})  I(2) : L(1^{2})
L(2)  L(2)
L(1^{2})

The only simple kΣ_{2}-module is D_{(1^{2})} = eL(1^{2}). According to Corollary 3.3, the structure of G_{Hom}(D_{(1^{2})}) is given by

G_{Hom}(D_{(1^{2})}) : L(2)
L(1^{2})

Hence, one can conclude by Theorem 2.5 that

Ext_{kΣ_{2}}^{1}(D_{(1^{2})}, D_{(1^{2})}) ≅ Hom_{S(2,2)}(L(1^{2}), I(1^{2})/G_{Hom}(D_{(1^{2})})) ≅ k.

6.2. [kΣ_{3}, p = 3]. Now let p = 3 and A = S(3, 3). Up to isomorphism, the simple A-modules are L(1^{3}), L(2, 1) and L(3). It is easy to calculate the structures of ∇(λ) for λ ∈ Λ^{+}(3): ∇(1^{3}) = L(1^{3}) is simple; ∇(2, 1) has two composition factors, namely L(2, 1) and L(1^{3}); ∇(3) also has two composition factors, L(3) and L(2, 1). This information determines the structure of the injective S(3, 3)-modules, displayed below.

I(1^{3}) : L(1^{3})  I(2, 1) :  L(2, 1)  I(3) : L(2, 1)
L(2, 1)  L(3) L(1^{3})  L(3)
L(1^{3})  L(2, 1)

The simple kΣ_{3}-modules are D_{(1^{3})} and D_{(2,1)}. Their images under the functor G_{Hom} can be determined by Corollary 3.3, and are shown below.

G_{Hom}(D_{(1^{3})}) : L(1^{3})  G_{Hom}(D_{(2,1)}) : L(3)
L(2, 1)

Therefore, by Theorem 2.5 we obtain the isomophisms

Ext_{kΣ_{3}}^{1}(D_{λ}, D_{μ}) ≅ Hom_{S(3,3)}(L(λ), I(μ)/G_{Hom}(D_{μ}))
= \begin{cases}0 & λ = μ \\ k & λ ≠ μ \end{cases}

for λ, μ ∈ {(1^{3}), (2, 1)} (the set of regular weights).

6.3. [kΣ_{p}, p ≥ 2]. We now consider the case when A = S(p, p) and eAe = kΣ_{p}, for p > 2 prime. In this situation there is only one block component which is not semi-simple, and this contains precisely the weights which are hook partitions. We can now employ the information given in Section 5.7.

The middle in (5.7.2) is projective, since it is induced from a subgroup of p′-order. As a consequence we get the composition factors for the modules in (5.7.2). The middle term is the projective cover, P say, of D_{(p−a,1^{a})}. It is self-dual, and it follows that rad(P)/soc(P) is the direct sum of one or two simples. Hence,

(6.3.1)  Ext_{kΣ_{p}}^{1}(D_{(p−a,1^{a})}, D_{(p−b,1^{b})}) = \{ \begin{matrix}k & |a − b| = 1 \\ 0 & otherwise. \end{matrix}

¿From Theorem 5.6, we know that G_{⊗}(eL(λ)) ≅ L(λ) if λ is regular but ≠ (p − 1, 1). On the other hand for λ = (p − 1, 1) we get from Theorem 5.5 that G_{⊗}(eL(λ)) ≅ ∇(p). This has length two [Dot1] (this can be verified directly). Moreover, eL(λ) = k. So we obtain the following from the spectral sequence in Theorem 2.2. Let λ = (p − a, 1^{a}) and μ = (p − b, 1^{b}) be restricted weights and μ ≠ (p − 1, 1). Then

(6.3.2) Ext_{A}^{1}(L(λ), L(μ)) = \{ \begin{matrix}k & |a − b| = 1 \\ 0 & otherwise. \end{matrix}

(Apply Theorem 2.2 with M = L(λ) and N = eL(μ)). Moreover, let λ be as above and μ = (p − 1, 1) then

(6.3.3) Ext_{A}^{1}(L(λ), ∇(p)) = \{ \begin{matrix}k & a = 2 \\ 0 & otherwise. \end{matrix}

(Take M = L(λ) and N = L(μ), here G_{⊗}(eN) = ∇(p)).

6.4. [kΣ_{4}, p = 2]. The representation theory of the symmetric group Σ_{4} over fields k of characteristic 2 is well understood. We will summarize some of the results and then discuss how to deduce information on the representation theory for the Schur algebra A = S(4, 4) when p = 2.

(6.4.1) The simple kΣ_{4}-modules are k = eL(1^{4}) and D := D_{(2,1^{2})} = eL(2, 1^{2}), of dimension 2. Clearly Ext_{kΣ_{4}}^{1}(k, k) ≅ k, and it is wellknown that Ext_{kΣ_{4}}^{1}(k, D) and Ext_{kΣ_{4}}^{1}(D, D) are both k as well. (This can be proved in various ways, but it can be done entirely with information from the alternating and symmetric groups of degree 4. See [E1].)

(6.4.2) Now consider representations of S(4, 4). The Weyl module Δ(4) is the dual of the symmetric power, so it is completely understood [Dot1]. It has length four and is uniserial; the composition factors are (from top to bottom)

L(4), L(2^{2}), L(3, 1), L(1^{4}).

The structure of the Weyl module Δ(3, 1) can also be determined from the sequence (5.7.2). One sees that it has length four. The socle is L(2, 1^{2}) and the remaining composition factors are L(2^{2}) and L(1^{4}).

Similarly, from 5.7 we see that the Weyl module Δ(2, 1^{2}) has length two and socle L(1^{4}). (In (5.7.1), take the part

0 → ∇(1^{4}) → Λ^{3}(E) ⊗ E → ∇(2, 1^{2}) → 0

This is non-split: the argument in 5.7 works here as well; the relevant Specht modules for hook partitions have the same properties as for odd primes. The middle is self-dual and hence the cokernel has a simple top isomorphic to L(1^{4}). The composition factors of ∇(2, 1^{2}) (other than the socle ) can only have highest weight (1^{4}). But L(1^{4}) does not have self-extensions and then ∇(2, 1^{2}) must have length two.)

Moreover, the Weyl module Δ(2, 2) has dimension 20 and the simple module L(2^{2}) is 6-dimensional. All other composition factors of Δ(2, 2) must have weights < (2^{2}), and by dimensions (and weight spaces), we deduce that Δ(2, 2) has length two; its socle is L(2, 1^{2}). Clearly, Δ(1^{4}) = L(1^{4}).

(6.4.3) By Theorem 5.5, we know that G_{⊗}(eL(1^{4})) ≅ ∇(4). Moreover, we claim that G_{⊗}(eL(2, 1^{2})) ≅ ∇(2, 2). By Theorem 3.2, Ae ⊗_{eAe} eL(2, 1^{2}) is the largest quotient of the projective P (2, 1^{2}) whose radical has only singular composition factors. Clearly ∇(2, 2) is such a quotient, and hence this is an epimorphic image of the required module.

Since (2, 1^{2}) is regular we know from general theory that P (2, 1^{2}) ≅ I(2, 1^{2}) because it is a tilting module, so by the reciprocity for highest weight categories (namely [I(λ) : ∇(μ)] = [Δ(μ) : L(λ)], see [CPS, 3.11]) it has a ∇-filtration with quotients ∇(3, 1), ∇(2, 2) and ∇(2, 1^{2}). The last two quotients have regular tops, and it follows that Ae ⊗ eL(2, 1^{2}) must be a quotient of ∇(3, 1). The claim now follows from (6.4.2).

As a consequence of our result on spectral sequences we deduce the following:

Proposition . Suppose μ = (2, 2) or (4). Then we have

Ext_{S(4,4)}^{1}(L(λ), Δ(μ)) = \{ \begin{matrix}k & if λ is regular; \\ 0 & otherwise. \end{matrix}

Proof. This follows from (6.4.3), Theorem 2.4 and from (6.4.1).

The proposition above shows that one has the following isomorphisms

Ext_{S(4,4)}^{1}(L(1^{4}), Δ(2, 2)) ≅ Ext_{S(4,4)}^{1}(∇(2, 2), L(1^{4})) ≅ k
Ext_{S(4,4)}^{1}(L(2, 1^{2}), Δ(2, 2)) ≅ Ext_{S(4,4)}^{1}(∇(2, 2), L(2, 1^{2})) ≅ k

The indecomposable tilting module T (2, 2) corresponding to the simple module L(2, 2) has good filtration with factors of the form ∇(1^{4}), ∇(2, 2) and ∇(2, 1) and Weyl filtration with factors of the form Δ(1^{4}), Δ(2, 2) and Δ(2, 1). The module diagram showing the structure of the tilting module T (2, 2) is given in the figure below.

[figure: L(2, 1^{2}); L(2, 2); L(2, 1^{2}); L(1^{4}); L(1^{4})] Figure 1

Observe that the structure agrees with the preceding Ext^{1} calculations. Interestingly enough, this is the smallest known case where an indecomposable tilting module does not have simple head (or socle).

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Mathematical Sciences, Loyola University Chicago, Chicago, Illinois 60626 U.S.A.

E-mail address: doty@math.luc.edu

Mathematical Institute, Oxford University, 24-29 St. Giles, Oxford, OX1 3LB, England

E-mail address: erdmann@maths.ox.ac.uk

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322 U.S.A.

E-mail address: nakano@sunfs.math.usu.edu