Some recent developments in the representation theory of general linear and symmetric groups

Daniel K. Nakano

1. Introduction

1.1. In his pioneering 1901 dissertation [S1], Schur exploited a fundamental connection between the polynomial representation theory of the general linear groups GL_{n}(ℂ) and the symmetric groups Σ_{d} by using a functor known in present day language as the “Schur functor”. Schur used this connection to compute the irreducible complex characters of general linear groups, based on Frobenius’ earlier computation [Fr] of the irreducible complex characters of symmetric groups. Later, Schur returned to this subject with his classic paper [S2], in which he used the dual actions of GL_{n}(ℂ) and Σ_{d} on the d-th tensor power V^{⊗d} of the natural n-dimensional module V = ℂ^{n} to rederive all the results of his dissertation. This new method was publicized and extended by Weyl in his influential treatise [W]. The important “double centralizer property” which Schur developed is now often referred to as “Schur-Weyl duality”.

The account of J.A. Green [Gr1] extends this set-up to an arbitrary infinite field k (instead of ℂ), where the representation theory need not be semisimple. Green showed that the category of polynomial representations of GL_{n}(k) of fixed homogeneous degree d is equivalent to the module category of a certain finite-dimensional algebra he called a Schur algebra, denoted by S(n, d). In this setting, for n ≥ d, the Schur functor is constructed by multiplying by an appropriate idempotent e of S(n, d) and identifying eS(n, d)e with the group algebra kΣ_{d}. This idempotent version of the Schur functor also has analogues for arbitrary finite-dimensional algebras [Gr1, §6.2]. In particular, there is also a Schur functor between the q-Schur algebras of Dipper and James [DJ1, DJ2] and Hecke algebras of Type A, and also between other types of algebras which include the infinitesimal Schur algebras [DNP1].

For the sake of simplicity we will only consider the relationship between the general linear group GL_{n}(k) and the symmetric group Σ_{d} over a field k in this paper. Many of the results stated do have quantum analogues and these results may be mentioned at certain times. The purpose of this paper is two-fold. First, in an article of this size it would be unimaginable to adequately survey the development of a field which has been in existence nearly 90 years. For accounts of this type, the reader is referred to [Gr1] and [Mar]. In this article we will primarily focus on new developments in the area since S. Martin’s book on Schur algebras [Mar] was published in 1994. Secondly, the connections between the representation theory and cohomology theories for the general linear group and the symmetric group over fields of characteristic p > 0 are still not well understood. At the end of each section, open problems have been outlined with the hope of making this area accessible to both graduate students and researchers interested in representation theory.

1991 Mathematics Subject Classification. Primary 20G.

Research of the author was supported in part by NSF grant DMS-9800960.

1.2. Acknowledgements. I would like to thank Brian Parshall, Stephen Doty, Karin Erdmann and Alexander Kleshchev for many enlightening and stimulating conversations about the connections between the representations of the general linear group and the symmetric group. The author also acknowledges the organizers of the International Conference on Representation Theory for their efforts in hosting a very successful meeting.

2. Schur algebras

2.1. Definition: The basic references for this section are [Gr1] and [Mar]. Let k be an algebraically closed field. Moreover, let GL_{n}(k) be the general linear group over k and kΣ_{d} be the group algebra of the symmetric group on d letters Σ_{d}. Let V be the n-dimensional natural representation of GL_{n}(k). The group Σ_{d} acts on V^{⊗d} by place permutation while group GL_{n}(k) acts diagonally. The Schur algebra S(n, d) is the finite-dimensional associative k-algebra End_{kΣ_{d}}(V^{⊗d}).

The Schur algebra also manifests itself in the following way. Let M = M_{n}(k) be the monoid of n × n matrices over the field k. As an affine algebraic variety, M can be regarded as an affine space of dimension n^{2}. Therefore, the coordinate algebra k[M] ≅ k[c_{ij} : 1 ≤ i, j ≤ n]. The algebra k[M] has a bialgebra structure given by Δ : k[M] → k[M] ⊗ k[M] and ϵ : k[M] → k where

Δ(c_{ij}) = ∑_{ℓ=1}^{n} c_{iℓ} ⊗ c_{ℓj}; ϵ(c_{ij}) = \{ \begin{matrix}1 & i = j \\ 0 & i ≠ j \end{matrix} . (2.1.1)

Let A(n, d) be the set of homogeneous elements of k[c_{ij}] of total degree d in the generators {c_{ij}}. The vector space A(n, d) is a subcoalgebra of k[M] and it turns out that one can identify S(n, d) with the linear dual of A(n, d)^{∗}. This construction of the Schur algebra shows that the category of rational M-modules is equivalent with the category of polynomial GL_{n}(k)-modules. Every object in either category splits into a direct sum of homogeneous subobjects of degree d. Furthermore, the subcategory of those subobjects is equivalent with the category of S(n, d)-modules. In this way the category of polynomial GL_{n}(k)-modules of a fixed degree d ≥ 0 is equivalent to the category of modules for S(n, d).

2.2. Representation type: The double-centralizer property mentioned in the introduction also holds over arbitrary infinite fields k. This has many useful applications and generalizations. One such application involves determining the basic algebras for certain Schur algebras to provide information essential to classifying the representation type of Schur algebras.

A finite-dimensional algebra A has finite representation type if A has finitely many indecomposable modules up to isomorphism. Otherwise the algebra is of infinite representation type. By using the representation theory of the symmetric group to compute the quiver and relations for certain Schur algebras, Erdmann in 1993 was able to classify the Schur algebras of finite representation type.

Theorem (A). [E1] The Schur algebra S(n, d) has finite representation type if and only if one of the following occurs.

(i) p ≥ 2, n ≥ 3, d < 2p; (ii) p ≥ 2, n = 2, d < p^{2}; (iii) p = 2, n = 2, d = 5, 7.

It was somewhat surprising that the question of semisimplicity of the Schur algebras was settled much later by Doty and the author in 1996.

Theorem (B). [DN1] Let p be the characteristic of k. The Schur algebra S(n, d) is semisimple if and only if

(i) p = 0;

(ii) p > d; (iii) p = 2, n = 2, d = 3.

Finite-dimensional algebras of infinite representation type fall into two mutually exclusive categories: tame or wild. Loosely speaking, an algebra A has tame representation type if for each dimension there exists finitely many one-parameter families of indecomposable objects in mod(A). In the case of tame algebras one has a good chance to provide some classification of the indecomposable modules. On the other hand, the algebras of wild representation type are those whose module theory has enough complexity where classification of the indecomposable module is nearly an impossible task. For precise definitions of these concepts the reader is referred to [DEMN, 2.1]. The tame Schur algebras were classified by Doty, Erdmann, Martin and the author.

Theorem (C). [DEMN] The Schur algebras S(n, d) have tame representation type if and only if

(i) p = 3, n = 3, d = 7, 8; (ii) p = 3, n = 2, d = 9, 10, 11; (iii) p = 2, n = 2, d = 4 or d = 9.

It is interesting to note that algebraic group results for GL_{n}(k) were employed to compute certain basic algebras for the Schur algebras for this result. This illustrates the dual nature of the Schur algebras.

2.3. Global dimension: The global dimension of an algebra A is the maximum non-negative integer i such that Ext_{A}^{i}(M, N) ≠ 0 for M and N in Mod(A). Donkin [Don2, (2.2)], Parshall [P] and Green [Gr2] have shown that the Schur algebra S(n, d) is a quasi-hereditary algebra, and thus must have finite global dimension. Recently, Totaro computed the global dimension of S(n, d) when n ≥ d. His result is stated as follows.

Theorem . [T] Let n ≥ d and let d = d_{0} + d_{1}p + ⋯ + d_{s}p^{s} be the p-adic expansion of d. The global dimension of S(n, d) is 2(d − ∑_{j=0}^{s} d_{j}).

2.4. Open Problems:

(2.4.1) Compute the global dimension of S(n, d) for n < d. In view of Theorem 2.2B, this might be a subtle question because S(2, 3) for p = 2 is semisimple so the algebra has global dimension zero.

(2.4.2) Determine the representation type of the blocks for the Schur algebra S(n, d).

3. Infinitesimal Schur algebras

3.1. Definition: As we have seen in Section 2, a standard approach in studying the representation theory of the general linear group G = GL_{n}(k) over an algebraically closed field has been to use the equivalence between the polynomial representation theory for G and the rational representation theory of the reductive monoid M = M_{n}(k) of n × n matrices. Let k be an algebraically closed field of characteristic p > 0. Now regard G = GL_{n} as an affine algebraic group scheme, and M = M_{n} as an affine monoid scheme over k. Throughout this section we will assume k is an algebraically closed field of characteristic p > 0. Let F : M → M be the Frobenius morphism, D be the subscheme of M determined by the diagonal matrices, and let M_{r}D be the inverse image of D under the rth iteration of F.

Another classical approach, initiated by Jantzen, has been to study representations of G via the group schemes G_{r}T where T is a maximal torus of G. Doty, Peters, and the author [DNP1] observed that the module category for a monoid scheme M_{r}D combines facets of both aforementioned theories.

The coordinate algebra k[M_{r}D] may be identified with the quotient algebra k[c_{ij}]/J_{r} where J_{r} is the ideal generated by {c_{ij}^{p^{r}} : i ≠ j}. Since M_{r}D is a monoid scheme it follows that J_{r} is a coideal in k[M] and that k[M_{r}D] inherits a coalgebra structure from k[M]. Let A(n, d)_{r} be span of the homogeneous polynomials of degree d in k[M_{r}D]. Then A(n, d)_{r} is a subcoalgebra of k[M_{r}D], so S(n, d)_{r} = A(n, d)_{r}^{∗} is a finite-dimensional algebra. These are the infinitesimal Schur algebras introduced in [DNP1]. The category of rational M_{r}D-modules is equivalent with the category of polynomial G_{r}T -modules. Furthermore, for fixed n and r the module category for {S(n, d)_{r} | d ≥ 0} is categorically equivalent to the polynomial representation theory of G_{r}T. Finally, one should observe that for each n and d with 0 ≤ d < p^{r}, S(n, d) = S(n, d)_{r} and S(n, d) = lim_{r^{⟶}} S(n, d)_{r}. A theory

for a quantum version of the infinitesimal Schur algebra was recently developed by A. Cox [Cox] in his thesis.

3.2. Simple modules: The set of isomorphism classes of simple G_{r}T -modules is in one-to-one correspondence with the set of weights X(T). The correspondence is given by λ → L̂_{r}(λ) = soc_{G_{r}T} (ind_{B_{r}T}^{G_{r}T} λ) [Jan, II 9.5 Prop.]. Let X(T)_{+} be the set of dominant weights and P (D) be the set of polynomial (but not necessarily dominant) weights. Let

P_{r}(D) = {λ ∈ P (D) : 0 ≤ λ_{i} − λ_{i+1} ≤ p^{r} − 1, for 1 ≤ i ≤ n}.

By convention λ_{n+1} = 0. Moreover, let Λ_{r}(D) = P_{r}(D) + p^{r}P (D) and Λ_{r}^{d}(D) = {λ ∈ Λ_{r}(D) : λ_{1}+λ_{2}+…+λ_{n} = d}. The set of isomorphism classes of simple M_{r}Dmodules is obtained by taking the simple G_{r}T -modules L̂_{r}(λ) such that λ ∈ Λ_{r}(D) [DNP1, Cor. 3.1]. Finally, the set of isomorphism classes of simple S(n, d)_{r}-modules is obtained by taking the simple G_{r}T -modules with weights in Λ_{r}^{d}(D).

3.3. Representation type: The three theorems listed below provide a complete classification of the representation type for the infinitesimal Schur algebras.

Theorem (A). [DN1] Let k be a field of characteristic p > 0. The infinitesimal Schur algebra S(n, d)_{r} is semisimple if and only if

(i) p ≥ 2, n ≥ 2, d < p, r ≥ 1; (ii) p = 2, n = 2, d = 3, r ≥ 1; (iii) p = 2, n = 2, d odd, r = 1.

Theorem (B). [DNP3] The infinitesimal Schur algebra S(n, d)_{r} has finite representation type if and only if one of the following occurs.

(i) p ≥ 2, n ≥ 3, d < 2p, r ≥ 2;

(ii) p ≥ 2, n ≥ 3, d < p, r = 1; (iii) p = 3, n = 3, d = 4, 5, r = 1; (iv) p = 2, n = 3, d = 2, 3, r = 1;

(v) p ≥ 2, n = 2, d < p^{2}, r ≥ 2; (vi) p = 2, n = 2, d = 5, 7, r ≥ 3; (vii) p = 2, n = 2, d odd, r = 2; (viii) p ≥ 2, n = 2, r = 1.

Theorem (C). [DEMN] The infinitesimal Schur algebras S(n, d)_{r} have tame representation type if and only if one of the following occurs.

(i) p ≥ 5, n = 3, p ≤ d ≤ 2p − 1, r = 1; (ii) p = 3, n = 3 d = 3, r = 1; (iii) p = 3, n = 4, d = 3, 4, 5, r = 1; (iv) p = 3, n = 3, d = 7, 8, r ≥ 2;

(v) p = 3, n = 2, d = 9, 10, 11, r ≥ 3; (vi) p = 2, n = 4, d = 2, 3, r = 1; (vii) p = 2, n = 2, d = 4, r ≥ 2; (viii) p = 2, n = 2, d = 9, r ≥ 3.

3.4. Truncation functors: There exist two natural truncation functors on Mod(G_{r}T). The first, F_{M_{r}D}, maps a G_{r}T -module to its largest M_{r}D-submodule. Take π to be the set of simple M_{r}D-modules. The second functor O_{π} is defined as follows. For N ∈ Mod(G_{r}T), O_{π}(N) is the unique maximal G_{r}T -submodule all of whose composition factors lie in π. Jantzen [N, Thm. 5.3] proved that these two functors are indeed equivalent.

Theorem (A). [N] F_{M_{r}D} is equivalent to O_{π}.

For a finite-dimensional algebra A, an arbitrary module M and a simple Amodule S, let [M : S] denote the number of times S appears as a composition factor in M. The next result provides a formula for the number of composition factors of the higher right derived functors of F_{M_{r}D}, namely R^{j}F_{M_{r}D}.

Theorem (B). Let λ ∈ Λ_{r}^{d}(D) and let P̂_{r}(λ) be the projective cover of L̂_{r}(λ) in Mod(M_{r}D). Then for j ≥ 0,

[R^{j}F_{M_{r}D}(N) : L̂_{r}(λ)] = dim_{k} Ext_{G_{r}T}^{j} (P̂_{r}(λ), N).

Proof. Let P̂_{r}(λ) be the projective cover of L̂_{r}(λ) in Mod(M_{r}D). There exists a spectral sequence of the following form

E_{2}^{i,j} = Ext_{M_{r}D}^{i}(N′, R^{j}F_{M_{r}D}(N)) ⇒ Ext_{G_{r}T}^{i+j} (N′, N)

where N′ ∈ Mod(M_{r}D) and N ∈ Mod(G_{r}T). Set N′ = P̂_{r}(λ). The spectral sequence collapses and yields the desired isomorphism.

3.5. Complexity and Support Varieties: Alperin first introduced the notion of complexity of a module for a finite-dimensional algebra as a generalization of periodicity in cohomology. The reader is referred to [Ben] for an introduction to this subject. Let A be a finite-dimensional algebra and N ∈ mod(A). Moreover, let

0 → N → Q_{0} → Q_{1} → Q_{2} → ⋯

be the minimal injective resolution of N. The minimal injective resolution is a resolution such that the cokernels of the maps have no injective summands. The complexity of the module N is the smallest integer s ≥ 0 such that dim_{k} Q_{t} ≤ C·t^{s−1} for some constant C ≥ 0 and all t ∈ ℕ. If no such integer exists then N has infinite complexity.

According to [FS], H^{2•}(G_{r}, k) is a finitely generated commutative k-algebra. Furthermore, R := H^{2•}(G_{r}, k)^{T} ≅ Ext_{G_{r}T}^{2•} (k, k) is a finitely generated k-algebra because T is reductive. Let N ∈ mod(G_{r}T). The commutative ring R acts on Ext_{G_{r}T}^{•} (N, N) so let J(N) be the annihilator ideal in R of this action. Set V_{G_{r}T} (N) = Spec(Ext_{G_{r}T}^{2∗} (k, k)/J(N)). In [N], it was shown that these varieties have many of the nice properties proved earlier for cohomological varieties of finite groups and restricted Lie algebras. Under suitable conditions this variety can also be used to determine the complexity for S(n, d)_{r}-modules.

Theorem . [DNP1] [N] Suppose N ∈ mod(S(n, d)_{r}) is F_{M_{r}D}-acyclic. Moreover, let

0 → N → Q_{0} → Q_{1} → ⋯

be the minimal injective resolution of N in mod(G_{r}T). Then

(a) c_{S(n,d)_{r}}(N) = dim V_{G_{r}T} (N);

(b) 0 → N → F_{M_{r}D}(Q_{0}) → F_{M_{r}D}(Q_{1}) → ⋯ is the minimal injective resolution of N in mod(M_{r}D).

3.6. Open problems:

(3.6.1) Find a rule for determining whether two simple modules L̂(λ) and L̂(μ) are in the same block of S(n, d)_{r}. Donkin [Don4] has shown that a generalized Nakayama rule determines the simple modules lying in a block for the ordinary Schur algebra.

(3.6.2) Given a parametrization of blocks of S(n, d)_{r}, determine the representation type of each block.

(3.6.3) When does the infinitesimal Schur algebra S(n, d)_{r} have finite global dimension? Does this occur only when 0 ≤ d < p^{r}?

(3.6.4) Let S = ⊕_{λ∈Λ_{r}^{d}(D)}L̂_{r}(λ). Compute c_{S(n,d)_{r}}(S). Note that this number would provide an upper bound for the complexity of any module in mod(S(n, d)_{r}).

(3.6.5) Provide necessary and sufficent conditions on λ ∈ Λ_{r}^{d}(D) such that

dim V_{G_{r}T} (L̂_{r}(λ)) = c_{S(n,d)_{r}}(L̂_{r}(λ)).

This is only known for S(2, d)_{1} [DNP1, Thm. 7.5].

4. The Schur functor and its adjoints

4.1. Basis for S(n, d)_{r}: Recall that A(n, d)_{r} is the span of homogeneous polynomials of degree d in k[c_{ij} : 1 ≤ i, j ≤ n]/J_{r}. Therefore, a basis for A(n, d)_{r} consists of monomials of the form

{c^{α} = ∏_{1≤i,j≤n} c_{ij}^{α_{ij}} : ∑ α_{ij} = d, and α_{i,j} < p^{r} for i ≠ j}. (4.1.1)

Let I(n, d) denote the set of multi-indices of the form i = (i_{1}, …, i_{d}), i_{j} ∈ {1, …, n} for j = 1, …, d. The symmetric group Σ_{d} acts on I(n, d) on the right by iπ = (i_{π(1)}, …, i_{π(d)}). Furthermore, Σ_{d} acts on I(n, d) × I(n, d). Let O(n, d) be the set of Σ_{d}-orbits in I(n, d) × I(n, d). The set O(n, d) indexes the monomial basis of A(n, d). This correspondence is given as follows. If (i, j) ∈ I(n, d) × I(n, d) is an orbit representative then the corresponding monomial is

c_{i,j} = c_{i_{1},j_{1}} c_{i_{2},j_{2}} ⋯ c_{i_{d},j_{d}}.

Observe that A(n, d)_{r} is a subspace of A(n, d), so one can let O(n, d)_{r} be the subset of O(n, d) corresponding to A(n, d)_{r}.

For (i, j) ∈ I(n, d) × I(n, d) let ξ_{i,j} denote the element of S(n, d) such that

ξ_{i,j}(c_{k,l}) = \begin{cases}1 & if (i, j) and (k, l) lie in the same Σ_{d}-orbit \\ 0 & otherwise. \end{cases}

Then a basis of S(n, d)_{r} dual to the monomial basis of A(n, d)_{r} is given by

{ξ_{i,j} : (i, j) ∈ R(n, d)_{r}} (4.1.2)

where R(n, d)_{r} denotes a set of orbit representatives for O(n, d)_{r}.

4.2. According to [Gr1, p.22] the distinct ξ_{i,i} (i ∈ I(n, d)) form a set of mutually orthogonal idempotents in S(n, d). Observe that ξ_{i,i} lies in the subalgebra S(n, d)_{r} for any r ≥ 1 and any i ∈ I(n, d). Hence, the above idempotent decomposition remains valid for S(n, d)_{r}. Set e = ζ_{(1,2,...,d)(1,2,...,d)} in S(n, d)_{r}. For n ≥ d, one can use this idempotent to identify the group algebra of the symmetric group in S(n, d)_{r}.

Proposition . [DNP1] For n ≥ d, eS(n, d)_{r}e ≅ kΣ_{d} and S(n, d)_{r}e ≅ V^{⊗d}.

The Schur functor H is the covariant exact functor from Mod(S(n, d)_{r}) to Mod(kΣ_{d}) defined by H(M) = eM. This functor can be expressed in several different ways:

H(M) = eM ≅ eS(n, d)_{r} ⊗_{S(n,d)_{r}} M ≅ Hom_{S(n,d)_{r}}(V^{⊗d}, M). (4.2.1)

Because of these identifications, H admits two adjoint functors. These adjoint functors, denoted by G_{Hom} and G_{⊗}, are defined on objects N ∈ mod(kΣ_{d}) by

(4.2.2) G_{Hom}(N) = Hom_{kΣ_{d}}(V^{⊗d}, N), G_{⊗}(N) = V^{⊗d} ⊗_{kΣ_{d}} N

and on morphisms in the obvious way. The functor G_{⊗} was considered in [Gr1] for the Schur algebra S(n, d). Also, for any object N ∈ mod(kΣ_{d}) we have isomorphisms:

H ∘ G_{Hom}(N) ≅ N, H ∘ G_{⊗}(N) ≅ N. (4.2.3)

Therefore, H is a left inverse to both its adjoint functors. We remark that Cline, Parshall and Scott [CPS2] have studied endomorphism algebras by using the functors similar to the ones defined in (4.2.2).

4.3. The simple S(n, d)-modules are parametrized by the set partitions of d with at most n parts. This set will be denoted by Λ = Λ^{+}(n, d) and the corresponding simple S(n, d)-module by L(λ) for λ ∈ Λ. Observe that the set Λ can be identified as the set of dominant polynomial weights of GL_{n}(k) of degree d. For λ ∈ Γ, set H^{0}(λ) = ind_{B}^{G}λ where B is the lower triangular Borel subgroup of GL_{n}(k) and let V (λ) be the Weyl module. Moreover, for λ ∈ Λ, let I(λ) be the injective hull of L(λ), P (λ) be the projective cover of L(λ) and T (λ) be the corresponding tilting module. If λ ∈ Λ with λ = (λ_{1}, λ_{2}, …, λ_{n}), then let S^{λ}(V) = S^{λ_{1}}(V) ⊗ S^{λ_{2}}(V) ⊗ ⋯ ⊗ S^{λ_{n}}(V).

4.4. Correspondences under H. Let D_{λ} = eL(λ). Then either D_{λ} = (0) or D_{λ} is a simple kΣ_{d}-module. A partition (λ_{1}, λ_{2}, …) is called p-restricted if λ_{i} − λ_{i+1} ≤ p − 1 for all i. A complete set of non-isomorphic simple kΣ_{d}-modules is given by

{D_{λ} | λ ∈ Λ is p-restricted}.

Thus, the simple kΣ_{d}-modules will be indexed by the p-restricted partitions λ ∈ Λ. Let Λ_{reg} denote the p-restricted partitions, and Λ_{sing} be the partitions λ such that eL(λ) = 0. For λ ∈ Λ_{reg}, let

D^{λ′} = D_{λ} ⊗ sgn.

Then a complete set of non-isomophic simple modules is given by

{D^{λ′} | λ ∈ Λ is p-restricted}.

The set of weights {λ′ | λ ∈ Λ is p-restricted} are the p-regular partitions.

The Specht module corresponding to λ ∈ Λ over kΣ_{d} is denoted by S^{λ}. For λ ∈ Λ, let M^{λ} = ind_{kΣ^{λ}}^{kΣ_{d}}k be the permutation module obtained from inducing up the trivial module k from the Young subgroup Σ^{λ}. The Young module Y^{λ} is the unique indecomposable summand of M^{λ} which contains S^{λ} as a submodule [Mar, Def 4.6.1].

Let sgn be the one-dimensional sign representation of Σ_{d}. Also, if λ is a partition of d, let λ′ be the dual partition. One has following correspondences between S(n, d)-modules and kΣ_{d}-modules under H [Gr1, § 6] and [Don3, 3.5, 3.6]:

(4.4.1)  H(H^{0}(λ)) = S^{λ};
(4.4.2)  H(V (λ)) = (S^{λ})^{∗};
(4.4.3)  H(I(λ)) = F(P (λ)) = Y^{λ};
(4.4.4)  H(T (λ)) = Y^{λ′} ⊗ sgn
(4.4.5)  H(S^{λ}(V)) = M^{λ}.

Finally, it is not always easy to recover modules for S(n, d) from modules for kΣ_{d}. However, for tilting modules this is possible (see [CPS2, (5.2.4) Thm.] and [DN2, Thm. 6.2]).

Theorem . Let p ≥ 3 and λ ∈ Γ. Then T (λ) = G_{Hom}(Y^{λ′} ⊗ sgn).

4.5. Open Problems:

(4.5.1) Let Ẑ_{r}(λ) = ind_{B_{r}T}^{G_{r}T} λ for λ ∈ X(T). Then F_{M_{r}D}(Ẑ_{r}(λ)) is in mod(M_{r}D). What types of modules for kΣ_{d} does one produce by looking at H(F_{M_{r}D}(Ẑ_{r}(λ)))?

(4.5.2) Let Î_{r}(λ) be the injective hull of L̂(λ) in mod(S(n, d)_{r}). What is H(Î_{r}(λ)) in mod(kΣ_{d})? Is this ever zero? This construction may produce “Young” like modules.

(4.5.3) On the other hand, let p ≥ 3 and consider G_{Hom}(H(Î_{r}(λ)) ⊗ sgn). Does this produce a module like a tilting module for S(n, d)_{r} (i.e. does this module admit some nice filtrations)?

5. Decomposition numbers

5.1. Results of James and Erdmann. The problem of computing the irreducible modular characters remains open for general linear groups and symmetric groups, although some notable progress has been made. Some results on these questions can be found in [E2 − 5]. This problem is equivalent to computing the decomposition numbers for Weyl modules and Specht modules.

The Schur functor can be used to derive a result of G. James, which states that the matrix of decomposition numbers for Σ_{d} is a submatrix of the matrix of decomposition numbers of homogeneous GL_{d}(k)-modules in degree d (see [Gr1, (6.6e)]).

Theorem (A). Let λ ∈ Λ and μ ∈ Λ_{reg}. Then [V (λ) : L(μ)] = [S^{λ} : D_{μ}].

For d ∈ ℕ, let t(d) = pd + (p − 1)\binom{n}{2}. Moreover, if μ ∈ Λ, then define t(μ) ∈ Λ^{+}(n, t(d)) as

t(μ) = pμ + (p − 1)(n − 1, n − 2, …, 1, 0).

Erdmann showed that the theory of tilting modules for S(n, d) can be used to prove the following result.

Theorem (B). [E4, 2.3 Prop.] Let λ, μ ∈ Λ. Then

[V (λ) : L(μ)] = [S^{t(λ)} : D^{t(μ)}].

This result shows that one can compute decomposition multiplicities of Weyl modules in terms of decomposition numbers for symmetric groups Σ_{d} assuming that one allows d to vary [E4, 2.4 Thm.].

5.2. Branching results. Determining the decomposition multiplicities of Weyl or Specht modules is equivalent to the following open problem.

Problem: For λ a p-regular partition of d and μ a p-regular partition of d − 1, determine [D^{λ}|_{Σ_{d−1}} : D^{μ}].

This has become known as the Branching Problem. For well-written surveys on this and other related problems we refer the reader to [K8] and [BK]. In this subsection we will highlight some of the significant progress made toward solving the Branching Problem.

We will closely follow the definitions given in [K8, §2]. Let (λ_{1}, λ_{2}, …, λ_{d}) be a partition of d. The Young diagram of λ is given by

Y [λ] = {(i, j) ∈ ℕ × ℕ : j ≤ λ_{j}}.

Each element of Y [λ] is called a node. A node is removable (resp. addable) if it can be removed from (resp. added to) Y [λ] to produce another Young diagram. If A = (i, λ_{j}) is a removable node then set

λ_{A} = (λ_{1}, λ_{2}, …, λ_{i−1}, λ_{i} − 1, λ_{i+1}, …).

The residue of a node (i, j) ∈ Y [λ] is defined as res (i, j) = (j − i) (mod p).

We say that a removable node A is normal if given an addable node B above A such that res A = res B, there exists a removable node C_{B} between A and B with the property that

(i) res A = res C_{B};
(ii) if B ≠ B′ then C_{B} ≠ C_{B′}.

A removable node is good if it is the lowest one among the normal nodes of a given residue. The following two theorems summarize several of the results due to Kleshchev on the Branching Problem (see [K1 − 7]).

Theorem (A). Let λ be a p-regular partition.

(i) Let A be a removable node such that λ_{A} is p-regular and normal. Then [D^{λ}|_{Σ_{d−1}} : D^{λ_{A}}] is equal to the number of normal nodes B above A satisfying res A = res B. Furthermore, the multiplicity of D^{λ_{A}} is zero if A is not normal.

(ii) soc(D^{λ}|_{Σ_{d−1}}) = ⊕_{A}D^{λ_{A}} where A runs over all good nodes.

Theorem (B). Let λ be a p-regular partition.

(i) Let {A_{j} : j = 1, 2, …, s} be the set of good nodes of λ. Then

D^{λ}|_{Σ_{d−1}} = I_{1} ⊕ I_{2} ⊕ ⋯ ⊕ I_{s}

where I_{j} is self dual and soc_{kΣ_{d−1}} I_{j} = hd_{kΣ_{d−1}} I_{j} ≅ D^{λ_{A_{j}}} for j = 1, 2, …, s. (ii) The module D|_{Σ_{d−1}} is completely reducible if and only if all normal nodes of

λ have distinct residues. In that case

D^{λ}|_{Σ_{d−1}} = D^{λ_{A_{1}}} ⊕ D^{λ_{A_{2}}} ⊕ ⋯ ⊕ D^{λ_{As}} .

Branching rules for symmetric groups have been used by Ford and Kleshchev to solve the Mullineux conjecture [FK]. For the general linear group the analogous Branching Problem is more difficult and remains open. Brundan, Kleshchev and Suprunenko provide a criterion for when L(λ) when restricted to GL_{n−1}(k) is completely reducible and describe the semisimple factors [BKS] (see [K8, Thm 7.7]).

5.3. Open problems.

(5.3.1) Prove a version of Erdmann’s result for quantum GL_{n} and the Hecke algebra of type A.

(5.3.2) What can be said about the structure and composition factors of the indecomposable modules I_{j} which appear in the decomposition of D^{λ}|_{Σ_{d−1}}? Kleshchev has some interesting results on the structure of the endomorphism algebras of these indecomposables [K8, Thm. 4.1].

(5.3.3) Determine the socle of L(λ) upon restriction to GL_{n−1}(k).

6. Cohomology theory

6.1. Spectral sequences: The intimate connections between the representation theories of GL_{n}(k) and kΣ_{d} makes it tantalizing to find relationships between the cohomology theories of these two groups. Doty, Erdmann and the author discovered a method for accomplishing such a task by considering the higher derived functors of G_{Hom} and G_{⊗}. Recall that

G_{Hom}(N) = Hom_{kΣ_{d}}(V^{⊗d}, N) and G_{⊗}(N) = V^{⊗d} ⊗_{kΣ_{d}} N

for N ∈ Mod(kΣ_{d}). The higher right (resp. left) derived functors of G_{Hom} (resp. G_{⊗}) are given by

R^{j}G_{Hom}(N) = Ext_{kΣ_{d}}^{j}(V^{⊗d}, N) and L^{j}G_{⊗}(N) = Tor_{j}^{kΣ_{d}}(V^{⊗d}, N)

for j ≥ 0 and N ∈ Mod(kΣ_{d}). One should observe how the actions of V^{⊗d} are being used in this setting. The theorem below demonstrates how one can relate the cohomology theories for these two groups via two spectral sequences.

Theorem (A). [DEN] Let M ∈ Mod(S(n, d)), N ∈ Mod(kΣ_{d}) with n ≥ d. There exists two first-quadrant Grothendieck spectral sequences, with E_{2}-pages given by

(6.1.1) E_{2}^{i,j} = Ext_{S(n,d)}^{i}(M, Ext_{kΣ_{d}}^{j}(V^{⊗d}, N)) ⇒ Ext_{kΣ_{d}}^{i+j}(H(M), N),
(6.1.2) E_{2}^{i,j} = Ext_{S(n,d)}^{i}(Tor_{j}^{kΣ_{d}}(V^{⊗d}, N), M) ⇒ Ext_{kΣ_{d}}^{i+j}(N, H(M)).

The next result shows how one can compute the composition factors of the higher right derived functor of G_{Hom} in mod(S(n, d)).

Theorem (B). [DEN] Let N ∈ mod(kΣ_{d}), and μ ∈ Λ.

(i) [R^{j}G_{Hom}(N) : L(μ)] = dim_{k} Ext_{kΣ_{d}}^{j}(Y^{μ}, N) for j ≥ 0.

(ii) In particular, H(R^{j}G_{Hom}(N)) = 0 for j > 0.

(iii) [G_{Hom}(N) : L(μ)] = dim_{k} Hom_{kΣ_{d}}(Y^{μ}, N).

6.2. Stability of extensions. For ν ∈ Λ, let D^{ν} = D_{ν′} ⊗sgn. For a partition λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{s} > 0) define h(λ) = s. A simple kΣ_{d} module is completely splittable if it is completely reducible when restricted to any Young subgroup Σ^{λ} of Σ_{d}. The spectral sequence in the preceding section was used to prove the following two results [KN, Thm. 5.1, Cor. 5.1]. These results show that the extensions over GL_{n}(k) and kΣ_{d} coincide in a range of degrees.

Theorem . [KN] Let D_{λ} = D^{ν} be a non-trivial completely splittable module and let N ∈ Mod(S(n, d)).

(a) For 0 ≤ i ≤ h(ν) − 2 we have Ext_{S(n,d)}^{i}(N, L(λ)) ≅ Ext_{kΣ_{d}}^{i}(eN, D_{λ}).

(b) In particular for any μ ∈ Λ and 0 ≤ i ≤ h(ν) − 2 we have:

(i) Ext_{S(n,d)}^{i}(L(μ), L(λ)) ≅ \begin{cases}Ext_{kΣ_{d}}^{i}(D_{μ}, D_{λ}) & if μ ∈ Λ_{reg}; \\ 0 & otherwise. \end{cases}

(ii) Ext_{S(n,d)}^{i}(∇(μ), L(λ)) ≅ Ext_{kΣ_{d}}^{i}(S^{μ}, D_{λ}).

(iii) Ext_{S(n,d)}^{i}(Δ(μ), L(λ)) ≅ Ext_{kΣ_{d}}^{i}((S^{μ})^{∗}, D_{λ}).

(iv) Ext_{S(n,d)}^{i}(T (μ), L(λ)) ≅ Ext_{kΣ_{d}}^{i}(Y^{μ′}, D^{λ′}).

(v) Ext_{kΣ_{d}}^{i}(Y^{μ}, D_{λ}) ≅ \begin{cases}k & if λ = μ and i = 0; \\ 0 & otherwise. \end{cases}

Corollary . [KN] Let N ∈ mod(S(d, d)) and δ be the determinant representation for S(d, d).

(a) For 0 ≤ i ≤ p − 3, H^{i}(Σ_{d}, eN ⊗ sgn) ≅ Ext_{S(d,d)}^{i}(δ, N).

(b) In particular for any μ ∈ Λ and 0 ≤ i ≤ p − 3 we have: (i) H^{i}(Σ_{d}, D^{μ′}) ≅ Ext_{S(d,d)}^{i}(δ, L(μ)) for μ ∈ Λ_{reg}.

(ii) H^{i}(Σ_{d}, S^{μ′}) ≅ Ext_{S(d,d)}^{i}(δ, V (μ)).

(iii) H^{i}(Σ_{d}, (S^{μ′})^{∗}) ≅ \begin{cases}k & if μ = (1^{d}) and i = 0; \\ 0 & otherwise. \end{cases}

(iv) H^{i}(Σ_{d}, Y^{μ} ⊗ sgn) ≅ \begin{cases}k & if μ = (1^{d}) and i = 0; \\ 0 & otherwise. \end{cases}

6.3. Self extensions: Let H be an algebraic group scheme over k and S be a simple H-module. If H is a reductive algebraic group scheme, then Ext_{H}^{1}(S, S) = 0 [Jan, II 2.12]. Andersen [And] proved that if H = G_{r} where G is a reductive algebraic group scheme, then Ext_{G_{r}}^{1}(S, S) = 0 as long as G does not contain a component of type C when p = 2. These results along with some strong circumstantial evidence prompted Kleshchev and Martin to make the following conjecture for symmetric groups.

Conjecture . Let p ≥ 3 and S be a simple kΣ_{d}-module. Then Ext_{kΣ_{d}}^{1}(S, S) = 0.

This conjecture has been verified by Kleshchev and the author in the case when S is a completely splittable kΣ_{d}-module [KN, Cor. 3.4]. Also, Kleshchev and Sheth have proved the conjecture when the partition corresponding to the simple module has at most p − 1 parts.

6.4. Support varieties: Let A = kG where G is a finite group. Evens proved that H^{2•}(A, k) is finitely generated. Given M ∈ mod(kG), let J(M) be the annhilator of H^{2•}(A, k) on Ext_{A}^{•}(M, M). As in Section 3.5, one can let V_{kG}(M) = Spec(H^{2•}(A, k)/J(M)). The cohomological varieties for finite groups have been well studied in the last 20 years. However, not many specific computations of these varieties have been made. For symmetric groups the problem of computing these varieties and complexity for simple, Specht, and Young modules is open.

6.5. Friedlander and Suslin cohomology classes: For n ≥ d, Mod(S(n, d)) is categorically equivalent to the category of strict polynomial functors of degree d [FS, Thm. 3.2]. This fact was used in Friedlander and Suslin’s proof of the finite generation of cohomology. A major ingredient in their proof was the construction of certain cohomology classes e_{r} ∈ Ext_{S(n,p^{r})}^{2p^{r−1}}(V^{(r)}, V^{(r)}) where V^{(r)} is the the natural representation of GL_{n}(k) twisted by the Frobenius map r times [FS, Thm. 1.2]. This construction is encompassed by a more general result involving the calculation of Ext_{S(n,p^{r})}^{•}(V^{(r)}, V^{(r)}).

Theorem . [FS, Cor 4.11] Let n ≥ p^{r}. Then

Ext_{S(n,p^{r})}^{m}(V^{(r)}, V^{(r)}) ≅ \begin{cases}0 & m odd or m ≥ 2p^{r} \\ k & m = 2l < 2p^{r}. \end{cases}

We will show how this theorem can be recovered in the case when r = 1. The cohomology for Σ_{p} is given as follows (see [KN, Thm. 4.1]).

H^{j}(Σ_{p}, k) ≅ \begin{cases}k & j = 2m(p − 1), 2m(p − 1) − 1, m ∈ ℤ^{+} \\ 0 & otherwise. \end{cases}

According to Theorem 6.1B, we have

[R^{j}G_{Hom}(k) : V^{(1)}] = dim_{k} Ext_{kΣ_{p}}^{j}(Y^{(p)}, k) = dim_{k} Ext_{kΣ_{p}}^{j}(k, k) = dim_{k} H^{j}(Σ_{p}, k).

Note that for S(n, p) the only simple module S such that H(S) = 0 is V^{(1)}. Therefore, for j > 0,

R^{j}G_{Hom}(k) ≅ \begin{cases}V^{(1)} & j = 2m(p − 1), 2m(p − 1) − 1, m ∈ ℤ^{+} \\ 0 & otherwise. \end{cases}

Now consider the spectral sequence given in Theorem 6.1A with M = V^{(1)} and N = k:

E_{2}^{i,j} = Ext_{S(n,p)}^{i}(V^{(1)}, R^{j}G_{Hom}(k)) ⇒ 0.

Observe that the only non-zero rows of the E_{2}-page occur when j = 0, j = 2m(p−1), j = 2m(p − 1) − 1, m ∈ ℤ^{+}. Moreover, by Totaro’s result (Theorem 2.3), we have E_{2}^{i,j} = 0 for i > 2p. The differentials d_{r} have bidegree (r, 1 − r) so it is really enough to look at the bottom three non-zero rows (i.e. j = 0, j = 2p − 3 and 2p − 2). We must have E_{2}^{1,2p−3} = Ext_{S(n,p)}^{1}(V^{(1)}, V^{(1)}) = 0 because the spectral sequence abuts to zero. Therefore, E_{2}^{1,j} = Ext_{S(n,p)}^{1}(V^{(1)}, V^{(1)}) = 0. Therefore, E_{2}^{3,2p−3} = Ext_{S(n,p)}^{3}(V^{(1)}, V^{(1)}) = 0 because the spectral sequnece abuts to zero and the only possible term which might hit E_{2}^{3,2p−3} comes from E_{2}^{1,2p−2} = 0. Once again E_{2}^{3,j} = 0. Continuing this process we see that Ext_{S(n,p)}^{m}(V^{(1)}, V^{(1)}) = 0 for m odd. On the other hand, E_{2}^{0,2p−2} = Hom_{S(n,p)}(V^{(1)}, V^{(1)}) ≅ k. Since the spectral sequence abuts to zero, d_{2} maps E_{2}^{0,2p−2} isomorphically onto E_{2}^{2,2p−3} = Ext_{S(n,p)}^{2}(V^{(1)}, V^{(1)}). Therefore, E_{2}^{2,j} ≅ k for j = 2m(p − 1), j = 2m(p − 1) − 1, m ∈ ℤ^{+}. One can continue this argument recursively because E_{2}^{2,2p−2} ≅ k must be mapped isomorphically onto E_{2}^{4,2p−3} = Ext_{S(n,p)}^{4}(V^{(1)}, V^{(1)}). Hence, we must have Ext_{S(n,p)}^{m}(V^{(1)}, V^{(1)}) ≅ k for m = 2l < 2p.

6.6. Open problems:

(6.6.1) Are there results like the ones stated in Section 6.2 which relate the extensions over S(n, d)_{r} with kΣ_{d}? One might hope to extend the range of degrees because both S(n, d)_{r} and kΣ_{d} can have infinite global dimension.

(6.6.2) Prove the conjecture stated in Section 6.3 due to Kleshchev and Martin.

(6.6.3) Compute the complexity and support varieties for the Young modules and Specht modules in mod(kΣ_{d}).

(6.6.4) Can one provide an alternative proof for Friedlander and Suslin’s result [FS, Cor. 4.11] by using cohomological infomation about symmetric groups? This may lead to a different proof of their finite generation of cohomology result. We remark that [FS, Cor 4.11] can be proved by using the same argument given in Section 6.5, if there exists N ∈ mod(kΣ_{p^{r}}) such that

R^{j}G_{Hom}(N) ≅ \begin{cases}V^{(r)} & j = 2m(p^{r} − 1), 2m(p^{r} − 1) − 1, m ∈ ℤ^{+} \\ 0 & otherwise. \end{cases}

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Department of Mathematics and Statistics, Utah St. University, Logan, UT 84322

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