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Relating the Cohomology of General Linear Groups and Symmetric Groups

Stephen R. Doty and Daniel K. Nakano

1991 Mathematics Subject Classification: 20C, 20G.

1. Introduction

1.1 In their pioneering work Frobenius and Schur demonstrated that there is an intimate connection between the representation theory for the general linear group and complex representations of the symmetric groups Σ_{d}. During the last twenty years, much effort has been made in relating the modular representation theory of these two groups. For a detailed surveys of results in this area, we refer the reader to [Gr] and [Mar]. Recently, the authors with K. Erdmann discovered a method for relating the cohomology theories of the general linear group and the symmetric group over fields with non-zero characteristic. This approach is based on constructing a Grothendieck spectral sequence with E_{2}-page involving extensions of modules over the general linear group and converging to extensions of modules over the symmetric group. The construction of this spectral sequence along with the basic results is given in [DEN].

The purpose of this paper is to demonstrate how this new approach in relating the extensions for these two groups can be used to obtain previously known results in the area. Many representation theoretic problems dealing with these groups can be phrased in the language of homological algebra. With an appropriate translation our methods involving the spectral sequence become useful in solving these problems in a systematic way.

1.2 The paper is organized as follows. In Section 2, elementary properties of the Schur functor are reviewed. Furthermore, several of the main results of [DEN] which will be used in this paper are stated. The following section is devoted to showing how the spectral sequence can be used to derive representation theoretic results about the decomposition numbers and blocks for the Schur algebras S(n, d). Old results of James and Donkin are proved by using this new approach. The results in Sections 4 and 5 demonstate the computational power of our approach. We first compute R^{i}G(sgn) where i = 1, 2. Here R^{i}G(−) denotes the ith higher right derived functor of a right adjoint, G, to the Schur functor. This explicit computation allows us to recover the main results of Burichenko, Kleshchev, and Martin [BKM]. These results entail computing the first and second cohomology of Σ_{d} with coefficients in a dual Specht module. Finally, in the last section, results pertaining to the realization of tilting modules for GL_{n}(k) as images of the functor G are used to gain insight about the structure of tilting modules.

Research of the second author was supported in part by NSF grant 9800960

1.3 Acknowledgements This work was completed during the conference on modular representation theory held at the University of Virginia in May 1998. Both authors thank Michael Collins, Brian Parshall and Leonard Scott for their efforts in organizing a productive meeting. The authors also acknowledge Stephen Donkin and Brian Parshall for several useful discussions involving the realization of tilting modules.

2. The Schur Functor

2.1 Notation Let k be an algebraically closed field of characteristic p > 0. Let A be a finite-dimensional algebra over k and let Mod(A) be the category of modules over A. The subcategory of finite-dimensional modules over A will be denoted by mod(A). For M ∈ mod(A) and S a simple A-module let [M : S] be the multiplicity of S as a composition factor of M.

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}. The Schur algebra S(n, d) is the finite-dimensional associative k-algebra End_{kΣ_{d}}(V^{⊗d}) where V is the natural representation of GL_{n}(k). The category of homogeneous polynomial GL_{n}(k)-modules of a fixed degree d ≥ 0 is equivalent to the category of modules for S(n, d). The basic references for this section are [Gr] and [Mar].

Throughout the paper we assume that n ≥ d. Let e = ζ_{(1,2,...,d)(1,2,...,d)} be the idempotent in S(n, d) described in [Gr, (6.1)]. Then eS(n, d)e ≅ kΣ_{d} for n ≥ d. The Schur functor F is the covariant exact functor from Mod(S(n, d)) to Mod(kΣ_{d}) defined by F(M) = eM.

The simple S(n, d)-modules are in one-to-one correspondence with the set of partitions of d. We will denote this set by Λ := Λ^{+}(n, d) and the corresponding simple S(n, d)-module by L(λ) for λ ∈ Λ. The set Λ can be identified as the set of dominant polynomial weights of GL_{n}(k) of degree d. The standard dominance order on partitions will be denoted by ▷ [Mar, Def. 1.4.2]. For λ ∈ Λ, 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. Furthermore, if 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. There exists a duality on mod(S(n, d)) which fixes simple modules called the transposed dual. This duality will be denoted by τ.

Set D_{λ} = eL(λ). 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}. Therefore, the simple kΣ_{d}-modules will be labelled by the p-restricted partitions λ ∈ Λ. Let Λ_{reg} denote the set of p-restricted partitions, and Λ_{sing} be the set of partitions λ such that eL(λ) = 0. We will ofter say that L(λ) is regular if λ ∈ Λ_{reg} and singular otherwise. The Specht module corresponding to a partition λ ∈ Λ over kΣ_{d} is denoted by S^{λ}. For λ = (λ_{1}, λ_{2}, …, λ_{d}) ∈ 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} = Σ_{λ_{1}} × Σ_{λ_{2}} × … × Σ_{λ_{d}}. 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 kΣ_{d}. Also, if λ is a partition of d, let λ′ be the dual partition. The usual linear dual on mod(kΣ_{d}) will be denoted by ∗. One has following correspondences between S(n, d)-modules and kΣ_{d}-modules under F (see [Gr, § 6], [Mar, Lem. 4.6.2] and [Don3, 3.6]):

\begin{aligned}F(H^{0}(λ)) & = S^{λ}; \\ F(V (λ)) & = (S^{λ})^{∗}; \\ F(I(λ)) & = F(P (λ)) = Y^{λ}; \\ F(T (λ)) & = Y^{λ′} ⊗ sgn. \end{aligned}

2.2 Spectral sequences The Schur functor F can be represented as a tensor functor: F(M) ≅ eM ≅ eS(n, d)⊗_{S(n,d)} M, because V^{⊗d} ≅ eS(n, d) is a projective right S(n, d)-module. The functor F admits a natural right adjoint functor G defined as follows:

G(N) = Hom_{eS(n,d)e}(eS(n, d), N),

for N ∈ Mod(eS(n, d)e) ≅ Mod(kΣ_{d}). Furthermore, F is also a left inverse to G, and G takes injective kΣ_{d}-modules to injective S(n, d)-modules. One can use these facts to construct a Grothendieck spectral sequence [DEN, Thm. 2.2].

Theorem [A]. Let M ∈ Mod(S(n, d)), N ∈ Mod(kΣ_{d}) with n ≥ d. There exists a first-quadrant spectral sequence, with E_{2}-page given by

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

Let R^{•}G(−) := Ext_{kΣ_{d}}^{•}(V^{⊗d}, −) denote the higher right derived functors of G. These functors are of central importance. We will show later that knowledge about R^{•}G(N) as an S(n, d)-module can be used to directly compute cohomology for the symmetric group. The following result provides information about the composition factors of the higher right derived functors of G [DEN, Prop. 5.1].

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

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

(ii) In particular, e(R^{j}G(N)) = 0 for j > 0. (iii) [G(D_{λ}) : L(μ)] = dim_{k} Hom_{kΣ_{d}}(Y^{μ}, D_{λ}).

Finally, we will need some results describing the image of G on simple kΣ_{d}-modules [DEN, Thm. 2.4A, Thm. 2.5A, Cor. 3.3].

Theorem [C]. Let D_{λ} be a simple kΣ_{d}-module.

(i) If I_{λ} is the injective hull of D_{λ} then G(I_{λ}) = I(λ). (ii) The module G(D_{λ}) is isomorphic to the largest submodule of I(λ) whose socle

quotient contains only composition factors of the form L(μ) where μ ∈ Λ_{sing}. (iii) If λ, μ ∈ Λ_{reg} then

Ext_{kΣ_{d}}^{1}(D_{λ}, D_{μ}) ≅ Ext_{S(n,d)}^{1}(L(λ), G(D_{μ})).

3. Decomposition numbers and blocks

3.1 James’ theorem In this section we will indicate how the spectral sequence in Theorem 2.2(A) encodes information about decomposition numbers for GL_{n}(k) and kΣ_{d} by proving a well-known result of James.

Theorem. Let λ ∈ Λ and μ ∈ Λ_{reg}. Then [H^{0}(λ) : L(μ)] = [S^{λ} : D_{μ}].

Proof. Consider the spectral sequence in Theorem 2.2(A) with M = H^{0}(λ) and N = I_{μ} where I_{μ} is the injective hull of D_{μ}. By using the fact that E_{0} = E^{0,0} (or adjointness) and Theorem 2.2(C), we have

[H^{0}(λ) : L(μ)] = dim_{k} Hom_{S(n,d)}(H^{0}(λ), I(μ))
= dim_{k} Hom_{S(n,d)}(H^{0}(λ), G(I_{μ}))
= dim_{k} Hom_{kΣ_{d}}(S^{λ}, I_{μ})
= [S^{λ} : D_{μ}].
⊓ ⊔

3.2 Blocks for S(n, d), n ≥ d Let ∼ be the equivalence relation on Λ = Λ^{+}(n, d) given by λ = (λ_{1}, λ_{2}, …, λ_{d}) ∼ μ = (μ_{1}, μ_{2}, …, μ_{d}) if and only if there exists σ ∈ Σ_{d} such that λ_{i} − i = μ_{σ(i)} − σ_{i} (mod p) for i = 1, 2, …, d. The following theorem was proved by Donkin in 1987 [Don2, (2.12)]. Donkin constructed a “double Schur functor” to prove this theorem. The proof presented here employs Theorem 2.2(C), the Nakayama Rule and the Linkage Principle.

Theorem. Let n ≥ d and λ, μ ∈ Λ, the simple modules L(λ) and L(μ) are in the same block of S(n, d) if and only if λ ∼ μ.

Proof. Let L(λ) and L(μ) be in the same block of S(n, d). Then there exists λ_{0}, λ_{1}, …, λ_{t} ∈ Λ such that λ = λ_{0}, μ = λ_{n} with Ext_{S(n,d)}^{1}(L(λ_{i}), L(λ_{i+1})) ≠ 0. By the Linkage Principle, λ_{i} ∼ λ_{i+1} for all i, thus λ ∼ μ.

On the other hand, suppose that λ ∼ μ and consider the Young modules Y^{λ} = F(P (λ)) and Y^{μ} = F(P (μ)). Even if one (or both) of the weights λ or μ is not regular, there exists λ̂ and μ̂ in Λ_{reg} such that Dλ̂ (resp. Dμ̂) is a composition factor of Y^{λ} = F(P (λ)) (resp. Y^{μ} = F(P (μ))) because the Young modules are indecomposable and non-zero modules for kΣ_{d}. By the exactness of the Schur functor it now follows that L(λ) and L(λ̂) (resp. L(μ) and L(μ̂)) must be S(n, d)- composition factors of I(λ) (resp. I(μ)). Therefore, L(λ) (resp. L(μ)) and L(λ̂) (resp. L(μ̂)) lie in the same block for S(n, d). Moreover, note that λ ∼ λ̂ and μ ∼ μ̂ by the argument given in the preceding paragraph. Therefore, it suffices to show that L(λ̂) and L(μ̂) are in the same block of S(n, d).

Since λ̂ ∼ μ̂, it follows by the Nakayama Rule [Mar, Thm. 5.1.1] that Dλ̂ and Dμ̂ are in the same block of kΣ_{d}. Thus, there exists σ_{0}, σ_{1}, …, σ_{s} in Λ_{reg} such that λ̂ = σ_{0}, μ̂ = σ_{s} with Ext_{kΣ_{d}}^{1}(D_{σ_{i}}, D_{σ_{i+1}}) ≠ 0. According to Theorem 2.3(C),

0 ≠ Ext_{kΣ_{d}}^{1}(D_{σ_{i}}, D_{σ_{i+1}}) ≅ Ext_{S(n,d)}^{1}(L(σ_{i}), G(D_{σ_{i+1}})).

We also know that soc_{S(n,d)}G(D_{σ_{i+1}}) = L(σ_{i+1}), so G(D_{σ_{i+1}}) is indecomposable and belongs to the same block containing L(σ_{i+1}). Therefore, L(σ_{i}) and L(σ_{i+1}) are in the same block for all i, so L(λ̂) and L(μ̂) are in the same block. ⊓ ⊔

4. The calculation of first cohomology groups

4.1 In this and the following section, we will take n = d. The main purpose of this section will be to calculate R^{1}G(sgn) for p ≥ 3 as an S(d, d)-module. This result will then be applied in the next section to compute cohomology for duals of Specht modules over Σ_{d}.

Theorem Let p ≥ 3.

R^{1}G(sgn) ≅ \{ \begin{matrix}L(3) & d = 3 and p = 3 \\ L(4, 1^{d−4}) & d ≥ 4 and p = 3 \\ 0 & otherwise \end{matrix}

Proof. Let λ′ be the transposed partition of λ. We have

[R^{1}G(sgn) : L(λ′)] = dim_{k} Ext_{kΣ_{d}}^{1}(Y^{λ′}, sgn)
≤ dim_{k} Ext_{kΣ_{d}}^{1}(M^{λ′}, sgn)
= dim_{k} Ext_{kΣ^{λ′}}^{1} (k, sgn).

One can express λ′ = (σ_{1}, σ_{2}, …, σ_{t}). By the Kunneth formula,

H^{1}(Σ^{λ′}, sgn) ≅ ⨁_{i=1}^{t} H^{0}(Σ_{σ_{1}}, sgn) ⊗ … ⊗ H^{1}(Σ_{σ_{i}}, sgn) ⊗ … ⊗ H^{0}(Σ_{σ_{t}}, sgn).

An easy computation shows that for p ≥ 3;

H^{1}(Σ_{d}, sgn) ≅ \{ \begin{matrix}0 & d ≥ 5 \\ k & d = 3 or 4. \end{matrix}

This shows that [R^{1}G(sgn) : L(λ′)] ≠ 0 implies that λ′ = (3, 1^{d−3}) or (4, 1^{d−4}) when p = 3. In these cases [R^{1}G(sgn) : L(λ′)] ≤ 1. Since (3, 1^{d−3}) ∈ Λ_{reg} for d ≥ 4, Y^{(3,1^{d−3})} is a projective kΣ_{d}-module for d ≥ 4. In this case

[R^{1}G(sgn) : L(3, 1^{d−3})] = dim_{k} Ext_{kΣ_{d}}^{1}(Y^{(3,1^{d−3})}, sgn) = 0.

On the other hand,

[R^{1}G(sgn) : L(3)] = dim_{k} Ext_{kΣ_{3}}^{1}(k, sgn) = 1

because Y^{(3)} = eI(3) ≅ k.

Now consider the case λ′ = (4, 1^{d−4}) for p = 3. We have M^{λ′} ≅ Y^{λ′} ⊕ N where N is a direct sum of Young modules Y^{μ′} with μ′ ▷ λ′. Therefore,

Ext_{kΣ_{d}}^{1}(M^{λ′}, sgn) ≅ Ext_{kΣ_{d}}^{1}(Y^{λ′}, sgn) ⊕ Ext_{kΣ_{d}}^{1}(N, sgn).

However, Ext_{kΣ_{d}}^{1}(M^{λ′}, sgn) ≅ k and Ext_{kΣ_{d}}^{1}(N, sgn) = 0. Hence,

[R^{1}G(sgn) : L(4, 1^{d−4})] = dim_{k} Ext_{kΣ_{d}}^{1}(Y^{(4,1^{d−4})}, sgn) = 1.
⊓ ⊔

4.2 The previous result can be used in conjuction with the spectral sequence to prove a result first obtained by Burichenko, Kleshchev and Martin [BKM, Thm. 2.4].

Corollary. Let p ≥ 3.

H^{1}(Σ_{d}, (S^{λ})^{∗}) ≅ \{ \begin{matrix}k & p = 3 and λ = (1^{3}) or (d − 3, 1^{3}) \\ 0 & otherwise. \end{matrix}

Proof. Let M = V (λ′) and N = sgn in Theorem 2.2(A). Observe that

Ext_{kΣ_{d}}^{•}((S^{λ′})^{∗}, sgn) ≅ Ext_{kΣ_{d}}^{•}(k, (S^{λ})^{∗}) ≅ H^{•}(Σ_{d}, (S^{λ})^{∗}).

Consider the five term exact sequence 0 → E^{1,0} → E^{1} → E^{0,1} → E^{2,0} → E^{2}. For i > 0 we have

E^{i,0} = Ext_{S(d,d)}^{i}(V (λ′), δ) = 0.

From the five term sequence one obtains the isomorphism

H^{1}(Σ_{d}, (S^{λ})^{∗}) ≅ Hom_{S(d,d)}(V (λ′), R^{1}G(sgn)).

The result follows immediately from Theorem 4.1 and the fact that V (λ′) has simple head L(λ′). ⊓ ⊔

5. The calculation of second cohomology groups

5.1 We will now compute R^{2}G(sgn) for p ≥ 3 as an S(d, d)-module. The main result of this section is the following theorem.

Theorem. Let p ≥ 3.

R^{2}G(sgn) ≅ \{ \begin{matrix}L(3) & d = 3 and p = 3 \\ L(4) & d = 4 and p = 3 \\ L(4, 1) & d = 5 and p = 3 \\ L(4, 1^{2}) ⊕ L(6) & d = 6 and p = 3 \\ L(4, 1^{d−4}) ⊕ L(7, 1^{d−7}) & d ≥ 7 and p = 3 \\ 0 & otherwise. \end{matrix}

Proof. Once again let λ′ be the transposed partition of λ. We have

[R^{2}G(sgn) : L(λ′)] = dim_{k} Ext_{kΣ_{d}}^{2}(Y^{λ′}, sgn)
≤ dim_{k} Ext_{kΣ_{d}}^{2}(M^{λ′}, sgn)
= dim_{k} Ext_{kΣ^{λ′}}^{2} (k, sgn).

By [BKM, Lem. 2.3], Ext_{kΣ^{λ′}}^{2} (k, sgn) ≅ k whenever λ′ = (3, 1^{d−3}), (4, 1^{d−4}), (6, 1^{d−6}), (7, 1^{d−7}), (3^{2}, 1^{d−6}), (4, 3, 1^{d−7}), (4^{2}, 1^{d−8}) with p = 3 and 0 otherwise. One can now eliminate the cases when Y^{λ′} is projective (i.e. if λ′ is p-restricted) to conclude that [R^{2}G(sgn) : L(λ′)] ≠ 0 implies that λ′ = (3), (4, 1^{d−4}), (6, 1^{d−6}), (7, 1^{d−7}), (3^{2}), (4, 3), (4^{2}, 1^{d−8}).

We have M^{λ′} ≅ Y^{λ′} ⊕ N where N is a direct sum of Young modules Y^{μ′} with μ′ ▷ λ′. Therefore,

Ext_{kΣ_{d}}^{2}(M^{λ′}, sgn) ≅ Ext_{kΣ_{d}}^{2}(Y^{λ′}, sgn) ⊕ Ext_{kΣ_{d}}^{2}(N, sgn).

For λ′ = (3), (4), (4, 1), (6), (7, 1^{d−7}), Ext_{kΣ_{d}}^{2}(N, sgn) = 0. Therefore,

[R^{2}G(sgn) : L(λ′)] = dim_{k} Ext_{kΣ_{d}}^{2}(Y^{λ′}, sgn) = 1

for λ′ = (3), (4), (4, 1), (6), (7, 1^{d−7}).

A partial decomposition of M^{λ′} for λ′ = (3^{2}), (4, 3), (4^{2}, 1^{d−8}), and (6, 1, 1^{d−7}) is given as follows. One can obtain this decomposition by using the fact that the number of times Y^{μ′} appears in as a direct summand of M^{λ′} is the dimension of the λ′ weight space of L(μ′) [Mar, p. 126]. The character of L(μ′) can be determined for these cases by using Steinberg’s tensor product theorem.

\begin{aligned}M^{(3^{2})} & ≅ Y^{(3^{2})} ⊕ Y^{(6)} ⊕ N \\ M^{(4,3)} & ≅ Y^{(4,3)} ⊕ Y^{(7)} ⊕ N \\ M^{(4^{2},1^{d−8})} & ≅ Y^{(4^{2},1^{d−8})} ⊕ Y^{(7,1^{d−7})} ⊕ N \\ M^{(6,1,1^{d−7})} & ≅ Y^{(6,1,1^{d−7})} ⊕ Y^{(7,1^{d−7})} ⊕ N. \end{aligned}

Recall that Ext_{kΣ_{d}}^{2}(Y^{λ′}, sgn) ≅ k for λ′ = (6), (7, 1^{d−7}) and Ext_{kΣ_{d}}^{2}(M^{λ′}, sgn) ≅ k for λ′ = (3^{2}), (4, 3), (4^{2}, 1^{d−8}), and (6, 1, 1^{d−7}). Hence, Ext_{kΣ_{d}}^{2}(Y^{λ′}, sgn) = 0 for λ′ = (3^{2}), (4, 3), (4^{2}, 1^{d−8}), and (6, 1, 1^{d−7}).

Finally we need to show that Ext_{kΣ_{d}}^{2}(Y^{λ′}, sgn) ≅ k for λ′ = (4, 1^{d−4}). One can show by using the fact given above about the decomposition of M^{λ′} in terms of Young modules that Y^{(7,1^{d−7})} is not a summand of M^{(4,1^{d−4})} and Y^{(6)} is not a summand of M^{(4,1^{2})}. Since M^{λ′} ≅ Y^{λ′} ⊕ N where N is a direct sum of Young modules Y^{μ′} with μ′ ▷ λ′, it follows that

[R^{2}G(sgn) : L(λ′)] = dim_{k} Ext_{kΣ_{d}}^{2}(Y^{λ′}, sgn) = dim_{k} Ext_{kΣ_{d}}^{2}(M^{λ′}, sgn) = 1

for λ′ = (4, 1^{d−4}). ⊓ ⊔

5.2 The preceding calculation can be used to recover another result by Burichenko, Kleshchev and Martin [BKM, Thm. 4.1].

Corollary. Let p ≥ 3.

H^{2}(Σ_{d}, (S^{λ})^{∗}) ≅ \{ \begin{matrix}k & p = 3 and λ = (1^{3}), (2, 1), (2^{2}), (1^{6}), (d − 3, 1^{3}) \\ & (d − 3, 2, 1) or (d − 6, 1^{6}) \\ 0 & otherwise. \end{matrix}

Proof. Set M = V (λ′) and N = sgn in Theorem 2.2(A). Then E^{2} = H^{2}(Σ_{d}, (S^{λ})^{∗}) and E^{3,0} = Ext_{S(d,d)}^{3}(V (λ′), δ) = 0. Also, note that E^{2,0} = Ext_{S(d,d)}^{2}(V (λ′), δ) = 0. Since the differentials d_{r} in the spectral sequence have bidegree (r, 1 − r), it immediately follows that E^{1,1} is a vector space direct summand of E^{2}.

Consider the term E^{2,1} = Ext_{S(d,d)}^{2}(V (λ′), R^{1}G(sgn)). According to Theorem 4.1, E^{2,1} = 0 for p > 3. Now let p = 3 and for any σ ∈ Λ, consider the short exact sequence

0 → L(σ) → H^{0}(σ) → H^{0}(σ)/L(σ) → 0.

Once can apply the long exact sequence in cohomology to conclude that

Ext_{S(d,d)}^{2}(V (λ′), L(σ)) ≅ Ext_{S(d,d)}^{1}(V (λ′), H^{0}(σ)/L(σ));
Ext_{S(d,d)}^{1}(V (λ′), L(σ)) ≅ Hom_{S(d,d)}(V (λ′), H^{0}(σ)/L(σ)).

Therefore, E^{2,1} ≠ 0 implies that λ′ ◁ σ where σ = (3), (4, 1^{d−4}). Now let us look at E^{0,2} = Hom_{S(d,d)}(V (λ′), R^{2}G(sgn)). By Theorem 5.1, and the preceding observation, one has E^{0,2} ≠ 0 implies that E^{2,1} = 0. This analysis shows that

H^{2}(Σ_{d}, (S^{λ})^{∗}) ≅ E^{0,2} ⊕ E^{1,1}

≅ Hom_{S(d,d)}(V (λ′), R^{2}G(sgn)) ⊕ Ext_{S(d,d)}^{1}(V (λ′), R^{1}G(sgn)).

Next observe that E^{1,1} = 0 for p > 3. For p = 3, E^{1,1} = Ext_{S(d,d)}^{1}(V (λ′), L(σ)) where σ = (3) or (4, 1^{d−4}). The decomposition of H^{0}(σ) for σ = (3) and (4, 1^{d−4}) is given by

H^{0}(3) :  L(2, 1)  H^{0}(4) :  L(2^{2})  H^{0}(4, 1^{d−4}) :  L(3, 2, 1^{d−5})
L(3)  L(4)  L(4, 1^{d−4})

Therefore,

E^{1,1} = Ext_{S(d,d)}^{1}(V (λ′), R^{1}G(sgn)) ≅ \{ \begin{matrix}0 & p > 3 \\ k & p = 3, λ′ = (2, 1), (2^{2}), (3, 2, 1^{d−5}). \end{matrix}

The result now follows by Theorem 5.1. ⊓ ⊔

6. Tilting modules

6.1 Let A = S(n, d). For an A-module M, let S(M) = {m ∈ M : eAm = 0}. It is easy to see that S(M) is the unique largest A-submodule of M containing composition factors of the form L(μ) such that μ ∈ Λ_{sing}. In this section we will first look at conditions when G(Y^{λ}) = I(λ). Our first result shows that this happens only when λ ∈ Λ_{reg}.

Proposition. Let λ ∈ Λ and n ≥ d. The following statements are equivalent.

(a) S(I(λ)) = 0;

(b) soc_{S(n,d)}I(λ) = L(λ) is regular.

(c) G(Y^{λ}) = I(λ).

Proof. (a) ⇔ (b). This follows by the definition of S(−). (b) ⇒ (c). Suppose that soc_{S(n,d)}I(λ) = L(λ) is regular. Then Y^{λ} = eI(λ) is injective and by Theorem 2.2(C), I(λ) ≅ G(Y^{λ}). (c) ⇒ (b). Suppose that G(Y^{λ}) ≅ I(λ). Then by adjointness,

Hom_{S(n,d)}(L(μ), I(λ)) ≅ Hom_{S(n,d)}(L(μ), G(Y^{λ}))
≅ Hom_{kΣ_{d}}(eL(μ), Y^{λ}).

Therefore, soc_{S(n,d)}I(λ) is regular. ⊓ ⊔

6.2 Unlike the injective modules, the tilting modules can often be realized as the image of “twisted Young modules” under the functor G. The following result is dual to the one proved by Cline, Parshall and Scott [CPS, Thm. 5.2.4].

Theorem. Let p ≥ 3. For all λ ∈ Λ one has G(Y^{λ′} ⊗ sgn) = T (λ).

Proof. Let λ ∈ Λ. We have T (λ) ≅ Hom_{S(n,d)}(S(n, d), T (λ)). Since eS(n, d) ≅ V^{⊗d} and eT (λ) ≅ Y^{λ′} ⊗ sgn, there exists a map

e : Hom_{S(n,d)}(S(n, d), T (λ)) → Hom_{kΣ_{d}}(V^{⊗d}, Y^{λ′} ⊗ sgn).

induced by the Schur functor.

Let M and N be in mod(S(n, d)). We have the natural vector space isomorphisms

f : Hom_{S(n,d)}(M, N) → Hom_{S(n,d)}(N^{τ} , M^{τ})
g : Hom_{kΣ_{d}}(eM, eN) → Hom_{kΣ_{d}}((eN)^{∗}, (eM)^{∗})

Furthermore, by using the fact that eM^{τ} ≅ (eM)^{∗} and eN^{τ} ≅ (eN)^{∗}, we have the following maps induced by the Schur functor

e_{1} : Hom_{S(n,d)}(M, N) → Hom_{kΣ_{d}}(eM, eN)
e_{2} : Hom_{S(n,d)}(N^{τ} , M^{τ}) → Hom_{kΣ_{d}}((eN)^{∗}, (eM)^{∗}))

These maps are compatible in the sense that e_{2} ∘ f = g ∘ e_{1}.

Now set M = S(n, d) and N = T (λ). Then M^{τ} and N^{τ} admit good filtrations, so by [Mar, 4.5.6], e_{2} is injective. Therefore, e_{1} must also be injective. Hence, e = e_{1} gives an embedding T (λ) ↪ G(Y^{λ′} ⊗ sgn). Finally, we have

dim_{k} G(Y^{λ′} ⊗ sgn) = dim_{k} Hom_{kΣ_{d}}(V^{⊗d}, Y^{λ′} ⊗ sgn)
= dim_{k} Hom_{kΣ_{d}}((Y^{λ′} ⊗ sgn)^{∗}, (V^{⊗d})^{∗})
= dim_{k} Hom_{kΣ_{d}}(Y^{λ′} ⊗ sgn, V^{⊗d})
= dim_{k} T (λ).

by [CPS, Thm. 5.2.4]. Hence, T (λ) ≅ G(Y^{λ′} ⊗ sgn) as desired. ⊓ ⊔

Corollary [A]. Let p ≥ 3 and λ, μ ∈ Λ.

(i) [T (λ) : L(μ)] = dim_{k} Hom_{kΣ_{d}}(Y^{μ}, Y^{λ′} ⊗ sgn)

(ii) [soc_{S(n,d)}T (λ) : L(μ)] = \{ \begin{matrix}[soc_{kΣ_{d}}Y^{λ′} ⊗ sgn : D_{μ}] & μ ∈ Λ_{reg} \\ 0 & μ ∈ Λ_{sing}. \end{matrix}

Proof. (i) Since p ≥ 3, T (λ) ≅ G(Y^{λ′} ⊗ sgn). Then by Theorem 2.2(B), we have

[T (λ) : L(μ)] = dim_{k} Hom_{kΣ_{d}}(Y^{μ}, Y^{λ′} ⊗ sgn).

(ii) From adjointness, one has

[soc_{S(n,d)}T (λ) : L(μ)] ≅ dim_{k} Hom_{S(n,d)}(L(μ), T (λ))
≅ dim_{k} Hom_{S(n,d)}(L(μ), G(Y^{λ′} ⊗ sgn))
≅ dim_{k} Hom_{kΣ_{d}}(eL(μ), Y^{λ′} ⊗ sgn).

Observe that if eL(μ) = 0, then the last quantity is zero, otherwise eL(μ) = D_{μ}.

⊓ ⊔

Corollary [B]. Let p ≥ 3 and λ ∈ Λ. Then

(i) soc_{S(n,d)}T (λ) = hd_{S(n,d)}T (λ) has only regular factors. (ii) S(T (λ)) = 0.

Proof. By Corollary 6.2(A), the socle of T (λ) contains only regular composition factors. Since T (λ) is self-dual, it follows that hd_{S(n,d)}T (λ) has only regular factors. Once again by the definition of S(−), we have S(T (λ)) = 0 if and only if soc_{S(n,d)}T (λ) has composition factors with regular highest weights. ⊓ ⊔

6.3 We can now look at the question of computing extensions between tilting modules and simple modules for S(n, d).

Theorem. Let p ≥ 3 and λ, μ ∈ Λ.

(a) Ext_{S(n,d)}^{1}(L(λ), T (μ)) ≅ \{ \begin{matrix}Ext_{kΣ_{d}}^{1}(D_{λ}, Y^{μ′} ⊗ sgn) & λ ∈ Λ_{reg} \\ 0 & λ ∈ Λ_{sing}. \end{matrix}

(b) Ext_{kΣ_{d}}^{1}(Y^{λ}, Y^{μ}) = 0.

Proof. (a) If λ is in Λ_{reg}, then by [DEN, Thm. 2.4A]

Ext_{S(n,d)}^{1}(L(λ), T (μ)) ≅ Ext_{S(n,d)}^{1}(L(λ), G(Y^{μ′} ⊗ sgn))
≅ Ext_{kΣ_{d}}^{1}(D_{λ}, Y^{μ′} ⊗ sgn).

On the other hand, by using the five term exact sequence obtained from the spectral sequence we have an embedding

Ext_{S(n,d)}^{1}(L(λ), T (μ)) ↪ Ext_{kΣ_{d}}^{1}(eL(λ), Y^{μ′} ⊗ sgn) → …

for all λ ∈ Λ. In particular, if λ ∈ Λ_{sing}, then Ext_{S(n,d)}^{1}(L(λ), T (μ)) = 0.

(b) The tilting modules admit both a good and Weyl filtration. Moreover, the tilting modules when p ≥ 3 have only regular composition factors in the head. Therefore, by Theorem 6.2 and [DEN, Thm. 2.4A], we have

0 = Ext_{S(n,d)}^{1}(T (λ′), T (μ′))
≅ Ext_{kΣ_{d}}^{1}(Y^{λ} ⊗ sgn, Y^{μ} ⊗ sgn)
≅ Ext_{kΣ_{d}}^{1}(Y^{λ}, Y^{μ}).
⊓ ⊔

The preceding result can be used to provide an alternative proof of the following well-known result (see [CPS, (4.6.1)]).

Corollary. Let p ≥ 3, Ext_{kΣ_{d}}^{1}(V^{⊗d}, V^{⊗d}) = 0.

Proof. The module V^{⊗d} is a direct sum of permutation modules each of which is a direct sum of Young modules. The result now follows by Theorem 6.3(b). ⊓ ⊔

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