ON EXTENSIONS OF SIMPLE MODULES OVER SYMMETRIC AND ALGEBRAIC GROUPS

A.S. KLESHCHEV AND J. SHETH

Introduction

Let F be an algebraically closed field of characteristic p > 0. If G is a (connected) reductive algebraic group over F, L(λ) the irreducible G-module with highest weight λ, and Δ(λ) the Weyl module with highest weight λ, then, as shown in [13, II.2.14(4)],

Ext_{G}^{1}(L(λ), L(μ)) ≅ Hom_{G}(rad Δ(λ), L(μ)) if μ does not strictly dominate λ.

(1)

Here and below the cohomology of algebraic groups is the rational cohomology, see [13, I.4]. This result is quite useful for calculating some Ext-groups between simple G-modules. For example, it immediately implies that Ext_{G}^{1}(L(λ), L(λ)) = 0.

We would be interested to have a result similar to (1) for the symmetric group Σ_{n}. For a partition λ of n we denote by S^{λ} the corresponding Specht module, and, if λ is p-regular, we write D^{λ} for the corresponding irreducible module over F Σ_{n}, see [11]. It is known that the Specht modules play a role similar to that of Weyl modules for algebraic groups. So we are interested in results connecting Ext_{Σ_{n}}^{1}(D^{λ}, D^{μ}) and Hom_{Σ_{n}}(rad S^{λ}, D^{μ}). The first main result of the paper is

Theorem 2.10. Let p > 2, and let λ and μ be partitions of n with at most p − 1 non-zero parts. If λ does not strictly dominate μ then

(2) Ext_{Σ_{n}}^{1}(D^{λ}, D^{μ}) ≅ Hom_{Σ_{n}}(rad S^{λ}, D^{μ}).

Remarks. (i) If p = 2 then the result is wrong since S^{(n)} ≅ D^{(n)} ≅ 1_{Σ_{n}}, the trivial Σ_{n}-module, and Ext_{Σ_{n}}^{1}(1_{Σ_{n}}, 1_{Σ_{n}}) ≅ F.

(ii) The assumption that λ and μ have at most p−1 parts also cannot be removed. The easiest counterexample is as follows. Take n = p > 2, λ = (1^{p}), μ = (3, 1^{p−3}). Then S^{λ} ≅ sgn_{n}, the sign representation of Σ_{n}, so the right hand side of (2) is 0, while the left hand side is H^{1}(Σ_{p}, D^{μ} ⊗ sgn_{n}) ≅ H^{1}(Σ_{p}, D^{(p−1,1)}) ≠ 0.

(iii) If we take μ = (n), the result yields H^{1}(Σ_{n}, D^{λ}) ≅ Hom_{Σ_{n}}(rad S^{λ}, 1_{Σ_{n}}). We note that this result is valid for p > 2 without any restrictions on the (p-regular) λ, see [6, 5.1(i)].

1991 Mathematics Subject Classification. 20C30, 20J06.

The first author partially supported by the NSF, grant DMS-9600124

(iv) It is well known that if D^{μ} appears in rad S^{λ} then μ strictly dominates λ. So Theorem 2.10 immediately implies that Ext_{Σ_{n}}^{1}(D^{λ}, D^{λ}) = 0, provided p > 2 and λ has at most p − 1 parts.

(v) In a recent paper [8] one can find some general conditions for the vanishing of Ext^{1}(S^{λ}, D^{μ}). Under those conditions the equality (2) is also true.

(vi) We are grateful to Karin Erdmann and to the referee for the following remark. Let A := F Σ_{n}/I, where I is the annihilator of the action of F Σ_{n} on V_{h}^{⊗n} by place permutations, where V_{h} is the natural h-dimensional F GL_{h}(F)-module. Then A is a quasi-hereditary algebra with respect to the opposite dominance order on partitions with at most h parts, see [9, Theorem 4.4]. This immediately implies that Ext_{A}^{1}(D^{λ}, D^{μ}) ≅ Hom_{A}(rad S^{λ}, D^{μ}) for partitions λ and μ with at most h parts such that λ does not strictly dominate μ. This does not imply Theorem 2.10. However, if h ≤ p − 2, one can deduce Theorem 2.10, using the fact that I is generated by an idempotent, which follows from [7]. On the other hand, the exceptional cases in Theorem 2.9 show that this approach will not work in a straightforward way for h > p − 2. We note that our approach here is very different and the arguments also simplify greatly if h ≤ p − 2.

As the main application of Theorem 2.10 we calculate the groups Ext_{Σ_{n}}^{1}(D^{λ}, D^{μ}) for λ, μ arbitrary two row partitions and p > 2, see Theorem 3.5(iv). We note that the case p = 2 has been treated in [15] using different methods. To apply Theorem 2.10, we need a description of the submodule structure of the corresponding Specht modules. We deduce one from the results of Adamovich on the Weyl modules over GL_{n}(F) corresponding to two column partitions by applying Schur functors. We also calculate the groups Ext_{G}^{1}(L(λ), L(μ)) for G = GL_{n}(F) or SL_{n}(F) and λ, μ arbitrary two column partitions or G = Sp_{2n}(F) and λ, μ arbitrary fundamental weights, see Theorem 3.5(i),(ii),(iii).

1. Preliminaries

We recall some basic facts on representations of Σ_{n}, referring the reader to [11] for more details.

In what follows ℕ = {1, 2, 3, … }, and λ = (λ_{1}, λ_{2}, …) ⊢ n means that λ is a partition of n. We do not distinguish between a partition λ and its Young diagram, which is the set of all nodes (i, j) ∈ ℕ × ℕ satisfying j ≤ λ_{i}. For any λ ⊢ n, k ∈ ℕ we have the following partial sums:

σ_{k}(λ) = ∑_{i=1}^{k} λ_{i}.

Let λ, μ ⊢ n. We say λ dominates μ, written λ ⊵ μ, if σ_{k}(λ) ≥ σ_{k}(μ) for every k ∈ ℕ. If λ ⊵ μ and λ ≠ μ we say λ strictly dominates μ and write λ ▷ μ.

A node A = (i, λ_{i}) is called removable (for λ) if

λ_{A} := λ \ {A} = (λ_{1}, …, λ_{i−1}, λ_{i} − 1, λ_{i+1}, …)

is a partition (of n − 1). A node B = (j, λ_{j} + 1) is called addable (for λ) if

λ^{B} := λ ∪ {B} = (λ_{1}, …, λ_{j−1}, λ_{j} + 1, λ_{j+1}, …)

is a partition (of n + 1).

If A = (i, j) then we define its p-residue to be res A := j −i (mod p). The residue content of the Young diagram λ is defined to be the p-tuple

cont(λ) = (c_{0}, c_{1}, …, c_{p−1})

where

c_{α} = cont_{α}(λ) := #{nodes in λ of residue α}, α = 0, 1, …, p − 1.

We write λ ∼ μ whenever cont(λ) = cont(μ).

In the following technical lemma we are concerned with the dominance order when certain nodes are removed.

Lemma 1.1. Let λ, μ be two partitions of n, with λ being p-regular, λ ∼ μ, and λ ▷̸ μ. Let A be the top removable node of λ, and B be any removable node of μ such that λ_{A} ∼ μ_{B}. Then λ_{A} ▷̸ μ_{B}.

Proof. Assume λ_{A} ▷ μ_{B}, and prove that λ ⊵ μ. This is enough as λ = μ implies λ_{A} ▷̸ μ_{B}.

Let A (resp. B) belong to row i (resp. j) of λ (resp. μ). Then

σ_{k}(λ) = \{ \begin{matrix}σ_{k}(λ_{A}) & if k < i \\ σ_{k}(λ_{A}) + 1 & if k ≥ i \end{matrix} and σ_{k}(μ) = \{ \begin{matrix}σ_{k}(μ_{B}) & if k < j \\ σ_{k}(μ_{B}) + 1 & if k ≥ j. \end{matrix}

So σ_{k}(λ) ≥ σ_{k}(μ) for any k ≥ i and any k < j. Assume j ≤ k < i. Then σ_{k}(μ) > σ_{k}(λ) implies σ_{k}(λ_{A}) = σ_{k}(μ_{B}). But σ_{k}(λ_{A}) = kλ_{1} as A is the top removable node of λ, and σ_{k}(μ_{B}) < kμ_{1} as k ≥ j. If j > 1 then λ_{A} ▷ μ_{B} implies λ_{1} ≥ μ_{1}, whence σ_{k}(λ_{A}) > σ_{k}(μ_{B}). If j = 1, then λ_{A} ▷ μ_{B} implies λ_{1} ≥ μ_{1} − 1. If λ_{1} = μ_{1} − 1 then res A ≠ res B since λ is p-regular, and so λ_{A} ≁ μ_{B}.

If G is a group we denote by F G -mod the category of all finite dimensional F G-modules. For M ∈ F G -mod we write soc M, rad M and head M for the socle, radical and head of M, respectively. If D is an irreducible F G-module, [M : D] denotes the multiplicity of D in a composition series of M. If M has a filtration,

0 = M_{0} ⊆ M_{1} ⊆ M_{2} ⊆ ⋯ ⊆ M_{N} = M

with sections S_{i} = M_{i}/M_{i−1}, we write M ∼ S_{1}|S_{2}| … |S_{N}.

We denote by S^{λ} the Specht module over F Σ_{n} corresponding to λ ⊢ n, and by D^{μ} the irreducible F Σ_{n}-module corresponding to a p-regular μ ⊢ n. A filtration of an F Σ_{n}-module is called a Specht filtration if all consecutive factors are Specht modules.

Lemma 1.2. [11, 12.2] If D^{γ} is a composition factor of S^{λ} then γ ⊵ λ. Moreover, if λ is p-regular then [S^{λ} : D^{λ}] = 1 and S^{λ}/ rad S^{λ} ≅ D^{λ}.

Represent n in the form n = k(p − 1) + d, with k ∈ ℤ, 0 ≤ d < p − 1, and let ε_{n} denote the partition ((k + 1)^{d}, k^{p−1−d}). We shall use the fact that

D^{ε_{n}} ≅ S^{(1^{n})} ≅ sgn_{n}, (3)

where sgn_{n} denotes the sign representation of Σ_{n}, see [11, 24.5(iii)].

The following result is known as the ‘Nakayama Conjecture’.

Theorem 1.3. [12, 6.1.21, 2.7.41] D^{λ} and D^{μ} belong to the same block of F Σ_{n} if and only if λ ∼ μ. Moreover, any Specht module belongs to a fixed block of F Σ_{n}, and S^{λ} and S^{μ} belong to the same block if and only if λ ∼ μ.

Fix a residue α ∈ ℤ/pℤ. We define functors

Ind^{α} : F Σ_{n}-mod → F Σ_{n+1}-mod  and  Res_{α} : F Σ_{n}-mod → F Σ_{n−1}-mod,

by defining them first on a module M in any fixed block, and then extending additively to all of F Σ_{n}-mod. Assume M belongs to the block corresponding to the residue content (c_{0}, c_{1}, …, c_{p−1}). We let Ind^{α} M (resp. Res_{α} M) denote the largest submodule of the induced module M↑^{Σ_{n+1}} (resp. restricted module M↓_{Σ_{n−1}}) all of whose composition factors are of the form D^{μ} with cont_{α}(μ) = c_{α} + 1 (resp. cont_{α}(μ) = c_{α} −1), and cont_{β}(μ) = c_{β} for all β ≠ α. Given a morphism θ : M → N, Ind^{α} θ is just the restriction to Ind^{α} M of the natural map θ̂ : M↑^{Σ_{n+1}} → N↑^{Σ_{n+1}} induced by θ, and similarly for Res_{α}. The functors Res_{α} and Ind^{α} are exact, and

M↑^{Σ_{n+1}} ≅ ⨁_{α∈ℤ/pℤ} Ind^{α} M and M↓_{Σ_{n−1}} ≅ ⨁_{α∈ℤ/pℤ} Res_{α} M.

The functors just defined are called Robinson’s α-induction and α-restriction functors, cf. [12, 6.3.16], [5].

Lemma 1.4. Let i ∈ ℕ, α ∈ ℤ/pℤ, M ∈ F Σ_{n} -mod, and N ∈ F Σ_{n+1} -mod. Then

Hom_{Σ_{n+1}}(Ind^{α} M, N) ≅ Hom_{Σ_{n}}(M, Res_{α} N),
Ext_{Σ_{n+1}}^{i}(Ind^{α} M, N) ≅ Ext_{Σ_{n}}^{i}(M, Res_{α} N).

Proof. This follows from Frobenius reciprocity, Shapiro’s Lemma, and Theorem 1.3.

Lemma 1.5. Let λ ⊢ n, and

{A_{1}, A_{2}, …, A_{l}} = {A | A removable for λ, res A = α}
{B_{1}, B_{2}, …, B_{m}} = {B | B addable for λ, res B = α}.

Assume A_{1}, A_{2}, …, A_{l} are counted from top to bottom (i.e. if A_{k} = (i_{k}, j_{k}) then k < l implies i_{k} < i_{l}) and B_{1}, B_{2}, …, B_{m} are counted from bottom to top. Then

Res_{α} S^{λ} ∼ S^{λ_{A_{1}}} |S^{λ_{A_{2}}} | … |S^{λ_{A_{l}}} and Ind^{α} S^{λ} ∼ S^{λ^{B_{1}}} |S^{λ^{B_{2}}} | … |S^{λ^{Bm}} .

Proof. This follows from [11, 9.3, 17.14] by considering the block decompositions S^{λ}↓_{Σ_{n−1}} = ⨁ Res_{α} S^{λ} and S^{λ}↑^{Σ_{n+1}} = ⨁ Ind^{α} S^{λ}.

Lemma 1.6. If D^{γ} is a composition factor of Res_{α} D^{μ} then there exists a removable node A (for μ) with res A = α such that γ ⊵ μ_{A} and γ ∼ μ_{A}.

Proof. By Lemma 1.2, Res_{α} D^{μ} is a quotient of Res_{α} S^{μ}. Now the result follows from Lemmas 1.5 and 1.2 and Theorem 1.3.

Let A be a removable node for λ, α = res A. We call A normal if for any addable node B of residue α above A there exists a removable node C(B) of residue α strictly between B and A, and C(B) ≠ C(B′) if B ≠ B′. We make use of the following result (where normal always means normal for λ).

Theorem 1.7. [5, Theorem E′] Let λ be a p-regular partition, and α ∈ ℤ/pℤ.

(i) Res_{α} D^{λ} is irreducible if and only if there is exactly one normal node A of residue α, in which case Res_{α} D^{λ} ≅ D^{λ_{A}}.

(ii) Let μ ⊢ n − 1 be p-regular. Then Hom_{Σ_{n−1}}(S^{μ}, Res_{α} D^{λ}) = 0 unless μ = λ_{A} for some normal A of residue α, in which case the Hom-space is 1dimensional.

(iii) Res_{α} D^{λ} = 0, unless there is at least one normal node A of residue α, in which case soc Res_{α} D^{λ} ≅ head Res_{α} D^{λ} ≅ D^{λ_{B}} where B is the bottom normal node of residue α.

(iv) Let A be a removable node such that λ_{A} is p-regular. Then the multiplicity [Res_{α} D^{λ} : D^{λ_{A}}] is 0 unless A is normal and res A = α, in which case [Res_{α} D^{λ} : D^{λ_{A}}] is equal to

1 + |{the number of normal nodes B (strictly) above A with res B = α}|.

2. A cohomology vanishing result for symmetric groups

We start from a result which follows easily from Lemma 1.5.

Lemma 2.1. Let λ be a partition of n and A be a removable node for λ and set α := res A. If there are no addable nodes C (for λ) above A with res C = α then there exists a short exact sequence

0 → N → Ind^{α} S^{λ_{A}} → S^{λ} → 0, (4)

such that N has a Specht filtration with quotients S^{λ_{A}^{B}} where B runs over the addable nodes of λ_{A} below A with res B = α.

For a partition λ ⊢ n and 1 ≤ i < j ≤ n we set

d_{ij}(λ) = (λ_{1}, …, λ_{i} + 1, …, λ_{j} − 1, …, λ_{n}),

which is a partition of n if λ_{j} > λ_{j+1} and λ_{i} < λ_{i−1} (or i = 1).

Recall the partition ε_{n} defined in (3). For p > 2, n ≥ p − 1 we define

δ_{n} = \{ \begin{matrix}d_{1,p−1}(d_{1,p−1}(ε_{n})) & if n = c(p − 1) some c ≥ 2 \\ d_{1,p−1}(ε_{n}) & otherwise. \end{matrix}

Lemma 2.2. Let p > 2, n ≥ p. There exists a removable node A of δ_{n} such that δ_{n−1} = (δ_{n})_{A}. If α = res A then there exists an exact sequence

0 → N → Ind^{α} S^{δ_{n−1}} → S^{δ_{n}} → 0 (5)

where N is either (0) or has a Specht filtration with quotients of the form S^{μ} for some p-regular partitions μ. If n > p, none of these μ is equal to ε_{n}. For n = p, N = S^{ε_{p}}.

Proof. If n = p then δ_{p} = (3, 1^{p−3}), δ_{p−1} = (2, 1^{p−3}), and the result follows from Lemma 2.1.

Let n > p and n ≡ 1 (mod p − 1). Then

α α+1 α α+1 δ_{n}=  and δ_{n−1}=  α α+1 α α+1

Now take A to be the bottom (labelled) node with residue α and use Lemma 2.1. For the case n ≡ 0 (mod p − 1) we have

α-1 α α-1 α δ_{n}=  and δ_{n−1}=  α-1 α α-1 α

Here we take A to be the top (labelled) node with residue α, and the result follows from Lemma 2.1 again.

In the remaining cases

α δ_{n}=  and δ_{n−1}=  α

Take A to be the labelled node with residue α, and apply Lemma 2.1 again.

Lemma 2.3. Let p > 2 and n ≥ p. Then dim_{F} Ext_{Σ_{n}}^{1}(S^{δ_{n}}, sgn_{n}) ≤ 1.

Proof. Applying the functor Hom_{Σ_{n}}(−, sgn_{n}) to (5) yields the exact sequence

Hom_{Σ_{n}}(N, sgn_{n}) → Ext_{Σ_{n}}^{1}(S^{δ_{n}}, sgn_{n}) → Ext_{Σ_{n}}^{1}(Ind^{α} S^{δ_{n−1}}, sgn_{n}).

By Lemmas 2.2 and 1.2, the first term is zero if n > p and is 1-dimensional if n = p. By Lemma 1.4, the third term is isomorphic to Ext_{Σ_{n−1}}^{1}(S^{δ_{n−1}}, Res_{α} sgn_{n}). If Res_{α} sgn_{n} = 0, the result is clear. Otherwise Res_{α} sgn_{n} = sgn_{n−1}, and the result follows by induction.

Lemma 2.4. Hom_{Σ_{2p}}(S^{(2^{p})}, sgn_{2p}) = 0.

Proof. By [11, 8.15], we have S^{(2^{p})∗} ≅ S^{(p^{2})} ⊗ sgn_{2p} . So

Hom_{Σ_{2p}}(S^{(2^{p})}, sgn_{2p}) ≅ Hom_{Σ_{2p}}(sgn_{2p}, S^{(2^{p})∗})
≅ Hom_{Σ_{2p}}(sgn_{2p}, S^{(p^{2})} ⊗ sgn_{2p}) ≅ Hom_{Σ_{2p}}(1_{Σ_{2p}}, S^{(p^{2})}) = 0

thanks to [11, 24.4].

Lemma 2.5. Let A be the top removable node of ε_{n} and set α = res A. Then Res_{α} D^{ε_{n}} ≅ D^{ε_{n−1}} and Res_{β} D^{ε_{n}} = 0 for any β ≠ α.

Proof. This follows from the fact that D^{ε_{n}} ≅ sgn_{n}.

Lemma 2.6. Let p > 2, n > p, B be the bottom addable node for ε_{n−1}, and A be the top removable node of ε_{n}. Set μ = (ε_{n−1})^{B}. Then Ext_{Σ_{n}}^{1}(S^{μ}, D^{ε_{n}}) = 0.

Proof. We may assume res A = res B as otherwise the lemma follows from Theorem 1.3. Set α := res A. Let C be the top removable node of μ. There are two possibilities for μ, as shown in the following picture.

C A B D  or  C A B D

In both cases, res C = α + 1. Note that n must satisfy n ≥ 2p, for otherwise res A ≠ res B. By Lemma 2.1, there exists the exact sequence

0 → N → Ind^{α+1} S^{μ_{C}} → S^{μ} → 0

where N = S^{μ_{C}^{D}} for D as shown. We claim that Hom_{Σ_{n}}(N, D^{ε_{n}}) = 0. This is clear by Lemma 1.2 if n > 2p, since in this case μ_{C}^{D} is p-regular. When n = 2p we use Lemma 2.4 instead. Therefore, the exact sequence above yields the exact sequence

0 → Ext_{Σ_{n}}^{1}(S^{μ}, D^{ε_{n}}) → Ext_{Σ_{n}}^{1}(Ind^{α+1} S^{μ_{C}}, D^{ε_{n}}).

By Lemma 1.4, it suffices to note that Res_{α+1} D^{ε_{n}} = 0, but this follows from Lemma 2.5.

Lemma 2.7. Let p > 2 and n ≥ p. Then dim_{F} Ext_{Σ_{n}}^{1}(S^{ε_{n}}, D^{ε_{n}}) = 1.

Proof. Apply induction on n. If n = p then ε_{n} = (2, 1^{p−2}). In this case the exact sequence (4) looks like

0 → S^{(1^{p})} → Ind^{1} S^{(1^{p−1})} → S^{ε_{p}} → 0.

The induced long exact sequence is

0 → Hom_{Σ_{p}}(S^{ε_{p}}, D^{ε_{p}}) → Hom_{Σ_{p}}(Ind^{1} S^{(1^{p−1})}, D^{ε_{p}})
→ Hom_{Σ_{p}}(S^{(1^{p})}, D^{ε_{p}}) → Ext_{Σ_{p}}^{1}(S^{ε_{p}}, D^{ε_{p}}) → Ext_{Σ_{p}}^{1}(Ind^{1} S^{(1^{p−1})}, D^{ε_{p}}).

The last term is isomorphic to Ext_{Σ_{p−1}}^{1}(S^{(1^{p−1})}, Res_{1} D^{ε_{p}}) = 0. Note that the first term is 1-dimensional by Lemma 1.2, and the second term is isomorphic to

Hom_{Σ_{p−1}}(S^{(1^{p−1})}, Res_{1} D^{ε_{p}}) ≅ Hom_{Σ_{p−1}}(sgn_{p−1}, sgn_{p−1}) ≅ F

as S^{(1^{n})} ≅ sgn_{n} and D^{ε_{n}} ≅ sgn_{n} . Similarly the third term is also F, and the result follows.

Now, let n > p, and let α be the residue of the top removable node A of ε_{n}. Observe that (ε_{n})_{A} = ε_{n−1}. So (4) looks like

0 → N → Ind^{α} S^{ε_{n−1}} → S^{ε_{n}} → 0

where N is either (0) or S^{(ε_{n−1})^{B}}, with B the bottom addable node for ε_{n−1}. In any case, Hom_{Σ_{n}}(N, D^{ε_{n}}) = 0. So we get the exact sequence

0 → Ext_{Σ_{n}}^{1}(S^{ε_{n}}, D^{ε_{n}}) → Ext_{Σ_{n}}^{1}(Ind^{α} S^{ε_{n−1}}, D^{ε_{n}}) → Ext_{Σ_{n}}^{1}(N, D^{ε_{n}}),

the last term of which is zero by Lemma 2.6. The second term is isomorphic to Ext_{Σ_{n−1}}^{1}(S^{ε_{n−1}}, D^{ε_{n−1}}) by Lemmas 1.4 and 2.5. So the inductive assumption applies.

Lemma 2.8. Let n > p, n ≡ 1 (mod p − 1), and δ_{n} ▷ γ ▷ ε_{n}. Then γ ≁ δ_{n}.

Proof. We have

α α δ_{n}=  ▷ γ ▷ ε_{n}=  α α

Since γ ◁ δ_{n} the first row of γ must equal the first row of ε_{n}. Now γ ▷ ε_{n} implies that γ has one less node of residue α than ε_{n}, so γ ≁ ε_{n} ∼ δ_{n}.

Theorem 2.9. Let p > 2, and λ, μ be partitions of n with at most p − 1 parts, and ε_{n} be as defined in ( 3). If λ ▷̸ μ then

Ext_{Σ_{n}}^{1}(S^{λ}, D^{μ}) ≅ \{ \begin{matrix}F & if λ = μ = ε_{n} and n ≥ p \\ 0 & otherwise \end{matrix}

Proof. By Lemma 2.7, we may assume that (λ, μ) ≠ (ε_{n}, ε_{n}). Moreover, by Theorem 1.3 we may assume λ ∼ μ. We prove the result by induction on n. The result is clearly true for n < p so the induction starts. Let n ≥ p. Set A to be the top removable node of λ, α := res A. Applying the functor Hom_{Σ_{n}}(−, D^{μ}) to (4), we get the exact sequence

Hom_{Σ_{n}}(N, D^{μ}) → Ext_{Σ_{n}}^{1}(S^{λ}, D^{μ}) → Ext_{Σ_{n}}^{1}(Ind^{α} S^{λ_{A}}, D^{μ}).

By Lemma 2.1, N has a filtration with quotients S^{λ_{A}^{B}}, where B runs over the addable nodes for λ_{A} below A with res B = α. But μ ◁̸ λ and λ_{A}^{B} ◁ λ, so μ ≠ λ_{A}^{B}. If n ≠ p then λ_{A}^{B} is p-regular, and Lemma 1.2 implies Hom_{Σ_{n}}(N, D^{μ}) = 0. If n = p then λ_{A}^{B} is not p-regular only if λ = ε_{p}, when λ_{A} = (1^{p−1}) and so N = S^{(1^{p})} ≅ sgn_{p} . In this case μ ≠ ε_{p} by our assumption, and so again Hom_{Σ_{p}}(N, D^{μ}) = 0.

It remains to prove that Ext_{Σ_{n}}^{1}(Ind^{α} S^{λ_{A}}, D^{μ}) = 0, which, by Lemma 1.4, is equivalent to Ext_{Σ_{n}}^{1}(S^{λ_{A}}, Res_{α} D^{μ}) = 0. By Lemma 1.6, for any composition factor D^{γ} of Res_{α} D^{μ} we have γ ⊵ μ_{C} for some removable node C of μ with res C = α. By Lemma 1.1, μ_{C} ◁̸ λ_{A}, hence γ ◁̸ λ_{A}. So the inductive hypothesis applies as long as λ_{A} ≠ ε_{n−1} or γ ≠ ε_{n−1}.

So we may assume that λ_{A} = ε_{n−1} and γ = ε_{n−1} for some composition factor D^{γ} in Res_{α} D^{μ}.

For any i > 1 we denote κ_{i} = (ε_{i−1})^{E}, where E is the top addable node of ε_{i−1} (i.e. E belongs to the first row). Note that κ_{n} = ε_{n} if n ≡ 1 (mod p − 1). Observe also that λ_{A} = ε_{n−1} implies λ = ε_{n} or λ = κ_{n}. Moreover, in view of Lemma 1.6, γ = ε_{n−1} implies μ = ε_{n} or μ = κ_{n}. So if n ≡ 1 (mod p − 1) then λ = μ = ε_{n}, giving a contradiction as we have assumed (λ, μ) ≠ (ε_{n}, ε_{n}). In the other cases, ε_{n} ≁ κ_{n}, and, in view of Theorem 1.3 and the inequality (λ, μ) ≠ (ε_{n}, ε_{n}), we may assume λ = μ = κ_{n} (in which case E = A). We distinguish between two cases:

Case 1: n ≢ 2 (mod p − 1) (and n ≢ 1 (mod p − 1)). Then κ_{n} looks as follows:

κ_{n}=  A B D

Choose B as shown, and set β = res B. Then (κ_{n})_{B} = κ_{n−1}, and Lemma 2.1 yields the exact sequence

0 → N → Ind^{β} S^{κ_{n−1}} → S^{κ_{n}} → 0

where N is either (0) or S^{κ_{n−1}^{D}}, with D the bottom addable node for κ_{n}. In any case Hom_{Σ_{n}}(N, D^{κ_{n}}) = 0, so we get the exact sequence

0 → Ext_{Σ_{n}}^{1}(S^{κ_{n}}, D^{κ_{n}}) → Ext_{Σ_{n}}^{1}(Ind^{β} S^{κ_{n−1}}, D^{κ_{n}}).

The last term is isomorphic to Ext_{Σ_{n−1}}^{1}(S^{κ_{n−1}}, Res_{β} D^{κ_{n}}). However, Res_{β} D^{κ_{n}} is D^{κ_{n−1}} or (0), thanks to Theorem 1.7, and Ext_{Σ_{n−1}}^{1}(S^{κ_{n−1}}, D^{κ_{n−1}}) = 0 by the inductive assumption.

Case 2: n ≡ 2 (mod p − 1). Then

A B κ_{n}= D

Choose B as shown. Then res B = res A = α, and (κ_{n})_{B} = δ_{n−1}. By Lemma 2.1 we have the exact sequence

0 → N → Ind^{α} S^{δ_{n−1}} → S^{κ_{n}} → 0

where N is either (0) or S^{δ_{n−1}^{D}}, where D is as shown. In both cases Hom_{Σ_{n}}(N, D^{κ_{n}}) = 0 so we get the exact sequence

0 → Ext_{Σ_{n}}^{1}(S^{κ_{n}}, D^{κ_{n}}) → Ext_{Σ_{n}}^{1}(Ind^{α} S^{δ_{n−1}}, D^{κ_{n}}),

the last term of which is isomorphic to Ext_{Σ_{n−1}}^{1}(S^{δ_{n−1}}, Res_{α} D^{κ_{n}}). By Theorem 1.7 there exists the short exact sequence

0 → Q → Res_{α} D^{κ_{n}} → D^{δ_{n−1}} → 0,

and Hom_{Σ_{n−1}}(S^{δ_{n−1}}, Q) ≅ Hom_{Σ_{n−1}}(S^{δ_{n−1}}, Res_{α} D^{κ_{n}}) ≅ F. So in the long exact sequence

0 → Hom_{Σ_{n−1}}(S^{δ_{n−1}}, Q) → Hom_{Σ_{n−1}}(S^{δ_{n−1}}, Res_{α} D^{κ_{n}}) → Hom_{Σ_{n−1}}(S^{δ_{n−1}}, D^{δ_{n−1}})
→ Ext_{Σ_{n−1}}^{1}(S^{δ_{n−1}}, Q) → Ext_{Σ_{n−1}}^{1}(S^{δ_{n−1}}, Res_{α} D^{κ_{n}}) → Ext_{Σ_{n−1}}^{1}(S^{δ_{n−1}}, D^{δ_{n−1}})

the first three terms are isomorphic to F. Moreover, the last term is zero by the inductive hypothesis, and so it suffices to show that the fourth term has dimension at most one. By Theorem 1.7, D^{ε_{n−1}} ≅ sgn_{n−1} appears in Q with multiplicity one. Moreover, by Lemmas 1.6 and 2.8, we have δ_{n−1} ▷̸ γ for any composition factor D^{γ} of Q. By Lemma 2.3 and the inductive hypothesis, we get dim_{F} Ext_{Σ_{n−1}}^{1}(S^{δ_{n−1}}, Q) ≤ 1, which completes the proof.

Theorem 2.10. Let p > 2 and λ, μ be p-regular partitions of n with at most p − 1 non-zero parts. If λ ▷̸ μ then

Ext_{Σ_{n}}^{1}(D^{λ}, D^{μ}) ≅ Hom_{Σ_{n}}(rad S^{λ}, D^{μ}).

Proof. The long exact sequence corresponding to 0 → rad S^{λ} → S^{λ} → D^{λ} → 0 is

0 → Hom_{Σ_{n}}(D^{λ}, D^{μ}) → Hom_{Σ_{n}}(S^{λ}, D^{μ}) → Hom_{Σ_{n}}(rad S^{λ}, D^{μ})
→ Ext_{Σ_{n}}^{1}(D^{λ}, D^{μ}) → Ext_{Σ_{n}}^{1}(S^{λ}, D^{μ}) → ⋯

By Theorem 2.9, Ext_{Σ_{n}}^{1}(S^{λ}, D^{μ}) = 0, unless λ = μ = ε_{n}, and then the result follows. In the exceptional case Ext_{Σ_{n}}^{1}(D^{ε_{n}}, D^{ε_{n}}) = Ext_{Σ_{n}}^{1}(sgn_{n}, sgn_{n}) = H^{1}(Σ_{n}, F) = 0 as p > 2. So the result claimed is still valid.

3. Results of Adamovich and some extension groups

We first state for future reference some results obtained by Adamovich [1],[2],[3]. These results deal with the submodule structure of Weyl modules over GL_{n}(F) corresponding to 2-column partitions and Weyl modules over Sp_{2n}(F) with fundamental highest weights.

Let G be a group and M be an F G-module. Denote by F(N) the set of the composition factors of M. Let us assume that M is multiplicity free, i.e. every composition factor in M occurs with multiplicity one. If S ∈ F(M), we set M_{S} to be the minimal submodule of M having S as its composition factor. Write T ≤_{M} S if M_{T} ⊆ M_{S}. Then the partially ordered set (F(M), ≤_{M}) determines the submodule structure of M. Indeed, if N_{1}, N_{2} are submodules of M then N_{1} ⊂ N_{2} if and only if F(N_{1}) ⊂ F(N_{2}), and N_{1} = N_{2} if and only if F(N_{1}) = F(N_{2}). Moreover, F(N_{1} + N_{2}) = F(N_{1}) ∪ F(N_{2}). Now the claim follows since every submodule of M is a sum of modules of the form M_{S} and F(M_{S}) = {T ∈ F(M) | T ≤_{M} S}.

Proposition 3.1. Let G be a (connected) reductive algebraic group, and V := Δ(λ) be the Weyl module over G with highest weight λ. Assume that μ does not strictly dominate λ and V is multiplicity free. Then

Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(μ), L(λ)) ≅ F

if and only if L(μ) ∈ F(V), μ ≠ λ and there is no ν with L(μ) <_{V} L(ν) <_{V} L(λ). In all other cases these Ext^{1} spaces vanish.

Proof. Put J := rad V. In view of [13, II.2.12(4)] and [13, II.2.14(4)], we just have to prove that L(μ) belongs to the head of J if and only if L(μ) ∈ F(V), μ ≠ λ and there is no ν with L(μ) <_{V} L(ν) <_{V} L(λ).

We first show that

J = ∑{V_{L(ν)} | L(ν) ∈ head J}. (6)

By definition, any V_{L(ν)} ⊆ J. Denote the right hand side of (6) by X, and assume X ≠ J. Pick some L(ν_{0}) ∈ head(J/X). Then L(ν_{0}) ∈ head J, so L(ν_{0}) ∈ F(X) and L(ν_{0}) ∈ F(J/X), contrary to the fact that V is multiplicity-free.

Now, suppose L(μ) ∈ F(V), μ ≠ λ and there is no ν with L(μ) <_{V} L(ν) <_{V} L(λ). If L(μ) ∉ head J then by (6) we can find ν ≠ μ with L(ν) ∈ head J, L(μ) ∈ F(V_{L(ν)}), which gives a contradiction. Conversely, take L(μ) ∈ head J. Then L(μ) ∈ F(V), μ ≠ λ, and for any submodule J′ ⊆ J either L(μ) ∈ head J′ or L(μ) ∉ F(J′). Since head V_{L(ν)} = L(ν), we conclude that L(μ) ∉ F(V_{L(ν)}) for ν ≠ μ, λ. So there is no ν with L(μ) <_{V} L(ν) <_{V} L(λ).

Proposition 3.2. Let p > 2 and λ, μ be partitions of n, having at most p − 1 non-zero parts. If λ does not strictly dominate μ and V := S^{λ} is multiplicity free then

Ext_{Σ_{n}}^{1}(D^{λ}, D^{μ}) ≅ Ext_{Σ_{n}}^{1}(D^{μ}, D^{λ}) ≅ F

if and only if D^{μ} ∈ F(V), μ ≠ λ and there does not exist ν with D^{μ} <_{V} D^{ν} <_{V} D^{λ}. In all other cases these Ext^{1} spaces vanish.

Proof. The proof is entirely similar to that of Proposition 3.1 but uses Theorem 2.10 instead of [13, II.2.14(4)].

Recall that the (polynomial) dominant weights for GL_{n}(F) can be identified with partitions having at most n parts. We denote by {ω_{1}, …, ω_{n−1}} and {ω_{1}, …, ω_{n}} the fundamental highest weights for the groups SL_{n}(F) and Sp_{2n}(F) respectively, using the labelling as in [4]. In what follows, the weights ω_{0}, ω_{n} for SL_{n}(F) and the weight ω_{0} for Sp_{2n}(F) are interpreted as 0.

Adamovich shows that the combinatorics related to the submodule structure of Weyl modules over GL_{n} corresponding to the two column partitions and Weyl modules over Sp_{2n} corresponding to the fundamental highest weights is essentially the same. So from now on we consider the following four situations simultaneously:

\begin{aligned}(a) & G & = GL_{n}(F), & λ = (2^{u}, 1^{v−u}), & (0 ≤ u ≤ v ≤ n); \\ (b) & G & = SL_{n}(F), & λ = ω_{u} + ω_{v}, & (0 ≤ u ≤ v ≤ n); \\ (c) & G & = Sp_{2n}(F), & λ = ω_{u}, & (0 ≤ u ≤ n); \\ (d) & G & = Σ_{n}, & λ = (v, u), & (0 ≤ u ≤ v ≤ n, u + v = n). \end{aligned}

Define the integer a_{λ} by setting

a_{λ} := \{ \begin{matrix}v − u + 1 & if G = GL_{n}(F), SL_{n}(F) or Σ_{n} \\ n − u + 1 & if G = Sp_{2n}(F) \end{matrix}

Let

a_{λ} = ∑_{i≥0} a_{i}p^{i}

be the p-adic expansion. Put

B_{λ}^{−} := {i | a_{i} ≠ 0}, B_{λ}^{+} := {i | a_{i} ≠ p − 1}.

Note that B_{λ}^{+} contains all i ≫ 0. For i, j ∈ ℤ, set [i, j) := {k ∈ ℤ | i ≤ k < j}. Let Â_{λ} be the family of sets which comprises the empty set along with any set I of the form

I = [i_{1}, i_{2}) ∪ [i_{3}, i_{4}) ∪ ⋯ ∪ [i_{2t−1}, i_{2t})

with

i_{1} < i_{2} < ⋯ < i_{2t}, and i_{2j−1} ∈ B_{λ}^{−}, i_{2j} ∈ B_{λ}^{+} for any j = 1, 2, …, t.

For every I ∈ Â_{λ} define

δ_{I} := ∑_{i∈I}(p − 1 − a_{i})p^{i} + ∑_{j=1}^{t} p^{i_{2j−1}}.

(note that δ_{∅} = 0). Now set

\begin{aligned}A_{λ} & := \{I ∈ Â_{λ} | δ_{I} ≤ u, δ_{I} ≤ n − v\} & (for GL_{n}, SL_{n}, Σ_{n}) \\ A_{λ} & := \{I ∈ Â_{λ} | 2δ_{I} ≤ u\} & (for Sp_{2n}) \end{aligned}

For I ∈ A_{λ} define

ν_{I}(λ) := (2^{u−δ_{I}}, 1^{v−u+2δ_{I}}) (for GL_{n})
ν_{I}(λ) := ω_{u−δ_{I}} + ω_{v+δ_{I}}  (for SL_{n})
ν_{I}(λ) := ω_{u−2δ_{I}}  (for Sp_{2n})
ν_{I}(λ) := (v + δ_{I}, u − δ_{I}) (for Σ_{n})

Now we are able to state the main result of Adamovich [1, 2].

Theorem 3.3. [1, 2] Let V = Δ(λ) be the Weyl module for GL_{n}(F), SL_{n}(F) or Sp_{2n}(F) with λ as in (a), (b) or (c) above, respectively. Then:

(i) V is multiplicity-free, and F(V) = {L(ν_{I}(λ)) | I ∈ A_{λ}}.

(ii) L(ν_{J}(λ)) ≥_{V} L(ν_{I}(λ)) if and only if J ⊆ I.

Corollary 3.4. Let λ be a p-regular 2-row partition as above, and S = S^{λ} be the Specht module corresponding to the partition of λ as above. Then:

(i) S is multiplicity-free, and F(S) = {D^{ν_{I}(λ)} | I ∈ A_{λ}};

(ii) D^{ν_{J} (λ)} ≥_{S} D^{ν_{I}(λ)} if and only if J ⊆ I.

Proof. We adopt the notation of [14, section 2]. In particular, M_{F} (n, n) is the category of the polynomial GL_{n}(F)-modules which are homogeneous of degree n, ϕ : M_{F} (n, n) → F Σ_{n}-mod is the Schur functor. Then for μ ⊢ n we have ϕ(Δ(μ)) ≅ S^{μ′} ⊗sgn_{n}, while ϕ(L(μ)) ≅ D^{μ′} ⊗sgn_{n}, provided μ′ is p-regular [10, 6.3,6.4]. Now (i) follows from the fact that ϕ is exact. To prove (ii) we need to employ the ‘truncated inverse Schur functor’, F Σ_{n}-mod→ M_{F} (n, n)^{c}, W ↦ \overline{W} , as defined in [14, section 2]. Here M_{F} (n, n)^{c} is the full subcategory of M_{F} (n, n), which consists of all modules with p-restricted socle and head. By assumption, all composition factors of Δ(λ) are p-restricted. So the two functors yield an isomorphism of the two submodule structures, see [14, 2.12].

Theorem 3.5.

(i) Assume G = GL_{n}(F), λ = (2^{u}, 1^{v−u}), μ = (2^{r}, 1^{s−r}), 0 ≤ u ≤ v ≤ n, 0 ≤ r ≤ s ≤ n, u ≥ r, and let v − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then the space Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(μ), L(λ)) is isomorphic to

\{ \begin{matrix}F & if s − v = u − r = (p − a_{i})p^{i} for some i with a_{i} > 0 \\ 0 & otherwise. \end{matrix}

(ii) Assume G = SL_{n}(F), λ = ω_{u} + ω_{v}, μ = ω_{r} + ω_{s}, 0 ≤ u ≤ v ≤ n, 0 ≤ r ≤ s ≤ n, u ≥ r, (ω_{0}, ω_{n} are interpreted as 0), and let v − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then the space Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(μ), L(λ)) is isomorphic to

\{ \begin{matrix}F & if s − v = u − r = (p − a_{i})p^{i} for some i with a_{i} > 0 \\ 0 & otherwise. \end{matrix}

(iii) Assume G = Sp_{2n}(F), λ = ω_{u}, μ = ω_{r}, 0 ≤ r ≤ u ≤ n, (ω_{0} is interpreted as 0) and let n − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then the space Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{G}^{1}(L(μ), L(λ)) is isomorphic to

\{ \begin{matrix}F & if u − r = 2(p − a_{i})p^{i} for some i with a_{i} > 0 \\ 0 & otherwise. \end{matrix}

(iv) Assume p > 2, G = Σ_{n} and λ = (v, u), μ = (s, r) be partitions of n with 0 ≤ u ≤ v, 0 ≤ r ≤ s, and u ≥ r. Let v − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then the space Ext_{G}^{1}(D^{λ}, D^{μ}) ≅ Ext_{G}^{1}(D^{μ}, D^{λ}) is isomorphic to

\{ \begin{matrix}F & if u − r = (p − a_{i})p^{i} for some i with a_{i} > 0 \\ 0 & otherwise. \end{matrix}

Proof. (i) By Proposition 3.1 and Theorem 3.3 we have Ext_{G}^{1}(L(λ), L(μ)) = 0 unless μ = ν_{I}(λ) for a minimal non-empty I in A_{λ}, in which case the Ext^{1} group is 1-dimensional. To describe the minimal non-empty sets I in A_{λ} recall that any I ∈ A_{λ} is a union of the form [i_{1}, i_{2}) ∪ [i_{3}, i_{4}) ∪ ⋯, where [i_{1}, i_{2}) clearly belongs to A_{λ}. So we may assume that I = [i_{1}, i_{2}), with a_{i_{1}} ≠ 0 and a_{i_{2}} ≠ p − 1. We want to prove that i_{2} = i_{1} + 1. Otherwise, if a_{j} ≠ 0 for some j ∈ (i_{1}, i_{2}) then [j, i_{2}) belongs to A_{λ} so I is not minimal. Hence we may assume that a_{j} = 0 for any j ∈ (i_{1}, i_{2}) but then [i_{1}, i_{1} + 1) belongs to A_{λ}. Thus I = {i_{1}}. This proves (i).

The proof of (ii) and (iii) is entirely similar, as is (iv), except that in (iv) we use Proposition 3.2 in place of Proposition 3.1 and Corollary 3.4 in place of Theorem 3.3.

Corollary 3.6.

(i) Let G = SL_{n}(F), λ = ω_{u}+ω_{v} for 0 ≤ u ≤ v ≤ n, and let v−u+1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then

H^{1}(G, L(λ)) = \{ \begin{matrix}F & if n − v = u = (p − a_{i})p^{i} for some i with a_{i} > 0 \\ 0 & otherwise \end{matrix}

(ii) Let G = Sp_{2n}(F), λ = ω_{u} for 0 ≤ u ≤ n, and let n − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then

H^{1}(G, L(λ)) = \{ \begin{matrix}F & if u = 2(p − a_{i})p^{i} for some i with a_{i} > 0 \\ 0 & otherwise \end{matrix}

(iii) Let p > 2, G = Σ_{n}, λ = (v, u) be a partition of n, and v − u + 1 = ∑_{i≥0} a_{i}p^{i} be the p-adic expansion. Then

H^{1}(Σ_{n}, D^{λ}) = \{ \begin{matrix}F & if u = (p − a_{i})p^{i} for some i with a_{i} > 0 \\ 0 & otherwise \end{matrix}

Proof. Take μ = ω_{0} + ω_{n} = 0 in Theorem 3.5(ii), μ = ω_{0} in 3.5(iii) and μ = (n, 0) in 3.5(iv).

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Department of Mathematics, University of Oregon, Eugene OR 97403, USA E-mail address: klesh@math.uoregon.edu, sheth@noether.uoregon.edu