The Ext^{1} quiver for hook representations of the symmetric group

David Hemmer ^{∗}

March 6, 2000

Abstract The simple Σ_{n} modules in characteristic p correspond to p-regular partitions of n. Let λ and μ be p-regular partitions of n of the form (n − r, 1^{r}), and let D^{λ} and D^{μ} be the corresponding simple modules. We determine Ext^{1}(D^{λ}, D^{μ}) for odd characteristic p. In particular we verify, for these hook partitions, a conjecture of Kleshchev and Martin that Ext^{1}(D^{λ}, D^{λ}) = 0 for any p-regular λ.

1 Introduction

Let Σ_{n} be the symmetric group on n letters. The complex irreducible Σ_{n} modules correspond to partitions λ of n bijectively, and we denote by S^{λ} the irreducible corresponding to λ. In positive characteristic the simple modules are indexed by p-regular partitions, and we denote the corresponding simple by D^{λ}. For a comprehensive treatment of the theory see [4].

There is considerable interest in determining Ext^{1}(D^{λ}, D^{μ}), as this would be a first step toward determining the indecomposable projectives corresponding to each simple module. Erdmann, Martin and Russell have determined the Ext^{1}- quiver for principal blocks of small defect, see [2, 9, 10] for example, but little is known in general.

Kleshchev and Martin have conjectured that Ext^{1}(D^{λ}, D^{λ}) = 0 for odd characteristic. The results for blocks of small defect and the results of [7] for partitions with at most p − 1 parts provide evidence for this conjecture.

In this paper we give further evidence by verifying the conjecture for simple modules corresponding to hook partitions. We also determine Ext^{1}(D^{λ}, D^{μ}) for arbitrary hook partitions λ and μ.

2 Preliminaries

We follow the notation of Kleshchev’s excellent survey paper [6]. We write λ ⊢ n for λ = (λ_{1}, λ_{2}, …, λ_{k}) a partition of n. Let N = {0, 1, 2, …}. We do not distinguish between λ and its Young diagram:

∗Work supported by an NSF graduate fellowhip

λ = {(i, j) ∈ N × N | j ≤ λ_{i}}.

For λ ⊢ n let

σ_{k}(λ) = ∑_{i=1}^{k} λ_{k}.

Then we say λ dominates μ, denoted λ ⊵ μ, if σ_{k}(λ) ≥ σ_{k}(μ) for every k ∈ N. If λ ≠ μ we write λ ▷ μ.

The removable (resp. addable) nodes are those which can be removed (resp. added) to the diagram λ to produce a diagram for a partition of n − 1 (resp. n + 1). For a removable node A, we write λ_{A} for λ with the node A removed, so λ_{A} ⊢ n − 1. Similarly for λ^{B} ⊢ n + 1, where B is an addable node. The p-residue of a node A = (i, j) is defined as resA = j − i (mod p). The residue content of λ is

contλ = (c_{0}, c_{1}, …, c_{p−1})

where c_{α} is the number of nodes of λ with p-residue α.

The Nakayama conjecture says that D^{λ} and D^{μ} belong to the same p-block if and only if cont(λ) = cont(μ), and similarly for S^{λ} and S^{μ}. This allows us to define Robinson’s α-induction and α-restriction functors. For a module M in a fixed block corresponding to the residue content (c_{0}, c_{1}, …, c_{p−1}), and a residue α ∈ Z/pZ, we define Ind^{α}M as the summand of IndM in the block corresponding to residue content (c_{0}, … c_{α} + 1, c_{α+1}, … c_{p−1}). Res_{α} is defined similarly. For full details see [4, 6.3.16].

We need Kleshchev’s notion of normal and good nodes. A removable node A of λ is normal if for every addable node B above A with resB =resA, there exists a removable node C(B) strictly between A and B with resC(B) = resA, and B ≠ B′ implies C(B) ≠ C(B′). A removable node is called good if it is the lowest among the normal nodes of a fixed residue. An addable node B is called good addable if it is good as a removable node of λ^{B}.

We will make extensive use of the following easy application of the classical branching theorem:

Lemma 1 [7, Lemma 2.1] Let λ ⊢ n and let A be a removable node of λ with α := resA. If there are no addable nodes C (for λ) above A with resC = α then there exists a short exact sequence:

0 → N → Ind^{α}S^{λ_{A}} → S^{λ} → 0,

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 resB = α.

For convenience we collect here the versions of Kleshchev’s branching theorems which we will need. A full list appears in [6].

Theorem 2 (i) Let A_{1}, …, A_{d} be all the good nodes of λ. Then

Res_{Σ_{n−1}}D^{λ} ≅ I_{1} ⊕ ⋯ ⊕ I_{d}

where

soc I_{j} = head I_{j} = D^{λ_{A_{j}}} , j = 1, 2, …, d,

and the modules I_{j} are indecomposable, self-dual and all belong to distinct blocks. A similar result (with good addable nodes) holds for Ind^{Σ_{n+1}}D^{λ}.

(ii) Res_{Σ_{n−1}}D^{λ} is completely reducible if and only if all the normal nodes of λ are good in which case

Res_{Σ_{n−1}}D^{λ} = ⨁_{A good} D^{λ_{A}}.

(iii) Ind^{Σ_{n+1}}D^{λ} is completely reducible if and only if the number of normal nodes of λ is one less than the number of good addable nodes for λ, in which case

Ind^{Σ_{n+1}}D^{λ} = ⨁_{B good addable} D^{λ^{B}}.

We will frequently use the following theorem of Peel [11]:

Theorem 3 Let λ = (n − r, 1^{r}) be a p-regular hook with n − r > 1 and r > 1 (so S^{λ} is not one-dimensional). Then:

S^{λ} = \{ \begin{matrix}D^{λ} & if p ∤ n \\ \begin{matrix}D^{λ} \\ D^{(n−r+1,1^{r−1})} \end{matrix} & if p | n \end{matrix} .

3 Main result

We can now state our main result:

Theorem 4 Let λ = (n − r, 1^{r}) and μ = (n − s, 1^{s}) be p-regular. Then:

dimExt^{1}(D^{λ}, D^{μ}) = \{ \begin{matrix}1 & if p | n, |r − s| = 1 \\ 0 & otherwise \end{matrix}

Proof. We first note that for λ = μ the result holds by [7] except in the case λ = (n − p + 1, 1^{p−1}), since all other p-regular hook partitions have at most p − 1 parts. Also, unless p | n, the p-regular hook representations lie in distinct blocks by the Nakayama conjecture, so the Ext groups are clearly zero.

We prove the theorem by induction on n. The base case is well known. For instance Martin [9] has determined the Ext^{1} quiver for n ≤ 2p and the theorem holds there, so we will assume n > 2p and that the theorem holds for Σ_{k}, k < n. We handle self extensions first, so let λ = (n − p + 1, 1^{p−1}). We must show Ext^{1}(D^{λ}, D^{λ}) = 0. Since n > p, all p-regular hooks have two removable nodes. Throughout this proof we let A denote the higher removable node and B denote the lower. If the partition is ambiguous, we write A_{λ} to denote the higher removable node of λ.

Case 1: n ≢ 0, 1 mod p.

In this case resB = 1, resA > 1, and both A and B are normal and good. Since p ∤ n and p ∤ n − 1, we know S^{λ} ≅ D^{λ}, S^{λ_{A}} ≅ D^{λ_{A}} and S^{λ_{B}} ≅ D^{λ_{B}}. Theorem 2(ii) (or the classical branching rule) gives:

ResD^{λ} ≅ D^{λ_{A}} ⊕ D^{λ_{B}}  (1)

where λ_{A} and λ_{B} are in different p-blocks. Since the addable nodes of λ_{A} below A have residue 0, and resA ≠ 0, the module N in Lemma 1 is 0, so we get by Lemma 1:

0 → Ind^{resA}S^{λ_{A}} → D^{λ} → 0.

Thus:

Ext^{1}(D^{λ}, D^{λ}) = Ext^{1}(Ind^{resA}S^{λ_{A}}, D^{λ})
= Ext^{1}(D^{λ_{A}}, Res_{resA}D^{λ}) by Shapiro’s Lemma
= Ext^{1}(D^{λ_{A}}, D^{λ_{A}}) by equation 1
= 0 by inductive hypothesis.

Case 2: n ≡ 1 mod p.

We have λ_{B} = (n − p + 1, 1^{p−2}). Notice that λ_{B} has its top removable node normal of p-residue 1 and its bottom removable node of p-residue 2 is not normal. The lower two (of the three) addable nodes for λ_{B} are good addable nodes so by Theorem 2(iii) we have:

IndD^{λ_{B}} = D^{λ} ⊕ D^{(n−p+1,2,1^{p−3})}  (2)

By Theorem 3, D^{λ} ≅ S^{λ} so the classical branching theorem yields:

0 → S^{λ_{A}} → ResD^{λ} → S^{λ_{B}} → 0.

Theorem 3 tells us that:

S^{λ_{A}} ≅ \begin{matrix}D^{λ_{A}} \\ D^{λ_{B}} \end{matrix} , S^{λ_{B}} ≅ \begin{matrix}D^{λ_{B}} \\ D^{μ} \end{matrix}  (3)

where μ = (n − p + 2, 1^{p−3}).

Combining the classical branching theorem with Theorem 2(i), plus the selfduality of D^{λ}, gives the structure of ResD^{λ}:

ResD^{λ} := M ≅ \begin{matrix}D^{λ_{B}}. \\ D^{λ_{A}}D^{μ} \\ D^{λ_{B}} \end{matrix}  (4)

We will see that the module M defined above as a restriction can also be viewed as an induced module from Σ_{n−2}. Define a partition of n − 2 by:

τ = (n − p, 1^{p−2}).

Since λ_{B} has a unique good node, and removing it gives τ, by Theorem 2(ii) we get

D^{τ} ≅ S^{τ} ≅ Res_{Σ_{n−2}}D^{λ_{B}}.  (5)

In the same manner that we determined the structure of M, (applying the classical and Kleshchev branching theorems to IndD^{τ} ≅ IndS^{τ}), we obtain:

Ind^{Σ_{n−1}}D^{τ} ≅ S^{(n−p,2,1^{p−3})} ⊕ M  (6)

where the two summands are in different blocks. Thus:

M ≅ Ind^{1}D^{τ}  (7)

Combining these results gives:

Ext^{1}(D^{λ}, D^{λ}) = Ext^{1}(Ind^{1}D^{λ_{B}}, D^{λ}) by (2)
= Ext^{1}(D^{λ_{B}}, Res_{1}D^{λ}) by Schapiro
= Ext^{1}(D^{λ_{B}}, M) by (4)
= Ext^{1}(D^{λ_{B}}, Ind^{1}D^{τ}) by (7)
= Ext^{1}(Res_{1}D^{λ_{B}}, D^{τ}) by Schapiro
= Ext^{1}(D^{τ} , D^{τ}) by (5)
= 0 by inductive hypothesis

Case 3: n ≡ 0 mod p.

We begin with a preliminary lemma:

Lemma 5

S^{(n−p,1^{p})} = \begin{matrix}D^{ϕ} \\ D^{λ} \end{matrix}

where ϕ is not a hook partition.

Proof. By a theorem of Peel (see [3, 24.1]), S^{(n−p,1^{p})} has the correct composition factors. The Specht modules are indecomposable for p ≠ 2, so it is enough to prove socS^{(n−p,1^{p})} ≅ D^{λ}.

Peel proved in [11, Thm. 1] that:

Res_{Σ_{n−1}}socS^{(n−p,1^{p})} ≅ S^{(n−p,1^{p−1})} ≅ D^{λ_{A}}.

By Theorem 2(ii), ResD^{λ} ≅ D^{λ_{A}}. Also, since no other node of λ_{A} is good addable, Theorem 2(ii) implies ResD^{ϕ} ≇ D^{λ_{A}}. Thus the structure of S^{(n−p,1^{p})} is as desired. ■

In this case resA = 0 and λ has two addable nodes below A of residue 0. Thus the module N has composition factors S^{(n−p,1^{p})} and S^{(n−p,2,1^{p−2})}. Since p ≠ 2 the Specht modules are indecomposable. S^{(n−p,2,1^{p−2})} has simple head isomorphic to D^{(n−p,2,1^{p−2})}. By Lemma 5, S^{(n−p,1^{p})} has a simple head that is not isomorphic to D^{λ}. In particular Hom(N, D^{λ}) = 0, so the exact sequence in Lemma 1 yields:

0 → Ext^{1}(S^{λ}, D^{λ}) → Ext^{1}(S^{λ_{A}}, Res_{0}D^{λ}).  (8)

But p ∤ n − 1 so S^{λ_{A}} ≅ D^{λ_{A}} and ResD^{λ} = D^{λ_{A}} since A is the only normal node of λ. Thus

Ext^{1}(S^{λ_{A}}, Res_{0}D^{λ}) = Ext^{1}(D^{λ_{A}}, D^{λ_{A}}) = 0 by induction hypothesis.

So equation (8) gives Ext^{1}(S^{λ}, D^{λ}) = 0.

Finally we apply Hom(−, D^{λ}) to the short exact sequence:

0 → D^{(n−p+2,1^{p−2})} → S^{λ} → D^{λ} → 0

to obtain

0 → Ext^{1}(D^{λ}, D^{λ}) → Ext^{1}(S^{λ}, D^{λ})

so Ext^{1}(D^{λ}, D^{λ}) = 0, completing case three.

We now proceed to determine Ext^{1}(D^{λ}, D^{μ}) for hooks λ ≠ μ. We call hooks (a, 1^{b}) and (a+1, 1^{b−1}) adjacent. Unless p | n, the p-regular hooks are in distinct blocks. When p | n we know adjacent hooks have nonsplit extensions by each other as they occur in the Specht modules. So to complete the proof of the main theorem it is enough to show:

Lemma 6 Let p | n and let λ ▷ μ be p-regular hooks. Then dimExt^{1}(D^{λ}, D^{μ}) ≤ 1, and is 0 unless λ and μ are adjacent.

Proof. We first handle the case where λ ▷ μ are nonadjacent and p | n. We write ▷^{a} to denote adjacent hooks. In this case we have:

λ_{B} ▷^{a} λ_{A} ▷ μ_{B} ▷^{a} μ_{A}.

Also, since λ and μ are both p-regular, we know resA_{λ} ≠ resA_{μ}. Let C be the bottom addable node of λ_{A}. Then Lemma 1 gives:

0 → N → Ind^{resA_{λ}}S^{λ_{A}} → S^{λ} → 0  (9)

where N has composition factors S^{(λ_{A})^{C}} and perhaps S^{(−,2,1^{−})}. Both these are p-regular partitions, and neither is equal to μ. Thus Hom(N, D^{μ}) = 0 so applying Hom(−, D^{μ}) to (9) gives:

0 → Ext^{1}(S^{λ}, D^{μ}) → Ext^{1}(S^{λ_{A}}, Res_{resA_{λ}}D^{μ}).

But A_{μ} is the only normal node of μ and since resA_{λ} ≠ resA_{μ}, Theorem 2(ii) implies Res_{resA_{λ}}D^{μ} = 0. Thus Ext^{1}(S^{λ}, D^{μ}) = 0 which implies Ext^{1}(D^{λ}, D^{μ}) = 0 by applying Hom(−, D^{μ}) to

0 → D^{ψ} → S^{λ} → D^{λ} → 0.

Finally we handle the case of adjacent hooks. We have

S^{λ} = \begin{matrix}D^{λ} \\ D^{ψ} \end{matrix} , S^{μ} = \begin{matrix}D^{μ} \\ D^{λ} \end{matrix} , and ψ ▷ λ ▷ μ.

For both λ and μ, resA = resB −1, A is normal and B is not. Thus Theorem 2(ii) gives

ResD^{λ} = D^{λ_{A}}, ResD^{μ} = D^{μ_{A}}.

Also observe that resA_{λ} = resA_{μ} +1, so μ has no normal nodes of residue resA_{λ}. Consider

0 → N → Ind^{resA_{λ}}S^{λ_{A}} → S^{λ} → 0

from Lemma 1. N has as composition factors S^{μ} and perhaps S^{(−,2,1^{−})} if resA_{λ} = 0. Hence dim Hom(N, D^{μ}) ≤ 1. We have:

0 → Hom(N, D^{μ}) → Ext^{1}(S^{λ}, D^{μ}) → Ext^{1}(S^{λ_{A}}, Res_{resA_{λ}}D^{μ})  (10)

But Res_{resA_{λ}}D^{μ} is 0 since μ has no normal nodes of residue resA_{λ}. Thus the last term in (10) is 0 so dimExt^{1}(S^{λ}, D^{μ}) ≤ 1. But applying Hom(−, D^{μ}) to

0 → D^{ψ} → S^{λ} → D^{λ} → 0

gives

0 → Ext^{1}(D^{λ}, D^{μ}) → Ext^{1}(S^{λ}, D^{μ}) → Ext^{1}(D^{ψ}, D^{μ}).  (11)

But ψ and μ are not adjacent so the last term in (11) is 0 by the proof of the nonadjacent case above. Hence dimExt^{1}(D^{λ}, D^{μ}) ≤ 1 as desired. ■

4 Final Remarks

This paper was inspired by [7] where Kleshchev and Sheth made extensive use of Lemma 1 to relate Ext^{1}(D^{λ}, D^{μ}) to the second Loewy layer of S^{λ} when λ and μ are incomparable partitions with at most p − 1 parts. In the hook case, λ and μ are comparable, so Hom(N, D^{λ}) was not always be zero. Further results may be available in other cases where the module N is not too complicated so that Hom(N, D^{λ}) can be determined and Lemma 1 applied.

The main theorem was proven for n = p in [1]. The entire Ext^{1} quiver was already known for Σ_{p}, but the proof in [1] is very different, using spectral sequences. The module M we encounter in case 2 above occurs as the middle term in (5.7.2) of [1], also as an induced module, this time from a Young subgroup. However when n = p the Young subgroups are p′ so the module M is the projective cover of D^{λ_{B}}, so Ext^{1}(D^{λ_{B}}, D^{λ_{A}}) can be deduced immediately.

References

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