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

David J. Hemmer ^{*}

University of Chicago Department of Mathematics, 5734 University Avenue, Chicago, Illinois 60637 E-mail: hemmer@math.uchicago.edu

Kleshchev has recently [7] classified those modules for the symmetric group which have semisimple restriction to any Young subgroup. We determine Ext^{1}(D^{λ}, D^{μ}) where D^{λ} and D^{μ} are modules of this type, called completely splittable. As a corollary of this and recent work of Kleshchev and Nakano, we can determine Ext_{GL_{n}(K)}^{1}((L(λ), L(μ)) for certain simple GL_{n}(K)− modules L(λ) and L(μ).

0. INTRODUCTION

Let Σ_{d} denote the symmetric group on d letters. The complex irreducible Σ_{d}-modules correspond bijectively with partitions λ of d, and we denote by S^{λ} the irreducible corresponding to λ. We work over an algebraically closed field K of positive characteristic p > 2. The simple KΣ_{d}-modules are indexed by p-regular partitions, and we denote the corresponding simple module by D^{λ}. These modules can also be indexed by column p-regular partitions, and for λ column p-regular we denote the corresponding simple module by D_{λ}. For a comprehensive treatment of the theory see [4].

For any composition μ = (μ_{1}, μ_{2}, … μ_{k}) of d there is a standard Young subgroup defined as:

Σ_{μ} = Σ_{μ_{1}} × Σ_{μ_{2}} × ⋯ × Σ_{μ_{k}} < Σ_{d}.

We will consider the following class of modules:

Definition 0.1. An irreducible KΣ_{d}− module D^{λ} is called completely splittable if and only if the restriction D^{λ} ↓_{Σ_{μ}} to any Young subgroup Σ_{μ} < Σ_{d} is semisimple. We will also say λ is completely splittable.

* Work supported by an NSF graduate fellowship

In [8], Kleshchev and Nakano obtained several results about the cohomology of completely splittable modules, suggesting that obtaining the Ext^{1}-quiver for these modules should be possible. In this paper we apply Kleshchev’s branching theorems to prove:

Theorem 0.1. Let D^{λ} and D^{μ} be completely splittable and in the same p-block of KΣ_{d}. Then

Ext_{KΣ_{d}}^{1}(D^{λ}, D^{μ}) = \{ \begin{matrix}K & if h_{11}(λ) ≥ p, h_{11}(μ) ≥ p \\ & h_{21}(λ) < p, h_{21}(μ) < p \\ & and | h(λ) − h(μ) |= 1 \\ 0 & else \end{matrix}

where h(λ) is the height of λ and

h_{ij}(λ_{1}, λ_{2}, …, λ_{k}) = λ_{i} + λ′_{j} + 1 − i − j

is the (i, j) hook length.

This theorem plus the work of [8] gives a corresponding result for GL_{n}(K) for n ≥ d. Let L(λ) denote the simple, polynomial GL_{n}(K)-module with highest weight λ. Let m denote the Mullineaux map on p-regular partitions defined by:

D^{λ} ⊗ sgn ≅ D^{m(λ)}.

Then:

Corollary 0.1. Let D^{λ} and D^{μ} be completely splittable. Then:

Ext^{1}(L(m(λ)′), L(m(μ)′)) = \{ \begin{matrix}K & if h_{11}(λ) ≥ p, h_{11}(μ) ≥ p \\ & h_{21}(λ) < p, h_{21}(μ) < p \\ & and | h(λ) − h(μ) |= 1 \\ 0 & else \end{matrix}

The Ext^{1} in Corollary 0.1 is taken in the category M(n, d) of polynomial GL_{n}(K) modules of homogeneous degree d.

1. NOTATION AND PRELIMINARIES

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

λ = {(i, j) ∈ N × N | j ≤ λ_{i}}.

The conjugate partition of λ, denoted λ′, is obtained by interchanging the rows and columns in the Young diagram for λ.

For λ ⊢ d let

σ_{s}(λ) = ∑_{i=1}^{s} λ_{s}.

Then we say λ dominates μ, denoted λ ⊵ μ, if σ_{s}(λ) ≥ σ_{s}(μ) for every s ∈ N. If λ ≠ μ we write λ ▷ μ. This is the usual dominance order on partitions.

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 d − 1 (resp. d + 1). For a removable node A, we write λ_{A} for λ with the node A removed, so λ_{A} ⊢ d − 1. Similarly for λ^{B} ⊢ d + 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

cont(λ) = (c_{0}, c_{1}, …, c_{p−1})

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

The Nakayama conjecture (now a theorem) 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}. We denote this by λ ∼ μ. We assume familiarity with this and the equivalent characterization that λ ∼ μ if and only if λ and μ have the same p-core. See [4] for details.

The Nakayama conjecture allows us to define Robinson’s α-induction and α-restriction functors. For a KΣ_{d}-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 direct summand of IndM in the block of KΣ_{d+1} corresponding to residue content (c_{0}, … c_{α}+1, c_{α+1}, … c_{p−1}). Res_{α}M 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 C(B) ≠ C(B′) for B ≠ 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 now recall the main results of [7] on completely splittable modules. For a partition λ = (λ_{1}, λ_{2}, …, λ_{k}) define:

h(λ) := k and χ(λ) := λ_{1} − λ_{k} + h(λ).

Then:

Theorem 1.1. [7] D^{λ} is completely splittable if and only if χ(λ) ≤ p.

We remark than χ(λ) is the length of the hook with foot at (k, λ_{k}). We further remark that Theorem 1.1 holds as well for p = 2, but since there are no nontrivial completely splittable modules in this case, we can assume p > 2 for this paper.

It is clear from the definition that restricting a completely splittable module to Σ_{d−1} will give a direct sum of completely splittable Σ_{d−1}− modules. Kleshchev determined this decomposition:

Theorem 1.2. [7] Let D^{λ} be completely splittable. Then:

D^{λ} ↓_{Σ_{d−1}}= ⨁ D^{λ_{A}}

where the sum is over all removable nodes A with χ(λ_{A}) ≤ p.

We will make extensive use of induced modules, and the next several results describe how some of the simple and Specht modules induce. The first two are easy applications of the classical branching theorem.

Lemma 1.1. Let {B_{1}, B_{2}, …, B_{s}} be the addable nodes for λ of residue α. Then Ind^{α}S^{λ} has a filtration with composition factors {S^{λ^{B_{i}}} }.

Proof. This follows from the classical branching theorem [3, 9.3] and the fact that Ind^{α}S^{λ} has a filtration by Specht modules [3, 17.14].

Lemma 1.1 has an immediate corollary:

Lemma 1.2. S^{λ} ≅ Ind^{resA}S^{λ_{A}} if and only if A is the unique addable node of λ_{A} of residue resA.

Inducing the simple modules is not as easy, but Kleshchev [5] has determined the head and socle of IndD^{λ} and a criterion for when the induced module is semisimple [6]. We state these results below:

Lemma 1.3. [5] Let B be a good addable node for λ. Then:

soc(Ind^{resB}D^{λ}) ≅ head(Ind^{resB}D^{λ}) ≅ D^{λ^{B}}.

If there is no good addable node of λ of residue α then Ind^{α}D^{λ} = 0.

Because there is at most one good addable node of each residue, Lemma 1.3 makes sense. Kleshchev has determined exactly when IndD^{λ} is completely reducible:

Lemma 1.4. [6] Ind^{Σ_{d+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_{d+1}}D^{λ} = ⨁_{B good addable} D^{λ^{B}}.

If the Specht module S^{λ} is α-induced from Σ_{d−1} as in Lemma 1.2 and if D^{λ} is completely splittable, then D^{λ} is also α-induced from Σ_{d−1}:

Lemma 1.5. Let χ(λ) ≤ p. If S^{λ} ≅ Ind^{resA}S^{λ_{A}}, then D^{λ} ≅ Ind^{resA}D^{λ_{A}}.

Proof. Since χ(λ) ≤ p, λ_{A} must be p-regular. Also, there are no removable or addable nodes of λ of residue resA except A. Thus A is good addable for λ_{A}. We apply Ind^{resA} to:

0 ⟶ R ⟶ S^{λ_{A}} ⟶ D^{λ_{A}} ⟶ 0

to give:

0 ⟶ Ind^{resA}R ⟶ S^{λ} ⟶ Ind^{resA}D^{λ_{A}} ⟶ 0.

But [S^{λ} : D^{λ}] = 1 so Lemma 1.3 applied to λ_{A} implies the result.

2. MINIMAL MODULES

We wish to determine Ext^{1}(D^{λ}, D^{μ}) for completely splittable D^{λ} and D^{μ} by using induction and the Eckmann-Shapiro lemma. It is not surprising then that it is useful to determine which completely splittable modules arise by α-inducing a completely splittable from Σ_{d−1} to Σ_{d}. This motivates the following definition:

Definition 2.1. A completely splittable module D^{λ} is minimal if there does not exist μ ⊢ d − 1, with D^{μ} completely splittable, such that D^{λ} ≅ Ind^{α}D^{μ} for some α. If D^{λ} is minimal, we also say λ is minimal.

Notice that if D^{λ} and D^{μ} are completely splittable and D^{λ} is not minimal then there is some removable node A of λ such that D^{λ_{A}} is completely splittable and D^{λ} ≅ Ind^{resA}D^{λ_{A}}. In this case:

Ext^{1}(D^{λ}, D^{μ}) = Ext^{1}(Ind^{resA}D^{λ_{A}}, D^{μ}) ≅ Ext^{1}(D^{λ_{A}}, Res_{resA}D^{μ}) (1)

Res_{resA}D^{μ} is either 0 or completely splittable, so Eq. 1 implies that unless both partitions are minimal, we can determine Ext^{1}(D^{λ}, D^{μ}) from knowledge of the Ext^{1}-quiver for completely splittable modules of Σ_{d−1}. Recall here that the D^{λ} are self-dual so

Ext^{1}(D^{λ}, D^{μ}) ≅ Ext^{1}(D^{μ}, D^{λ}).

Thus our next goal will be to classify the minimal modules and study their extensions.

We begin classifying the minimal modules with the following lemma:

Lemma 2.1. If D^{λ} is minimal then λ is either of the form (λ_{1}^{a_{1}}) or (λ_{1}^{a_{1}}, λ_{2}^{a_{2}}).

Proof. Suppose λ = (λ_{1}^{a_{1}}, λ_{2}^{a_{2}}, …, λ_{t}^{a_{t}}) is minimal and t > 2. Then the diagram of λ looks like:

λ =  a A b B c d

where A and B are the top two of the at least three removable nodes and a, b and c are the top three of the at least four addable nodes. Note that the residues of a, b, c, A, and B are all distinct because χ(λ) ≤ p. Thus at least one of A or B, (whichever has residue not equal to resd) will be the unique addable node of its residue when it is removed from λ. Suppose, for example, that resA ≠ resd. Since χ(λ_{A}) ≤ χ(λ_{B}) = χ(λ) ≤ p, we know D^{λ_{A}} is completely splittable. Then Lemmas 1.2 and 1.5 imply:

D^{λ} ≅ Ind^{resA}D^{λ_{A}}

so λ is not minimal, contradicting our assumption. The same argument works with D^{λ_{B}} if resB ≠ resd. Thus the assumption that t > 2 contradicts the assumption that D^{λ} is minimal.

We can now completely classify the minimal modules:

Theorem 2.1. A completely splittable KΣ_{d}-module D^{λ} is minimal if and only if p | d and D^{λ} is in the principal block B_{0}(KΣ_{p}).

Proof. Let D^{λ} be completely splittable and minimal. By Lemma 2.1, we can assume λ has at most two distinct parts. Suppose first that λ = (λ_{1}^{a_{1}}, λ_{2}^{a_{2}}) for λ_{1} ≠ λ_{2}. We label the addable and removable nodes of λ as follows:

λ = ^{c}  ^{B} ^{b}  ^{A}^{a}

We calculate (congruences are mod-p):

resA ≡ λ_{1} − a_{1}
resB ≡ λ_{2} − a_{1} − a_{2}
resa ≡ λ_{1}
resb ≡ λ_{2} − a_{1}
resc ≡ −a_{1} − a_{2}
χ(λ) = λ_{1} − λ_{2} + a_{1} + a_{2}.

(2)

Notice that Lemmas 1.2 and 1.5 imply D^{λ} ≅ Ind^{resA}D^{λ_{A}} unless resA = resc. But λ is minimal and λ_{A} is completely splittable so we can assume resA = resc. Because χ(λ) ≤ p we know resA ≠ resB, and hence resB ≠ resc. Using Lemmas 1.2 and 1.5 again, we conclude that resB = resa (else D^{λ} ≅ Ind^{resB}D^{λ_{B}}).

Setting resB = resa and resA = resc, a simple calculation using Eq. 2 shows that χ(λ) = p and resb = 0.

But resb = 0 if and only if λ_{2} ≡ a_{1}. And resA = resc if and only if λ_{1} ≡ −a_{2}. So D^{λ} being minimal implies λ_{2} ≡ a_{1} and λ_{1} ≡ −a_{1} which yields:

d = a_{1}λ_{1} + a_{2}λ_{2} ≡ −a_{1}a_{2} + a_{1}a_{2} = 0

so p | d.

Before proceeding further, we verify that p | d, χ(λ) = p and resb = 0 are sufficient conditions for the partition (λ_{1}^{a_{1}}, λ_{2}^{a_{2}}) to be minimal:

Lemma 2.2. Let (λ_{1}^{a_{1}}, λ_{2}^{a_{2}}) ⊢ mp satisfy χ(λ) = p and resb = 0 (i.e. λ_{2} ≡ a_{1}). Then λ is minimal.

Proof. First suppose m = 1 so λ must be a hook. Then D^{λ_{A}} ≅ S^{λ_{A}} and D^{λ_{B}} ≅ S^{λ_{B}} (because λ_{A} and λ_{B} partition p−1, and KΣ_{p−1} is semisimple). Then λ_{A} has two addable nodes of residue resA so Ind^{resA}D^{λ_{A}} is not simple. Similarly Ind^{resB}D^{λ_{B}} is not simple, so λ must be minimal.

Next suppose m > 1. Then λ_{B} is not completely splittable, so to prove that λ is minimal we must verify that D^{λ} ≅ I̸nd^{resA}D^{λ_{A}}.

If λ_{1} − λ_{2} = 1 then λ_{A} has one normal node, the top removable, and one good addable node, A. Thus Ind^{resA}D^{λ_{A}} ≅ IndD^{λ_{A}} is not simple by Lemmas 1.3 and 1.4.

If λ_{1} − λ_{2} > 1 then λ_{A} has two normal nodes, the top two removables, and two good addable nodes, A and b. The node b has residue 0 ≠ resA and Ind^{0}D^{λ_{A}} is simple by Lemmas 1.2 and 1.5. But IndD^{λ_{A}} is not semisimple by Lemma 1.4, so Ind^{resA}D^{λ_{A}} is not simple. This completes the proof that λ is minimal.

The rim hook with foot at B has p nodes and removing it preserves χ = p and preserves resb = 0, so preserves minimality by Lemma 2.2. Of course, it also preserves membership in B_{0}. Thus λ ∈ B_{0} by induction.

Now suppose λ is rectangular, i.e. λ = (λ_{1}^{a_{1}}) with a_{1} < p, and let A be the one removable node of λ. Unless p | λ_{1}, the partition λ_{A} will have a unique addable node of residue resA and, as in the previous case, D^{λ} ≅ Ind^{resA}D^{λ_{A}}. So the minimality of λ implies p | λ_{1} so p | d and λ ∈ B_{0}. An argument as in Lemma 2.2 confirms that λ = (cp^{a_{1}}) is indeed minimal.

To complete the proof of Theorem 2.1 we must show that a completely splittable module D^{λ} ∈ B_{0}(KΣ_{mp}) must be minimal. This too follows by induction. The base case m = 1 is just λ = (p − r, 1^{r}) for 0 ≤ r ≤ p − 2. In this case D^{λ} is minimal by Lemma 2.2. Suppose then that m > 1 and D^{λ} is completely splittable and in B_{0}(KΣ_{mp}). Since χ(λ) ≤ p and the rim of λ must have at least p nodes (else λ is a p-core, so not in the principal block), there must be a rim p-hook with head at (1, λ_{1}). Strip off this rim hook to get \overline{λ}. Clearly χ(\overline{λ}) ≤ p so D\overline{λ} is completely splittable and in B_{0}(KΣ_{(m−1)p}), so is minimal by induction. We leave it to the reader to verify that there is only one way to add a hook to a minimal partition such that the result is completely splittable and such that the head of the hook ends up in the first row. (See for example the diagram following Lemma 2.3). Doing so preserves χ(λ) = p and resb = 0, and hence produces another minimal partition. So we conclude that λ must be minimal, completing the proof of Theorem 2.1.

We now determine how many minimal modules there are, and some information we will need to determine the extensions between them.

Lemma 2.3. Let d = pm. There are p−1 minimal partitions of d. They have 1, 2, … p − 1 parts. The top removable node of the one with i parts has residue p − i.

Proof. The result is clear for m = 1, the partitions are just the pregular hooks. Now observe that removing the rim p-hook with head (1, λ_{1}) or adding a rim p-hook with foot at (k, λ_{k} + 1) gives a bijection between minimal partitions of mp and (m + 1)p that preserves the number of parts and the residue of the top removable node. The bijection for p = 5 and n = 5, 10, 15 is illustrated below, with the p-residues labelled.

[figure: Minimal partitions for p = 5 and n = 5, 10, 15; 2; 3; 4; 0; 1; 3; 4; 0; 1; 2; 4; 0; 1; 2; 3; 0; 1; 2; 3; 4; 2; 3; 3; 4; 4; 0; 1; 0; 1; 2; 3; 4; 4; 0; 1; 2; 0; 1; 2; 3; 4; 0; 1; 2; 3; 0; 1; 2; 3; 4; 0; 1; 2; 3; 4; 0; 1; 2; 3; 4; 0; 1; 2; 3; 4; 0; 1; 2; 3; 4; 0; 1; 2; 3; 4; 2; 3; 4; 3; 4; 0; 1; 4; 0; 1; 2; 0; 1; 2; 3; 3; 4; 0; 1; 2; 4; 0; 1; 2; 3; 0; 1; 2; 3; 4; 4; 0; 1; 2; 3; 4; 0; 1; 2; 3; 4; 0; 1; 2; 3]

We will need a few more technical results about minimal partitions before we can begin determining the Ext^{1}-quiver.

Lemma 2.4. Let λ ▷ μ be minimal partitions of d = mp with m > 1. Then μ ⋭λ_{A}^{B} for any removable node A and addable node B of λ.

Proof. First we recall the well known fact [4, 1.4.10] that partitions ρ▷τ are neighbors in the dominance order if and only if τ is obtained from ρ by moving single node down to the next available location. There are few cases to check, since minimal partitions have at most two removable nodes and at most three addable nodes. First note that m > 1 implies λ_{A}^{B} is never

minimal. Let ⊵^{a} denote partitions which are neighbors in the dominance order. The first case if for A the top removable node of λ. There are two possibilities for B, and we have illustrated them below.

[figure: a; a; λ; λ_{A}^{B}; λ_{A}^{B′}]

Since the three partitions drawn are all adjacent in the dominance order, and only λ is minimal, there can not be a minimal partition μ with λ ▷ μ ⊵ λ_{A}^{B}. The other case, when A is the lower removable node of λ is similar.

The final observation about minimal partitions we need is clear:

Lemma 2.5. The minimal partitions are totally ordered by ⊵ and λ ▷ μ if and only if λ has fewer parts.

3. Ext^{1} FOR MINIMAL MODULES

Recall from Eq. 1 that if we can determine Ext^{1}(D^{λ}, D^{μ}) for minimal modules D^{λ} and D^{μ}, we should be able to determine it for all completely splittables. Thus we prove:

Theorem 3.1. Let λ ≠ μ be minimal partitions of mp and m > 1. Then Ext^{1}(D^{λ}, D^{μ}) = 0.

Proof. By Lemma 2.5 we can assume λ▷μ. Let A be the top removable node of λ. By Lemma 2.3, the top removable node of μ has residue different from resA. But, since m > 1, removing the bottom removable node of μ (if it has two), leaves a partition with χ = p + 1. Thus by Lemma 1.2,

Res_{resA}D^{μ} = 0 (3)

We assumed λ is minimal so D^{λ} ≅ I̸nd^{resA}D^{λ_{A}}. However, A is good addable for λ_{A}, so by Lemma 1.3:

Ind^{resA}D^{λ_{A}} = \begin{matrix}D^{λ} \\ H \\ D^{λ} \end{matrix} . (4)

Equation (3) plus the Eckmann-Shapiro lemma proves:

Ext^{1}(Ind^{resA}D^{λ_{A}}, D^{μ}) = 0 (5)

Consider the exact sequence:

0 ⟶ D^{λ} ⟶ Ind^{resA}D^{λ_{A}} ⟶ \begin{matrix}D^{λ} \\ H \end{matrix} ⟶ 0.

Apply Hom(−, D^{μ}) to this to obtain a long exact sequence:

0 ⟶ Ext^{1}( \begin{matrix}D^{λ} \\ H \end{matrix} , D^{μ}) ⟶ Ext^{1}(Ind^{resA}D^{λ_{A}}, D^{μ}) ⟶

then use (5) to conclude that:

Ext^{1}( \begin{matrix}D^{λ} \\ H \end{matrix} , D^{μ}) = 0 (6)

Similarly we apply Hom(−, D^{μ}) to

0 ⟶ H ⟶ \begin{matrix}D^{λ} \\ H \end{matrix} ⟶ D^{λ} ⟶ 0

to obtain a long exact sequence and conclude, by (6), that:

Ext^{1}(D^{λ}, D^{μ}) ≅ Hom(H, D^{μ}) (7)

But H is a subquotient of Ind^{resA}D^{λ_{A}} so it is a subquotient of Ind^{resA}S^{λ_{A}}. However Ind^{resA}S^{λ_{A}} has a Specht series with factors S^{λ} and perhaps S^{λ_{A}^{B}}if resA = resB, by Lemma 1.1. (Where B is the lowest addable node of λ.) Since a Specht module S^{τ} has a composition factor D^{ρ} only if ρ⊵τ, Lemma 2.4 implies there are no composition factors D^{μ} in H. Thus Hom(H, D^{μ}) = 0, and the proof is complete by Eq. 7.

Often a completely splittable module D^{λ} can be obtained by α-induction from more than one completely splittable KΣ_{d−1} module. The next lemma determines when this happens:

Lemma 3.1. Let D^{λ} be completely splittable, and A a removable node of λ. Then D^{λ} ≅ Ind^{resA}D^{λ_{A}} unless:

(i) χ(λ) = p and A is the lowest removable node of λ or:

(ii) resA is equal to the residue of the lowest addable node of λ.

Proof. Suppose we are not in case (i) or (ii). Since we are not in case (i), A is not the lowest removable node of λ, so χ(λ_{A}) ≤ p, and D^{λ_{A}} is completely splittable. Since we are not in case (ii), A is the unique addable node for λ_{A} with residue resA. So Lemmas 1.2 and 1.5 prove D^{λ} ≅ Ind^{resA}D^{λ_{A}}.

We remark that if A satisfies (i) or (ii) from Lemma 3.1, then a calculation similar to the proof of Lemma 2.2 shows that D^{λ} ≅ I̸nd^{resA}D^{λ_{A}}.

Since we plan to reduce the calculation of Ext^{1}(D^{λ}, D^{μ}) to the case where λ and μ are minimal, the next definiton is natural:

Definition 3.1. Given D^{λ} completely splittable, the minimal core of λ, denoted λ̂, is obtained by successively removing nodes from λ which do not satisfy Lemma 3.1 (i) or (ii) until no such nodes can be removed. The remark following Lemma 3.1 implies that λ̂ is indeed minimal.

Let λ = (λ_{1}, λ_{2}, …, λ_{k}) be completely splittable. It is easy to see from a few examples that λ̂ is well defined and easy to determine. Starting with λ, simply remove nodes, never removing any of residue equal to the residue of the bottom addable node, and never allowing χ to be > p.

More explicitly, suppose the bottom addable node of λ has residue α. Let (u, v) be the node of residue α farthest to the right. This node is uniquely determined because h(λ) ≤ p − 1. If there is no such node, λ̂ is empty. If u = k then λ̂ = (v^{k}). If u < k then λ̂ is the partition with rim consisting of the nodes: (1, v), (2, v), ⋯, (u, v), (u, v−1), (u, v−2), ⋯, (u, v−p+k), (u+ 1, v − p + k), ⋯ (k, v − p + k), (k, v − p + k − 1), ⋯ (k, 1). In particular, note that h(λ) = h(λ̂) when λ̂ ≠ ∅.

Two examples are given below, with the node (u, v) in bold, and the minimal core drawn inside the partition. The residues of all nodes in λ and the bottom addable node of λ are labeled.

Two examples of minimal 7-cores^{λ = (9, 8)}_{λ̂ = (7, 7)}  ^{λ = (6, 6, 5, 4)}_{λ̂ = (5, 5, 2, 2)}^{5}^{6} ^{0} ^{1} ^{2} ^{3} ^{4} ^{5} ^{6}^{0} ^{1} ^{2} ^{3} ^{4} ^{5} ^{6} ^{0} ^{1}  ^{3}^{4} ^{5} ^{6} ^{0}^{5} ^{6} ^{0} ^{1} ^{2}^{6} ^{0} ^{1} ^{2} ^{3} ^{4}^{0} ^{1} ^{2} ^{3} ^{4} ^{5}

We now use the minimal cores and Eq. 1 to formalize the reduction of determining Ext^{1} between two minimal modules:

Theorem 3.2. Let D^{λ} and D^{μ} be completely splittable and in the same block. Then λ̂ and μ̂ partition the same number and:

Ext^{1}(Dλ̂, Dμ̂) ≅ Ext^{1}(D^{λ}, D^{μ}).

Proof. We first remark that since λ ∼ μ, by Theorem 2.1 either λ and μ are both minimal or neither is. So we will assume λ and μ are not minimal, since otherwise λ̂ = λ and μ̂ = μ.

To prove this claim we will use the abacus notation for partitions and the characterization of blocks by p-cores. See [4] for details. We make a series of observations about the abacus representations of λ and μ.

(i). Since λ has less than p parts, we will represent it on an abacus with p beads. We label the runners 0, 1, …, p − 1 from left to right.

(ii). Notice that χ(λ) can be read off the abacus display. It is equal to s − t + 1 where t is the position of the first bead after the first gap and s is the position of the last bead.

(iii). Since χ(λ) ≤ p, by (ii), the core of λ (also represented with p beads) has at most two beads on any runner and must have a bead on runner 0.

(iv). λ is not minimal, so not in the principal block of Σ_{mp}. Thus λ does not have an empty p-core, so some runner has two beads. Since χ(λ) ≤ p, if runner i has no beads, then runners to the right of i have at most one bead by (ii).

Of course, (i)-(iv) hold for μ as well. Also λ ∼ μ, so the abacus displays for λ and μ have the same number of beads on each runner.

Now suppose the lowest addable node for λ has residue i. Then in the abacus display for λ, the first gap occurs on runner i so (ii) implies that runner i has at most one bead. Similarly, if the lowest addable node of μ has residue j then runner j has at most one bead.

Consider the abacus diagram for the p-core λ̃ = μ̃. Note that (i) and (iv) imply runner p − 1 can not have two beads. We consider separately the case where it has no beads and where it has one bead.

Case 1 is if runner p − 1 has no beads. Let runner k < p − 1 be the leftmost runner with two beads. Then (even if k=0), both λ and μ must have a removable node of residue k. Furthermore, k is not equal to i or j since runners i and j have at most one bead.

Case 2 is when runner p − 1 has one bead. There must be some runner with no beads, and by (iv) all runners to the right of it have at most one bead. Thus we can choose a runner k such that runner k has one bead and runner k − 1 has no beads. Again both λ and μ must have a removable node of residue k. Since runner k − 1 has no beads, the partitions have no addable nodes of residue k so k is not equal to i or j. The two cases are illustrated below:

[figure: k; p-1; Case 1; Case 2; ?; ?; k]

In both cases λ and μ must have a removable node of residue k. Suppose that χ(λ) = p and the node of residue k is the bottom removable node for λ. Then removing this node will increase χ(λ) from p to p + 1. Then λ will have an abacus configuration:

[figure: ?; k]

which is impossible. In Case 1 above, runner k has another bead so χ(λ) > p, contradicting λ being completely splittable. In Case 2 above, runner k − 1 has no beads so this configuration cannot occur. The same applies for μ.

We have found removable nodes A_{λ} and A_{μ}, both with residue k not equal to the residue of the bottom addable node of λ or μ respectively. Also, χ(λ_{A_{λ}}) ≤ p and χ(μ_{A_{μ}}) ≤ p so these nodes are not the lowest removable nodes for λ and μ. Thus A_{λ} is the unique addable node of λ_{A_{λ}} of residue resA_{λ}, and similarly for A_{μ}.

Lemma 3.1 implies:

D^{λ} ≅ Ind^{k}D^{λ_{A_{λ}}} and D^{μ} ≅ Ind^{k}D^{μ_{Aμ}} . (8)

Also,

Res_{k}D^{λ} ≅ D^{λ_{A_{λ}}} and Res_{k}D^{μ} ≅ D^{μ_{Aμ}} . (9)

Equations (8) and (9) give us:

Ext^{1}(D^{λ}, D^{μ}) = Ext^{1}(D^{λ_{A_{λ}}} , D^{μ_{Aμ}}). (10)

But clearly λ̂ = λ̂_{A_{λ}} and μ̂ = μ̂_{A_{μ}} so the theorem follows from (10) by induction.

Before we state our main theorem, we need to deal with self extensions and with the base case of the induction. Self extensions have been handled previously:

Lemma 3.2. Ext^{1}(D^{λ}, D^{λ}) = 0 for D^{λ} completely splittable.

Proof. This was proved in [9] for partitions with < p parts, which clearly holds for λ since χ(λ) ≤ p and λ is p-regular.

We have determined Ext^{1}(D^{λ}, D^{μ}) = 0 for λ and μ minimal partitions of mp with m > 1. The minimal partitions for d = p are exactly the p-regular hooks. So we need to handle Ext^{1}(D^{λ}, D^{μ}) for λ and μ hook partitions of p. But this is just the principal block of Σ_{p}, and it is well know (see e.g. [10]) that:

Lemma 3.3. Let λ = (p − r, 1^{r}) and μ = (p − s, 1^{s}) for 0 ≤ r ≤ p − 2. Then:

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

We call the two hooks adjacent when |r − s| = 1. Thus for completely splittable modules, Ext^{1}(D^{λ}, D^{μ}) will be zero unless λ̂ and μ̂ are adjacent hook partitions of p as described in Lemma 3.3 above.

Everything is now in place to prove Theorem 0.1. We know Ext^{1}(D^{λ}, D^{μ}) = 0 unless λ̂ and μ̂ are adjacent hook partitions of p. Let the bottom addable node of λ have residue α. It is not difficult to see from our description of the minimal core that λ̂ = ∅ exactly when λ has no nodes of residue α. This is equivalent to h_{11}(λ) < p. For λ̂ to be a hook partition of p we need λ to have exactly one node of residue α. This is equivalent to the condition h_{21}(λ) < p. We know h(λ̂)=h(λ) if it is nonempty, so the criterion for λ̂ and μ̂ to be adjacent is | h(λ) − h(μ) |= 1. Thus we have proven Theorem 0.1.

4. APPLICATION TO GL_{N}(K)

We describe here how to prove Corollary 0.1. For a comprehensive discussion of background for this section see [2].

Let M(n, d) denote the polynomial representations of GL_{n}(K) of homogeneous degree d, for d ≤ n. The simple modules in M(n, d) are indexed by partitions λ of d and denoted L(λ). The Schur functor F is a covariant, exact functor from M(n, d) to KΣ_{d}-mod such that:

F(L(λ)) = \{ \begin{matrix}D_{λ} & if λ is column p-regular \\ 0 & else \end{matrix} (11)

Because Eq. 11 is in terms of D_{λ} and Theorem 0.1 is in terms of D^{λ}, we recall here how the two parametrizations of simple KΣ_{d} modules are related:

\begin{aligned}D_{λ} & ≅ & D^{λ′} ⊗ sgn & ≅ & D^{m(λ′)} \\ D^{λ} & ≅ & D_{λ′} ⊗ sgn & ≅ & D_{(m(λ))′} \end{aligned}  (12)

F has a right adjoint functor G, and this adjointness sometimes extends to Ext^{1}, as in the following:

Theorem 4.1. [1]

Ext_{Σ_{d}}^{1}(D_{λ}, D_{μ}) ≅ Ext_{M(n,d)}^{1}(L(λ), G(D_{μ}))

This theorem is especially useful for completely splittable modules, as the functor G behaves well on them:

Theorem 4.2. [8] Let D_{λ} be nontrivial and completely splittable. Then

G(D_{λ}) = L(λ).

Theorems 0.1, 4.1, and 4.2 and Eq. 12 immediately imply Corollary 0.1. Although Theorem 4.2 applies only to nontrivial modules, Corollary 0.1 still holds when D^{λ} = D^{μ} ≅ Kif D^{λ} = D^{μ} = K, by the well known fact that Ext_{M(n,d)}^{1}(L(λ), L(λ)) = 0.

REFERENCES

1. S.R. Doty, K. Erdmann, D.K. Nakano, “Extensions of modules over Schur algebras, symmetric groups and Hecke algebra,” preprint, 2000.

2. J.A. Green, “Polynomial Representations of GL_{n},” Lectures Notes in Mathematics 830, Springer-Verlag, New York, 1980.

3. G.D. James, “The representation theory of the symmetric groups,” Springer Lecture Notes 682, Berlin, Heidelberg, New York, 1978.

4. G.D. James and A. Kerber, “The representation theory of the symmetric group,” Addison-Wesley, London, 1981.

5. A.S. Kleshchev, “Branching rules for modular representations of symmetric groups III: some corollaries and a problem of Mullineaux,” J. London Math. Soc. 54 (1996), 25-38.

6. A.S. Kleshchev, Branching rules for symmetric groups and applications, in “Algebraic groups and their representations,” NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., 517, Cambridge 1997, 103-130.

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8. A.S. Kleshchev and D.K. Nakano, On Comparing the Cohomology of General Linear and Symmetric Groups, accepted, Pacific Journal of Mathematics, 2000.

9. A.S. Kleshchev and J. Sheth, On extensions of simple modules over symmetric and algebraic groups, J. Algebra 222 (1999), 705-722.

10. S. Martin, “Ext^{1} spaces for general linear and symmetricg roups,” Proc. Roy. Soc. Edinburgh Sect. A 119, (1991), 301-310.