The Ext-quiver for completely splittable representations of the symmetric group

Kleshchev has recently [7] classi®ed those modules for the symmetric group which have semisimple restriction to any Young subgroup. We determine Ext K Sd …D ;D m† where D l and D m are K Sd -modules of this type, called completely splittable. As a corollary of this and recent work of Kleshchev and Nakano, we can determine ExtGL n…K†……L…l†;L…m†† for certain simple GLn…K†-modules L…l† and L…m†. 1 Introduction Let Sd denote the symmetric group on d letters. The complex irreducible Sd -modules correspond bijectively with partitions l of d, and we denote by S l the irreducible module corresponding to l. We work over an algebraically closed ®eld K of positive characteristic p > 2. The simple K Sd -modules are indexed by p-regular partitions, and we denote the corresponding simple module by D. These modules can also be indexed by column p-regular partitions, and for l column p-regular we denote the corresponding simple module by Dl. For a comprehensive treatment of the theory, see [4]. For any composition m ˆ …m1; m2; . . . mk† of d there is a standard Young subgroup de®ned by Sm ˆ Sm1 Sm2 Smk < Sd : We will consider the following class of modules: De®nition 1.1. An irreducible K Sd -module D l is called completely splittable if and only if the restriction D#Sm to any Young subgroup Sm < Sd is semisimple. We will also say that l is completely splittable. In [8], Kleshchev and Nakano obtained several results about the cohomology of completely splittable modules, suggesting that obtaining the Ext-quiver for these * This work was supported by an NSF graduate fellowship modules should be possible. In this paper we apply Kleshchev's branching theorems to prove the following result: Theorem 1.2. Let D and D be completely splittable and in the same p-block of K Sd . Then Ext K Sd …D l;Dm† ˆ K if h11…l†d p; h11…m†d p h21…l† < p; h21…m† < p and jh…l† ÿ h…m†j ˆ 1 0 otherwise, 8<>: where h…l† is the height of l and hij…l1; l2; . . . ; lk† ˆ li ‡ l 0 j ‡ 1ÿ i ÿ j is the …i; j† hook length. This theorem together with the work of [8] gives a corresponding result for GLn…K† when nd d. Let L…l† denote the simple, polynomial GLn…K†-module with highest weight l. Let m denote the Mullineaux map on p-regular partitions de®ned by