The Ext1 -quiver for completely splittable representations of the symmetric group

Abstract

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 : where h l† is the height of l and hij l1; l2; . . . ; lk† ˆ li ‡ l 0 j ‡ 1y i y 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