Specht Modules and Simple Modules for Centralizer Algebras of the Symmetric Group

Abstract

Let Σ_n be the symmetric group on n letters. For l ≤ n identify Σ_l with a subgroup of Σ_n in the natural way. Let k be an algebraically closed field of characteristic p. This article begins to develop a theory for modules over the centralizer algebras kΣ_n^Σ_l that is analogous to James's theory of permutation modules, Specht modules, and simple modules over kΣ_n. We make a conjecture about how to construct all simple kΣ_n^Σ_l-modules, we develop tools to test the conjecture, and we prove that it is correct for all n when l < p.

Key Words:

• Centralizer algebra
• Simple modules
• Specht modules
• Symmetric group

2010 Mathematics Subject Classification:

• Primary: 20C05
• 20C20
• Secondary: 20C30
• 16S34