Hook modules for general linear groups
Stephen Doty, Stuart Martin

TL;DR
This paper provides an elementary proof describing the structure of blocks in Schur algebras over infinite fields of positive characteristic, leading to a universal character formula for certain simple GL_n(k)-modules.
Contribution
It offers a self-contained proof of known block structures in Schur algebras, enabling a general character formula for simple modules across all n and p.
Findings
Elementary proof of block structure in Schur algebras
Universal character formula for simple GL_n(k)-modules
Connection to existing character formulas by other researchers
Abstract
For an arbitrary infinite field k of characteristic p > 0, we describe the structure of a block of the algebraic monoid M_n(k) (all n x n matrices over k), or, equivalently, a block of the Schur algebra S(n,p), whose simple modules are indexed by p-hook partitions. The result is known; we give an elementary and self-contained proof, based only on a result of Peel and Donkin's description of the blocks of Schur algebras. The result leads to a character formula for certain simple GL_n(k)-modules, valid for all n and all p. This character formula is a special case of one found by Brundan, Kleshchev, and Suprunenko and, independently, by Mathieu and Papadopoulo.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Nonlinear Waves and Solitons
