Geometric interpretation of Murphy bases and an application

TL;DR

This paper explores the geometric structure of Murphy bases in Hecke algebras related to general linear group representations on flag spaces, providing new insights into their cellular decomposition and applications to module spaces over local rings.

Contribution

It offers a geometric interpretation of Murphy's cellular basis for Hecke algebras and applies this to decompose certain group representations over local rings.

Findings

Hecke algebra associated with general linear groups is cellular.

Murphy's basis has a geometric interpretation in this context.

Representation decomposition over principal ideal local rings is achieved.

Abstract

In this article we study the representations of general linear groups which arise from their action on flag spaces. These representations can be decomposed into irreducibles by proving that the associated Hecke algebra is cellular. We give a geometric interpretation of a cellular basis of such Hecke algebras which was introduced by Murphy in the case of finite fields. We apply these results to decompose representations which arise from the space of modules over principal ideal local rings of length two with a finite residue field.