Deformations of permutation representations of Coxeter groups

TL;DR

This paper introduces quasiparabolic subgroups of Coxeter groups, extending known properties of permutation representations, and explores their algebraic and combinatorial structures, with applications to symmetric groups acting on involutions.

Contribution

It defines quasiparabolic subgroups and demonstrates that their permutation representations share properties with classical Coxeter group actions.

Findings

Quasiparabolic subgroups inherit shellable Bruhat order

They admit flat deformations over Z[q]

Application to symmetric group actions on involutions

Abstract

The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over Z[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of ``quasiparabolic" subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.