A class of representations of Hecke algebras

TL;DR

This paper introduces W-digraphs, a new graph-theoretic model for certain Hecke algebra representations, providing characterizations, properties, and bounds, especially for finite Coxeter systems.

Contribution

It defines W-digraphs to model Hecke algebra representations and characterizes their structure and properties, extending Lusztig's results.

Findings

Complete characterization of W-digraphs via dihedral subgroups

Establishment of acyclicity under certain conditions

Bound on vertices of connected W-digraphs in finite Coxeter systems

Abstract

A type of directed multigraph called a W-digraph is introduced to model the structure of certain representations of Hecke algebras, including those constructed by Lusztig and Vogan from involutions in a Weyl group. Building on results of Lusztig, a complete characterization of W-digraphs is given in terms of subdigraphs for dihedral parabolic subgroups. Graph-theoretic properties of W-digraphs are established including, under certain assumptions, acyclicity. In case the Coxeter system is finite, a bound on the number of vertices of a connected W-digraph is obtained, and a graph-theoretic version of the usual duality operation is obtained.