Gelfand $W$-graphs for classical Weyl groups

TL;DR

This paper introduces perfect models for classical Weyl groups to construct Gelfand models and associated Gelfand W-graphs, revealing duality phenomena in types BC and D.

Contribution

It defines perfect models for classical Weyl groups, generalizes existing constructions to Iwahori-Hecke algebras, and introduces Gelfand W-graphs with duality properties.

Findings

Constructed Gelfand models for all classical Weyl groups except type D in even rank.

Established duality of Gelfand W-graphs in types BC and D.

Unified and extended previous models from type A to other classical types.

Abstract

A Gelfand model for an algebra is a module given by a direct sum of irreducible submodules, with every isomorphism class of irreducible modules represented exactly once. We introduce the notion of a perfect model for a finite Coxeter group, which is a certain set of discrete data (involving Rains and Vazirani's concept of a perfect involution) that parametrizes a Gelfand model for the associated Iwahori-Hecke algebra. We describe perfect models for all classical Weyl groups, excluding type D in even rank. The representations attached to these models simultaneously generalize constructions of Adin, Postnikov, and Roichman (from type A to other classical types) and of Araujo and Bratten (from group algebras to Iwahori-Hecke algebras). We show that each Gelfand model derived from a perfect model has a canonical basis that gives rise to a pair of related $W$-graphs, which we call Gelfand $W$-graphs. For types BC and D, we prove that these $W$-graphs are dual to each other, a phenomenon which does not occur in type A.