Gelfand $W$-graphs for classical Weyl groups

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.

arXiv comment35 pages, 5 figures; v2: minor corrections, a few additional references, several new figures

References in corpus (8)

• Canonical bases and higher representation theory
• Hecke algebras and involutions in Weyl groups
• A bar operator for involutions in a Coxeter group
• A natural representation model for symmetric groups
• Automorphisms and Generalized Involution Models of Finite Complex Reflection Groups
• Bar operators for quasiparabolic conjugacy classes in a Coxeter group
• Generalized Involution Models for Wreath Products
• Isomorphisms, automorphisms, and generalized involution models

Cited by in corpus (2)

• Perfect models for finite Coxeter groups
• Insertion algorithms for Gelfand $S_n$-graphs