Séminaire Lotharingien de Combinatoire, 85B.50 (2021), 12 pp.
Eric Marberg and Yifeng Zhang
Perfect Models and Gelfand W-Graphs
Abstract.
A Gelfand model for an algebra is a module isomorphic to a direct sum of irreducible modules,
with every isomorphism class of irreducible modules represented exactly once.
We introduce and study the notion of a perfect model for a finite Coxeter group;
such a model is a certain set of discrete data parametrizing a Gelfand model for the associated Iwahori-Hecke algebra. We classify which Coxeter groups have perfect models,
and then describe explicit Gelfand models for the classical finite Coxeter groups. This generalizes separate constructions of Adin, Postnikov, and Roichman
and of Araujo and Bratten.
Our Gelfand models have interesting "canonical bases" that give rise to associated W-graphs. We classify the molecules in these W-graphs when W is a symmetric group,
and conjecture that in type A every molecule is a cell.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
The following versions are available: