On Gelfand models for finite Coxeter groups

TL;DR

This paper provides a uniform criterion for when polynomial models of finite Coxeter groups serve as Gelfand models, specifically excluding groups with certain direct factors, simplifying previous case-by-case analyses.

Contribution

It offers a simplified, uniform approach to determine when polynomial models are Gelfand models for all finite Coxeter groups, extending prior results.

Findings

Polynomial models are Gelfand models iff the group has no direct factor of W(D_{2n}), W(E_7), or W(E_8).

Provides a unified criterion applicable to all finite Coxeter groups.

Simplifies the understanding of Gelfand models for Coxeter groups.

Abstract

A Gelfand model for a finite group $G$ is a complex linear representation of $G$ that contains each of its irreducible representations with multiplicity one. For a finite group $G$ with a faithful representation $V$, one constructs a representation which we call the polynomial model for $G$ associated to $V$. Araujo and others have proved that the polynomial models for certain irreducible Weyl groups associated to their canonical representations are Gelfand models. In this paper, we give an easier and uniform treatment for the study of the polynomial model for a general finite Coxeter group associated to its canonical representation. Our final result is that such a polynomial model for a finite Coxeter group $G$ is a Gelfand model if and only if $G$ has no direct factor of the type $W(D_{2n}), W(E_7)$ or $W(E_8)$.