On Gelfand models for finite Coxeter groups

Abstract

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 ðD 2n Þ, W ðE 7 Þ or W ðE 8 Þ.

Combinatorial Gelfand Models for Semisimple Diagram Algebras

“…As this Gelfand model is constructed in purely combinatorial terms on the set of involutions, it is sometimes also called an involutary or combinatorial Gelfand model. Similar models can be defined for many other finite groups, in particular, for all classical Weyl groups, see [APR2,Ar,ABi,ABr,Ca,CF1,CF2,GO] and references therein. The paper [KM] makes a step beyond the group theory and constructs Gelfand models for various semigroup algebras, in particular, for semigroup algebras of inverse semigroups in which all maximal subgroups are isomorphic to direct sums of symmetric groups.…”

“…As this Gelfand model is constructed in purely combinatorial terms on the set of involutions, it is sometimes also called an involutary or combinatorial Gelfand model. Similar models can be defined for many other finite groups, in particular, for all classical Weyl groups, see [APR2,Ar,ABi,ABr,Ca,CF1,CF2,GO] and references therein.…”

Groupoids, Geometric Induction and Gelfand Models

“…On the other hand, Garge and Oesterlé constructed in [9] for any finite group G with a faithful representation V , a representation which they called 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 Gel'fand models.…”

“…Araujo and others have proved that the polynomial models for certain irreducible Weyl groups associated to their canonical representations are Gel'fand models. In [9] it is proved that a polynomial model for a finite Coxeter group G is a Gel'fand model if and only if G has no direct factor of the type D 2n , E 7 or E 8 .…”

Gelfand models for classical Weyl groups

“…This second type of model is associated to a finite subgroup of the complex general linear group, and is shown to be a Gelfand model for reflection groups of type B n , D 2n+1 , I 2 (n) and G (m, 1, n) in [5], [6] and [7]. Garge and Oesterlé [11] study the polynomial model in a more general context and give a criteria for when it is a Gelfand model for a finite Coxeter group.…”