Gelfand models for classical Weyl groups

Abstract

In a recent preprint Kodiyalam and Verma give a particularly simple Gelfand model for the symmetric group that is built naturally on the space of involutions. In this manuscript we give a natural extension of Kodiyalam and Verma's model to a Gelfand model for Weyl groups of type Bn and D2n+1. Then we define an explicit isomorphism between this Gelfand model and the polynomial model using a technique we call telescopic decomposition.

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.…”