A Gelfand Model for Weyl Groups of Type D2n

Abstract

A Gelfand model for a finite group G is a complex representation of G, which is isomorphic to the direct sum of all irreducible representations of G. When G is isomorphic to a subgroup of GL n C , where C is the field of complex numbers, it has been proved that each G-module over C is isomorphic to a G-submodule in the polynomial ring C x 1 , . . . , x n , and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space N G in C x 1 , . . . , x n can be obtained, which contains all the simple G-modules over C. This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of type D 2n , E 7 , or E 8 . This paper presents a model of Gelfand for a Weyl group of type D 2n whose construction is based on the same principles as the polynomial model.

Generalized involution models for wreath products

“…Klyachko [12,13] and Inglis, Richardson, and Saxl [10] first constructed involution models for the symmetric group; additional models for S n and related Weyl groups appear in [3,4,5,6,16]. More recently, Adin, Postnikov, and Roichman [1] describe a simple combinatorial action to define a Gelfand model for the symmetric group.…”

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.…”

Gelfand models for classical Weyl groups

“…A second type of model, the polynomial model, was introduced in [3] and used to construct a Gelfand model for the symmetric group. 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.…”