A Gelfand Model for Weyl Groups of Type D2n

Abstract EN

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 GLn(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[x1,…,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space ?G in ℂ[x1,…,xn] 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 D2n, E7, or E8.

This paper presents a model of Gelfand for a Weyl group of type D2n whose construction is based on the same principles as the polynomial model.