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ISRN Algebra
Volume 2012 (2012), Article ID 658201, 16 pages
http://dx.doi.org/10.5402/2012/658201
Research Article

A Gelfand Model for Weyl Groups of Type 𝐷 2 𝑛

Departmento de Matemática, Facultad de Ciencias Exactas, UNICEN, B7000 GHG, Tandil, Argentina

Received 27 March 2012; Accepted 17 April 2012

Academic Editors: H. Airault, D. Sage, A. Vourdas, and H. You

Copyright © 2012 José O. Araujo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝐺 𝐿 𝑛 ( β„‚ ) , where β„‚ is the field of complex numbers, it has been proved that each 𝐺 -module over β„‚ is isomorphic to a 𝐺 -submodule in the polynomial ring β„‚ [ π‘₯ 1 , … , π‘₯ 𝑛 ] , and taking the space of zeros of certain 𝐺 -invariant operators in the Weyl algebra, a finite-dimensional 𝐺 -space 𝒩 𝐺 in β„‚ [ π‘₯ 1 , … , π‘₯ 𝑛 ] can be obtained, which contains all the simple 𝐺 -modules over β„‚ . 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 2 𝑛 , 𝐸 7 , or 𝐸 8 . This paper presents a model of Gelfand for a Weyl group of type 𝐷 2 𝑛 whose construction is based on the same principles as the polynomial model.