A Gelfand Model for Weyl Groups of Type D2n. (7th August 2012)

• Record Type: Journal Article
• Authors: Araujo, José O.; Maiarú, Luis C.; Natale, Mauro
• Other Names: Airault H. Academic Editor.; Sage D. Academic Editor.; Vourdas A. Academic Editor.; You H. Academic Editor.
• Is Part Of: ISRN algebra. Volume 2012(2012)
• Publication Date: 2012-08-07
• Subjects: Algebra -- Periodicals; Algebra; Periodicals; Electronic journals; 512
• DOI: 10.5402/2012/658201
• Languages: English
• ISSNs: 2090-6285

Abstract

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 ( 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 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.