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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    Gin   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 typeD    2  n,E    7, orE    8. This paper presents a model of Gelfand for a Weyl group of typeD    2  nwhose construction is based on the same principles as the polynomial model.

isrn algebra

A Gelfand Model for Weyl Groups of Type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>

A Gelfand Model for Weyl Groups of Type D2n