Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 44, No. 2, pp. 359373 (2003) 

A Gel'fand model for a Weyl group of type $B_{n}$J. O. AraujoFacultad de Ciencias Exactas  UNICEN, Paraje Arroyo Seco, 7000  Tandil, ArgentinaAbstract: A Gel'fand model for a finite group $G$ is a complex representation of $G$ which is isomorphic to the direct sum of all the irreducible representation of $G$ (see [S]). Gel'fand models for the symmetric group and the linear group over a finite field can be found in [A] and [K]. Using the same ideas as in [A], in this work we describe a Gel'fand model for a Weyl group of type $B_n$. When $K$ is a field of characteristic zero and $\mathfrak G$ is a Weyl group of type $B_n$, we give a finite dimensional $K$subspace $\mathcal N$ of the polynomial ring $K[x_1,\ldots,x_n]$. If $K$ is the field of complex numbers, then $\mathcal N$ provides a Gel'fand model for $\mathfrak G$. \noindent The space $\mathcal N$ can be defined in a more general way (see [AA]), obtained as the zeros of certain differential operators (symmetrical operators) in the Weyl algebra. However, in the case of a group $G$ of type $D_n$ ($n$ even), $\mathcal N$ is not a Gel'fand model for $G$. [A] Aguado, J. L.; Araujo, J. O.: A Gel'fand Model for the Symmetric Group. Communications in Algebra, {\bf 29}(4) (2001), 18411851. [AA] Araujo, J. O.; Aguado, J. L.: Representations of Finite Groups on Polynomial Rings. Actas, V Congreso de Matemática Dr. Antonio A. R. Monteiro, 3540, Bahía Blanca 1999. [K] Klyachko, A. A.: Models for the complex representations of the groups $G(n,q)$. Math. of the USSR  Sbornik {\bf 48} (1984), 365380. [S] SotoAndrade, J.: Geometrical Gel'fand Models, Tensor Quotients and Weil Representations. Proceedings of Symposia in Pure Mathematics {\bf 47}, part. 2 (1987), 306316. Full text of the article:
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