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Zbl 1081.20052
Araujo, J.O.; Bigeón, J.J.
A Gel'fand model for a Weyl group of type $D_n$ and the branching rules $D_n\hookrightarrow B_n$. (English)
[J] J. Algebra 294, No. 1, 97-116 (2005). ISSN 0021-8693

Summary: 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 representations of $G$ [see {\it J. Soto-Andrade}, Proc. Symp. Pure Math. 47, 305-316 (1987; Zbl 0652.20047)]. Gel'fand models for the symmetric group, Weyl groups of type $B_n$ and the linear group over a finite field can be found in [{\it C. W. Curtis} and {\it I. Reiner}, Representation theory of finite groups and associative algebras. Wiley, New York (1988; Zbl 0634.20001), {\it J. L. Aguado} and {\it J. O. Araujo}, Commun. Algebra 29, No. 4, 1841-1851 (2001; Zbl 1015.20009), {\it J. O. Araujo}, Beitr. Algebra Geom. 44, No. 2, 359-373 (2003; Zbl 1063.20008), {\it A. A. Klyachko}, Mat. Sb., N. Ser. 120(162), No. 3, 371-376 (1983; Zbl 0526.20033)]. When $K$ is a field of characteristic zero and $G$ is a finite subgroup of the linear group, we give a finite-dimensional $K$-subspace ${\cal N}_G$ of the polynomial ring $K[x_1,\dots,x_n]$. If $G$ is a Weyl group of type $A_n$ or $B_n$ [see {\it N. Bourbaki}, Éléments de mathématique. Groupes et algèbres de Lie. Chapitre IV, V et VI: Groupes de Coxeter et systèmes de Tits. Groupes engendrés par des réflexions. Systèmes de racines. Paris: Hermann (1968; Zbl 0186.33001)], ${\cal N}_G$ provides a Gel'fand model for these groups as shown in [{\it J. L. Aguado} and {\it J. O. Araujo}, loc. cit. and {\it J. O. Araujo}, loc. cit.]. In this work we show that if $G$ is a Weyl group of type $D_{2n+1}$, ${\cal N}_{D_{2n+1}}$ provides a Gel'fand model for this group. We also describe completely ${\cal N}_{D_{2n}}$ but this is not a Gel'fand model for a Weyl group of type $D_{2n}$, instead a subspace of ${\cal N}_{D_{2n}}$, $\widetilde{\cal N}_{D_{2n}}$ is a Gel'fand model. We also give simple proofs of the branching rules $D_n\hookrightarrow B_n$, a generator for each simple $D_n$-module and a formula for the dimension for all the simple $B_n$-modules and all the simple $D_n$-modules.

MSC 2000:: *20G05 Representation theory of linear algebraic groups
20F55 Coxeter groups

Keywords: Gelfand models; direct sums of irreducible representations; Weyl groups; branching rules; simple modules

Citations: Zbl 0652.20047; Zbl 0634.20001; Zbl 1015.20009; Zbl 1063.20008; Zbl 0526.20033; Zbl 0186.33001

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