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A Gel’fand model for a Weyl group of type ${D}_{n}$ and the branching rules ${D}_{n}↪{B}_{n}$. (English) Zbl 1081.20052
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 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 [C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras. Wiley, New York (1988; Zbl 0634.20001), J. L. Aguado and J. O. Araujo, Commun. Algebra 29, No. 4, 1841-1851 (2001; Zbl 1015.20009), J. O. Araujo, Beitr. Algebra Geom. 44, No. 2, 359-373 (2003; Zbl 1063.20008), 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 ${𝒩}_{G}$ of the polynomial ring $K\left[{x}_{1},\cdots ,{x}_{n}\right]$. If $G$ is a Weyl group of type ${A}_{n}$ or ${B}_{n}$ [see 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)], ${𝒩}_{G}$ provides a Gel’fand model for these groups as shown in [J. L. Aguado and J. O. Araujo, loc. cit. and J. O. Araujo, loc. cit.]. In this work we show that if $G$ is a Weyl group of type ${D}_{2n+1}$, ${𝒩}_{{D}_{2n+1}}$ provides a Gel’fand model for this group. We also describe completely ${𝒩}_{{D}_{2n}}$ but this is not a Gel’fand model for a Weyl group of type ${D}_{2n}$, instead a subspace of ${𝒩}_{{D}_{2n}}$, ${\stackrel{˜}{𝒩}}_{{D}_{2n}}$ is a Gel’fand model. We also give simple proofs of the branching rules ${D}_{n}↪{B}_{n}$, 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:
 20G05 Representation theory of linear algebraic groups 20F55 Reflection groups; Coxeter groups