A Gel'fand model for a Weyl group of type Dn and the branching rules Dn↪Bn

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, Geometrical Gel'fand models, tensor quotients and Weyl representations, in: Proc. Sympos. Pure Math., vol. 47 (2), Amer. Math. Soc., Providence, RI, 1987, pp. 306-316. [12]]). Gel'fand models for the symmetric group, Weyl groups of type Bn and the linear group over a finite field can be found in [C. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1988. [6]; J.L. Aguado, J.O. Araujo, A Gel'fand model for the symmetric group, Comm. Algebra 29 (4) (2001) 1841-1851; J.O. Araujo, A Gel'fand model for a Weyl group of type Bn, Beiträge Algebra Geom. 44 (2) (2003) 359-373; A.A. Klyachko, Models for the complex representations of the groups G(n,q), Math. USSR Sb. 48 (1984) 365-380. [10]]. 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 NG of the polynomial ring K[x1,…,xn]. If G is a Weyl group of type An or Bn (see [N. Bourbaki, Éléments de mathématique. Groupes et Algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines, vol. 34, Hermann, 1968. [4]]), NG provides a Gel'fand model for these groups as shown in [J.L. Aguado, J.O. Araujo, A Gel'fand model for the symmetric group, Comm. Algebra 29 (4) (2001) 1841-1851; J.O. Araujo, A Gel'fand model for a Weyl group of type Bn, Beiträge Algebra Geom. 44 (2) (2003) 359-373]. In this work we show that if G is a Weyl group of type D2n+1, ND2n+1 provides a Gel'fand model for this group. We also describe completely ND2n but this is not a Gel'fand model for a Weyl group of type D2n, instead a subspace of ND2n, N˜D2n is a Gel'fand model. We also give simple proofs of the branching rules Dn↪Bn, a generator for each simple Dn-module and a formula for the dimension for all the simple Bn-modules and all the simple Dn-modules.