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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[x 1 ,,x n ]. 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 , 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:
20G05Representation theory of linear algebraic groups
20F55Reflection groups; Coxeter groups