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A Gel’fand model for a Weyl group of type \(D_n\) and the branching rules \(D_n\hookrightarrow B_n\). (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 \({\mathcal N}_G\) of the polynomial ring \(K[x_1,\dots,x_n]\). If \(G\) is a Weyl group of type \(A_n\) or \(B_n\) [see 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)], \({\mathcal N}_G\) 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 \(D_{2n+1}\), \({\mathcal N}_{D_{2n+1}}\) provides a Gel’fand model for this group. We also describe completely \({\mathcal N}_{D_{2n}}\) but this is not a Gel’fand model for a Weyl group of type \(D_{2n}\), instead a subspace of \({\mathcal N}_{D_{2n}}\), \(\widetilde{\mathcal N}_{D_{2n}}\) 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:

20G05 Representation theory for linear algebraic groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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References:

[1] J.L. Aguado, J.O. Araujo, Representations of finite groups on polynomial rings, in: Actas V Congreso de Matemática Dr. Antonio R. Monteiro, Bahía Blanca, 1999, pp. 35-40; J.L. Aguado, J.O. Araujo, Representations of finite groups on polynomial rings, in: Actas V Congreso de Matemática Dr. Antonio R. Monteiro, Bahía Blanca, 1999, pp. 35-40
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