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- W1991819048 abstract "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 B n 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 B n , 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 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: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines, vol. 34, Hermann, 1968. [4] ]), N G 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 B n , Beiträge Algebra Geom. 44 (2) (2003) 359–373]. In this work we show that if G is a Weyl group of type D 2 n + 1 , N D 2 n + 1 provides a Gel'fand model for this group. We also describe completely N D 2 n but this is not a Gel'fand model for a Weyl group of type D 2 n , instead a subspace of N D 2 n , N ˜ D 2 n 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." @default.
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- W1991819048 date "2005-12-01" @default.
- W1991819048 modified "2023-09-26" @default.
- W1991819048 title "A Gel'fand model for a Weyl group of type <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.gif overflow=scroll><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> and the branching rules <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si2.gif overflow=scroll><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>↪</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>" @default.
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- W1991819048 doi "https://doi.org/10.1016/j.jalgebra.2005.09.001" @default.
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