×

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)
PDFBibTeX XMLCite
Full Text: DOI

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
[2] Aguado, J. L.; Araujo, J. O., A Gel’fand model for the symmetric group, Comm. Algebra, 29, 4, 1841-1851 (2001) · Zbl 1015.20009
[3] Araujo, J. O., A Gel’fand model for a Weyl group of type \(B_n\), Beiträge Algebra Geom., 44, 2, 359-373 (2003) · Zbl 1063.20008
[4] Bourbaki, N., 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 (1968), Hermann · Zbl 0186.33001
[5] Carter, R., Conjugacy classes in the Weyl group, Compositio Math., 25, 1, 1-59 (1972) · Zbl 0254.17005
[6] Curtis, C.; Reiner, I., Representation Theory of Finite Groups and Associative Algebras (1988), Wiley · Zbl 0634.20001
[7] James, G., The Representation Theory of the Symmetric Groups, Lecture Notes in Math., vol. 682 (1978), Springer-Verlag · Zbl 0393.20009
[8] James, G.; Kerber, A., The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl., vol. 16 (1981), Addison-Wesley · Zbl 0491.20010
[9] James, G.; Liebeck, M., Representations and Character of Groups (1995), Cambridge Univ. Press
[10] Klyachko, A. A., Models for the complex representations of the groups \(G(n, q)\), Math. USSR Sb., 48, 365-380 (1984) · Zbl 0543.20026
[11] MacDonald, I., Some irreducible representations of Weyl groups, Bull. London Math. Soc., 4, 148-150 (1972) · Zbl 0251.20043
[12] Soto-Andrade, J., Geometrical Gel’fand models, tensor quotients and Weyl representations, (Proc. Sympos. Pure Math., vol. 47 (2) (1987), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 306-316 · Zbl 0652.20047
[13] Springer, T., A construction of representations of Weyl groups, Invent. Math., 44, 279-293 (1978) · Zbl 0376.17002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.