Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1081.20052
Araujo, J.O.; Bigeón, J.J.
A Gel'fand model for a Weyl group of type $D_n$ and the branching rules $D_n\hookrightarrow B_n$.
(English)
[J] J. Algebra 294, No. 1, 97-116 (2005). ISSN 0021-8693

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 {\it 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 [{\it C. W. Curtis} and {\it I. Reiner}, Representation theory of finite groups and associative algebras. Wiley, New York (1988; Zbl 0634.20001), {\it J. L. Aguado} and {\it J. O. Araujo}, Commun. Algebra 29, No. 4, 1841-1851 (2001; Zbl 1015.20009), {\it J. O. Araujo}, Beitr. Algebra Geom. 44, No. 2, 359-373 (2003; Zbl 1063.20008), {\it 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 ${\cal N}_G$ of the polynomial ring $K[x_1,\dots,x_n]$. If $G$ is a Weyl group of type $A_n$ or $B_n$ [see {\it 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)], ${\cal N}_G$ provides a Gel'fand model for these groups as shown in [{\it J. L. Aguado} and {\it J. O. Araujo}, loc. cit. and {\it J. O. Araujo}, loc. cit.]. In this work we show that if $G$ is a Weyl group of type $D_{2n+1}$, ${\cal N}_{D_{2n+1}}$ provides a Gel'fand model for this group. We also describe completely ${\cal N}_{D_{2n}}$ but this is not a Gel'fand model for a Weyl group of type $D_{2n}$, instead a subspace of ${\cal N}_{D_{2n}}$, $\widetilde{\cal N}_{D_{2n}}$ 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 2000:
*20G05 Representation theory of linear algebraic groups
20F55 Coxeter groups

Keywords: Gelfand models; direct sums of irreducible representations; Weyl groups; branching rules; simple modules

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences