Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups

[Algèbre de descente généralisée et construction des caractères irréductibles des groupes hyperoctaédraux]

Bonnafé, Cédric ; Hohlweg, Christophe

Annales de l'Institut Fourier, Tome 56 (2006), p. 131-181 / Harvested from Numdam

Résumé

Nous construisons une sous-algèbre $\Sigma ^{\prime}(W_n)$ de dimension $2\cdot 3^{n-1}$ de l’algèbre du groupe de Weyl $W_n$ de type $B_n$ contenant son algèbre de Solomon usuelle ainsi que celle de ${\mathfrak{S}}_n$ : $\Sigma ^{\prime}(W_n)$ n’est autre que l’algèbre de Mantaci-Reutenauer mais notre point de vue nous permet de construire un morphisme d’algèbres surjectif $\Sigma ^{\prime}(W_n) \rightarrow {{\mathbf{Z}}}{\mathrm{Irr}}(W_n)$. La construction de Jöllenbeck des caractères irréductibles de ${\mathfrak{S}}_n$ à partir des classes d’équivalence coplaxique se transpose alors à $W_n$. Un appendice à cet article, écrit par P. Baumann et C. Hohlweg, donne le lien combinatoire explicite entre cette construction des caractères irréductibles de $W_n$ et celle obtenue par W. Specht en 1932.

Abstract

We construct a subalgebra $\Sigma ^{\prime}(W_n)$ of dimension $2\cdot 3^{n-1}$ of the group algebra of the Weyl group $W_n$ of type $B_n$ containing its usual Solomon algebra and the one of ${\mathfrak{S}}_n$: $\Sigma ^{\prime}(W_n)$ is nothing but the Mantaci-Reutenauer algebra but our point of view leads us to a construction of a surjective morphism of algebras $\Sigma ^{\prime}(W_n) \rightarrow {{\mathbf{Z}}}{\mathrm{Irr}}(W_n)$. Jöllenbeck’s construction of irreducible characters of the symmetric group by using the coplactic equivalence classes can then be transposed to $W_n$. In an appendix, P. Baumann and C. Hohlweg present in an explicit and combinatorial way the relation between this construction of the irreducible characters of $W_n$ and that of W. Specht.

Détail

Publié le : 2006-01-01

MR 2228684 | Zbl 1098.20011

DOI : https://doi.org/10.5802/aif.2176
Classification:  05E15
Mots clés: algèbre de descente, groupe hyperoctaédral, algèbre coplaxique

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