Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups

Bonnafé, Cédric; Hohlweg, Christophe

Annales de l'Institut Fourier, Tome 56 (2006) no. 1, pp. 131-181.

Nous construisons une sous-algèbre ${\Sigma }^{\prime }\left(W_n\right)$ de dimension $2·3^{n-1}$ de l’algèbre du groupe de Weyl $W_n$ de type $B_n$ contenant son algèbre de Solomon usuelle ainsi que celle de ${𝔖}_n$ : ${\Sigma }^{\prime }\left(W_n\right)$ n’est autre que l’algèbre de Mantaci-Reutenauer mais notre point de vue nous permet de construire un morphisme d’algèbres surjectif ${\Sigma }^{\prime }\left(W_n\right)\to \mathbf{Z}\mathrm{Irr}\left(W_n\right)$. La construction de Jöllenbeck des caractères irréductibles de ${𝔖}_n$ 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.

We construct a subalgebra ${\Sigma }^{\prime }\left(W_n\right)$ of dimension $2·3^{n-1}$ of the group algebra of the Weyl group $W_n$ of type $B_n$ containing its usual Solomon algebra and the one of ${𝔖}_n$: ${\Sigma }^{\prime }\left(W_n\right)$ is nothing but the Mantaci-Reutenauer algebra but our point of view leads us to a construction of a surjective morphism of algebras ${\Sigma }^{\prime }\left(W_n\right)\to \mathbf{Z}\mathrm{Irr}\left(W_n\right)$. 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.

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