Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups

Bonnafé, Cédric; Hohlweg, Christophe

doi:10.5802/aif.2176

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 ¹ ; Hohlweg, Christophe ²

¹ Université de Franche-Comté Département de Mathématiques 16 route de Gray 25000 Besançon (France)
² The Fields Institute 222 College Street Toronto, Ontario M5T 3J1 (Canada)

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

Résumé
Abstract

Nous construisons une sous-algèbre $Σ^{'} (W_{n})$ 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 $𝔖_{n}$ : $Σ^{'} (W_{n})$ n’est autre que l’algèbre de Mantaci-Reutenauer mais notre point de vue nous permet de construire un morphisme d’algèbres surjectif $Σ^{'} (W_{n}) \to Z Irr (W_{n})$ . 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 $Σ^{'} (W_{n})$ 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 $𝔖_{n}$ : $Σ^{'} (W_{n})$ is nothing but the Mantaci-Reutenauer algebra but our point of view leads us to a construction of a surjective morphism of algebras $Σ^{'} (W_{n}) \to Z Irr (W_{n})$ . 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.

MR Zbl

DOI : 10.5802/aif.2176

Classification : 05E15
Keywords: descent algebra, hyperoctahedral group, coplactic algebra
Mot clés : algèbre de descente, groupe hyperoctaédral, algèbre coplaxique

Affiliations des auteurs :

Bonnafé, Cédric ¹ ; Hohlweg, Christophe ²

¹ Université de Franche-Comté Département de Mathématiques 16 route de Gray 25000 Besançon (France)
² The Fields Institute 222 College Street Toronto, Ontario M5T 3J1 (Canada)

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     title = {Generalized descent algebra and construction of irreducible characters of~hyperoctahedral groups},
     journal = {Annales de l'Institut Fourier},
     pages = {131--181},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {1},
     year = {2006},
     doi = {10.5802/aif.2176},
     zbl = {1098.20011},
     mrnumber = {2228684},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2176/}
}

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Bonnafé, Cédric; Hohlweg, Christophe. Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups. Annales de l'Institut Fourier, Tome 56 (2006) no. 1, pp. 131-181. doi : 10.5802/aif.2176. https://www.numdam.org/articles/10.5802/aif.2176/

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