Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups
Annales de l'Institut Fourier, Volume 56 (2006) no. 1, p. 131-181

We construct a subalgebra Σ (W n ) 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 : 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 )ZIrr(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.

Nous construisons une sous-algèbre Σ (W n ) 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  : 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 )ZIrr(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.

DOI : https://doi.org/10.5802/aif.2176
Classification:  05E15
Keywords: descent algebra, hyperoctahedral group, coplactic algebra
@article{AIF_2006__56_1_131_0,
     author = {Bonnaf\'e, C\'edric and Hohlweg, Christophe},
     title = {Generalized descent algebra and construction of irreducible characters of~hyperoctahedral groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {1},
     year = {2006},
     pages = {131-181},
     doi = {10.5802/aif.2176},
     mrnumber = {2228684},
     zbl = {1098.20011},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_1_131_0}
}
Bonnafé, Cédric; Hohlweg, Christophe. Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups. Annales de l'Institut Fourier, Volume 56 (2006) no. 1, pp. 131-181. doi : 10.5802/aif.2176. http://www.numdam.org/item/AIF_2006__56_1_131_0/

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