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Algèbres de Hecke cyclotomiques : représentations, fusion et limite classique

Abstract : An inductive approach to the representation theory of the chain of the cyclotomic Hecke algebras of type G(m,1,n) is developed. This approach relies on the study of the spectrum of a maximal commutative family formed by the analogues of the Jucys-Murphy elements. The irreducible representations, labelled by the multi-partitions, are constructed with the help of a new associative algebra, whose underlying vector space is the tensor product of the cyclotomic Hecke algebra with the free associative algebra generated by the standard multi-tableaux. The analogue of this approach is presented for the classical limit, that is for the chain of complex reflection groups of type G(m,1,n). In a second part, a basis of the cyclotomic Hecke algebras is given and the flatness of the deformation is proved without using the representation theory. These results are extended to the affine Hecke algebras of type A. Then a fusion procedure is presented for the complex reflection groups and the cyclotomic Hecke algebras of type G(m,1,n). In both cases, a complete set of primitive orthogonal idempotents is obtained by successive evaluations of a rational fonction. In a third part, a new presentation is obtained for the alternating subgroups of all Coxeter groups. The generators are related to oriented edges of the Coxeter graph. This presentation is then extended, for all types, to the spinor extensions of the alternating groups, the alternating Hecke algebras and the alternating subgroups of braid groups.
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https://tel.archives-ouvertes.fr/tel-00748920
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Submitted on : Tuesday, November 6, 2012 - 12:00:22 PM
Last modification on : Friday, November 15, 2019 - 10:59:35 AM
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Loïc Poulain d'Andecy. Algèbres de Hecke cyclotomiques : représentations, fusion et limite classique. Théorie des représentations [math.RT]. Aix-Marseille Université, 2012. Français. ⟨tel-00748920⟩

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