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Catégorification de données Z-modulaires et groupes de réflexions complexes

Abstract : This work is a contribution to the categorification of mathbb{Z}-modular data and deals mainly with mathbb{Z}-modular data arising from complex reflection groups, as well as cellular characters for these groups. In his classification of representations of finite groups of Lie type, Lusztig defines a nonabelian Fourier transform, and associate a mathbb{N}-modular datum to each family of unipotent characters. In a generalization of Lusztig's theory to Spetses, Broué, Malle and Michel construct mathbb{Z}-modular data associated to some complex reflection groups. We first give a categorical explanation of some of these mathbb{Z}-modular data in terms of representation of the Drinfeld double of a finite group. We had to endow the category of representations with a non-spherical structure. The study of slightly degenerate categories shows that they naturally give rise to mathbb{Z}-modular data. In order to construct some examples, we consider an extension of the fusion categories associated to qgrroot{mathfrak{g}}, where mathfrak{g} is a simple Lie algebra and xi a root of unity. These categories are constructed as semisimplification of the category of tilting modules of qdblroot{mathfrak{g}}, which is a central extension of qgrroot{mathfrak{g}}. If mathfrak{s}=mathfrak{sl}_{n+1}, we show that this category is related to some mathbb{Z}-modular data associated to the complex reflection group Gleft(d,1,frac{n(n+1)}{2}right). Exceptional complex reflection groups are also considered and many different categories appear in the categorification of the associated mathbb{Z}-modular data : modules categories over twisted Drinfeld doubles as well as some subcategories of fusion categories of tilting modules over qdblroot{mathfrak{g}} in type A and B.
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https://tel.archives-ouvertes.fr/tel-02114250
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Submitted on : Monday, April 29, 2019 - 2:51:07 PM
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  • HAL Id : tel-02114250, version 1

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Abel Lacabanne. Catégorification de données Z-modulaires et groupes de réflexions complexes. Mathématiques générales [math.GM]. Université Montpellier, 2018. Français. ⟨NNT : 2018MONTS045⟩. ⟨tel-02114250⟩

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