Browsing by Author "Rossi, Damiano"

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    The Brown Complex in Non-Defining Characteristic and Applications
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Rossi, Damiano
    We study the Brown complex associated to the poset of ℓ-subgroups in the case of a finite reductive group defined over a field Fq of characteristic prime to ℓ. First, under suitable hypotheses, we show that its homotopy type is determined by the generic Sylow theory developed by Broué and Malle and, in particular, only depends on the multiplicative order of q modulo ℓ. This result leads to several interesting applications to generic Sylow theory, mod ℓ homology decompositions, and ℓ-modular representation theory. Then, we conduct a more detailed study of the Brown complex in order to establish an explicit connection between the local-global conjectures in representation theory of finite groups and the generic Sylow theory. This is done by isolating a family of ℓ-subgroups of finite reductive groups that corresponds bijectively to the structures controlled by the generic Sylow theory.
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    The Character Triple Conjecture for Maximal Defect Characters and the Prime 2
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Rossi, Damiano
    We prove that Späth's Character Triple Conjecture holds for every finite group with respect to maximal defect characters at the prime 2. This is done by reducing the maximal defect case of the conjecture to the so-called inductive Alperin–McKay condition whose verification has recently been completed by Ruhstorfer for the prime 2. As a consequence we obtain the Character Triple Conjecture for all 2-blocks with abelian defect groups by applying Brauer's Height Zero Conjecture, a proof of which is now available. We also obtain similar results for the block-free version of the Character Triple Conjecture at the prime 3.
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    The Simplicial Complex of Brauer Pairs of a Finite Reductive Group
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Rossi, Damiano
    In this paper we study the simplicial complex induced by the poset of Brauer pairs ordered by inclusion for the family of finite reductive groups. In the defining characteristic case the homotopy type of this simplicial complex coincides with that of the Tits building thanks to a well-known result of Quillen. On the other hand, in the non-defining characteristic case, we show that the simplicial complex of Brauer pairs is homotopy equivalen to a simplicial complex determined by generalised Harish-Chandra theory. This extends earlier results of the author on the Brown complex and makes use of the theory of connected subpairs and twisted block induction developed by Cabanes and Enguehard.