The Brown complex in non-defining characteristic and applications

2023/03/24 by Damiano Rossi, Rossi, Damiano · 2 citations
Mathematics · #20C20 #20G40 #20J05 #55P91 #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2303.13973

openalex publication_date 2023/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Brown complex associated to the poset of ℓ-subgroups in the case of a finite reductive group defined over a field \mathbbF_q 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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• Generic weights for finite reductive groups

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