Brou 'e's abelian defect group conjecture and 3-decomposition numbers of\n the sporadic simple Conway group Co₁

2013/09/27 by Shigeo Koshitani, Koshitani, Shigeo, Jürgen Müller +3
Computer Science · Mathematics · #20C20 #20C34 #20C40 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1309.7153

openalex publication_date 2013/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the representation theory of finite groups, Brou 'e's abelian defect group\nconjecture says that for any prime p if a p-block A of a finite group G has an\nabelian defect group P, then A and its Brauer corresponding block B of the\nnormaliser N_G(P) of P in G are derived equivalent. We prove that Brou 'e's\nconjecture, and even Rickard's splendid equivalence conjecture, are true for\nthe unique 3-block A of defect 2 of the sporadic simple Conway group Co₁,\nimplying that both conjectures hold for all 3-blocks of Co₁. To do so, we\ndetermine the 3-decomposition numbers of A, and we actually show that A is Puig\nequivalent to the principal 3-block of the symmetric group S₆ of degree 6.\n

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