Broué's abelian defect group conjecture and 3-decomposition numbers of the sporadic simple Conway group Co_1

Abstract

In the representation theory of finite groups, Broué's abelian defect group conjecture says that for any prime p if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the normaliser N_G(P) of P in G are derived equivalent. We prove that Broué's conjecture, and even Rickard's splendid equivalence conjecture, are true for the unique 3-block A of defect 2 of the sporadic simple Conway group Co_1, implying that both conjectures hold for all 3-blocks of Co_1. To do so, we determine the 3-decomposition numbers of A, and we actually show that A is Puig equivalent to the principal 3-block of the symmetric group S_6 of degree 6.

arXiv commentThis is another in our series of papers dealing with the sporadic simple groups cases of the abelian defect group conjecture, entailing some text overlap in the introduction with arXiv:1212.2775, arXiv:1206.0358, arXiv:1011.4429 and arXiv:0906.5124