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 NG(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 Co1, implying that both conjectures hold for all 3-blocks of Co1.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 S6 of degree 6.