authors:
Shigeo Koshitani, J\"urgen M\"uller, Felix Noeske

title:
Brou\'e'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,
Brou\'e'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 Brou\'e'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.

status:
submitted