# Darstellungstheorietage and Nikolaus Conference 2013

Speaker: Jürgen Müller (Jena)

Title: 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 $$6$$.

This is joint work with Shigeo Koshitani and Felix Noeske.