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.