The abelian defect group conjecture for sporadic groups

Let G be a finite group, let A be a prime block of G having
an abelian defect group D, let N be the normaliser in G of D,
and let B be the Brauer correspondent of A. Then the abelian
defect group conjecture says that the bounded derived categories
of the module categories of A and B equivalent as triangulated
categories. Although this conjecture is in the focus of intensive
studies since almost two decades now, it has only been verified
for certain cases and a general proof seems to be out of sight.

In this talk, we briefly introduce the notions to state the
abelian defect group conjecture, report on the current state
of knowledge, and on the strategies to prove it for explicit
examples. Then we show how these strategies are pursued and
combined with techniques from computational representation theory
to prove the abelian defect group conjecture for the sporadic simple
Harada-Norton group; this is joint work with Shigeo Koshitani.