DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 121-128

Jeremy Rickard

Title: The Abelian Defect Group Conjecture

Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. If $B$ is a block of the group algebra $kG$ with defect group $D$, the Brauer correspondent of $B$ is a block $b$ of $kN_G(D)$. When $D$ is abelian, the blocks $B$ and $b$, although they are rarely isomorphic or even Morita equivalent, seem to be very closely related. For example, Alperin's Weight Conjecture predicts that they should have the same number of simple modules. Brou\'{e}'s Abelian Defect Group Conjecture gives a more precise prediction of the relationship between $B$ and $b$: their module categories should have equivalent derived categories. In this article we survey this conjecture, some of its consequences, and some of the recent progress that has been made in verifying it in special cases.

1991 Mathematics Subject Classification: 20C20, 18E30

Keywords and Phrases: modular representation theory, derived category, abelian defect group conjecture

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