Open problems in the representation theory of finite groups. (Offene Probleme in der Darstellungstheorie endlicher Gruppen.) (German) Zbl 0768.20005

Külshammer, B.

Jahresber. Dtsch. Math.-Ver. 94, No. 2, 98-104 (1992).

This article summarizes the contents of the author’s survey lecture on open problems in representation theory of finite groups given at the Bavarian Mathematical Colloquium 1991 (Weihenstephan). The author gives a self-contained explanation of the Alperin-McKay conjecture and Alperin’s weight conjecture on \(p\)-blocks of finite groups. He describes the modern approaches by Knörr and Robinson to relate these conjectures. In the second part he considers the special case of blocks with abelian defect groups. He states the following theorem which is a consequence of a deep result of E. C. Dade: Let \(G\) be a solvable group. Let \(B\) be a \(p\)-block of \(G\) with an abelian defect group \(D\) and Brauer correspondent \(b\) in \(N_ G(D)\). Then the block algebras of \(B\) and \(b\) are Morita equivalent.
For non-solvable groups \(G\) such a strong theorem does not hold. Nevertheless, Broué’s conjecture asserts that for arbitrary \(p\)-blocks \(B\) with abelian defect groups \(D\) and Brauer correspondents \(b\) in \(N_ G(D)\) the block algebras \(B\) and \(b\) have equivalent derived categories. If true, Broué’s conjecture would imply both the conjectures by Alperin-McKay and by Alperin (weight conjecture) for blocks with abelian defect groups. This article is very readable. Even non-experts will recognize the beauty of the mathematical ideas described here.

Reviewer: G.Michler (Essen)

MSC:

20C20	Modular representations and characters
20-02	Research exposition (monographs, survey articles) pertaining to group theory

Keywords:

Alperin-McKay conjecture; Alperin’s weight conjecture; \(p\)-blocks of finite groups; blocks with abelian defect groups; Brauer correspondent; block algebras; Morita equivalent; non-solvable groups; Broué’s conjecture; derived categories