Perfect isometries for blocks with abelian defect groups and Klein four inertial quotients
Let \(G\) be a finite group, and let \(B\) be a block of \(G\) with abelian defect group \(P\). Moreover, let \(\beta\) be a block of \(C_ G(P)\) such that \(\beta^ G = B\), and suppose that the inertial quotient \(N_ G(P,\beta)/C_ G(P)\) is a Klein four group. In this situation the authors construct a perfect isometry between the group of generalized characters in \(B\) and the group of generalized characters in the Brauer correspondent \(b\) of \(B\) in \(N_ G(P)\). This confirms a conjecture of M. Broué for this case. It follows that \(B\) and \(b\) have the same number of irreducible Brauer characters and that the block algebras of \(B\) and \(b\) have isomorphic centers.
scientific article; zbMATH DE number 436487