2025/09/18 by Robert Boltje, Boltje, Robert, Serge Bouc +3
Mathematics · #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2509.14682
For any block of a finite group over an algebraically closed field of characteristic 2 which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal p-permutation functor over an algebraically closed field \mathbbF of characteristic 0 into a direct sum of simple functors. As a consequence we see that two blocks with dihedral, semidihedral, or generalized quaternion defect groups are functorially equivalent over \mathbbF if and only if their fusion systems are isomorphic. It is an open question if two blocks (with arbitrary defect groups) that are functorially equivalent over \mathbbF must have isomorphic fusion systems. The converse is wrong in general.