Abstract

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 $\mathbb{𝔽}$ 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 $\mathbb{𝔽}$ 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 $\mathbb{𝔽}$ must have isomorphic fusion systems. The converse is wrong in general.

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