Perfect isometries and Murnaghan-Nakayama rules

Abstract:

This article is concerned with perfect isometries between blocks of finite groups. Generalizing a method of Enguehard to show that any two $p$-blocks of (possibly different) symmetric groups with the same weight are perfectly isometric, we prove analogues of this result for $p$-blocks of alternating groups (where the blocks must also have the same sign when $p$ is odd), of double covers of alternating and symmetric groups (for $p$ odd, and where we obtain crossover isometries when the blocks have opposite signs), of complex reflection groups $G(d,1,n)$ (for $d$ prime to $p$), of Weyl groups of type $B$ and $D$ (for $p$ odd), and of certain wreath products. In order to do this, we need to generalize the theory of blocks in a way which should be of independent interest.