Perfect Isometries and Murnaghan-Nakayama Rules

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.