Perfect isometries and Murnaghan-Nakayama rules

TL;DR

This paper explores perfect isometries between blocks of various finite groups, generalizing existing methods to include symmetric, alternating, reflection, Weyl, and wreath product groups, with new theoretical developments.

Contribution

It extends the theory of perfect isometries to a broad class of groups, including symmetric, alternating, reflection, Weyl, and wreath product groups, generalizing previous results.

Findings

Proves that p-blocks of symmetric groups with the same weight are perfectly isometric.

Establishes analogues of this result for blocks of alternating groups and their covers.

Introduces generalized block theory applicable to multiple group classes.

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.