Perfect Isometries Between Blocks of Complex Reflection Groups

TL;DR

This paper proves that blocks of complex reflection groups G(de,e,r) and G(de,e,r') with the same p-weight are perfectly isometric, revealing a deep symmetry in their modular representation theory.

Contribution

It establishes a general perfect isometry between blocks of complex reflection groups with the same p-weight, extending known results to a broader class of groups.

Findings

Blocks with the same p-weight are perfectly isometric.

The result applies to all integers d, e, r, r' and primes p not dividing de.

Provides a new understanding of the structure of blocks in complex reflection groups.

Abstract

In this paper, we prove that, given any integers $d$, $e$, $r$ and $r'$, and a prime $p$ not dividing $de$, any two blocks of the complex reflection groups $G(de,e,r)$ and $G(de,e,r')$ with the same $p$-weight are perfectly isometric.