Local Functions on Blocks

TL;DR

This paper introduces a block-by-block approach to the Alperin-McKay conjecture, establishing equivalences and extending key results from Isaacs and Navarro's work to this new framework.

Contribution

It defines a new block-by-block version of the chain local condition and proves its equivalence to the Alperin-McKay conjecture, extending prior results.

Findings

Proves the equivalence of the Alperin-McKay conjecture with a block-by-block local condition.

Establishes several new block-by-block versions of existing results from Isaacs and Navarro.

Provides a framework for analyzing the conjecture through localized group functions.

Abstract

We define a block-by-block version of Isaacs and Navarro's chain local condition and then prove that the Alperin-McKay conjecture is equivalent to a certain function on groups having this property. We then go on to prove several other block-by-block versions of results from Isaacs and Navarro's paper.