Alperin's Main Problem of Block Theory

TL;DR

This paper introduces a new conjectural framework for Alperin's Main Problem in Block Theory, emphasizing character sets defined by nonvanishing at elements and connecting to McKay's conjecture.

Contribution

It proposes a novel perspective on local objects governing character values and verifies main conjectures in specific group families.

Findings

Main conjectures verified for simple groups with TI Sylow p-subgroups

New character sets based on nonvanishing at elements are introduced

Reorganization of classical questions in character theory

Abstract

This paper proposes a conjectural framework for Alperin's Main Problem of Block Theory from 1976. The character sets considered here are defined by nonvanishing at given elements, not only by degree conditions. From this point of view, McKay's conjecture is usually recovered as a first degree-level consequence. The guiding idea is that the right local objects governing character values are not, in general, the sets ${\rm Irr}_{p'}(G)$ and the normalizers of Sylow $p$-subgroups, but rather the sets ${\rm Irr}^x(G)$ of irreducible characters not vanishing at a given element $x$, together with the subnormalizer subgroup ${\rm Sub}_G(x)$. I state the basic conjectures of this theory, propose stronger versions, and verify the main conjectures in several families, including the simple groups with TI Sylow $p$-subgroups. I also show how this perspective reorganizes several classical questions in character theory.