The Main Problem of Block Theory: Picky Elements and Subnormalizers

TL;DR

This paper discusses a research program in local representation theory, focusing on Alperin's main problem and proposing conjectures involving picky elements and subnormalizers as key objects.

Contribution

It introduces conjectures on picky elements and subnormalizers, extending classical restrictions in local representation theory and addressing Alperin's main problem.

Findings

Proposes conjectures linking picky elements and subnormalizers to local character rules.

Extends classical restrictions beyond characters of p'-degree and abelian defect blocks.

Provides a new perspective on Alperin's main problem in block theory.

Abstract

This article is essentially an English translation of a paper of mine, published in \emph{La Gaceta de la RSME}. Its aim is to present, for a broad mathematical audience, a research programme in local representation theory that goes beyond the classical restrictions to characters of $p'$-degree, characters of height zero, and blocks of abelian defect. The final and most recent part of this programme concerns Alperin's main problem of block theory: the search for local rules for character values. In that direction I describe the conjectures on picky elements and subnormalizers, which suggest that the sets ${\rm Irr}^x(G)$ and the subgroups ${\rm Sub}_G(x)$ are the natural objects attached to a $p$-element $x$.