The Main Problem of Block Theory: Picky Elements and Subnormalizers

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 $x$.