The $p$-rationality of height-zero characters

TL;DR

This paper investigates a conjectural relationship between the $p$-rationality of height-zero characters in finite groups and their defect groups, suggesting that local subgroup properties reflect global character properties.

Contribution

It introduces a conjecture linking the $p$-rationality of height-zero characters to their normalizers, providing evidence for a global-local phenomenon in representation theory.

Findings

Proposes a conjecture relating $p$-rationality to normalizers of defect groups.

Provides evidence supporting the conjectural global-local relationship.

Highlights the significance of local subgroup structure in understanding global character properties.

Abstract

We propose and present evidence for a conjectural global-local phenomenon concerning the $p$-rationality of $p$-height-zero characters. Specifically, if $\chi$ is a height-zero character of a finite group $G$ and $D$ is a defect group of the $p$-block of $G$ containing $\chi$, then the $p$-rationality of $\chi$ can be captured inside the normalizer $N_G(D)$.