The $p$-rationality of height-zero characters

Abstract

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