The continuity of $p$-rationality of characters and the principal block

TL;DR

This paper investigates the continuity of $p$-rationality of characters in finite groups, showing it can be detected in the principal $p$-block for prime 2 and proposing a conjecture for all primes.

Contribution

It refines previous results by demonstrating the detectability of 2-rationality continuity in the principal 2-block and introduces a conjectural criterion for the continuity gap at level 1.

Findings

Continuity of 2-rationality can be detected in the principal 2-block.

Proposes a conjectural criterion for the continuity gap at level 1 for all primes.

Refines recent results by N. N. Hung.

Abstract

We study rationality properties of irreducible characters of finite groups. We show that the continuity of $2$-rationality is a phenomenon that can be detected in the principal $2$-block, thus refining a recent result of N. N. Hung. We also propose a conjectural group theoretical criterion for the continuity gap at level $1$ for all primes