Rational conjugacy classes and rational characters for some finite simple groups

TL;DR

The paper investigates when rational conjugacy classes coincide with rational-valued irreducible characters in finite groups, focusing on $PSL_2(q)$ and ATLAS simple groups. It develops a detailed case analysis by parity and congruence of $q$, leveraging known PSL$_2(q)$ character tables and Galois-action rationality criteria to enumerate rational classes and rational characters. The main result is an exact equality between the two counts for $PSL_2(q)$, with explicit RC$(G)$ values depending on $q$, and a verification that the same equality holds for all ATLAS simple groups except the Tits group. The work supports a broader conjecture that finite simple groups of Lie type may exhibit this equality and provides exhaustive computational confirmation for ATLAS groups via GAP CTblLib.

Abstract

If $G$ is a finite group, an irreducible complex-valued character $χ$ is called rational if $χ(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $χ$, the value $χ(x^G)$ is rational. We prove that for $q$, a power of prime, the group $\mathrm{PSL}_2(q)$ has same number of rational characters and rational conjugacy classes. Furthermore, we verify that this equality holds for all finite simple groups whose character tables appear in the $\textit{ATLAS of Finite Groups}$, except for the Tits group.

Theorems & Definitions (18)

• Theorem A
• Lemma 2.1
• proof
• Lemma 2.2
• Proposition 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4