# Irreducible characters of $3'$-degree of finite symmetric, general linear and unitary groups

## TL;DR

This paper constructs a canonical correspondence between irreducible characters of degree coprime to 3 in finite symmetric, general linear, and unitary groups and their Sylow 3-subgroup normalizers, preserving fields of values.

## Contribution

It introduces a new canonical bijection linking characters of degree coprime to 3 with those of Sylow 3-subgroup normalizers in key finite groups, maintaining Galois action compatibility.

## Key findings

- Established a canonical correspondence between characters of degree coprime to 3 and Sylow 3-subgroup normalizers.
- Proved that the fields of values of corresponding characters are identical.
- The bijections commute with Galois group actions, ensuring field invariance.

## Abstract

Let $G$ be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to $3$. We construct a canonical correspondence between irreducible characters of degree coprime to $3$ of $G$ and those of $N_{G}(P)$, where $P$ is a Sylow $3$-subgroup of $G$. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.