Restriction of odd degree characters and natural correspondences

TL;DR

This paper explores natural correspondences of odd-degree irreducible characters in finite groups like $GL_n(q)$, $SL_n(q)$, and symmetric groups, revealing structural insights and Galois invariance of character fields.

Contribution

It establishes new bijections between odd-degree irreducible characters across various finite groups and subgroups, extending known correspondences and confirming Galois invariance.

Findings

Constructs canonical bijections between odd-degree characters of $S_n$ and its maximal subgroups.

Shows these bijections commute with Galois group actions, preserving fields of values.

Provides answers to questions posed by R. Gow regarding character correspondences.

Abstract

Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a natural correspondence of characters, both for $GL_n(q)$ and $SL_n(q)$. A similar result is established for certain finite groups with self-normalizing Sylow $p$-subgroups. We also construct a canonical bijection between the odd-degree irreducible characters of $S_n$ and those of $M$, where $M$ is any maximal subgroup of $S_n$ of odd index; as well as between the odd-degree irreducible characters of $G = GL_n(q)$ or $GU_n(q)$ with $q$ odd and those of $N_{G}(P)$, where $P$ is a Sylow $2$-subgroup of $G$. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that the fields of values of character correspondents are the same. We use this to answer some questions of R. Gow.