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On a conjecture of Navarro and Tiep on character fields

Marco Albert

TL;DR

The paper addresses Navarro and Tiep's conjecture on character fields for finite quasi-simple groups in the odd-prime setting by developing a sums-of-roots-of-unity theory that yields actionable criteria (notably Theorem 2.5) to compare p-parts of conductors via $p$-elements. It then demonstrates Property 1.3 for several infinite families with known character tables and for finite general linear and special linear groups using Green’s parametrization and Clifford theory, producing partial but broad coverage for GL$_n(q)$, SL$_n(q)$, and PSL$_n(q)$ under favorable conditions. The work provides both theoretical criteria and concrete data (tables) to verify the conjecture in large families, contributing to a clearer understanding of which abelian fields can arise as character fields of finite groups. The results place the Navarro–Tiep conjecture within reach for many classical groups and offer practical tools for extending it to additional families.

Abstract

In 2021, Navarro and Tiep proposed a conjecture on character fields of finite quasi-simple groups. We develop some theory on sums of roots of unity and use this theory to prove the conjecture for some infinite families of finite quasi-simple groups with known character table. We then use the classification of the irreducible complex characters of the finite general linear groups developed by Green to obtain some partial results about the conjecture for the finite general and special linear groups in arbitrary dimension.

On a conjecture of Navarro and Tiep on character fields

TL;DR

The paper addresses Navarro and Tiep's conjecture on character fields for finite quasi-simple groups in the odd-prime setting by developing a sums-of-roots-of-unity theory that yields actionable criteria (notably Theorem 2.5) to compare p-parts of conductors via -elements. It then demonstrates Property 1.3 for several infinite families with known character tables and for finite general linear and special linear groups using Green’s parametrization and Clifford theory, producing partial but broad coverage for GL, SL, and PSL under favorable conditions. The work provides both theoretical criteria and concrete data (tables) to verify the conjecture in large families, contributing to a clearer understanding of which abelian fields can arise as character fields of finite groups. The results place the Navarro–Tiep conjecture within reach for many classical groups and offer practical tools for extending it to additional families.

Abstract

In 2021, Navarro and Tiep proposed a conjecture on character fields of finite quasi-simple groups. We develop some theory on sums of roots of unity and use this theory to prove the conjecture for some infinite families of finite quasi-simple groups with known character table. We then use the classification of the irreducible complex characters of the finite general linear groups developed by Green to obtain some partial results about the conjecture for the finite general and special linear groups in arbitrary dimension.
Paper Structure (5 sections, 16 theorems, 115 equations)

This paper contains 5 sections, 16 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. General theory
  3. Series of finite groups with known character table
  4. The finite general and special linear groups
  5. Tables of $p$-elements

Key Result

Theorem 1.1

Let $\chi\in\textnormal{Irr}_{2'}(G)$ have conductor $2^am$ for some $a\in\mathbb{Z}_{\geq0}$ and odd $m$. Then $\mathbb{Q}_{2^a}\subseteq\mathbb{Q}(\chi)$.

Theorems & Definitions (28)

  • Theorem 1.1: NaTi
  • Theorem 1.2: NaTi
  • Conjecture 1.4: NaTi
  • Theorem 1.5: NaTi
  • Theorem 1.6: NaTi
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3: LaLe
  • ...and 18 more