The Alperin weight conjecture and the Glauberman correspondence via character triples

Martínez, J. Miquel; Rizo, Noelia; Rossi, Damiano

doi:10.2140/ant.2026.20.333

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The Alperin weight conjecture and the Glauberman correspondence via character triples

J. Miquel Martínez, Noelia Rizo and Damiano Rossi

Vol. 20 (2026), No. 2, 333–382

DOI: 10.2140/ant.2026.20.333

Abstract

In 2017, G. Navarro introduced a new conjecture that unifies the Alperin weight conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple groups and prove it for several classes of groups and blocks. Our reduction can be divided into two steps. First, we show that by assuming the so-called inductive (blockwise) Alperin weight condition for finite simple groups, we obtain an analogous statement for arbitrary finite groups, that is, an automorphism-equivariant version of the Alperin weight conjecture inducing isomorphisms of modular character triples. Then, we show that the latter implies Navarro’s conjecture for each finite group.

Keywords

Alperin weight conjecture, Glauberman correspondence, block theory, isomorphisms of character triples

Mathematical Subject Classification

Primary: 20C20

Secondary: 20C15

Milestones

Received: 30 May 2024

Revised: 25 November 2024

Accepted: 20 January 2025

Published: 16 February 2026

Authors

J. Miquel Martínez
Departament de Matemàtiques Universitat de València 46100 Burjassot (Valencia) Spain	Dipartamento di Matematica e Informatica “Ulisse Dini” 50134 Firenze Italy

Noelia Rizo
Departament de Matemàtiques Universitat de València 46100 Burjassot (Valencia) Spain

Damiano Rossi
Department of Mathematics Rutgers University Piscataway, NJ 08854 United States

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Vol. 20, No. 2, 2026