A Murnaghan--Nakayama rule for values of unipotent characters in classical groups

TL;DR

This paper develops a Murnaghan--Nakayama type formula for unipotent character values in finite classical groups, enabling new insights into character vanishing, module classification, and block invariants.

Contribution

It introduces a novel formula for unipotent character values and applies it to classify endotrivial modules and analyze Cartan invariants in classical groups.

Findings

Most complex irreducible characters vanish on some $\ell$-singular element for certain primes $\ell$

Classifies simple endotrivial modules of finite quasi-simple classical groups

Shows the first Cartan invariant exceeds 2 unless Sylow $\ell$-subgroups are cyclic

Abstract

We derive a Murnaghan--Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai's explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some $\ell$-singular element for certain primes $\ell$. As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes $\ell\ge3$ the first Cartan invariant in the principal $\ell$-block is larger than~2 unless Sylow $\ell$-subgroups are cyclic.