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

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- by Frank Lübeck and Gunter Malle
- Represent. Theory
**20**(2016), 139-161 - DOI: https://doi.org/10.1090/ert/480
- Published electronically: March 4, 2016
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Corrigendum: Represent. Theory

**21**(2017), 1-3.

## 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 $\ell \ge 3$ the first Cartan invariant in the principal $\ell$-block is larger than 2 unless Sylow $\ell$-subgroups are cyclic.

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## Bibliographic Information

**Frank Lübeck**- Affiliation: Lehrstuhl D für Mathematik, RWTH Aachen, Pontdriesch 14/16, 52062 Aachen, Germany.
- MR Author ID: 362381
- Email: Frank.Luebeck@math.rwth-aachen.de
**Gunter Malle**- Affiliation: FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany.
- MR Author ID: 225462
- Email: malle@mathematik.uni-kl.de
- Received by editor(s): August 11, 2015
- Received by editor(s) in revised form: January 11, 2016
- Published electronically: March 4, 2016
- Additional Notes: The second author gratefully acknowledges financial support by ERC Advanced Grant 291512.
- © Copyright 2016 American Mathematical Society
- Journal: Represent. Theory
**20**(2016), 139-161 - MSC (2010): Primary 20C20; Secondary 20C33, 20C34
- DOI: https://doi.org/10.1090/ert/480
- MathSciNet review: 3466537