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


Authors: Frank Lübeck and Gunter Malle
Journal: Represent. Theory 20 (2016), 139-161
MSC (2010): Primary 20C20; Secondary 20C33, 20C34
DOI: https://doi.org/10.1090/ert/480
Published electronically: March 4, 2016
Corrigendum: Represent. Theory 21 (2017), 1-3.
MathSciNet review: 3466537
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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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Additional Information

Frank Lübeck
Affiliation: Lehrstuhl D für Mathematik, RWTH Aachen, Pontdriesch 14/16, 52062 Aachen, Germany.
Email: Frank.Luebeck@math.rwth-aachen.de

Gunter Malle
Affiliation: FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany.
Email: malle@mathematik.uni-kl.de

DOI: https://doi.org/10.1090/ert/480
Keywords: Murnaghan--Nakayama rule, classical groups, simple endotrivial modules, Loewy length, zeroes of characters
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.
Article copyright: © Copyright 2016 American Mathematical Society