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ON HIGHER FROBENIUS–SCHUR INDICATORS

Part of: Representation theory of groups

Published online by Cambridge University Press:  12 July 2021

YANJUN LIU Open the ORCID record for YANJUN LIU [Opens in a new window]  and
WOLFGANG WILLEMS Open the ORCID record for WOLFGANG WILLEMS [Opens in a new window]
YANJUN LIU*
Affiliation:
School of Mathematics and Statistics, Jiangxi Normal University, Nanchang330022, China
WOLFGANG WILLEMS
Affiliation:
Universität Magdeburg, Magdeburg, Germany and Universidad del Norte, Barranquilla, Colombia e-mail: willems@ovgu.de
*
e-mail: liuyanjun@pku.edu.cn
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Abstract

Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

Keywords

irreducible characterhigher Frobenius–Schur indicatorBrauer characterpermutation module

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 20C20: Modular representations and characters
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 2 , April 2022 , pp. 267 - 277
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Both authors were supported by NSFC (11661042 and 11761034) and the Natural Science Foundation of Jiangxi Province (20192ACB21008).

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