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COMMON ZEROS OF IRREDUCIBLE CHARACTERS

Part of: Representation theory of groups

Published online by Cambridge University Press:  11 December 2023

NGUYEN N. HUNG Open the ORCID record for NGUYEN N. HUNG [Opens in a new window] ,
ALEXANDER MORETÓ  and
LUCIA MOROTTI
NGUYEN N. HUNG*
Affiliation:
Department of Mathematics, The University of Akron, Akron, OH 44325, USA
ALEXANDER MORETÓ
Affiliation:
Departamento de Matemáticas, Universidad de Valencia, 46100 Burjassot, Valencia, Spain e-mail: alexander.moreto@uv.es
LUCIA MOROTTI
Affiliation:
Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany e-mail: lucia.morotti@uni-duesseldorf.de
*
e-mail: hungnguyen@uakron.edu
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Abstract

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $\mathsf {S}_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for $\mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.

Keywords

zeros of charactersirreducible characters

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 20C30: Representations of finite symmetric groups 20D06: Simple groups 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 25
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The second author thanks Noelia Rizo for helpful conversations. The authors also thank Silvio Dolfi for checking one result. The first author is grateful for the support of the UA Faculty Research Grant FRG 1747. The research of the second author is supported by Ministerio de Ciencia e Innovación (Grants PID2019-103854GB-I00 and PID2022-137612NB-I00 funded by MCIN/AEI/10.13039/501100011033 and ‘ERDF A way of making Europe’) and Generalitat Valenciana CIAICO/2021/163. While working on the revised version the third author was working at the Department of Mathematics of the University of York, supported by the Royal Society grant URF$\setminus$R$\setminus$221047.

Communicated by Michael Giudici

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