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Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups with Symmetric Groups

Part of: Representation theory of groups Algebraic combinatorics

Published online by Cambridge University Press:  13 April 2020

Omar Tout
Omar Tout*
Affiliation:
Department of Mathematics, Faculty of Sciences III, Lebanese University, Tripoli, Lebanon
*
e-mail: omar.tout1988@gmail.com
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Abstract

It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

Keywords

Gelfand pairsrepresentation theory of finite groupswreath product of finite abelian groups with symmetric groups

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory
Secondary: 20C30: Representations of finite symmetric groups
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 91 - 97
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Copyright
© Canadian Mathematical Society 2020

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