Year: 2025
Type: article
Abstract: A strong Gelfand pair (πΊ, π») consists of a group πΊ and its subgroup π», characterized by the property that the induced representation Ind π πΊ π» of the permutation representation associated with the action of πΊ on the set of cosets πΊπ» is multiplicity-free. In this study, the conditions under which πΊ forms a strong Gelfand pair with certain subgroups are investigated, with emphasis on two main cases: i) the wreath product πΊ β π π with its subgroup πΊ β π πβ1 , ii) the group π π = πΊ β π π = πΊ π β π π with its subgroup πΆ π = πΈ π Γ π π , where πΈ π is the diagonal subgroup of the direct product of π copies of πΊ. While such pairs are always Gelfand for abelian groups, in general they do not satisfy the strong Gelfand property. Specifically, there exists a positive integer 2 β€ ππ(πΊ) < |πΊ| such that (πΊ β π πβ1 , πΊ β π π ) and (π π , πΆ π ) are strong Gelfand pairs for π < ππ(πΊ), but not for π β₯ ππ(πΊ). This result holds for both abelian and non-abelian finite groups. A method is developed to compute this critical number ππ(πΊ), and explicit examples are provided for specific groups.
Author: Saad Bedaiwi
Institution: Mustansiriyah University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4416877184
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