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Exploring strong Gelfand pairs in wreath products of specific finite groups | ||
| Journal of University of Anbar for Pure Science | ||
| Volume 19, Issue 2, November and December 2025, Pages 216-220 PDF (317.67 K) | ||
| Document Type: Research Paper | ||
| DOI: 10.37652/juaps.2025.157548.1358 | ||
| Author | ||
| Saad O. Bedaiwi* | ||
| Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq | ||
| 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 π−1 , ii) the group π π = πΊ β π π = πΊ π β π π with its subgroup πΆ π = πΈ π × 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 ≤ ππ(πΊ) < |πΊ| such that (πΊ β π π−1 , πΊ β π π ) and (π π , πΆ π ) are strong Gelfand pairs for π < ππ(πΊ), but not for π ≥ 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. | ||
| Keywords | ||
| Computer algebra system GAP; Multiplicity-free; Strong Gelfand pair; Wreath product | ||
| References | ||
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