Exploring strong Gelfand pairs in wreath products of specific finite groups

A strong Gelfand pair (𝐺, 𝐻) consists of a group 𝐺 and its subgroup 𝐻, characterized by the property that the induced representation Ind 𝑑 𝐺 𝐻 of the permutation representation associated with the action of 𝐺 on the set of cosets 𝐺𝐻 is multiplicity-free. In this study, the conditions under which 𝐺 forms a strong Gelfand pair with certain subgroups are investigated, with emphasis on two main cases: i) the wreath product 𝐺 ≀ 𝑆 𝑛 with its subgroup 𝐺 ≀ 𝑆 𝑛−1 , ii) the group 𝑊 𝑛 = 𝐺 ≀ 𝑆 𝑛 = 𝐺 𝑛 ⋊ 𝑆 𝑛 with its subgroup 𝐶 𝑛 = 𝐸 𝑛 × 𝑆 𝑛 , where 𝐸 𝑛 is the diagonal subgroup of the direct product of 𝑛 copies of 𝐺. 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 ≤ 𝑆𝑁(𝐺) < |𝐺| such that (𝐺 ≀ 𝑆 𝑛−1 , 𝐺 ≀ 𝑆 𝑛 ) and (𝑊 𝑛 , 𝐶 𝑛 ) are strong Gelfand pairs for 𝑛 < 𝑆𝑁(𝐺), but not for 𝑛 ≥ 𝑆𝑁(𝐺). This result holds for both abelian and non-abelian finite groups. A method is developed to compute this critical number 𝑆𝑁(𝐺), and explicit examples are provided for specific groups.