Let be a finite group. Consider the wreath product and the subgroup , where is the symmetric group and is the diagonal subgroup of . For certain values of (which depend on the group ), the pair is a Gelfand pair. It is not known for all finite groups which values of result in Gelfand pairs. Building off the work of Benson–Ratcliff , we obtain a result which simplifies the computation of multiplicities of irreducible representations in certain tensor product representations, then apply this result to show that for , , is a Gelfand pair exactly when .
"Finite Gelfand pairs and cracking points of the symmetric groups." Rocky Mountain J. Math. 50 (5) 1807 - 1812, October 2020. https://doi.org/10.1216/rmj.2020.50.1807