The Okounkov–Vershik approach to the representation theory of $G\sim S_n$

Abstract

In the Okounkov–Vershik approach to the complex irreducible representations of $S_n$ and $G\sim S_n$, we parametrize the irreducible representations and their bases by spectral objects rather than combinatorial objects and then, at the end, give a bijection between the spectral and combinatorial objects. The fundamental ideas are similar in both cases, but there are additional technicalities involved in the $G\sim S_n$ case. This was carried out by Pushkarev (J Math Sci 96:3590–3599, 1999). The present work gives a fully detailed exposition of Pushkarev’s theory incorporating the following new elements: (i) Our definition of a Gelfand–Tsetlin subspace, based on a multiplicity-free chain of subgroups, leads to a more natural development of the theory and (ii) Ceccherini-Silberstein et al. (Adv Math 206:503–537, 2006) defined the “generalized Johnson scheme,” a certain multiplicity-free $G\sim S_n$ permutation module. We give an algorithm to explicitly write down the Gelfand–Tsetlin subspaces of this module. This gives the simplest nontrivial example of the Okounkov–Vershik theory.