Markov Measures on Young Tableaux and Induced Representations of the Infinite Symmetric Group

Abstract

We show that the class of so‐called Markov representations of the infinite symmetric group ${\mathfrak G}_{\bf N}$, associated with Markov measures on the space of infinite Young tableaux, coincides with the class of simple representations, i.e., inductive limits of representations with simple spectrum. The spectral measure of an arbitrary representation of ${\mathfrak G}_{\bf N}$ with simple spectrum is equivalent to a multi‐Markov measure on the space of Young tableaux. We also show that the representations of ${\mathfrak G}_{\bf N}$ induced from the identity representations of two‐block Young subgroups are Markov and find explicit formulas for the transition probabilities of the corresponding Markov measures. The induced representations are studied with the help of the tensor model of two‐row representations of the symmetric groups; in particular, we deduce explicit formulas for the Gelfand–Tsetlin basis in the tensor models.