Stochastic Monotonicity in Young Graph and Thoma Theorem

Abstract

We show that the order on probability measures, inherited from the dominance order on the Young diagrams, is preserved under natural maps reducing the number of boxes in a diagram by $1$. As a corollary we give a new proof of the Thoma theorem on the structure of characters of the infinite symmetric group. We present several conjectures generalizing our result. One of them (if it is true) would imply the Kerov's conjecture on the classification of all homomorphisms from the algebra of symmetric functions into $\mathbb R$ which are non-negative on Hall--Littlewood polynomials.