Stochastic monotonicity in Young graph and Thoma theorem

TL;DR

This paper demonstrates that the order on probability measures derived from Young diagram dominance is maintained under certain natural maps, leading to a new proof of Thoma's theorem and proposing conjectures with broader implications.

Contribution

It introduces a novel approach to proving Thoma's theorem using stochastic monotonicity in Young graphs and proposes conjectures extending these results.

Findings

Order preservation under maps reducing diagram size

New proof of Thoma's theorem

Conjectures on generalizations and implications for symmetric functions

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.