One-dimensional central measures on numberings of ordered sets

TL;DR

This paper characterizes one-dimensional central measures on numberings of ideals in posets, especially multidimensional Young tableaux, providing a combinatorial proof of Thoma's theorem for these measures.

Contribution

It offers a complete description of one-dimensional central measures on poset ideals and introduces a new combinatorial proof of Thoma's theorem for these measures.

Findings

Complete description of one-dimensional central measures on poset ideals.

Proof that ergodic measures are determined by frequencies.

First purely combinatorial proof of Thoma's theorem for certain measures.

Abstract

We describe one-dimensional central measures on numberings (tableaux) of ideals of partially ordered sets (posets). As the main example, we study the poset ${\Bbb Z}_+^d$ and the graph of its finite ideals, multidimensional Young tableaux; for $d=2$, it is the ordinary Young graph. The central measures are stratified by dimension; in the paper we give a complete description of the one-dimensional stratum and prove that every ergodic central measure is uniquely determined by its frequencies. The suggested method, in particular, gives the first purely combinatorial proof of E.~Thoma's theorem for one-dimensional central measures different from the Plancherel measure (which is of dimension~$2$).