One-Dimensional Central Measures on Numberings of Ordered Sets

We describe one-dimensional central measures on numberings (tableaux) of ideals of partially ordered sets (posets). As the main example, we study the poset \(\mathbb{Z}_+^d\) and the graph of its finite ideals, multidimensional Young tableaux; for \(d=2\), this 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 \(2\)).