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 \(\mathbb{Z}_+^d\) 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 \(2\)).
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Notes
Possibly, such a reduction might be performed for arbitrary posets, but this would require a long digression into the general theory of posets (provided that such a theory exists).
References
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Supported by the RSF grant 21-11-00152.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 17–24 https://doi.org/10.4213/faa4048.
Translated by N. V. Tsilevich
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Vershik, A.M. One-Dimensional Central Measures on Numberings of Ordered Sets. Funct Anal Its Appl 56, 251–256 (2022). https://doi.org/10.1134/S0016266322040025
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DOI: https://doi.org/10.1134/S0016266322040025