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 Tr. Mat. Inst. Steklova, 2019, Volume 305, Pages 71–85 (Mi tm3993)

Three Theorems on the Uniqueness of the Plancherel Measure from Different Viewpoints

A. M. Vershikabc

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, nab. Fontanki 27, St. Petersburg, Russia
b Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504 Russia
c Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bol'shoi Karetnyi per. 19, str. 1, Moscow, 127051 Russia

Abstract: We consider three uniqueness theorems: one from the theory of meromorphic functions, another from asymptotic combinatorics, and the third concerns representations of the infinite symmetric group. The first theorem establishes the uniqueness of the function $\exp z$ in a class of entire functions. The second is about the uniqueness of a random monotone nondegenerate numbering of the two-dimensional lattice $\mathbb Z^2_+$, or of a nondegenerate central measure on the space of infinite Young tableaux. And the third theorem establishes the uniqueness of a representation of the infinite symmetric group $\mathfrak S_\mathbb N$ whose restrictions to finite subgroups have vanishingly few invariant vectors. However, in fact all the three theorems are, up to a nontrivial rephrasing of conditions from one area of mathematics in terms of another area, the same theorem! Up to now, researchers working in each of these areas have not been aware of this equivalence. The parallelism of these uniqueness theorems on the one hand and the difference of their proofs on the other call for a deeper analysis of the nature of uniqueness and suggest transferring the methods of proof between the areas. More exactly, each of these theorems establishes the uniqueness of the so-called Plancherel measure, which is the main object of our paper. In particular, we show that this notion is general for all locally finite groups.

 Funding Agency Grant Number Russian Science Foundation 17-71-20153 This work is supported by the Russian Science Foundation under grant 17-71-20153 and performed at the St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences.

DOI: https://doi.org/10.4213/tm3993

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 305, 63–77

Bibliographic databases:

UDC: 512.542.74+517.987.5
Revised: December 24, 2018
Accepted: March 13, 2019

Citation: A. M. Vershik, “Three Theorems on the Uniqueness of the Plancherel Measure from Different Viewpoints”, Algebraic topology, combinatorics, and mathematical physics, Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday, Tr. Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 71–85; Proc. Steklov Inst. Math., 305 (2019), 63–77

Citation in format AMSBIB
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