Virtual permutations and polymorhisms

2022/02/25 by Neretin, Yury A. · 1 citation
Mathematics · #05E10 #05E45 #22D10 #60J50 #60K35 #Analytic Number Theory Research #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2202.12978

openalex publication_date 2022/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There is a natural map from a symmetric group S_n to a smaller symmetric group S_n-1, we write a decomposition of a permutation into a product of disjoint cycles and remove the element n from this expression. For this reason there exists the inverse limit \mathfrakS of sets S_n. We equip S_n with the uniform distribution (or more generally with an Ewens distribution) and get a structure of a measure space on \mathfrakS (it is called 'virtual permutations' or 'Chinese restaurant process'), a double S_∞× S_∞ of an infinite symmetric group acts on \mathfrakS by left and right 'multiplications'. We discuss the closure of S_∞× S_∞ in the semigroup of polymorphisms (spreading maps with spreaded Radon--Nikodym derivatives) of \mathfrakS. We get formulas for some polymorphisms, in particular for the center of the closure. Expressions are sums of multiple convolutions of Dirichlet distributions, summation sets are certain collections of dessins d'enfant.

Cited by

• On algebras of conjugacy classes of symmetric groups with respect to Young subgroups

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