Virtual permutations and polymorhisms

TL;DR

This paper explores the structure of virtual permutations derived from symmetric groups, analyzing their measure-theoretic properties and the action of infinite symmetric groups, with connections to polymorphisms and dessins d'enfant.

Contribution

It introduces the closure of the infinite symmetric group action in the semigroup of polymorphisms on the space of virtual permutations and provides explicit formulas involving Dirichlet distributions.

Findings

Formulas for polymorphisms and their centers.

Descriptions of the closure of $S_ abla imes S_ abla$ in the semigroup of polymorphisms.

Connections between virtual permutations, dessins d'enfant, and Dirichlet distributions.

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 $\mathfrak{S}$ 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 $\mathfrak{S}$ (it is called 'virtual permutations' or 'Chinese restaurant process'), a double $S_\infty\times S_\infty $ of an infinite symmetric group acts on $\mathfrak{S}$ by left and right 'multiplications'. We discuss the closure of $S_\infty\times S_\infty $ in the semigroup of polymorphisms (spreading maps with spreaded Radon--Nikodym derivatives) of $\mathfrak{S}$. 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.