虚置换与多形性
Virtual permutations and polymorhisms
摘要：
存在从对称群$S_n$到更小的对称群$S_{n-1}$的自然映射，我们将置换分解为不相交圈的乘积，并从该表达式中删除元素$n$。由于这个原因，存在集合$S_n$的逆极限$\mathfrak{S}$。我们给$S_n$赋上均匀分布（或更一般地赋上Ewens分布），得到$\mathfrak{S}$上的测度空间的结构（称为“虚置换”或“中餐厅过程”），无限对称群的二重$S_\infty\times S_\infty $通过左、右“乘法”作用于$\mathfrak{S}$.本文讨论了$\mathfrak{S}$的多态性半群（具有扩展Radon-Nikodym导数的扩展映射）中$S_\infty\times S_\infty $的闭包.我们得到了一些多态性的公式，特别是闭包的中心。表达式是狄利克雷分布的多重卷积的和，求和集是某些dessins d 'enfant的集合。
英文摘要：
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.
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