On Characters of Inductive Limits of Symmetric Groups

TL;DR

This paper fully characterizes the characters of certain infinite groups formed as limits of symmetric groups, linking them to ergodic measures and fixed points on associated Bratteli diagrams.

Contribution

It provides a complete description of characters for inductive limits of symmetric groups, connecting group actions, ergodic measures, and fixed point sets.

Findings

Characters are uniquely determined by ergodic measures and fixed point data.

Explicit formula for characters in terms of ergodic measures and fixed points.

Application to the group of rational permutations of the unit interval.

Abstract

In the paper we completely describe characters (central positive-definite functions) of simple locally finite groups that can be represented as inductive limits of (products of) symmetric groups under block diagonal embeddings. Each such group $G$ defines an infinite graph (Bratteli diagram) that encodes the embedding scheme. The group $G$ acts on the space $X$ of infinite paths of the associated Bratteli diagram by changing initial edges of paths. Assuming the finiteness of the set of ergodic measures for the system $(X,G)$, we establish that each indecomposable character $\chi :G \rightarrow \mathbb C$ is uniquely defined by the formula $\chi(g) = \mu_1(Fix(g))^{\alpha_1}...\mu_k(Fix(g))^{\alpha_k}$, where $\mu_1,...,\mu_k$ are $G$-ergodic measures, $Fix(g) = \{x\in X: gx = x\}$, and $\alpha_1,...,\alpha_k\in \{0,1,...,\infty\}$. We illustrate our results on the group of rational permutations of the unit interval.