On Characters of Inductive Limits of Symmetric Groups

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 $χ:G \rightarrow \mathbb C$ is uniquely defined by the formula $χ(g) = μ_1(Fix(g))^{α_1}...μ_k(Fix(g))^{α_k}$, where $μ_1,...,μ_k$ are $G$-ergodic measures, $Fix(g) = \{x\in X: gx = x\}$, and $α_1,...,α_k\in \{0,1,...,\infty\}$. We illustrate our results on the group of rational permutations of the unit interval.

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