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 graded graph that encodes the embedding scheme. The group G acts on the space X of infinite paths of the associated graph by changing initial edges of paths. Assuming the finiteness of the set of ergodic measures for the system (X,G), we establish that each non-regular indecomposable character χ:G→C is uniquely determined by the formula χ(g)=μ1(Fix(g))α1⋯μk(Fix(g))αk, where μ1,…,μk are G-ergodic measures, Fix(g)={x∈X:gx=x}, and α1,…,αk∈{0,1,…}. We illustrate our results on the group of rational permutations of the unit interval.