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))k , 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. MSC: 20C32, 20B27, 37B05.