Positive Definite Functions of Diagonal Limits of Finite Alternating Groups

Abstract

The normalized positive definite class functions CG −→ C are determined for all those direct limits G of finite alternating groups Alt(Ω i ) for which the embeddings Alt(Ω i ) −→ Alt(Ω j ) are natural in the sense that every non-trivial Alt(Ω i )-orbit in Ω j is natural.

Invariant random subgroups of strictly diagonal limits of finite symmetric groups

“…In earlier work, Leinen-Puglisi [8] and Dudko-Medynets [6] classified the characters of the SDS-groups; and combining their classification and Theorem 1.1, we obtain that if G is a simple SDS-group, then the indecomposable characters of G are precisely the associated characters of the ergodic IRS's of G. However, it should be stressed that our work neither makes use of nor implies the classification theorem of Leinen-Puglisi and Dudko-Medynets. (See Vershik [12] for some fascinating conjectures concerning the relationship between invariant random subgroups and characters.)…”

On characters of inductive limits of symmetric groups

“…We mention that the description of characters in terms of measures of fixed points also appears in [6,16,29], 4 see also [17,Chapter 9] for characters of certain finitely generated groups. The description of characters taking positive values of the group S(∞) as measures of fixed points can be found in [39].…”

“…Characters of groups similar to S(2 ∞ ) were also studied by Leinen and Puglisi in [29], who applied a version of the ergodic approach to describe characters for inductive limits of alternating groups A(X n ) with the embedding scheme allowing the set X n+1 to be covered by multiple copies of X n .…”

“…Many groups covered in [29] can be also seen as groups of permutations of Bratteli diagrams (corresponding to odometers).…”

“…Formally, the characters in[29, Theorem 3.2] are given as functions that depend on the cardinality of the set of fixed points. However, these functions can be interpreted as powers of the measure of fixed points.…”