Measures for the colored circle

TL;DR

This paper computes measures for the automorphism groups of n-colored circles, expanding understanding of oligomorphic groups and their associated tensor categories, which was previously limited to few cases.

Contribution

It provides the first explicit measure calculations for the automorphism groups of n-colored circles, a significant class of oligomorphic groups.

Findings

Measures for automorphism groups of n-colored circles are explicitly determined.

The results facilitate the construction of tensor categories from these groups.

The work extends the class of oligomorphic groups with known measures.

Abstract

In recent work with Harman, we introduced a new notion of measure for oligomorphic groups, and showed how they can be used to produce interesting tensor categories. Determining the measures for an oligomorphic group is (in our view) an important and difficult combinatorial problem, which has only been solved in a handful of cases. The purpose of this paper is to solve this problem for a certain infinite family of oligomorphic groups, namely, the automorphism group of the $n$-colored circle (for each $n \ge 1$).