On the representation theory of the symmetry group of the Cantor set

TL;DR

This paper explores the representation theory of the symmetry group of the Cantor set, revealing exactly two measures and connecting their associated representations to well-studied algebraic categories, providing new insights and counterexamples.

Contribution

It characterizes the measures on the Cantor set symmetry group and links their representations to known algebraic categories, offering new perspectives and counterexamples.

Findings

Exactly two measures on the symmetry group of the Cantor set

Representation of (G, μ) relates to F2-vector spaces

Representation of (G, ν) relates to Boolean semi-ring vector spaces

Abstract

In previous work with Harman, we introduced a new class of representations for an oligomorphic group $G$, depending on an auxiliary piece of data called a measure. In this paper, we look at this theory when $G$ is the symmetry group of the Cantor set. We show that $G$ admits exactly two measures $\mu$ and $\nu$. The representation theory of $(G, \mu)$ is the linearization of the category of $\mathbf{F}_2$-vector spaces, studied in recent work of the author and closely connected to work of Kuhn and Kov\'acs. The representation theory of $(G, \nu)$ is the linearization of the category of vector spaces over the Boolean semi-ring (or, equivalently, the correspondence category), studied by Bouc--Th\'evenaz. The latter case yields an important counterexample in the general theory.