Atomic representations of R. Thompson's groups and Cuntz's algebra

TL;DR

This paper investigates atomic and diffuse components of Pythagorean unitary representations of Richard Thompson's groups and their extension to the Cuntz algebra, revealing a detailed structure of the atomic part.

Contribution

It provides a complete description of the atomic part of these representations, showing it as a finite sum of irreducible monomial representations from specific parabolic subgroups.

Findings

Atomic part decomposes into irreducible monomial representations.

Diffuse part is Ind-mixing and contains no finite-dimensional induced representations.

Complete classification of atomic components in these representations.

Abstract

We continue to study Pythagorean unitary representation of Richard Thompson's groups $F,T,V$ and their extension to the Cuntz(-Dixmier) algebra. Any linear isometry from a Hilbert space to its direct sum square produces such. We focus on those arising from a finite-dimensional Hilbert space. We show that they decompose as a direct sum of a so-called diffuse part and an atomic part. We previously proved that the diffuse part is Ind-mixing: it does not contain induced representations of finite-dimensional ones. In this article, we fully describe the atomic part: it is a finite direct sum of irreducible monomial representations arising from a precise family of parabolic subgroups.