Decomposition of Pythagorean representations of R. Thompson's groups

TL;DR

This paper analyzes Pythagorean unitary representations of Richard Thompson's groups, decomposing them into diffuse and atomic parts using diagrammatic techniques, and characterizes the atomic part through monomial representations from specific subgroups.

Contribution

It introduces a diagrammatic method to decompose Pythagorean representations of Thompson's groups into diffuse and atomic components, providing a detailed classification of the atomic part.

Findings

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

Atomic part is fully decomposed into monomial representations from parabolic subgroups.

Develops diagrammatic techniques for representation analysis.

Abstract

We continue to study Pythagorean unitary representation of Richard Thompson's groups $F$, $T$ and $V$ that are built from a single isometry from a Hilbert space to its double. By developing powerful diagrammatically based techniques we show that each such representation splits into a diffuse and an atomic parts. We previously proved that the diffuse part is Ind-mixing: it does not contain induced representations of finite-dimensional ones. We fully decompose the atomic part: the building blocks are monomial representations arising from a precise family of parabolic subgroups of $F$.