Representations of Higman-Thompson groups from Cuntz algebras

TL;DR

This paper explores how permutative representations of Cuntz algebras derived from interval maps induce irreducible unitary representations of Higman-Thompson groups, with equivalence characterized by orbit coincidence.

Contribution

It introduces a family of permutative representations from interval maps and characterizes their irreducibility and equivalence in the context of Higman-Thompson groups.

Findings

Representations are irreducible.

Two representations are equivalent iff their orbits coincide.

Provides a classification of these representations based on orbits.

Abstract

Every representation of the Cuntz algebra $\mathcal{O}_n$ leads to a unitary representation of the Higman-Thompson group $V_n$. We consider the family $\{\pi_x\}_{x\in [0,1[}$ of permutative representations of $\mathcal{O}_n$ that arise from the interval map $f(x)=nx$ (mod 1) acting on the Hilbert space that underlies each orbit, and then study the unitary equivalence and the irreducibility of the corresponding family $\{\rho_x\}_{x\in [0,1[}$ of representations of Higman-Thompson group $V_n$, showing that that these representations are indeed irreducible and moreover $\rho_x$ and $\rho_y$ are equivalent if and only if the orbits of $x$ and $y$ coincide.