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

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.