Decomposition of Pythagorean representations of R. Thompson's groups

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 $F$.