  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 $F$.
