Jones’ representations of R. Thompson’s groups not induced by finite-dimensional ones

Given any linear isometry from a Hilbert space to its square one can explicitly construct a so-called Pythagorean unitary representation of Richard Thompson’s group F. We introduce a condition on the isometry implying that the associated representation does not contain any representation induced by finite-dimensional ones. This provides the first result of this kind. We illustrate this theorem via a family of representations parameterized by the real 3-sphere for which all of them have this property except on two sub-circles.