Irreducible Pythagorean representations of R. Thompson's groups and of the Cuntz algebra

We introduce the Pythagorean dimension: a natural number (or infinity) for all representations of the Cuntz algebra and certain unitary representations of the Richard Thompson groups called Pythagorean. For each finite Pythagorean dimension d we completely classify (in a functorial manner) all such representations using finite dimensional linear algebra. Their irreducible classes form a nice moduli space: a real manifold of dimension $2d^2+1$. Apart from a finite disjoint union of circles, each point of the manifold corresponds to an irreducible unitary representation of Thompson's group F (which extends to the other Thompson groups and the Cuntz algebra) that is not monomial. The remaining circles provide monomial representations which we previously fully described and classified. We translate in our language a large number of previous results in the literature. We explain how our techniques extend them.