Irreducible Pythagorean representations of Thompson’s groups

Abstract: Richard Thompson’s groups $F$, $T$ and $V$ are one of the most fascinating discrete infinite groups for their several unusual properties and their analytical properties have been challenging experts for many decades. One reason for this is because very little is known about its representation theory. Luckily, thanks to the novel technology of Jones, a rich family of so-called Pythagorean unitary representation of Thompson’s groups can be constructed by simply specifying a pair of finite-dimensional operators satisfying a certain equality. These representations can even be extended to the celebrated Cuntz algebra and carry a powerful diagrammatic calculus which we use to develop techniques to study their properties. This permits to reduce very difficult questions concerning irreducibility and equivalence of infinite-dimensional representations into problems in finite-dimensional linear algebra. This provides a new rich class of irreducible representations of $F$. Moreover, we introduce the Pythagorean dimension which is a new invariant for all representations of the Cuntz algebra and Pythagorean representations of $F,T,V$. For each dimension $d$, we show the irreducible classes form a moduli space of a real manifold of dimension $2d^2+1$.

group theory

Audience: researchers in the discipline

Symmetry in Newcastle