Real roots in the root system ${\mathsf{T}}_{2,p,q}$

Abstract

Motivated by the recent advances in the categorification of the cluster structure on the coordinate rings of Grassmannians of $k$-subspaces in $n$-space, we investigate a particular construction of root systems of type ${\mathsf{T}}_{2,p,q}$, including the type ${\mathsf{E}}_n$. This construction generalizes Manin's ``hyperbolic construction'' of ${\mathsf{E}}_8$ and reveals a lot of otherwise hidden regularities in this family of root systems.