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

TL;DR

This paper explores the structure of root systems of type T_{2,p,q}, generalizing Manin's hyperbolic construction of E8, and uncovers hidden regularities relevant to cluster algebra categorification.

Contribution

It introduces a new construction of root systems of type T_{2,p,q}, extending previous work and revealing new regularities, including for type E_n, linked to cluster structures.

Findings

Generalization of Manin's hyperbolic construction to T_{2,p,q}

Identification of hidden regularities in these root systems

Connections to categorification of Grassmannian coordinate rings

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.