Real roots in the root system $\mathsf{E}_{k,n}$.

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 $\mathsf{E}_{k,n}$, 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.