Rigid Indecomposable Modules in Grassmannian Cluster Categories

Understand

The coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb{C}^n$ has a cluster algebra structure with Pl\"ucker relations giving rise to exchange relations.

• In this paper, we study indecomposable modules of the corresponding Grassmannian cluster categories ${\rm CM}(B_{k,n})$.
• Jensen, King, and Su have associated a Kac-Moody root system $J_{k,n}$ to ${\rm CM}(B_{k,n})$ and shown that in the finite types, rigid indecomposable modules correspond to roots.
• In general, the link between the category ${\rm CM}(B_{k,n})$ and the root system $J_{k,n}$ remains mysterious and it is an open question whether indecomposables always give roots.