Rigid indecomposable modules in Grassmannian cluster categories

The coordinate ring of the Grassmannian variety of \(k\)-dimensional subspaces in \(\mathbb{C^n}\) has a cluster algebra structure with Plucker relations giving rise to exchange relations. We study indecomposable modules of the corresponding Grassmannian cluster categories of type \((k,n)\). Jensen, King, and Su have associated a Kac-Moody root system to the category and shown that in the finite types, rigid indecomposable modules correspond to roots. We provide evidence for this association in the infinite types: we show that every indecomposable rank 2 module corresponds to a root of the associated root system. We also study roots and indecomposable rank 3 modules for the case \((3,n)\).