Construction of rank 2 indecomposable modules in Grassmannian cluster categories

The category CM(Bk,n) of Cohen-Macaulay modules over a quotient Bk,n of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of k-dimensional subspaces in Cn, [13]. Among the indecomposable modules in this category are the rank 1 modules which are in bijection with k-subsets of {1,2,…,n}, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in CM(Bk,n) can be filtered by them. In this paper we give an explicit construction of rank 2 modules. With this, we give all indecomposable rank 2 modules in the cases when k=3 and k=4. In particular, we cover the tame cases and go beyond them. We also characterise the modules among them which are uniquely determined by their filtrations. For k≥4, we exhibit infinite families of non-isomorphic rank 2 modules having the same filtration.