Construction of rank 2 indecomposable modules in Grassmannian cluster categories

Abstract

The category $\mathrm{CM}(B_{k,n})$ of Cohen-Macaulay modules over a quotient $B_{k,n}$ 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 $\mathbb{C}^{n}$, [13]. Among the indecomposable modules in this category are the rank 1 modules which are in bijection with $k$-subsets of $\{1, 2, \dots, n\}$, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in $\mathrm{CM}(B_{k,n})$ 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 $k \geq 4$, we exhibit infinite families of non-isomorphic rank 2 modules having the same filtration.