Construction of Rank $2$ Indecomposable Modules in Grassmannian Cluster Categories

TL;DR

This paper constructs explicit rank 2 indecomposable modules in Grassmannian cluster categories, extending known results for rank 1 modules and covering cases for k=3 and 4, including tame and beyond.

Contribution

It provides explicit constructions of rank 2 modules in Grassmannian cluster categories, including classification for specific cases and analysis of filtration uniqueness.

Findings

Explicit construction of rank 2 modules for k=3 and 4.

Characterization of modules uniquely determined by filtrations.

Identification of infinite families of non-isomorphic modules with same filtrations.

Abstract

The category ${\rm 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 $\mathbb C^n$, \cite{JKS16}. 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 ${\rm 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 $k\ge 4$, we exhibit infinite families of non-isomorphic rank 2 modules having the same filtration.