Decomposable extensions between rank $1$ modules in Grassmannian cluster categories

TL;DR

This paper characterizes when rank 2 modules in a Cohen-Macaulay category are indecomposable or decomposable, focusing on modules built from tightly interlacing rank 1 modules related to Grassmannian cluster categories.

Contribution

It provides necessary and sufficient conditions for indecomposability of rank 2 modules with tightly interlacing layers and constructs all decomposable modules as extensions of rank 1 modules.

Findings

Criteria for indecomposability of rank 2 modules.

Explicit construction of decomposable modules as extensions.

Connection between module structure and combinatorial properties of subsets.

Abstract

Rank $1$ modules are the building blocks of the category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra of affine type $A$. Jensen, King and Su showed in \cite{JKS16} that the category ${\rm CM}(B_{k,n})$ provides an additive categorification of the cluster algebra structure on the coordinate ring $\mathbb C[{\rm Gr}(k, n)]$ of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$. Rank $1$ modules are indecomposable, they are known to be in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has been given in \cite{JKS16}. In this paper, we give necessary and sufficient conditions for indecomposability of an arbitrary rank 2 module in ${\rm CM}(B_{k,n})$ whose filtration layers are tightly interlacing. We give an explicit construction of all rank 2 decomposable modules that appear as extensions between rank 1 modules corresponding to tightly interlacing $k$-subsets $I$ and $J$.