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

Abstract

Rank $1$ modules are the building blocks of the category $\mathrm{CM}(B_{k,n})$ 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 $\mathrm{CM}(B_{k,n})$ provides an additive categorification of the cluster algebra structure on the coordinate ring $\mathbb{C}[\mathrm{Gr}(k,n)]$ 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 $\mathrm{CM}(B_{k,n})$ 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 $J$.

arXiv admin note: substantial text overlap with arXiv:2011.14176