Decomposable Extensions Between Rank 1 Modules in Grassmannian Cluster Categories - Bogdanic, Dusko; Boroja, Ivan-Vanja (Sarajevo Journal of Mathematics, 2024)
Rank 1 modules are the building blocks of the category 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 JKS16 that the category CM(B_k,n) provides an additive categorification of the cluster algebra structure on the coordinate ring C[Gr(k, n)] of the Grassmannian variety of k-dimensional subspaces in C^n. Rank 1 modules are indecomposable, they are known to be in bijection with k-subsets of {1,2,…,n}, and their explicit construction has been given in [8]. In this paper, we give necessary and sufficient conditions for indecomposability of an arbitrary rank 2 module in 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.