2021/07/07 by Dusko Bogdanic, Bogdanic, Dusko, Ivan-Vanja Boroja +1 · 1 citation
Mathematics · #05E10 #16G50 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:05E10 #msc:16G50
paper · pdf · doi:10.48550/arxiv.2107.03503
arXiv admin note: substantial text overlap with arXiv:2011.14176
arxiv created 2021/07/07 · arxiv updated 2021/07/09
Rank 1 modules are the building blocks of the category \rm CM(Bk,n) of Cohen-Macaulay modules over a quotient Bk,n of a preprojective algebra of affine type A. Jensen, King and Su showed in \citeJKS16 that the category \rm CM(Bk,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 Cn. 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 \citeJKS16. In this paper, we give necessary and sufficient conditions for indecomposability of an arbitrary rank 2 module in \rm CM(Bk,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.