Decomposable Extensions Between Rank 1 Modules in Grassmannian Cluster Categories

Authors

  • Dusko Bogdanic University of Banja Luka Faculty of Natural Sciences and Mathematics, Banja Luka
  • Ivan-Vanja Boroja University of Banja Luka Faculty of Electrical Engineering, Banja Luka

DOI:

https://doi.org/10.5644/SJM.18.02.10

Keywords:

Cohen-Macaulay modules, Grassmannian cluster categories, decomposable extensions

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 [8]. 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 $J$.

Downloads

Download data is not yet available.

Downloads

Published

16.01.2024

How to Cite

Bogdanic, D. ., & Boroja, I.-V. . (2024). Decomposable Extensions Between Rank 1 Modules in Grassmannian Cluster Categories. Sarajevo Journal of Mathematics, 18(2), 297–312. https://doi.org/10.5644/SJM.18.02.10

Issue

Section

Articles