Cluster categories from Grassmannians and root combinatorics

Abstract

The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander--Reiten sequences and study the Auslander--Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac-Moody algebra in the tame cases.

arXiv commentA construction of modules of arbitrary ranks is added

References in corpus (6)

• Rigid modules over preprojective algebras
• Dimer models and cluster categories of Grassmannians
• The representation type of Jacobian algebras
• A categorification of Grassmannian cluster algebras
• Ice quivers with potentials associated with triangulations and Cohen-Macaulay modules over orders
• Tubular Jacobian Algebras

Cited by in corpus (1)

• Friezes satisfying higher SL${}_k$-determinants