Cluster categories from Grassmannians and root combinatorics

TL;DR

This paper explores the structure of Cohen-Macaulay modules related to Grassmannians, establishing new connections with root systems and providing explicit module constructions to deepen understanding of cluster categories.

Contribution

It introduces canonical Auslander--Reiten sequences, studies translation periodicity, and constructs Cohen-Macaulay modules of arbitrary rank, linking modules to root systems.

Findings

Identified canonical Auslander--Reiten sequences

Established translation periodicity in the category

Connected rank 2 modules to real roots in Kac-Moody algebras

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.