Rigid Indecomposable Modules in Grassmannian Cluster Categories

TL;DR

This paper investigates the relationship between indecomposable modules in Grassmannian cluster categories and roots of associated Kac-Moody root systems, providing evidence for a conjectured correspondence especially in infinite types.

Contribution

It demonstrates that indecomposable rank 2 and 3 modules in Grassmannian cluster categories correspond to roots, and characterizes their profiles, supporting the link between modules and roots.

Findings

Rank 2 modules correspond to roots in the root system.

Rank 3 modules in ${ m CM}(B_{3,n})$ give rise to roots.

Confirmed the link in ${ m CM}(B_{3,9})$ with exactly 225 profiles.

Abstract

The coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb{C}^n$ has a cluster algebra structure with Pl\"ucker relations giving rise to exchange relations. In this paper, we study indecomposable modules of the corresponding Grassmannian cluster categories ${\rm CM}(B_{k,n})$. Jensen, King, and Su have associated a Kac-Moody root system $J_{k,n}$ to ${\rm CM}(B_{k,n})$ and shown that in the finite types, rigid indecomposable modules correspond to roots. In general, the link between the category ${\rm CM}(B_{k,n})$ and the root system $J_{k,n}$ remains mysterious and it is an open question whether indecomposables always give roots. In this paper, we provide evidence for this association in the infinite types: we show that every indecomposable rank 2 module corresponds to a root of the associated root system. We also show that indecomposable rank 3 modules in ${\rm CM}(B_{3,n})$ all give rise to roots of $J_{3,n}$. For the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to real roots, we show that their underlying profiles are cyclic permutations of a certain canonical one. We also characterize the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to imaginary roots. By proving that there are exactly 225 profiles of rigid indecomposable rank 3 modules in ${\rm CM}(B_{3,9})$ we confirm the link between the Grassmannian cluster category and the associated root system in this case. We conjecture that the profile of any rigid indecomposable module in ${\rm CM}(B_{k,n})$ corresponding to a real root is a cyclic permutation of a canonical profile.