A correspondence between additive and monoidal categorifications with application to Grassmannian cluster categories

TL;DR

This paper proposes a conjectural link between additive and monoidal categorifications of cluster algebras, supported by results in Grassmannian cluster categories, and constructs specific modules within these categories.

Contribution

It introduces a new conjectural correspondence connecting additive and monoidal categorifications, with applications to Grassmannian cluster categories and module construction.

Findings

Evidence from Grassmannian cluster algebras supports the conjecture.

Constructs rigid and non-rigid modules in Grassmannian cluster categories.

Provides a new perspective linking reachability conjectures in cluster theory.

Abstract

Building on work of Derksen-Fei and Plamondon, we formulate a conjectural correspondence between additive and monoidal categorifications of cluster algebras, which reveals a new connection between the additive reachability conjecture and the multiplicative reachability conjecture. Evidence for this conjecture is provided by results on Grassmannian cluster algebras and categories in the tame types. Moreover, we give a construction of the generic kernels introduced by Hernandez and Leclerc for type $\mathbb{A}$ via the Grassmannian cluster categories. As an application of the correspondence, we construct rigid indecomposable modules and indecomposable non-rigid modules in Grassmannian cluster categories.