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

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.