Cluster Categorification of Rank 2 Webs

Abstract

The homogeneous coordinate ring of the Grassmannian $\mathrm{Gr}(\mathrm{k},\mathrm{n})$ has a well-known cluster structure. There is a categorification of this cluster structure via a category of modules for a ring $A_{k,n}$ due to Jensen-King-Su, building on work of Geiss-Leclerc-Schröer, in which cluster variables correspond to indecomposable rigid modules. We give a combinatorial description of modules that correspond to rank $2$ cluster variables by using webs. This conjecturally gives the categorification of all rank $2$ cluster variables.