The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a well-known cluster structure. There is a categorification of this cluster structure via a category of modules for a ring Ak,n due to Jensen-King-Su, building on work of Geiss-Leclerc-Schr\"oer, 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.
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Le et al. (2024) studied this question.
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