Categorification and mirror symmetry for Grassmannians

Abstract

Abstract The homogeneous coordinate ring of the Grassmannian is a cluster algebra, with an additive categorification . In particular, every in has a cluster character . We work initially in a more general Frobenius 2‐CY subcategory of , where is an algebra defined simply relative to , but equivalent to a choice of Grassmann necklace. For any cluster tilting object in , with , we define two new cluster characters, a generalised partition function , whose leading exponent is a ‐vector/index of , and a generalised flow polynomial , whose leading exponent is , an invariant introduced in an earlier paper. These (formal) polynomials are related by applying a map to their exponents. When , in the ‐cluster chart corresponding to , we can show that the function becomes . Furthermore, when mutates, undergoes ‐mutation and undergoes tropical ‐mutation. We also show that the monoid of g‐vectors can be described by inequalities obtained by tropicalising Marsh–Rietsch's superpotential for Grassmannians and give a module‐theoretic interpretation of the inequalities. This provides a categorical incarnation of Grassmannian mirror symmetry, in the sense of Rietsch–Williams. In the process, we also prove that the Newton–Okounkov body constructed by Rietsch–Williams can be described using .