Categorification and mirror symmetry for Grassmannians

The homogeneous coordinate ring C[Gr(k,n)] of the Grassmannian is a cluster algebra, with an additive categorification CMC. Thus every M∈CMC has a cluster character ΨM∈C[Gr(k,n)]. For any cluster tilting object T, with A=End(T)op, we define two new cluster characters, a generalised partition function PTM∈C[K(CMA)], whose leading exponent is g-vector/index of M, and a generalised flow polynomial FTM∈C[K(fdA)], whose leading exponent is κ(T,M), an invariant introduced in earlier paper. These (formal) polynomials are related by applying a map wt:K(CMA)→K(fdA) to their exponents. In the X-cluster chart corresponding to T, the function ΨM becomes FTM. Further more when T mutates, FTM undergoes X-mutation and κ(T,M) undergoes tropical A-mutation. We show that the monoid of g-vectors is given by a rational polyhedral cone, which can be described, following Rietsch-Williams' mirror symmetry strategy, by tropicalisation of the Marsh-Reitsch superpotential~W and, from that, by module-theoretic inequalities. In the process, the NO-body of Rietsch--Williams can be described in terms of κ(T,M). This leads to a categorical incarnation of Grassmannian mirror symmetry, in the sense of Rietsch-Williams. Some of the machinery we develop works in a greater generality, which is relevant to the positroid subvarieties of Gr(k,n).