Categorification and mirror symmetry for Grassmannians

Abstract

The homogeneous coordinate ring $\mathbb{C}[\operatorname{Gr}(k,n)]$ of the Grassmannian is a cluster algebra, with an additive categorification $\operatorname{CM}C$. Thus every $M\in\operatorname{CM}C$ has a cluster character $\Psi_M\in\mathbb{C}[\operatorname{Gr}(k,n)]$. For any cluster tilting object $T$, with $A=\operatorname{End}(T)^{\mathrm{op}}$, we define two new cluster characters, a generalised partition function $\mathcal{P}^T_M\in\mathbb{C}[K(\operatorname{CM}A)]$, whose leading exponent is $g$-vector/index of $M$, and a generalised flow polynomial $\mathcal{F}^T_M\in\mathbb{C}[K(\operatorname{fd}A)]$, whose leading exponent is $\boldsymbol{\kappa}(T,M)$, an invariant introduced in earlier paper. These (formal) polynomials are related by applying a map $\operatorname{wt}\colon K(\operatorname{CM}A)\to K(\operatorname{fd}A)$ to their exponents. In the $\mathbb{X}$-cluster chart corresponding to $T$, the function $\Psi_M$ becomes $\mathcal{F}^T_M$. Further more when $T$ mutates, $\mathcal{F}^T_M$ undergoes $\mathbb{X}$-mutation and $\boldsymbol{\kappa}(T,M)$ undergoes tropical $\mathbb{A}$-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 $\boldsymbol{\kappa}(T,M)$. 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 $\operatorname{Gr}(k,n)$.