Categorification and mirror symmetry for Grassmannians

Abstract

The homogeneous coordinate ring C[Gr(k,n)]$\mathbb {C}[\operatorname{Gr}(k,n)]$ of the Grassmannian is a cluster algebra, with an additive categorification CMC$\operatorname{CM}C$ . In particular, every M$M$ in CMC$\operatorname{CM}C$ has a cluster character ΨM∈C[Gr(k,n)]$\Psi _M\in \mathbb {C}[\operatorname{Gr}(k,n)]$ . We work initially in a more general Frobenius 2‐CY subcategory GPB$\operatorname{GP}B$ of CMC$\operatorname{CM}C$ , where B$B$ is an algebra defined simply relative to C$C$ , but equivalent to a choice of Grassmann necklace. For any cluster tilting object T$T$ in GPB$\operatorname{GP}B$ , with A=End(T)op$A=\operatorname{End}(T)^{\mathrm{op}}$ , we define two new cluster characters, a generalised partition function PMT∈C[K(CMA)]$\mathcal {P}^T_M\in \mathbb {C}[K(\operatorname{CM}A)]$ , whose leading exponent is a g$g$ ‐vector/index of M$M$ , and a generalised flow polynomial FMT∈C[K(fdA)]$\mathcal {F}^T_M\in \mathbb {C}[K(\operatorname{fd}A)]$ , whose leading exponent is κ(T,M)$\bm{\kappa }(T,M)$ , an invariant introduced in an earlier paper. These (formal) polynomials are related by applying a map wt:K(CMA)→K(fdA)$\operatorname{\mathsf {wt}}\colon K(\operatorname{CM}A)\rightarrow K(\operatorname{fd}A)$ to their exponents. When B=C$B=C$ , in the X$\mathbb {X}$ ‐cluster chart corresponding to T$T$ , we can show that the function ΨM$\Psi _M$ becomes FMT$\mathcal {F}^T_M$ . Furthermore, when T$T$ mutates, FMT$\mathcal {F}^T_M$ undergoes X$\mathbb {X}$ ‐mutation and κ(T,M)$\bm{\kappa }(T,M)$ undergoes tropical A$\mathbb {A}$ ‐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 κ(T,M)$\bm{\kappa }(T, M)$ .