Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
| Autor: | Lauren Williams, Konstanze Rietsch |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
General Mathematics Schubert calculus 14J33 mirror symmetry Divisor (algebraic geometry) Langlands dual group Newton–Okounkov bodies 01 natural sciences 13F60 Cluster algebra Mathematics - Algebraic Geometry Grassmannian TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY 0103 physical sciences FOS: Mathematics Mathematics - Combinatorics 0101 mathematics Plucker Mathematics::Representation Theory Algebraic Geometry (math.AG) Mathematics Laurent polynomial 010102 general mathematics 52B20 Combinatorics (math.CO) 010307 mathematical physics Grassmannians Mirror symmetry 14M15 cluster algebra |
| Zdroj: | Duke Math. J. 168, no. 18 (2019), 3437-3527 Rietsch, K & Williams, L 2019, ' Newton-Okounkov bodies, cluster duality and mirror symmetry for Grassmannians ', Duke mathematical journal, vol. 168, no. 18, pp. 3437-3527 . https://doi.org/10.1215/00127094-2019-0028 |
| ISSN: | 3437-3527 |
| Popis: | We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, as well as the mirror dual Landau-Ginzburg model $(\check{X}^\circ, W_q:\check{X}^\circ \to \mathbb C)$, where $\check{X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{X} = Gr_k((\mathbb C^n)^*)$, and the superpotential W_q has a simple expression in terms of Pl\"ucker coordinates. Grassmannians simultaneously have the structure of an $\mathcal{A}$-cluster variety and an $\mathcal{X}$-cluster variety. Given a cluster seed G, we consider two associated coordinate systems: a $\mathcal X$-cluster chart $\Phi_G:(\mathbb C^*)^{k(n-k)}\to X^{\circ}$ and a $\mathcal A$-cluster chart $\Phi_G^{\vee}:(\mathbb C^*)^{k(n-k)}\to \check{X}^\circ$. To each $\mathcal X$-cluster chart $\Phi_G$ and ample `boundary divisor' $D$ in $X\setminus X^{\circ}$, we associate a Newton-Okounkov body $\Delta_G(D)$ in $\mathbb R^{k(n-k)}$, which is defined as the convex hull of rational points. On the other hand using the $\mathcal A$-cluster chart $\Phi_G^{\vee}$ on the mirror side, we obtain a set of rational polytopes, described by inequalities, by writing the superpotential $W_q$ in the $\mathcal A$-cluster coordinates, and then "tropicalising". Our main result is that the Newton-Okounkov bodies $\Delta_G(D)$ and the polytopes obtained by tropicalisation coincide. As an application, we construct degenerations of the Grassmannian to toric varieties corresponding to these Newton-Okounkov bodies. Additionally, when $G$ corresponds to a plabic graph, we give a formula for the lattice points of the Newton-Okounkov bodies, which has an interpretation in terms of quantum Schubert calculus. Comment: 55 pages, many figures; to appear in Duke |
| Databáze: | OpenAIRE |
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