Towards Landau-Ginzburg models for cominuscule spaces via the exceptional cominuscule family

Abstract

We present projective Landau-Ginzburg models for the exceptional cominuscule homogeneous spaces ${\mathbb{OP}}^2=E_6^{\mathrm{sc}}/P_6$ and $E_7^{\mathrm{sc}}/P_7$, known respectively as the Cayley plane and the Freudenthal variety. These models are defined on the complement $X_{\mathrm{can}}^{\vee }$ of an anti-canonical divisor of the "Langlands dual homogeneous spaces" ${\mathbb{X}}^{\vee }=P^{\vee }\backslash G^{\vee }$ in terms of generalized Plücker coordinates, analogous to the canonical models defined for Grassmannians, quadrics and Lagrangian Grassmannians in arXiv:1307.1085, arXiv:1404.4844, arXiv:1304.4958. We prove that these models for the exceptional family are isomorphic to the Lie-theoretic mirror models defined in arXiv:math/0511124 using a restriction to an algebraic torus, also known as the Lusztig torus, as proven in arXiv:1912.09122. We also give a cluster structure on $\mathbb{C}[{\mathbb{X}}^{\vee }]$, prove that the Plücker coordinates form a Khovanskii basis for a valuation defined using the Lusztig torus, and compute the Newton-Okounkov body associated to this valuation. Although we present our methods for the exceptional types, they generalize immediately to the members of other cominuscule families.

arXiv comment46 pages, 2 figures, code available at https://github.com/foxflo/ExceptionalCominusculeSpaces

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• Kac-Moody groups and cluster algebras
• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Skew hook formula for $d$-complete posets
• A Chevalley formula for the equivariant quantum K-theory of cominuscule varieties
• Representations of Lie algebras arising from polytopes
• Fukaya category of Grassmannians: rectangles

Cited by in corpus (1)

• A Laurent Phenomenon for the Cayley Plane