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Towards Landau-Ginzburg models for cominuscule spaces via the exceptional cominuscule family by Peter Spacek; Charles Wang is a Mathematics article available to read on EtoBox.

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We present projective Landau-Ginzburg models for the exceptional cominuscule homogeneous spaces OP 2 = E 6 sc / P 6 and E 7 sc / P 7 , known respectively as the Cayley plane and Freudenthal variety. These models are defined on the complement X can ∨ of an anti-canonical divisor of the “Langlands dual homogeneous spaces” X ∨ = P ∨ \ G ∨ in terms of generalized Plücker coordinates, analogous to the canonical models defined for Grassmannians, quadrics and Lagrangian Grassmannians in [20,25,23]. We prove that these models for the exceptional family are isomorphic to the Lie-theoretic mirror models defined in [26] using a restriction to an algebraic torus, also known as the Lusztig torus, as proven in [28]. We also give a cluster structure on C [ X ∨ ] , prove that the Plücker coordinates form a Khovankii 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.

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Author

Peter Spacek; Charles Wang

Publisher

Elsevier BV

Published

2023

Language

EN

Field

Mathematics (Physical Sciences)