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Tropical geometry and Newton–Okounkov cones for Grassmannian of planes from compactifications

Part of: Special varieties Representation theory of rings and algebras

Published online by Cambridge University Press:  12 October 2020

Christopher Manon  and
Jihyeon Jessie Yang
Christopher Manon
Affiliation:
University of Kentucky, Lexington, KY, USA e-mail: Christopher.Manon@uky.edu
Jihyeon Jessie Yang*
Affiliation:
Department of Mathematical and Computational Sciences, University of Toronto Mississauga, Mississauga, ON, Canada
*
e-mail: jessie.yang@utoronto.ca
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Abstract

We construct a family of compactifications of the affine cone of the Grassmannian variety of $2$ -planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.

Keywords

Compactificationtropical geometryNewton–Okounkov body

MSC classification

Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds 14M25: Toric varieties, Newton polyhedra 14M27: Compactifications; symmetric and spherical varieties 14T05: Tropical geometry
Secondary: 16G20: Representations of quivers and partially ordered sets
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 1 , February 2022 , pp. 199 - 231
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

C.M. was supported by the NSF (DMS 1500966) and a Simons Collaboration Grant.

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