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GRÖBNER THEORY AND TROPICAL GEOMETRY ON SPHERICAL VARIETIES

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Abstract

Let G be a connected reductive algebraic group. We develop a Gröbner theory for multiplicity-free G-algebras, as well as a tropical geometry for subschemes in a spherical homogeneous space G/H. We define the notion of a spherical tropical variety and prove a fundamental theorem of tropical geometry in this context. We also propose a definition for a spherical amoeba in G/H using Cartan decomposition. Our work partly builds on the previous work of Vogiannou on spherical tropicalization and in some ways is complementary.

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA

    KIUMARS KAVEH

  2. Department of Mathematics, University of Kentucky, Lexington, KY, USA

    CHRISTOPHER MANON

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  1. KIUMARS KAVEH
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  2. CHRISTOPHER MANON
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Correspondence to KIUMARS KAVEH.

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The first author is partially supported by a National Science Foundation Grant (Grant ID: DMS-1601303), Simons Foundation Collaboration Grant for Mathematicians, and Simons Fellowship.

The second author is partially supported by a National Science Foundation Grant (Grant ID: DMS-1500966).

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KAVEH, K., MANON, C. GRÖBNER THEORY AND TROPICAL GEOMETRY ON SPHERICAL VARIETIES. Transformation Groups 24, 1095–1145 (2019). https://doi.org/10.1007/s00031-019-09536-5

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