Gr\"obner theory and tropical geometry on spherical varieties

TL;DR

This paper develops a Gr"obner theory and tropical geometry framework for spherical varieties, introducing spherical tropical varieties and amoebas, and establishing foundational theorems in this specialized setting.

Contribution

It introduces the first comprehensive Gr"obner and tropical geometric theories tailored for spherical varieties, expanding the tools available for their study.

Findings

Defined spherical tropical varieties and proved a fundamental theorem of tropical geometry in this context.

Proposed a new definition for spherical amoebas in $G/H$.

Extended the framework of tropicalization to multiplicity-free $G$-algebras.

Abstract

Let $G$ be a connected reductive algebraic group. We develop a Gr\"obner 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 $G/H$. Our work partly builds on the previous work of Vogiannou on spherical tropicalization and in some ways is complementary.