Chevalley Polytopes and Newton-Okounkov Bodies

TL;DR

This paper introduces Chevalley polytopes associated with homogeneous spaces, demonstrating their role as Newton-Okounkov bodies in certain cases and exploring their combinatorial properties and connections to string polytopes.

Contribution

It constructs Chevalley polytopes for homogeneous spaces, relates them to Newton-Okounkov bodies, and proposes conjectures on their properties and decomposition, linking to representation theory.

Findings

Chevalley polytopes are Newton-Okounkov bodies for minuscule cases.

Plücker coordinates form a Khovanskii basis for these varieties.

Chevalley polytopes can have better combinatorial properties than string polytopes.

Abstract

We construct a family of polytopes, which we call Chevalley polytopes, associated to homogeneous spaces $X=G/P$ in their projective embeddings $X\hookrightarrow \mathbb{P}(V_{\varpi})$ together with a choice of reduced expression for the minimal coset representative $w^P$ of $w_0$ in $W/W_P$. When $X$ is minuscule in its minimal embedding, we describe our construction in terms of order polytopes of minuscule posets and use the associated combinatorics to show that minuscule Chevalley polytopes are Newton-Okounkov bodies for $X$ and that the Pl\"ucker coordinates on $X$ form a Khovanskii basis for $\mathbb{C}[X]$. We conjecture similar properties for general $X$ and general embeddings $X\hookrightarrow\mathbb{P}(V_\varpi)$, along with a remarkable decomposition property which we consider as a polytopal shadow of the Littlewood-Richardson rule. We highlight a connection between Chevalley polytopes and string polytopes and give examples where Chevalley polytopes possess better combinatorial properties than string polytopes. We conclude with several examples further illustrating and supporting our conjectures.