Abstract

Generalized alcoved polytopes are polytopes whose facet normals are roots in a given root system. We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it. The type $A$ alcoved polytopes are precisely the tropical polytopes that are also convex in the usual sense. In this case the tropical generators form a generating set. We show that for any root system other than $F_4$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the root system.

Highlights

  • We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it

  • We show that for any root system other than F4, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the root system

  • In this paper we investigate alcoved polytopes which are symmetric under the action of the Weyl group

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Summary

Introduction

In this paper we investigate alcoved polytopes which are symmetric under the action of the Weyl group. For a root system Φ, an alcoved polytope of type Φ is a polytope defined by inequalities of the form a, x c where a ∈ Φ and c ∈ Z They are unions of (faces of) alcoves in the affine Coxeter arrangement associated to Φ. For type A root systems, alcoved polytopes are precisely the tropical polytopes that are convex in the usual sense. They are named polytropes by Joswig and Kulas [Jo-Ku]. Our main result states that every alcoved polytope for an irreducible and reduced root system Φ not of type F4 that is symmetric under the action of the Weyl group can be generated by h vertices, where h is the Coxeter number of Φ.

Tropical convexity and alcoved polytopes
Generating sets of alcoved polytopes
Types B and C
Type F4
If μ λ or
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