Symmetric Alcoved Polytopes

TL;DR

This paper studies symmetric alcoved polytopes associated with root systems, establishing that most such polytopes have generating sets whose size equals the Coxeter number, revealing a deep connection between symmetry, root systems, and polytope structure.

Contribution

It proves that for all root systems except F4, symmetric alcoved polytopes have generating sets of size equal to the Coxeter number, linking symmetry and combinatorial properties.

Findings

Most symmetric alcoved polytopes have generating sets of size equal to the Coxeter number.

Type A alcoved polytopes are exactly the tropical convex polytopes with tropical generators.

The result excludes the F4 root system, highlighting unique properties of its polytopes.

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.