Lattice polytopes cut out by root systems and the Koszul property

TL;DR

This paper proves that lattice polytopes associated with classical root systems are normal and Koszul, extending known results from type A to other types, and explores related Cayley sums using combinatorial methods.

Contribution

It generalizes the normality and Koszul properties of lattice polytopes from type A to all classical root systems and introduces combinatorial techniques for these proofs.

Findings

Lattice polytopes cut out by classical root systems are normal.

Such polytopes are also Koszul.

Results extend known type A cases to other classical types.

Abstract

We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.