Algebraic aspects of unconditional lattice polytopes

TL;DR

This paper explores the algebraic properties of unconditional lattice polytopes, establishing equivalences with anti-blocking polytopes regarding normality and quadratic generation of toric ideals, with applications to graph theory.

Contribution

It proves that normality and quadratic generation of toric rings and ideals are equivalent for anti-blocking and unconditional lattice polytopes, providing new algebraic insights.

Findings

Toric ring normality of anti-blocking and unconditional polytopes are equivalent.

Toric ideal quadratic generation of anti-blocking and unconditional polytopes are equivalent.

Graph-theoretic characterization of quadratic generation of symmetric stable set ideals.

Abstract

Unconditional polytopes are convex polytopes that are symmetric with respect to all coordinate hyperplanes and arise naturally from anti-blocking polytopes by reflection. This paper investigates algebraic relations between an anti-blocking lattice polytope and its associated unconditional lattice polytope. We prove that the toric ring of an anti-blocking lattice polytope is normal if and only if the toric ring of the associated unconditional lattice polytope is normal. We also show that the toric ideal of an anti-blocking lattice polytope is generated by quadratic binomials if and only if the same holds for the associated unconditional lattice polytope. As an application, we obtain a graph-theoretic characterization of quadratic generation of symmetric stable set ideals.