Toric quiver cells

TL;DR

This paper investigates the algebraic properties of toric ideals associated with quiver polytopes, establishing degree bounds and conditions for quadratic generation and Gröbner bases, especially in low dimensions.

Contribution

It proves that up to dimension four, toric ideals of quiver polytopes are generated in degree two, confirming B{4}gvad's conjecture in these cases, and extends results to compressed polytopes in higher dimensions.

Findings

Toric ideals of quiver polytopes are generated in degree two up to dimension four.

B{4}gvad's conjecture holds for quiver polytopes of dimension at most four.

Compressed polytopes with no neighboring singular vertices have quadratic toric ideals.

Abstract

It is shown that up to dimension four, the toric ideal of a quiver polytope is generated in degree two, with the only exception of the four-dimensional Birkhoff polytope. As a consequence, B{\o}gvad's conjecture holds for quiver polytopes of dimension at most four. In arbitrary dimension, the toric ideal of a compressed polytope is generated in degree two if the polytope has no neighbouring singular vertices. Furthermore, the toric ideal of a compressed polytope with at most one singular vertex has a quadratic Gr\"obner basis.