On the polymatroidal property of monomial ideals with a view towards orderings of minimal generators

TL;DR

This paper characterizes polymatroidal monomial ideals generated in a single degree through linear quotients under lexicographical orderings and explores a conjecture relating to reverse lexicographical orderings.

Contribution

It establishes a characterization of polymatroidal ideals via linear quotients for all variable orderings and proposes a conjecture for reverse lexicographical orderings, proving it in special cases.

Findings

Polymatroidal ideals are characterized by linear quotients under lex orderings.

The paper conjectures a similar characterization for reverse lex orderings.

The conjecture is proved in several special cases.

Abstract

We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also conjecture that the polymatroidal ideals can be characterized with linear quotients property with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables. We prove our conjecture in many special cases.