The behaviors of expansion functor on monomial ideals and toric rings

TL;DR

This paper investigates how the expansion functor affects algebraic and combinatorial properties of monomial ideals and toric rings, showing preservation of key properties and exploring implications for White's conjecture.

Contribution

It demonstrates that properties like polymatroidalness and linear quotients are preserved under expansion, and studies the impact on toric ideals and White's conjecture.

Findings

Polymatroidalness preserved under expansion

Toric ideals of expansions maintain double swap generation

White's conjecture is preserved under expansion

Abstract

In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking the expansion functor. The main part of the paper is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid $P$, if toric ideal of $P$ is generated by double swaps then toric ideal of any expansion of $P$ has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gr\"{o}bner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated.