Alternative polarizations of Borel fixed ideals

TL;DR

This paper introduces a new type of polarization for Borel fixed ideals, expanding the theory of squarefree monomial ideals and refining existing results in shifting theory of simplicial complexes.

Contribution

It presents a non-standard polarization method for Borel fixed ideals, generalizing previous work and offering a novel approach distinct from prior methods.

Findings

Introduces non-standard polarization for Borel fixed ideals

Refines results on squarefree operation in shifting theory

Generalizes a result of Nagel and Reiner

Abstract

For a monomial ideal $I$ of a polynomial ring $S$, a "polarization" of $I$ is a \textit{squarefree} monomial ideal $J$ of a larger polynomial ring $S'$ such that $S/I$ is a quotient of $S'/J$ by a regular sequence (consisting of degree 1 elements). We show that a Borel fixed ideal admits a "non-standard" polarization. For example, while the standard polarization sends $xy^2 \in S$ to $x_1y_1y_2 \in S'$, ours sends it to $x_1y_2y_3$. Using this idea, we recover/refine the results on "squarefree operation" in the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, while our approach is very different from theirs.