Alternative polarizations of Borel fixed ideals, Eliahou-Kervaire type resolution and discrete Morse theory

TL;DR

This paper develops a new cellular minimal free resolution for Borel fixed ideals using discrete Morse theory, providing novel insights into their algebraic and combinatorial structures.

Contribution

It introduces an Eliahou-Kervaire-like resolution for the polarization of Borel fixed ideals, connecting algebraic resolutions with discrete Morse theory.

Findings

Constructed a cellular minimal free resolution for $b-pol(I)$.

Provided new descriptions for resolutions of $I$ and $I^sq$.

Connected algebraic resolutions with discrete Morse theory.

Abstract

We construct an Eliahou-Kervaire-like minimal free resolution of the alternative polarization $b-pol(I)$ of a Borel fixed ideal $I$. It yields new descriptions of the minimal free resolutions of $I$ itself and $I^sq$, where $(-)^sq$ is the squarefree operation in the shifting theory. These resolutions are cellular, and the (common) supporting cell complex is given by discrete Morse theory. If $I$ is generated in one degree, our description is equivalent to that of Nagel and Reiner.