Alternative polarizations of Borel fixed ideals

Abstract

For a monomial ideal $I$ of a polynomial ring $S$, a polarization of $I$ is a square-free monomial ideal $J$ of a larger polynomial ring $\tilde{S}$ such that $S/I$ is a quotient of $\tilde{S}/J$ by a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sends $x{y}^2\in S$ to $x_1{y}_1{y}_2\in \tilde{S}$, ours sends it to $x_1{y}_2{y}_3$. Using this idea, we recover/refine the results on square-free operation in the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.