For a monomial ideal of a polynomial ring , a polarization of is a square-free monomial ideal of a larger polynomial ring such that is a quotient of by a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sends to , ours sends it to . 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.
"Alternative polarizations of Borel fixed ideals." Nagoya Math. J. 207 79 - 93, September 2012. https://doi.org/10.1215/00277630-1630032