Alternative polarizations of Borel fixed ideals (Q2911017)

For a monomial ideal \(I\) of a polynomial ring \(S\), a polarization of \(I\) is a squarefree 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. The present paper defines a nonstandard polarization, called b-pol which is valid for Borel-fixed ideals but not for monomial ideals in general. For Borel fixed ideals it does not coincide with the usual polarization (see e.g. [\textit{J. Herzog} and \textit{T. Hibi}, Monomial ideals. Graduate Texts in Mathematics 260. London: Springer (2011; Zbl 1206.13001)]). The concept of b-pol is present in [\textit{U. Nagel} and \textit{V. Reiner}, Electron. J. Comb. 16, No. 2, Research Paper R3, 59 p. (2009; Zbl 1186.13022)]. The main result of the paper states that b-pol is in fact a polarization for Borel fixed monomial ideals and that it is faithful (Theorem 3.4). In section 4 of the paper b-pol is used to recover and refine results on shifting theory given in the work of \textit{S. Murai}, cf. [Adv. Math. 214, No. 2, 701--729 (2007; Zbl 1128.52009)].