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

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