Geodesic
Alternative polarizations of Borel fixed ideals, Eliahou-Kervaire type resolution and discrete Morse theory
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 2, pp. 407-436.

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Summary: We construct an Eliahou-Kervaire-like minimal free resolution of the alternative polarization $\operatorname{\mathsf{b-pol}}(I)$ 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.
Keywords: Borel fixed ideal, squarefree strongly stable monomial ideal, Eliahou-Kervaire resolution, discrete Morse theory 
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     author = {Okazaki, Ryota and Yanagawa, Kohji},
     title = {Alternative polarizations of {Borel} fixed ideals, {Eliahou-Kervaire} type resolution and discrete {Morse} theory},
     journal = {Journal of Algebraic Combinatorics},
     pages = {407--436},
     publisher = {mathdoc},
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     year = {2013},
     language = {en},
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Okazaki, Ryota; Yanagawa, Kohji. Alternative polarizations of Borel fixed ideals, Eliahou-Kervaire type resolution and discrete Morse theory. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 2, pp. 407-436. http://geodesic.mathdoc.fr/item/JAC_2013__38_2_a3/