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In this paper, we study different polarizations of powers of the maximalideal m^d, and polarizations of its related square-free version I_d. For n = 3,we show that every minimal free cellular resolution of m^d comes from a certainpolarization of the ideal m^d. When I is a square-free ideal, we show that theAlexander dual of any polarization of I is a polarization of the Alexander dualideal of I. We apply this theorem and study different polarizations of I_d andits Alexander dual I_{n-d+1} simultaneously. We do this by giving acombinatorial description of the polarizations, which has a natural duality. Westudy the case of d = 2 and d = n-1 in more detail. Here, we show that there isa one- to-one correspondence between spanning trees of the complete graph K_nand the maximal polarizations of these ideals.

The many polarizations of powers of maximal ideals