The many polarizations of powers of maximal ideals

In this paper, we study different polarizations of powers of the maximal ideal 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 certain polarization of the ideal m^d. When I is a square-free ideal, we show that the Alexander dual of any polarization of I is a polarization of the Alexander dual ideal of I. We apply this theorem and study different polarizations of I_d and its Alexander dual I_{n-d+1} simultaneously. We do this by giving a combinatorial description of the polarizations, which has a natural duality. We study the case of d = 2 and d = n-1 in more detail. Here, we show that there is a one- to-one correspondence between spanning trees of the complete graph K_n and the maximal polarizations of these ideals.