The many polarizations of powers of maximal ideals

TL;DR

This paper explores various polarizations of powers of maximal ideals and their square-free counterparts, revealing dualities and combinatorial structures, including a correspondence with spanning trees for specific cases.

Contribution

It provides a comprehensive study of polarizations of maximal ideals and their square-free versions, establishing duality results and a novel connection to spanning trees.

Findings

Every minimal free cellular resolution of m^d arises from a polarization when n=3.

The Alexander dual of any polarization of a square-free ideal is a polarization of its Alexander dual.

A one-to-one correspondence exists between spanning trees of K_n and maximal polarizations for d=2 and d=n-1.

Abstract

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.