Polarizations and Hook Partitions

Abstract

In this paper, we relate combinatorial conditions for polarizations of powers of the graded maximal ideal with rank conditions on submodules generated by collections of Young tableaux. We apply discrete Morse theory to the hypersimplex resolution introduced by Batzies-Welker to show that the L-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal is supported on a CW-complex. We then translate the "spanning tree condition" of Almousa-Fløystad-Lohne characterizing polarizations of powers of the graded maximal ideal into a condition about which sets of hook tableaux span a certain Schur module. As an application, we give a complete combinatorial characterization of polarizations of so-called "restricted powers" of the graded maximal ideal.

Detecting Golodness via Gröbner Degeneration

“…In Section 5, we consider a ubiquitous class of monomial ideals which have been referred to as rainbow monomial ideals or facet ideals associated to n-partite nuniform clutters. These ideals have already been studied for their connections to polarizations of Artinian monomial ideals ([AFL22], [AV21]), arithmetically Cohen-Macaulay sets of points in products of projective space, and the combinatorial structure of the homological invariants associated to such ideals ([Nem21], [Van21a]). We prove that any rainbow monomial ideal with linear resolution is Golod; this result encompasses many of the Golodness results known for monomial ideals in the literature, since numerous classes of monomial ideals are obtained by specializing rainbow monomial ideals with linear resolution (see the statement of Corollary 5.14 for a list of these classes of monomial ideals).…”