Monomial and toric ideals associated to Ferrers graphs

TL;DR

This paper studies Ferrers ideals from bipartite graphs, showing they have a 2-linear resolution, characterizing Ferrers graphs, and providing explicit descriptions of their algebraic and geometric properties using polyhedral complexes.

Contribution

It characterizes Ferrers graphs via 2-linear resolutions, describes minimal free resolutions explicitly, and analyzes their toric rings with new algebraic methods.

Findings

Ferrers ideals have 2-linear minimal free resolutions

Explicit description of resolution maps via polyhedral complexes

Formulas for Hilbert series, regularity, and multiplicity of toric rings

Abstract

Each partition $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$ determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution, i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.