Monomial and toric ideals associated to Ferrers graphs

Abstract

Abstract. Each partition λ = (λ 1 , λ 2 , . . . , λ n ) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a 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.

Specializations of Ferrers ideals

“…A Ferrers ideals is the edge ideal associated with a Ferrers graph. See [4] and [13] for additional details.…”

A Beginner’s Guide to Edge and Cover Ideals

“…Finally, if you are interested in learning more about the regularity of edge ideals, the following papers should be part of your reading list: [12,32,44,46,50,62,65,66,68].…”

“…Biermann [2] looked at the resolutions of I(C c n ), i.e., the edge ideal of the complement of cycles; Chen [6] and Horwitz [41] examined the case of ideals with linear resolutions, i.e., ideals of the form I(G) with G c chordal; Corso and Nagel [11,12] considered the case of Ferrers graph; and Dochtermann and Engström [16] studied the cellular resolutions of co-interval graphs. I do not know of any work on the structure of cover ideals of graphs.…”

Associated primes of monomial ideals and odd holes in graphs

“…Many researchers have been interested in using the edge ideal construction to build a dictionary between the fields of graph theory and commutative algebra. For general references, see [26,30,31]; for invariants encoded in the resolution, see [5,7,16,17,18,22,24]; for classes of (sequentially) Cohen-Macaulay graphs, see [9,12,20,29].…”