Monomial and toric ideals associated to Ferrers graphs

By A. Corso, Uwe Nagel. 2006

132 citations recorded by the source corpus.

Each partition A = (λ 1 , λ 2 ,..., An) 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/87ed83495913dfc246e140b2fac6b632d5178e43

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Citations in the corpus

Resolutions of square-free monomial ideals via facet ideals: a survey (2006, background): “Similar results can also be found in the paper of Corso and Nagel [5] in which they study the edge ideals of Ferrers graphs, a class of graphs which includes all complete bipartite graphs.”
A Beginner’s Guide to Edge and Cover Ideals (2013, background): “, ideals of the form I(G) with G 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.”
Edge ideals: algebraic and combinatorial properties (2010, background): “Several authors have studied the regularity of edge ideals of graphs and clutters [18, 20, 53, 74, 75, 77, 81, 90, 103, 106, 114].”
Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers (2006, background): “Of particular relevance to this paper, the minimal graded resolution of I(G) was investigated in [5, 7, 9, 19, 20, 21, 22, 26, 33] (see also [23] for a survey).”