Monomial and toric ideals associated to Ferrers graphs

Corso, Alberto; Nagel, Uwe

doi:10.1090/S0002-9947-08-04636-9

Monomial and toric ideals associated to Ferrers graphs
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by Alberto Corso and Uwe Nagel PDF

Trans. Amer. Math. Soc. 361 (2009), 1371-1395 Request permission

Abstract:

Each partition $\lambda = (\lambda _1, \lambda _2, \ldots , \lambda _n)$ 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 formulæ 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.

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