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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Monomial and toric ideals associated to Ferrers graphs


Authors: Alberto Corso and Uwe Nagel
Journal: Trans. Amer. Math. Soc. 361 (2009), 1371-1395
MSC (2000): Primary 05A15, 13D02, 13D40, 14M25; Secondary 05C75, 13C40, 13H10, 14M12, 52B05
Published electronically: October 17, 2008
MathSciNet review: 2457403
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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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Additional Information

Alberto Corso
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: corso@ms.uky.edu

Uwe Nagel
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: uwenagel@ms.uky.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04636-9
PII: S 0002-9947(08)04636-9
Received by editor(s): January 22, 2007
Published electronically: October 17, 2008
Additional Notes: The second author gratefully acknowledges partial support from the NSA under grant H98230-07-1-0065
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.