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

 

Trees, parking functions, syzygies, and deformations of monomial ideals


Authors: Alexander Postnikov and Boris Shapiro
Journal: Trans. Amer. Math. Soc. 356 (2004), 3109-3142
MSC (2000): Primary 05C05; Secondary 05A99, 13D02, 13P99
Published electronically: March 12, 2004
MathSciNet review: 2052943
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Abstract: For a graph $G$, we construct two algebras whose dimensions are both equal to the number of spanning trees of $G$. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to $G$-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.


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Additional Information

Alexander Postnikov
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: apost@math.mit.edu

Boris Shapiro
Affiliation: Department of Mathematics, University of Stockholm, Stockholm, S-10691, Sweden
Email: shapiro@matematik.su.se

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03547-0
PII: S 0002-9947(04)03547-0
Keywords: Spanning tree, parking function, abelian sandpile model, monomial ideal, deformation, minimal free resolution, order complex, Hilbert series
Received by editor(s): January 20, 2003
Published electronically: March 12, 2004
Additional Notes: The first author was supported in part by NSF grant DMS-0201494
Article copyright: © Copyright 2004 American Mathematical Society