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Minimal free resolutions of the $ G$-parking function ideal and the toppling ideal


Authors: Madhusudan Manjunath, Frank-Olaf Schreyer and John Wilmes
Journal: Trans. Amer. Math. Soc. 367 (2015), 2853-2874
MSC (2010): Primary 13D02; Secondary 05C25
DOI: https://doi.org/10.1090/S0002-9947-2014-06248-X
Published electronically: September 24, 2014
MathSciNet review: 3301884
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Abstract: The $ G$-parking function ideal $ M_G$ of a directed multigraph $ G$ is a monomial ideal which encodes some of the combinatorial information of $ G$. It is an initial ideal of the toppling ideal $ I_G$, a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy. A minimal free resolution for $ M_G$ was given by Postnikov and Shapiro in the case when $ G$ is saturated, i.e., whenever there is at least one edge $ (u,v)$ for every ordered pair of distinct vertices $ u$ and $ v$. They also raised the problem of an explicit description of the minimal free resolution in the general case. In this paper, we give a minimal free resolution of $ M_G$ for any undirected multigraph $ G$, as well as for a family of related ideals including the toppling ideal $ I_G$. This settles a conjecture of Manjunath and Sturmfels, as well as a conjecture of Perkinson and Wilmes.


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Additional Information

Madhusudan Manjunath
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: mmanjunath3@math.gatech.edu

Frank-Olaf Schreyer
Affiliation: Mathematik und Informatik, Universität des Saarlanes, 66123 Saarbrucken, Germany
Email: schreyer@math.uni-sb.de

John Wilmes
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: wilmesj@math.uchicago.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06248-X
Received by editor(s): January 8, 2013
Received by editor(s) in revised form: May 14, 2013
Published electronically: September 24, 2014
Additional Notes: Part of the work on this project was done while the first author was affiliated with Fachrichtung Mathematik, Universität des Saarlandes, Germany.
The third author was supported in part by NSF Grant No. DGE 1144082.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.