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 Laplacian ideals, arrangements, and resolutions

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The Laplacian matrix of a graph G describes the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The lattice ideal associated to the lattice generated by the columns of the Laplacian provides an algebraic perspective on this recently reemerging field. This ideal I G has a distinguished monomial initial ideal M G, characterized by the property that the standard monomials are in bijection with the G-parking functions of the graph G. The ideal M G was also introduced by Postnikov and Shapiro 2004 in the context of monotone monomial ideals. We study resolutions of M G and show that a minimal free cellular resolution is supported on the bounded subcomplex of a section of the graphical arrangement of G. This generalizes constructions from Postnikov and Shapiro for the case of the complete graph and connects to work of Manjunath and Sturmfels, and of Perkinson et al. on the commutative algebra of Sandpiles. As a corollary we verify a conjecture of Perkinson et al. regarding the Betti numbers of M G, and in the process provide a combinatorial characterization in terms of acyclic orientations.

Autor: Anton Dochtermann; Raman Sanyal


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