Laplacian ideals, arrangements, and resolutions

摘要

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 (re)emerging field. This binomial 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 considered by Postnikov and Shapiro (Trans Am Math Soc 356:3109–3142, 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 towork 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.