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Laplacian ideals, arrangements, and resolutions. (English) Zbl 1303.05217
Summary: 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 A. Postnikov and B. Shapiro [Trans. Am. Math. Soc. 356, No. 8, 3109–3142 (2004; Zbl 1043.05038)] 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 M. Manjunath and B. Sturmfels [J. Algebr. Comb. 37, No. 4, 737–756 (2013; Zbl 1272.13017)] and of D. Perkinson et al. [“Primer for the algebraic geometry of sandpiles”, Preprint, arXiv:1112.6163] on the commutative algebra of sandpiles. As a corollary, we verify a conjecture of Perkinson et al. [loc. cit.] regarding the Betti numbers of \(M_G\) and in the process provide a combinatorial characterization in terms of acyclic orientations.

MSC:
05E40 Combinatorial aspects of commutative algebra
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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