Laplacian ideals, arrangements, and resolutions

Dochtermann, Anton ; Sanyal, Raman

Journal of Algebraic Combinatorics, Tome 40 (2014) no. 3, pp. 805-822.

Résumé

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.

Détail

Classification : 05E40, 05C25, 05C50, 13D02, 52C35
Keywords: graph Laplacian, chip-firing, lattice ideal, initial ideal, $G$-parking function, cellular resolution, graphical arrangement, acyclic orientation