Skeleta of $G$-parking function ideals

Anton Dochtermann

Published 2017-08-15Version 1

Given a graph $G$, the $G$-parking function ideal $M_G$ is an artinian monomial ideal in the polynomial ring $S$ with the property that a linear basis for $S/M_G$ is provided by the set of $G$-parking functions. It follows that the dimension of $S/M_G$ is given by the number of spanning trees of $G$, which by the Matrix Tree Theorem is equal to the determinant of the reduced Laplacian of $G$. The ideals $M_G$ and related algebras were introduced by Postnikov and Shapiro where they studied their Hilbert functions and homological properties. In previous work it was shown that a minimal resolution of $M_G$ can be constructed from the graphical hyperplane arrangement associated to $G$, providing a combinatorial interpretation for the Betti numbers. Motivated by constructions in the theory of chip-firing on graphs, we study certain `skeleton' ideals $M_G^{(k)} \subset M_G$ generated by subsets of vertices of $G$ of size at most $k+1$. We study monomial bases of $M_G^{(k)}$ and provide formulas and combinatorial interpretations for the dimensions of $S/M_G^{(1)}$ and $S/M_G^{(n-2)}$ for the case that $G = K_{n+1}$ is the complete graph. These monomial bases have connections to various combinatorial objects including Cayley trees and determinants of the signless Laplacians, and in some cases lead to new enumerative formulas. We furthermore study resolutions of $M_G^{(1)}$ and show that for certain $G$ a minimal resolution is supported on decompositions of Euclidean space coming from the theory of tropical hyperplane arrangements. This leads to combinatorial interpretations of the Betti numbers of these ideals.