Minimal free resolutions of the 𝐺-parking function ideal and the toppling ideal

Abstract

Abstract. The G-parking function ideal M G of a directed multigraph G is a monomial ideal which encodes some of the combinatorial information of G. It is an initial ideal of the toppling ideal I G , a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy. A minimal free resolution for M G was given by Postnikov and Shaprio in the case when G is saturated, i. e., whenever there is at least one edge (u, v) for every ordered pair of distinct vertices u and v. They also raised the problem of an explicit description of the minimal free resolution in the general case. In this paper, we give a minimal free resolution of M G for any undirected multigraph G, as well as for a family of related ideals including the toppling ideal I G . This settles a conjecture of Manjunath and Sturmfels, as well as a conjecture of Perkinson and Wilmes.

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

“…We remark that the Cohen-Macaulay property of R/M q G and R/I G also follows from the results of [MS14] and [MSW15].…”

“…So one immediately obtains a combinatorial description of the (ungraded) Betti numbers in terms of acyclic partial orientations. This interpretation for the Betti numbers of I G was conjectured in [PPW13] and was proved in [MS14] and [MSW15].…”

Sandpiles and Dominos

“…Other work on ASM and statistical physics includes [12]. In addition, the ASM has been shown to have connections with a wide range of mathematics, including algebraic geometry and commutative algebra ( [2], [10], [7], [23], [22], [34]), pattern formation ( [30], [31], [33], [32], [36]), potential theory ( [3], [4], [20]), combinatorics ( [15], [16], [26], [35]), and number theory ( [28]). The citations here are by no means exhaustive.…”

Another proof of Wilmes’ conjecture

“…The G-parking function ideal was introduced by Cori, Rossin, and Salvy [4] and its resolutions have been studied by Postnikov and Shapiro [15] and Manjunath and Sturmfels [9]. At any rate, Wilmes' conjecture has now been proven several times: by Mohammadi and Shokrieh [12] (for both the monomial and binomial cases), by Manjunath, Schreyer, and Wilmes [8] (again for both the monomial and binomial cases), and by Dochtermann and Sanyal [5] (in the monomial case). Also, Mania [7] has a proof for β 1 in the binomial case.…”

“…The idea is to construct an ideal J, the connected cut-set ideal of G, whose lcm-lattice is dual to the connected 1 The G-parking function ideal can be constructed in various other, equivalent ways. See [7], [12], [8], or [5] for a proof that the x C are in fact the minimal generators of I. 2 The Betti numbers are topological invariants of an R-module that can be read off from a minimal free resolution of that module.…”