Restricted Chip-Firing: Toric Toppling Ideals, Picard Groups, and Cellular Resolutions

TL;DR

This paper explores algebraic structures from chip-firing games on pargraphs, revealing connections to toric ideals, resolutions, and Picard groups, with new conditions and constructions for these mathematical objects.

Contribution

It introduces new conditions for toppling ideals to be toric, constructs a Gr"obner basis, and analyzes the Picard group of pargraphs.

Findings

Toppling ideals of pargraphs can be toric under certain conditions.

Constructed a Gr"obner basis for the toppling ideal.

Established Cohen-Macaulay property for these ideals.

Abstract

We study certain groups and ideals arising from the chip-firing game on a generalisation of graphs called pargraphs. Several well-known families of toric ideals arise as toppling ideals of pargraphs. These include ideals defining rational normal curves, binomial edge ideals of complete graphs, and toric ideals of certain Fano polytopes. We provide sufficient conditions under which the toppling ideal of a pargraph to be toric. In addition, we construct a Gr\"obner basis for the toppling ideal, a minimal cellular free resolution for a distinguished initial ideal known as the $G$-parking function ideal, and establish Cohen-Macaulay property for these ideals. We also study the Picard group of a pargraph and provide sufficient conditions ensuring its freeness.