A chip-firing game on the product of two graphs and the tropical Picard group (Q2411499)

Summary: \textit{D. Cartwright} [``Combinatorial tropical surfaces'', Preprint, \url{arXiv:1506.02023}] introduced the notion of a weak tropical complex in order to generalize the theory of divisors on graphs from \textit{M. Baker} and \textit{S. Norine} [Adv. Math. 215, No. 2, 766--788 (2007; Zbl 1124.05049)]. A weak tropical complex \(\Gamma\) is a \(\Delta\)-complex equipped with algebraic data that allows it to be viewed as the dual complex to a certain kind of degeneration over a discrete valuation ring. Every graph has a unique tropical complex structure (which is the same structure studied by Baker and Norine) in which divisors correspond to states in the chip-firing game on that graph. Let \(G\) and \(H\) be graphs, and let \(\Gamma\) be a triangulation of \(G\times H\) obtained by adding in one diagonal of each resulting square. There is a particular weak tropical complex structure on \(\Gamma\) that Cartwright conjectured was closely related to the weak tropical complex structures on \(G\) and \(H\). The main result of this paper is a proof of Cartwright's conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.