A diagonal-lattice reduction for the tropical Picard group of a product of graphs

Definitions and mathematical statement

Dustin Cartwright asks for the tropical Picard group, in the sense of the source's reference [13], of a product of graphs, and mentions a guess when one factor is a tree. The intended Picard model matters: the aid uses ridge-supported Cartier classes on a triangulated product surface, not the real-torus Jacobian of a metric graph.

Let $G,H$ be finite connected loopless multigraphs, each with at least one edge so their cubical product is two-dimensional, and let $\Gamma$ be a triangulation choosing one diagonal in each square of $G\times H$, with the intended Cartwright tropical structure. For a graph $G$, put \[ \operatorname{Pic}(G)={\mathbb{Z}}^{V(G)}/L_G{\mathbb{Z}}^{V(G)},\qquad K(G)=\operatorname{Div}^0(G)/L_G{\mathbb{Z}}^{V(G)}, \] where $L_G$ is its graph Laplacian and $\operatorname{Div}^0(G)$ consists of coefficient vectors with sum zero. Thus $\operatorname{Pic}(G)\cong{\mathbb{Z}}\oplus K(G)$, with the splitting requiring a degree-one choice.

Write $\mathcal C_\Gamma\subseteq{\mathbb{Z}}^{E(\Gamma)}$ for the integral ridge-supported Cartier-divisor lattice, $\mathcal P_\Gamma\subseteq\mathcal C_\Gamma$ for its principal-divisor lattice, and $\operatorname{Pic}_{\mathrm{ridge}}(\Gamma)=\mathcal C_\Gamma/\mathcal P_\Gamma$. The factor map $\gamma:\operatorname{Pic}(G)\oplus\operatorname{Pic}(H)\to\operatorname{Pic}_{\mathrm{ridge}}(\Gamma)$ is induced by the two factor pullbacks. The aid attributes to Lazar (2017) its injectivity and its being an isomorphism if either factor is a tree $T$. In that case it reports \[ \operatorname{Pic}_{\mathrm{ridge}}(G\times T) \cong\operatorname{Pic}(G)\oplus{\mathbb{Z}} \cong{\mathbb{Z}}^2\oplus K(G). \] These are supplied literature claims, with product triangulation/structure conventions implicit.

For the aid's further reduction, let $D$ be the chosen diagonal edges, indexed by $E(G)\times E(H)$, and let $\pi_D$ restrict ridge coefficients to $D$. Define $\Lambda_\Gamma=\pi_D(\mathcal C_\Gamma)$ and $Q_\Gamma=\pi_D(\mathcal P_\Gamma)$. With incidence matrices $B_G\in{\mathbb{Z}}^{E(G)\times V(G)}$ and $B_H\in{\mathbb{Z}}^{E(H)\times V(H)}$, the aid identifies $Q_\Gamma$ with the image of $B_G\otimes B_H$ after a row sign change determined by diagonal choices, and claims that this image is primitive in ${\mathbb{Z}}^D$. It records the exact sequence \[ 0\longrightarrow\operatorname{Pic}(G)\oplus\operatorname{Pic}(H) \xrightarrow{\gamma}\operatorname{Pic}_{\mathrm{ridge}}(\Gamma) \longrightarrow\Lambda_\Gamma/Q_\Gamma\longrightarrow0. \] Accordingly its quotient is free abelian and the factor map identifies \[ \operatorname{Tor}\operatorname{Pic}_{\mathrm{ridge}}(\Gamma) \cong K(G)\oplus K(H). \] The remaining general computation is the free rank of $\Lambda_\Gamma/Q_\Gamma$. The local Cartier conditions and tropical structure constants needed to compute $\mathcal C_\Gamma$ are not supplied, so this is a reduction, not an explicit Picard-group formula for every product.

Short English statement

Compute the tropical Picard group of a graph product. In the ridge-supported surface model, the supplied aid reports a tree-factor formula and reduces the general case to a free diagonal-lattice quotient, with torsion given by the two graph critical groups.

Sources

• AIM workshop Generalizations of chip-firing and the critical group, problem 31, Dustin Cartwright: https://aimath.org/pastworkshops/chipfiringproblems.pdf. The intended tropical Picard group is referred to reference [13], whose full bibliographic entry is absent.
• The attached aid attributes the injective factor map and tree-factor theorem to Lazar (2017), and mentions Lazar's stronger conjecture, without a title, complete citation, or its full statement.
• Local registry data/unsolvedmath/problems.json, permanent ID 20000996, exact source code catalogue entry; canonical location aim-combinatorics-notes.json notes[120]. Attached attempt 1, partial result, problem status at run partially solved, rigor proved, low novelty confidence; machine-generated, not peer reviewed, requiring independent verification before citation.