A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group

In his preprint https://arxiv.org/abs/1308.3813, Cartwright introduced the notion of a weak tropical complex in order to generalize the concepts of divisors and the Picard group on graphs from Baker and Norine's paper Riemann-Roch and Abel-Jacobi Theory on a Finite Graph. A tropical complex $\Gamma$ is a $\Delta$-complex equipped with certain algebraic data. Divisors in a tropical complex are formal linear combinations of ridges, and piecewise-linear functions on a tropical complex give rise in a natural way to divisors. Divisors that arise from PL-functions are called principal, and divisors that are locally principal are called Cartier. Two divisors that differ by a principal divisor are said to be linearly equivalent. The linear equivalence classes of Cartier divisors on a tropical complex $\Gamma$ form a group called the Picard group of $\Gamma$, by analogy to the definition of the Picard group of a variety in algebraic geometry. Every graph has a unique tropical complex structure. If $G$ and $H$ are graphs, and $\Gamma$ is a triangulation of their product, then $\Gamma$ has a weak tropical complex structure that is compatible with the tropical complex structures on $G$ and $H$. Thus, divisors on $\Gamma$ can be thought of as states in a higher-dimensional chip-firing game on $\Gamma$. Cartwright conjectured that the Picard groups of $\Gamma$, $G$, and $H$ were closely related. Let $Pic(\Gamma)$ be the tropical Picard group of $\Gamma$, and $Pic(G)$ and $Pic(H)$ be the tropical Picard groups of $G$ and $H$. Then, it was conjectured that there is a map $\gamma: Pic(G) \times Pic(H) \to Pic(\Gamma)$ that is always injective and is surjective if at least one of $G$ or $H$ is a tree. In this paper, we prove the conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.


Introduction
The chip-firing game is a well-studied subject in combinatorics with deep connections to other areas of mathematics, including algebraic geometry. In [1], Baker and Norine demonstrated an analogy between the chip-firing game and linear equivalence of divisors on a Riemann surface. In the same paper, they proved a graph-theoretic analogue of the Riemann-Roch theorem from algebraic geometry. In fact, beyond these analogies, there is a fundamental connection between graph theory and algebraic geometry.
For example, let X t be a family of smooth curves over C, parametrized by t, that becomes a singular curve when t = 0 (this is an instance of a "degeneration" of curves). We can associate a graph to X 0 , and divisors on X 0 descend to chip-firing states on that graph, with chip-firing moves on the graph corresponding to linear equivalence of divisors on X 0 . Thus, one can address questions in algebraic geometry by studying the chip-firing game.
Furthermore, one can generalize this sort of reasoning to higher-dimensional objects. In his preprint [7], Cartwright introduced the notion of a weak tropical complex in order to generalize the concepts of divisors and the Picard group on graphs from [1]. A weak tropical complex Γ is a ∆-complex equipped with algebraic data that allows Γ to be viewed as the dual complex of a particular kind of degeneration over a discrete valuation ring.
Within the context of weak tropical complexes, the analogue of the chip-firing game is the theory of divisors. A divisor on a weak tropical complex is a formal linear combination of codimension-1 polyhedral subsets (which we can think of as a higher-dimensional "chip configuration"), and two divisors are linearly equivalent if they differ by a "tropical principal divisor" (which we can think of as encoding a chip-firing move). One important invariant in this theory is the tropical Picard group, which consists of a certain set of tropical divisors up to linear equivalence.
Every finite graph has a unique tropical complex structure. In this tropical complex structure, divisors correspond to states in a certain form of the chip-firing game on that graph. If G and H are graphs, and Γ is a triangulation of their product obtained by adding in a diagonal of each resulting square, then Γ has a weak tropical complex structure that is compatible with the tropical complex structures on G and H. Motivated by analogous results in algebraic geometry (e.g., [10, Theorem 1.7]), Cartwright conjectured that the Picard groups of Γ, G, and H were closely related. Theorem 1.1 (Main Theorem). Let Pic(Γ) be the tropical Picard group of Γ, and Pic(G) and Pic(H) be the tropical Picard groups of G and H. Then, there is a map γ : Pic(G) × Pic(H) → Pic(Γ) that is always injective and is surjective if at least one of G or H is a tree.
As we shall see, the proof of this theorem is independent of the choices made in constructing Γ, although the cokernel of the map γ may vary as the triangulation changes.
In this paper, we prove a seemingly weaker form of the conjecture where we restrict our attention to "ridge divisors" -divisors that are formal linear combinations of ridges of Γ. Due to computations in sheaf cohomology (see [6, Section 3]), the ridge divisor form of the conjecture implies the more general form.

Divisors on Tropical Complexes.
For the rest of this section, let (Γ, α) be a weak tropical complex. A divisor on Γ is a formal Z-linear combination of the ridges of Γ. If C is a divisor on Γ and r is a ridge, we write C(r) for the coefficient of r in C.
A piecewise linear (PL) function on Γ is a continuous piecewise linear function φ that restricts to a linear function with integer slope on each simplex in Γ. We can associate to each PL function φ a divisor Div(φ) as follows: Definition 2.2. A tropical principal divisor is a divisor that can be written as Div(φ) for some PL function φ. We define Prin(Γ) to be the group of principal divisors on Γ.
For any vertex v ∈ Γ, we define φ v to be the unique PL function that is 1 on v and 0 on all other vertices of Γ. We note that Prin(Γ) is generated by {Div(φ v ) | v ∈ V (Γ)}, since our PL functions are uniquely specified by their values on V (Γ). 2 We are interested in divisors that are "locally principal", which we make precise in the following sense: Definition 2.3. A tropical Cartier divisor is a formal Z-linear combination D of ridges of Γ such that for every v ∈ V (Γ), there exists a PL function φ such that D and Div(φ) agree on all ridges containing v. We let Cart(Γ) be the group of Cartier divisors on Γ.
Definition 2.4. A tropical Q-Cartier divisor is a divisor D such that mD is Cartier for some m ∈ Z \{0}. We let QCart(Γ) be the group of Q-Cartier divisors on Γ.
If two divisors differ by a principal divisor, they are said to be linearly equivalent. Thus, Pic(Γ) is the group of linear equivalence classes of Cartier divisors.
Note. As alluded to in the introduction, the definitions in this section (including that of a piecewise linear function on a weak tropical complex) are more restrictive than those in [7]. In [7], the divisors above are called ridge divisors, and the group of Cartier ridge divisors modulo principal ridge divisors is denoted Pic ridge (Γ). Since we are only dealing with ridge divisors in this paper, we omit the word "ridge".  We put a weak tropical complex structure α on Γ, which we write in the form of a matrix A whose (i, j)th entry is α(e i , v j ): We recall that any PL function φ is uniquely specified by its values on the vertices of Γ. Thus, Equation (1) tells us that we can write the homomorphism Div : Z V (Γ) → Z E(Γ) as the following matrix D: The group Prin(Γ) consists of the column-span of the matrix D. Of particular interest is the case of one-dimensional (weak) tropical complexes, i.e., graphs.
Let G be a loopless connected graph, possibly with multiple edges between a given pair of vertices, with vertex set V (G) and edge set E(G). Restricting Definition 2.1 to dimension 1, a 1-dimensional weak tropical complex structure on G is a function α : Thus, we see that G admits exactly one tropical complex structure. Furthermore, if we take the PL function φ v as in Section 2.1, we see that where adj(v, w) is the number of edges between v and w. Since {φ v | v ∈ V (G)} spans the module of PL functions on G, we can express the map Div as a |V (G)| × |V (G)| matrix L(G), with This matrix is exactly the Laplacian matrix of G.
. This condition is always true, so every divisor on G is Cartier.
Since Prin(G) is exactly the column-span of L(G), we see that Pic(G) ∼ = coker(L(G)). Therefore, by the Matrix-Tree Theorem, Pic(G) ∼ = Z ⊕K(G), where K(G) is a finite group called the critical group of G.

Balancing Conditions
For the rest of this paper, we assume without loss of generality that all graphs are connected. Given a pair of graphs G and H, the product G × H is a cubical complex -a cell complex whose 0-cells come from pairs of vertices, whose 1-cells come from vertex-edge pairs, and whose 2-cells are squares arising from pairs of edges. For much of this paper, we will consider triangulations Γ of G × H, obtained subdividing each square into two triangles. We call the edges of the form (edge of G)×(vertex of H) horizontal edges, edges of the form (vertex of G)×(edge of H) vertical edges, and new edges added in this triangulation diagonal edges. We define Diag(Γ) to be the set of diagonal edges of Γ. If σ is a square or a triangle, we define diag(σ) to be the unique diagonal edge contained in σ.
In the preceding figure, E 1 × b is a horizontal edge, 0 × E 2 is a vertical edge, and R is a diagonal edge.
We define a natural weak tropical complex structure on a triangulation Γ of G×H as follows (this construction is due to Cartwright [4]; see also [5, Example 6.2]). Let v ∈ V (Γ) and e ∈ E(Γ), and write F (Γ) for the set of 2-dimensional faces of Γ. Then, The main result of this section is the following criterion for a divisor on Γ to be Q-Cartier. For any graph G and any vertex a of G, define N G (a) to be the set of neighbors of a. Note that this is a special form of the balancing condition mentioned in [7, Section 5], but we prove it here for completeness.
is independent of the choice of y.
Proof. We recall that a tropical Cartier divisor D is precisely one that is locally a principal divisor at every vertex v in G × H. By "locally" we mean that there is some principal divisor that agrees with the restriction of D on the graph star E Γ (v) of v -the union of the collection of edges containing v.
So, suppose we have a Cartier divisor D on Γ. Fix an ordering v 1 , . . . , v n of V (Γ) and an ordering e 1 , . . . , e m of E(Γ). The set of principal divisors on Γ is precisely the column span of the (m × n) matrix M whose entries i, j are given by: The matrix M is the negation of the matrix of the map Div in Equation (1). The negation does not affect the column span, and is more convenient for the purposes of this proof.
We note that every triangle containing the edge (ab, xb) is either of the form {ab, xb, xc} or of the form {ab, xb, ac}, with c ∈ N H (b). Triangles of the former type must include the diagonal edge (ab, xc), while triangles of the latter type contain the diagonal edge (ac, xb). In other words, α((ab, xb), ab) counts the number of triangles containing (ab, xb) of the latter type, while every triangle of the former type containing (ab, xb) gives rise to a diagonal edge containing ab.
so we see that Case 2: w = a b . In this case, the edge (ab, a b ) is a diagonal edge. Thus, we can compute D explicitly: We note that for all of the cases above, analogous arguments would hold for Υ D ab (y). We will construct a submatrix of M v with δ(v) rows, and show that some δ(v) × δ(v) minor of this submatrix does not vanish. This will show that rank(M v ) ≥ δ(v). 7 We see that R v is a δ(v) × δ(v) square matrix, with the following block form: Let x be a vertex and Q be an edge. Then, On the other hand, I is an identity matrix. E i and E j share no common triangles when i = j (so E i is never in link(d j ) or vice-versa), and α(d i , E i ) = 1 for all i by definition of α for diagonal edges.
Since E is the only diagonal edge that shares a square with both U and H, we see that B has the following form: where {E 1 , . . . , E k } are the diagonal edges containing v that are in link(u) and {E k+1 , . . . , E k+ } are the diagonal edges containing v that are in link(h).

Now,
We see where k = #{i : E i ∈ link(u)}, and = #{i : E i ∈ link(h)}. Thus, i.e. the number of triangles of the form shown below. Case 2: v is not contained in any diagonal edges. In this case, δ(v) = 2, so we need to find some nonvanishing 2 × 2 minor of M v . Let U be a vertical edge containing v, and let H be a horizontal edge containing v. We write U = vu and H = vh, and we let R v be the 2 × 2 submatrix of M v whose columns are indexed by v and h respectively, and whose rows are indexed by U and H, respectively. Then, The characterization of Q-Cartier divisors given by balancing conditions is useful both as a technical tool (as will be demonstrated in later proofs), and as a means of making tropical Q-Cartier divisors more understandable. We have already seen that the principal divisors on a weak tropical complex are the vectors in the column span of an easily-constructed matrix, and the balancing conditions allow us to construct a matrix whose kernel consists of the Q-Cartier divisors. We rearrange all of these equations so that one side is equal to 0, and hence can express a Q-Cartier divisor D as a vector in the kernel of the 6 × 16 matrix C whose columns are labeled by the edges of Γ and whose rows are the characteristic vectors of the equations above.
We can also express the principal divisors on Γ in terms of a matrix: the ridges of Γ are the edges of Γ, and a P L-function on Γ is uniquely determined by the values it takes on the vertices of Γ. We define a matrix P with columns indexed by vertices of Γ and rows indexed by edges of Γ, with the column corresponding to a vertex v given by Div(φ v ).
Thus, the divisor class group of Γ can be viewed as ker(C)/Im(P ). Using a computer algebra system (in this instance, Sage [11]), we see that Cl(Γ) Z 2 . Note that Pic(G) × Pic(H) ∼ = Pic(Γ) ∼ = Z 2 , by the main theorem of this paper (1.1) since G and H are both trees. In fact, Proposition 4.3 implies that Cl(Γ) and Pic(Γ) coincide in this case. (for convenience, we write φ : G → H) is a set map such that A morphism φ : G → H is harmonic if, for all x ∈ V (G) and y ∈ V (H) such that φ(x) = y, the quantity |{e ∈ E(G) | x ∈ e, φ(e) = e }| is independent of the choice of edge e containing y. Now, let G and H be simple graphs, let Γ be a triangulation of G × H as in this section, and let S be the 1-skeleton of Γ (i.e. the graph consisting of the vertices of Γ and the edges of Γ). We observe that there is only one graph morphism from S to G that acts as projection onto the first coordinate when restricted to the vertex set V (S) = V (G) × V (H).
Indeed, suppose that v ∈ V (G), w ∈ V (H), and that φ G is a graph morphism from S to G with φ G (v, w) = v. If e ∈ E(S) is a vertical edge of Γ containing (v, w), then φ G (e) = φ(v, w) = v, since the other endpoint of e is also mapped to v under φ (and by assumption G contains no loops). If e ∈ E(S) is a horizontal or diagonal edge of Γ that contains (v, w), then the other endpoint of e is mapped by φ G to some neighbor v 10 of v, so e must be mapped to the edge vv ∈ E(G). Similarly, there is a unique graph morphism φ H from S to H that restricts to projection onto the second coordinate on the vertex set of S.
The morphisms φ G and φ H are not harmonic in general. However, the balancing conditions for a Q-Cartier divisor D on Γ are equivalent to saying that for any vertex v ∈ V (G) and w ∈ V (H), the sums are independent of the choice of e ∈ E(G) with v ∈ E and the choice of f ∈ E(H) with w ∈ f . Thus, we see that the divisor is Q-Cartier if and only if the maps φ G and φ H are harmonic.

Cartwright's Conjecture
Let G and H be graphs, and Γ a triangulation of G × H. Cartwright defined a map β : Div(G) × Div(H) → Div(Γ) [4] as follows. Let C be a divisor on G and D be a divisor on H. Then, where (v, r) and (e, w) are edges in Γ. We observe that β is injective. Fix some vertex v ∈ H. By the definition of β, we know that where adj(v, v ) is the number of edges between v and v in H.
We claim that

11
where P (w, v) = Div(φ wv ), and φ wv is the PL function that is 1 at vertex wv and 0 at all other vertices. We know that For every r ∈ E(Γ) we will show that the coefficient of [r] in β(0, L v ) is equal to the coefficient of [r] in the right-hand side of (4). We treat the cases of diagonal, vertical, and horizontal edges separately.
Case 1: r ∈ Diag(Γ). We know that the coefficient in can only occur in terms A or B. There is exactly one square containing r (since r is a diagonal edge), and it has two possible forms: Thus, we get a contribution of −[r] either from P (w 1 , v) (in the second case), or from P (w 2 , v) (in the first case), and a contribution of [r] from the other, so the coefficient of [r] on the right-hand side of (4) is 0.
Case 2: r is a vertical edge of Γ. We observe that β(0, L v )(r) = 0. For any fixed w in V (G), [r] can only occur in terms A or C in Equation (4).
Consider a square in Γ containing r. Again, it has exactly two possible forms: The left square, S, contributes 0 to the coefficient of [r], because w 1 v is on diag(S), so S not counted in α(r, w 1 v), and r / ∈ link(w 2 v). The right square, T , contributes 1 to α(r, w 1 v) in sum C of P (w 1 , v) since w 1 v is not on the diagonal of T , but it contributes −1 to α(r, w 2 v) in sum A of P (w 2 , v), since r ∈ link Γ (w 2 v).
Case 3: r is a horizontal edge of Γ.
Case 3b: wv / ∈ r, r ∈ link Γ (wv) for some w ∈ V (G). We can write r = (e, v ) for e ∈ E(G), v ∈ N H (v). Each square containing r can have one of two forms: For each w ∈ V (G) such that r ∈ link Γ (wv), the term [r] only appears in sum A of P (w, v). For every edge in H connecting v and v , there is a square of the form {w 1 v, w 2 v, w 1 v , w 2 v }, which contributes −1 to the coefficient of [r] -r is in exactly one of link Γ (w 1 v) or link Γ (w 2 v). Thus, the coefficient of [r] in the right-hand side of equation (4) is −adj(v, v ), as we desired.
Case 3c: w 1 v ∈ r for some w 1 ∈ V (G). Each square in Γ containing r has one of the two following forms: In equation (4) v is fixed, so [r] can appear only in P (w 1 , v) and P (w 2 , v). Furthermore, [r] appears only in sum C. Each square S containing r contributes 1 to exactly one of α(r, w 1 v) or α(r, w 2 v) -the diagonal edge of S contains exactly one of (w 1 , v) and (w 2 , v). Thus, the coefficient of [r] in Equation (4) (4) is deg(v), as desired.
Finally, it is clear that if we switch the roles of G and H, the same argument gives us that β(L w , 0) is a principal divisor on Γ. Proof. Fix a vertex (v, w) of Γ. We wish to show that there is some principal divisor P on Γ that agrees with L on all of the edges of Γ containing (v, w). By the definition of β, we know that for all e ∈ E(G) L(e, w) := D(w) and that for all f ∈ E(H), L(v, f ) := C(v). We also know that L ≡ 0 on all of the diagonal edges containing (v, w).
Now, since every divisor on a graph is Cartier, there exist principal divisors P 1 on G and P 2 on H such that P 1 (v) = C(v) and P 2 (w) = D(w). Clearly, P := β(P 1 , P 2 ) is a divisor on Γ that agrees with L on all of the edges of Γ containing (v, w). We have already showed that β : Prin(G) × Prin(H) → Prin(Γ), so we are done.
Since every divisor on a graph is a Cartier divisor, we see that β : Cart(G) × Cart(H) → Cart(Γ). Proof. The ⊆ direction is the result of the previous proposition. For the ⊇ direction, let P be a Cartier divisor on Γ that assigns the value 0 to every diagonal edge and let (a, b) be a vertex in Γ. Equation (2a) says Ξ P ab (x) = D(xb, ab) is independent of the choice of x ∈ N G (a). This tells us that P is constant on star G (a) × {b}. If a ∈ V (G), the same argument says that P is constant on star G (a ) × {b}. Since G is connected, this implies that P is constant on G × {b}. We call this quantity Horiz(b).
A similar argument using Υ P ab shows that P is constant on {a} × H. We call this quantity Vert(a). We write We now come to the main theorem. This was conjectured by Cartwright at the AIM workshop "Generalizations of chip-firing and the critical group" [4], and was motivated by analogous results in algebraic geometry -if X 1 and X 2 are varieties, Pic(X 1 ) × Pic(X 2 ) → Pic(X 1 × X 2 ), and Pic(X 1 × P 1 ) = Pic(X 1 ) (see, for instance, [10, Theorem 1.7]). On the other hand, if C is a principal divisor, we can write C = Div(φ) for some PL function φ. Consider a square S = {a, b, c, d} as pictured below, with diag(S) = r. By equation (1) If C ∈ ePrin(Γ), we have φ(a) + φ(c) = φ(b) + φ(d). Recall that every PL function φ can be viewed as a vector in Z V (Γ) . Let W be the vector space of PL functions on Γ that map to divisors in ePrin(Γ) under Div. Then, if φ ∈ W , we see that every square in Γ gives rise to a linear constraint on the entries in the vector φ. We observe that ker(Div) ⊆ W .
We claim that the codimension of W is at least (|V (G)| − 1)(|V (H)| − 1). To see this, let T 1 be a spanning tree in G, and T 2 be a spanning tree of H. Then T 1 × T 2 gives rise to a subcomplex of Γ with (|V (G)| − 1)(|V (H)| − 1) squares (since T 1 has |V (G)| − 1 edges and T 2 has |V (H)| − 1 edges). Fix a root v 1 of T 1 , and take some linear extension of the partial order on the vertices of T 1 arising from this choice of root. Fix a similar linear order on the vertices of T 2 . Now, these linear orders give rise to a lexicographic ordering on the squares in T 1 × T 2 , with the property that the ith square S i includes a vertex that is not in S 1 ∪ · · · ∪ S i−1 . This means that the relation arising from each square is linearly independent of all prior relations. Thus, codim(W ) ≥ (|V (G)| − 1)(|V (H)| − 1).