ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the $F$-polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the g-vector fan has the property that its nef cone is simplicial.


Introduction
Let Q be a Dynkin quiver. That is to say, Q is an orientation of a simplylaced Dynkin diagram, with vertices numbered 1 to n. Let B 0 be the matrix with the property that the entry (B 0 ) ij equals the number of arrows from i to j minus the number of arrows from j to i. Starting from the matrix B 0 , one can define a corresponding cluster algebra A(Q), which is a commutative ring with a distinguished set of generators, known as cluster variables, grouped together into overlapping sets of size n known as clusters. Because of our particular choice of Q, there are only finitely many cluster variables. Each cluster variable has an associated vector in Z n , its g-vector. The g-vectors are the rays of the g-vector fan, the maximal-dimensional cones of which correspond to clusters. This fan is proper, in the sense that the union of its cones is all of R n .
Considerable attention has been given to the problem of constructing polytopes whose outer normal fan is the g-vector fan. Such polytopes are called generalized associahedra. The question of whether the g-vector fan can be realized in this way was first raised by Fomin and Zelevinsky in [16], and first solved by Chapoton, Fomin, and Zelevinsky [10]. In fact, g-vectors had not yet been defined at the time of these two papers, but the fan which they study would subsequently be recognized as the g-vector fan associated to a particular orientation of each Dynkin diagram. Subsequently, [18] gave a construction which solves the problem as described here. [19] solves a more general problem, where the initial quiver is assumed only to be mutation-equivalent to a Dynkin quiver, and not necessarily Dynkin itself. The papers we have cited actually work in greater generality, in that they also treat Dynkin types which are not simply laced. In this paper we focus on the simplylaced case because it is technically easier.
The prehistory of this problem goes back much further. The combinatorics of the face lattice of a generalized associahedron is not sensitive to the orientation of Q. When Q is of type A n , the face lattice is that of the associahedron, as originally defined, as a cell complex, by Stasheff [31]. The first polytopal realization to appear in the literature is due to Lee [22], and many others followed. An excellent overview is provided by [8]. One particular realization of the associahedron has been studied by many authors [30,23,26,28], including Loday, whose name is most often associated to it. The associahedra constructed in this way turn out to be generalized associahedra corresponding to the linear orientation of the A n diagram. Recently, yet another construction of this associahedron was given in the physics literature by Arkani-Hamed, Bai, He, and Yan [1], in connection with scattering amplitudes for bi-adjoint scalar ϕ 3 theory. We refer to this as the ABHY construction.
In this paper, we extend the ABHY construction to arbitrary (simply-laced) Dynkin quivers. We further show that, quite surprisingly, this same construction realizes the Newton polytopes of the F -polynomials of the corresponding cluster algebras. (The F -polynomials are certain polynomials in n variables which are a reparameterization of the cluster variables; in particular, the cluster variables can be recovered from them.) In fact, the ABHY construction can be seen even more naturally as constructing Newton polytopes of certain universal F -polynomials, which we define. The universal F -polynomials bear the same relationship to the cluster algebra with universal coefficients as the usual F -polynomials do to the cluster algebra with principal coefficients. We also consider the toric variety associated to the g-vector fan. We show that our results on realizations of associahedra imply that the nef cone of this toric variety is simplicial.
After the appearance of the first version of this paper in 2018, Palu, Padrol, Pilaud, and Plamondon [25] showed that a very similar construction can be applied to construction of generalized associahedra for all seeds in all Dynkin type cluster algebras. The first author of the present paper also extended the techniques in the present paper in her thesis [5], giving another proof that the same construction can be applied to any seed of simply-laced Dynkin type, and that this construction also gives Newton polytopes of F -polynomials in the same generality. Fei [13,14] takes a less explicit approach but proves a very general result, to the effect that if A is any algebra with finitely many τ -rigid indecomposable modules, then a polytope dual to the τ tilting fan of A can be constructed as the Minkowski sum of the submodule polytopes of the τ -rigid indecomposables.
Our result that our polytopes also yield Newton polytopes of F -polynomials has subsequently been extended to non-simply laced types by Arkani-Hamed, He, and Lam [2], by using a folding argument to reduce to the case we consider.

Construction
In the interests of self-containedness, we will begin with a completely explicit, if somewhat unmotivated, description of our construction. We will then provide a more representation-theoretic description, which is needed for the proof of correctness. Statements which are not proved in this section will be proved in the following section (or will turn out to be equivalent to well-known facts from the theory of quiver representations).
We write Q 0 for the set of vertices of Q and Q 1 for the set of arrows. We assume that Q 0 = {1, . . . , n}. We write Q op for the opposite quiver of Q, all of whose arrows are reversed compared to Q.
Draw Z ≥0 many copies of Q. The vertices in this quiver are denoted (i, j) where i ∈ Z ≥0 and j ∈ Q 0 . We also add arrows between the copies of Q: if there is an arrow from j to k in Q, we put an arrow from (i, k) to (i + 1, j). This infinite quiver we denote by Z ≥0 Q. See Example 1 below for an example of the initial part of the quiver Z ≥0 Q for Q the quiver 1 . We associate to each vertex (i, j) a vector in Z n , which we call the dimension vector, and which we denote dim(i, j). (What exactly it is the dimension of will be explained in the following section. For now, it is simply an integer vector.) To (0, j), we associate the dimension vector dim(0, j) obtained by putting a 1 at every vertex that can be reached from j by following arrows of Q op (including the vertex j itself), and 0 at all other vertices. For (i, j) with i > 0, we associate the dimension vector which satisfies: Here, the sum on the righthand side runs over all vertices (i , j ) on a path of length two from (i − 1, j) to (i, j). Starting with the dimension vectors already defined for (0, j), these equations allow us to deduce the value of dim(i, j) for all It turns out that the dimension vectors calculated in this way have the property that they are non-zero and sign-coherent, in the sense that each dim(i, j) either has all entries non-negative or all entries non-positive. In these two cases we simply say that dim(i, j) is non-negative or non-positive, respectively.
A certain subset of the vertices of Z ≥0 Q are in natural correspondennce with the cluster variables of A(Q). For 1 ≤ j ≤ n, define i j to be the maximal index such that all of dim(0, j), dim(1, j), . . . , dim(i j , j) are non-negative. Define There is a natural bijection between the elements of I and the cluster variables of A(Q). We recall the details of the correspondence in Section 4 below. We write x ij for the cluster variable associated to (i, j) ∈ I. We say that two cluster variables are compatible if there is some cluster which contains both of them. We say that two elements of I are compatible if the corresponding cluster variables are.
Consider a real vector space with basis indexed by the elements of I, say V = R I . For (i, j) ∈ I, we write p ij for the coordinate function on V indexed by (i, j).
Fix a collection of positive integers c = (c ij ) ij∈I + . We construct an associahedron for each choice of c.
Consider the following collection of equations, one for each (i, j) ∈ I + .
We call these equations the c-deformed mesh relations. They define an ndimensional affine space E c inside V .
Write U c for the region inside E c all of whose coordinates p ij are non-negative. That is to say, U c is the intersection of the positive orthant in V with E c . Example 1. Let us consider the quiver Q : 1 The following shows the part of Z ≥0 Q whose vertices are in I, labelled by the elements of I.
The corresponding dimension vectors are: We fix a six-tuple of positive integers c = (c ij ) (i,j)∈I + . The region E c is cut out by the following equations: p 01 + p 11 = p 02 + c 01 p 03 + p 13 = p 02 + c 03 p 02 + p 12 = p 11 + p 13 + c 02 p 11 + p 21 = p 12 + c 11 p 13 + p 23 = p 12 + c 13 Notice that, by construction, the dimension vectors satisfy the 0-deformed mesh relations (which are generally called the mesh relations), that is to say, the deformed mesh relations with the deformation parameters set to zero. There is another important collection of vectors which satisfy these equations: the g-vectors. By definition, g(0, j) is the j-th standard basis vector, and the other g-vectors are determined by the mesh relations. The g-vector fan is the fan whose rays are the g-vectors, and such that a collection of rays generates a cone of the fan if and only if the corresponding collection of cluster variables is compatible.
Example 2. We continue Example 1. In this case the corresponding g-vectors are as follows: Note that, in the example, the g-vectors corresponding to the elements of I \ I + are the negative standard basis vectors. This is a general phenomenon, and allows us to define an important projection π : V → R n : the k-th coordinate of the projection to R n is given by p ij for (i, j) ∈ I \ I + such that g(i, j) = −e k . This projection defines a bijection between E c and R n . We define A c = π(U c ).
Given a full-dimensional polytope P in R n , there is fan associated to it called the outer normal fan of P , denoted Σ P . For each facet F of P , let ρ F be the ray pointing in the direction perpendicular to F and away from P . The collection of rays {ρ F | F is a facet of P } are the rays of Σ P ; the set of rays {ρ F1 , . . . , ρ Fj } generates a cone in Σ P if and only if there is a face G of P such that the facets of P containing G are exactly F 1 , . . . , F j .
The following is our main theorem about realizing generalized associahedra.

Theorem 1.
(1) Each facet of U c is defined by the vanishing of exactly one coordinate of V .
(2) If G is a face of U c , then the set {α ∈ I | G lies on the hyperplane p α = 0} is a compatible subset of I, and every compatible subset arises this way. (3) In particular, the vertices of U c correspond to clusters. (4) π is an isomorphism of affine spaces between E c and R n . Consequently, the faces of A c also correspond bijectively to compatible sets in I. (5) If F ij is the facet of U c given by p ij = 0, the normal to π(F ij ) oriented away from A c is the ray generated by g(i, j).
Here we show an illustration of A c in this case. At each vertex, we have drawn a small copy of the quiver with vertex set I (with the arrows omitted) on which we have marked the cluster corresponding to the vertex. To represent the vertices of U c in V , it is convenient to write the values of p ij at position (i, j), with the value c ij positioned between (i, j) and (i + 1, j). We write c ij in red to make clear the distinction between the two kinds of entries. To see A c , we forget everything in each row except the last entry.
In the above example, we set all the c ij = 1. The vertices are as follows:

Proof of correctness
Since the foundational work of [24,7], it has been clear that representations of quivers are extremely useful in understanding the combinatorics of cluster algebras.
We will use a setting that is inspired by [7], though described in somewhat different terms.
The quiver Z ≥0 Q restricted to the vertices of I + , gives the Auslander-Reiten quiver for the category of representations of Q op . If we restrict to I instead, we get n additional vertices added to the righthand end. We understand this as the Auslander-Reiten quiver of a full subcategory of D b (rep Q op ), whose vertices correspond to the indecomposable quiver representations together with the P j [1] where P j is the projective representation at vertex j and [1] is the shift functor. We write W ij for the object in D b (rep Q op ) corresponding to the vertex (i, j) ∈ I. The indecomposable projective P j is W 0j . As is well known, dim(i, j), as defined in the previous section, is the dimension vector of W ij if (i, j) ∈ I + ; to include the cases of (i, j) ∈ I \ I + as well, we can say that dim(i, j) is the class in the Grothendieck Because it will be useful later in the paper, we will begin by weakening the hypothesis on the tuple c: we will begin by assuming that its entries are nonnegative, rather than all being strictly positive. Let Representation theory gives us a natural point v c in V . The coordinates of this point are defined by: Proof. Suppose that (i, j) ∈ I + . Let E ij be the direct sum of all the W i j for (i , j ) on a path of length two between (i, j) and (i + 1, j). We therefore have an Auslander-Reiten triangle in D b (rep Q): We must verify, for each (i, j) ∈ I + that We know that the following sequence is exact except at the righthand end: Thus, the lefthand side of (1) is nothing but the dimension of the quotient of Hom(W ij , M c ) by the image of Hom(E ij , M c ). By the definition of Auslander-Reiten triangles, any map from W ij to a summand of M c which is not an isomorphism factors through E ij . The lefthand side of (1) is therefore the multiplicity of We can now say Here, by E 0 , we mean the points of V satisfying the (0-deformed) mesh relations. We see that E 0 is an n-dimensional vector space. For 1 ≤ k ≤ n, define a vector . . , d n are a basis for E 0 . Similarly, for 1 ≤ k ≤ n, and (i, j) ∈ I, let g k ij be the signed multiplicity of P k in a projective resolution for W ij . That is to say, we take a resolution and take g k ij to be the multiplicity of P k in P 0 minus its multiplicity in P 1 . Now define the vector g k = (g k ij ) ij∈I . Then the set of g k are also a basis for E 0 . This is clear because (2) implies that dimW ij = dimP 0 − dimP 1 , from which we see that we can calculate the g-vector of W ij by expressing dimW ij in the basis dimP 1 , dimP 2 , . . . , dimP n of Z n . It follows that the collection of vectors {g k } are related to the vectors {d k } by a change of basis.
As we have already mentioned, there is a relation of compatibilty of cluster variables, which we can take as defining a notion of compatibility for elements of The following is the key lemma. Proof. If α and β are incompatible, then Ext 1 (W α , W β ) = 0 or Ext 1 (W β , W α ) = 0. Without loss of generality, suppose the former. This implies, in particular, that β ∈ I + . Choose a non-split triangle .
Also note that we have the Hom long exact sequence But the quantity on the lefthand side is a just a sum of the coordinates of x evaluated at the summands of E, weighted by their multiplicities. Thus at least one of the coordinates of x is negative, so x is not in U c .
Proof. If x is in U c then, as in the previous proof, there is an n-tuple (m 1 , . . . , m n ) such that Consider what this equation says at some coordinate α ∈ I \ I + . Note first that (v c ) α = 0. Of course, x α ≥ 0, since we assumed that x ∈ U c . Since the g-vectors of the P j [1] are negative standard basis vectors, g k α is non-positive, and it follows that the m k must be non-positive.
Repeating the same argument with the projectives rather than the shifted projectives, and taking into account the fact that (v c ) 0j > 0, we conclude that each of the m k is bounded between 0 and some negative number B. Therefore U c is bounded.
Proof of Theorem 1. We begin by establishing that the vertices of U c correspond bijectively to maximal compatible sets in I, with the bijection sending the vertex to the indices of the hyperplanes p α = 0 on which it lies.
Since E c is n-dimensional, a vertex of U c must lie on at least n facets, and therefore on at least n hyperplanes of the form p α = 0. By Lemma 2, the collection of hyperplanes corresponding to a vertex must be compatible. The maximal compatible sets of I are exactly the compatible sets of size n. Thus, every vertex of U c corresponds to a cluster.
We now argue by induction that every cluster corresponds to a vertex of U c . Let us suppose that we have a collection of n compatible indices from I, such as T = {α, α 1 , . . . , α n−1 }, and another collection of n compatible indices T = {α , α 1 , . . . , α n−1 }. Suppose that U c has a vertex q T at the intersection of the facets corresponding to T . Consider moving along the ray from q T where the facets corresponding to α 1 , . . . , α n−1 intersect. Since U c is bounded by Lemma 3, we must eventually hit another hyperplane bounding U c . By Lemma 2, it must be a hyperplane which is compatible with α 1 , . . . , α n−1 ; the only one is the the hyperplane corresponding to α . This intersection is a vertex of U c corresponding to T .
We therefore know that T also corresponds to a vertex of U c . Since the clusters are connected by mutations, every cluster corresponds to a vertex of U c , and since by following edges, we only ever get to vertices that correspond to clusters (never to an edge that goes to infinity, or to a vertex that doesn't correspond to a cluster), all the vertices of U c correspond to clusters. Further, the one-skeleton of U c is the cluster exchange graph.
We have therefore established point (3) of the theorem. By Lemma 2, each face of U c corresponds to a compatible set (necessarily all different), and since all the maximal compatible sets correspond to vertices of U c , every compatible set does correspond to a face. This establishes (1) and (2). (4) is straightforward. Given a point in y ∈ R n , we can uniquely find an x such that π(x) = y by inductively solving the c-deformed mesh relations, working from right to left through the quiver for I.
Finally, we establish (5). Let G be the n × |I| matrix whose (i, j)-th column is the g-vector g(i, j). Let σ be the affine map from R n to E c ⊂ V which is a section of π. This map is given by right multiplying by −G and adding v c . This induces a map from the tangent vectors of V to tangent vectors of R n (which we identify with V and R n respectively, via our fixed bases for each of them). Linear algebra tells us that the map on tangent vectors is given by left multiplying by −G. Write e ij for the standard basis vector corresponding to (i, j) ∈ I. Since −e ij is orthogonal to the hyperplane p ij = 0, and points away from U c , it image under left multiplication by −G generates the outer normal ray corresponding to this facet. But clearly (−G)(−e ij ) = g(i, j), as desired.
We close the section by commenting on the implications of Theoem 1 for different assumptions on c.
Clearly, if we assume that the constants c are positive rational numbers, Theorem 1 still holds. By continuity, it also holds for positive real numbers. Now consider the case that the some of the constants c are zero. Proof. We can imagine what happens if we begin with c where all values are positive, and then deform gradually to a nearby vector c, where some values become zero. What can happen is that some vertices can merge, and some facets can collapse to something which is no longer codimension 1. The results are as described in the statement of the corollary.

Background on cluster algebras
Let Q be a quiver without loops or oriented two-cycles. As already mentioned, there is an associated cluster algebra A(Q), whose cluster variables are in natural bijection with the elements of I, as we now explain.
A slice in I is a subset T of I such that: • For each j with 1 ≤ j ≤ n, there is exactly one element of T of the form (i, j) for some i, and • if j and j are adjacent vertices of Q, and (i, j) and (i , j ) are the corresponding elements of T , then (i, j) and (i , j ) are adjacent in I. The collection of all (0, j) form a slice, corresponding to the projective representations, as does the collection I \ I + , corresponding to the shifted projectives. We refer to {(0, j)} as the initial slice and I \ I + as the final slice.
Note that any slice in I has at least one element (i, j) such that all the arrows between it and the other elements of the slice are oriented from (i, j). We call such an (i, j) a source in the slice. Any slice other than the final slice has a source which is in I + .
We assign the intial cluster variable x j to (0, j) ∈ I. Suppose we have a slice T which is not the final slice, and the elements of T are associated to the cluster variables of a cluster. Let (i, j) be a source in the slice which is in I + . We can therefore replace (i, j) by (i + 1, j) to obtain a new slice T . Mutate the cluster at the variable corresponding to (i, j), and associate the resulting cluster variable to the vertex (i + 1, j). This provides a cluster associated to the slice T . As we proceed from the initial slice to the final slice, we may well have choices of source. However, these choices do not matter. We have the following theorem, essentially from [7]: Theorem 2. The above procedure results in a well-defined map from I to the cluster variables of A(Q), independent of choices.
Example 4. In the setting of Example 1, the result is the following: x2+1 x3 Let us emphasize that any slice in I corresponds to a cluster, though there are also further clusters which are not slices. Note that if we restrict the quiver on I to the vertices of the initial slice, we recover Q. In fact, it is easy to show by induction that for any slice, the quiver associated to that cluster is given by restricting the quiver on I to that slice.
One way to construct a more complicated cluster algebra which is still governed by the same Dynkin combinatorics is to define a new ice quiver Q ice which is obtained from Q by adding some vertices which are designated as frozen. The new vertices may be connected to the vertices of Q in any way (provided that there are still no oriented two-cycles). It is standard to assume that there are also no arrows between frozen vertices, but this is not actually important since such arrows play no role. The cluster algebra associated to an ice quiver, A(Q ice ) is the algebra generated by the initial cluster variables and all cluster variables obtained by all sequences of mutations at non-frozen vertices only. Note that there is a cluster variable associated to each frozen vertex. To distinguish the frozen and unfrozen variables, we generally write x 1 , . . . , x n for the initial unfrozen variables and y 1 , . . . , y m for the frozen variables. A refined version of the Laurent Phenomenon says that every cluster variable is contained in Z[x ± 1 , . . . , x ± n , y 1 , . . . , y m ], see [15,Theorem 3.3.6]. The unfrozen cluster variables still correspond to the elements of I and can be calculated in the same way, by starting with the initial variables associated to the initial slice and then carrying out mutations as before.
There are two particular ice quivers obtained from Q which are of special importance, one of which we will explain now. For each vertex i of Q, add a frozen vertex i and an arrow from i to i. The resulting quiver is called the framed quiver of Q; we denote it Q prin . To the new vertex i we associate the frozen cluster variable y i . The cluster algebra associated to Q prin is called the cluster algebra with principal coefficients. We denote it A prin (Q). The cluster variables of the corresponding cluster algebra A prin (Q) are listed in Section 7.
The significance of this choice of coefficients will be explained further in Section 7. Briefly, it turns out that from the cluster variables for the cluster algebra with principal coefficients, one can immediately calculate the cluster variables for any system of coefficients. The cluster algebra with principal coefficients is also essential for defining F -polynomials, as we shall explain shortly.
An ice quiver with n unfrozen vertices and m frozen vertices can also be represented as an (n + m) × n matrix of integers, B 0 . We associate the n columns and the corresponding first n rows to the n unfrozen vertices of Q, and we associate the remaining rows to the frozen vertices. The entry ( B 0 ) ij is the number of arrows from i to j minus the number of arrows from j to i. We see that the first n rows of B 0 are simply the matrix B 0 which we have already seen. The matrix corresponding to Q prin consists of B 0 with an n × n identity matrix below it. Example 6. For our running example, the extended matrix B 0 corresponding to principal coefficients is the following:

Submodule polytopes and torsion classes
For the duration of this section, we work with an arbitrary finite-dimensional algebra A, thought of as a the path algebra of a quiver Q with relations, where the vertices of Q are numbered 1 to n. We can therefore still consider dimension vectors of such modules.
For X an A-module, we write P X for the polytope in R n which is the convex hull of the dimension vectors of subrepresentations of X. We call P X the submodule polytope of X. Submodule polytopes for representations of preprojective algebras play an important rôle in the paper by Baumann, Kamnitzer and Tingley [4] which we are essentially following in this section.
A torsion class in A-mod is a full subcategory closed under extensions and quotients. If T is a torsion class, any A-module M has a unique largest submodule which is contained in T . We call this the torsion part of M with respect to T . See [3, Chapter VI] for more background on torsion classes.
The key fact about submodule polytopes is that, in order to find the vertices of a submodule polytope, we do not need to consider all submodules: it suffices to consider those submodules which are torsion parts with respect to some torsion class. The following proposition is established in [4, Section 3]. Because we need rather less than is established in [4], we give the simple proof here. Proof. Let x be a vertex of P M . It follows that there must be at least one submodule of M whose dimension vector is x. Let N be such a submodule. We will show that there is a torsion class T such that the torsion part of M with respect to T is N .
Choose a linear form θ on the space of dimension vectors such that the unique point on P M maximizing θ is x. Define a torsion class by It is not hard to establish T θ is a torsion class ([4, Proposition 3.1]). Suppose L is a proper submodule of N . Since L is also a submodule of M , we know that θ(dimL) < θ(dimN ), so θ(dimN/L) > 0, and it follows that N ∈ T θ . Let us write N for the torsion part of M with respect to T θ . Since N is a submodule of M and is in T θ , we know that N must be a submodule of N . If it were a proper submodule, then θ(dimN ) < θ(dimN ), so θ(dimN /N ) < 0, contradicting the assumption that N ∈ T . Thus N = N .
For the final statement, suppose that there were two distinct modules N 1 , N 2 with dimension vector x. The above argument shows that both N 1 and N 2 are in T , so the maximal torsion part of M with respect to T is larger than either of them, which we have established is impossible.

Newton polytopes of F -polynomials
By definition, the F -polynomial F α for α ∈ I is obtained by taking x prin α , the cluster variable associated to α in A prin (Q), and setting all the x i to 1. It is therefore a polynomial in y 1 , . . . , y n . It turns out that the cluster variable x prin α can be recovered from the F -polynomial (see [17,Corollary 6.3]), so no information has been lost, and at the same time, the F -polynomial turns out to be convenient for another reason: for (i, j) ∈ I with i ≥ 1, the F -polynomial F (i,j) is the generating function for the submodules of the representation W (i−1,j) in the following sense: F (i,j) (y 1 , . . . , y n ) = e≤dim(W (i−1,j) ) χ(Gr e (W (i−1,j) )y e We refer to [12, Eq. (1.6)] for the formula. Here e = (e 1 , . . . , e n ) is a dimension vector in Z n ≥0 , Gr e (X) means the quiver Grassmannian of subrepresentations of X whose dimension vector is e, and χ is the Euler characteristic. We write y e for y e1 1 y e2 2 . . . y en n . Let f ∈ Z[y 1 , . . . , y n ] be a polynomial. Let P be the subset of Z n such that f can be written as f = p∈P f p y p with all f p non-zero integers. That is to say P is the collection of the the n-tuples corresponding to exponents of terms which appear in f . The Newton polytope of f is then the convex hull of the points in P . Newton polytopes of F -polynomials have been studied by Brodsky and Stump [6]. They give a description in type A n and a uniform conjecture for all finite types. Subsequent to the first appearance of the present paper, this conjecture was proved in [20], relying in part on our results.
Newton polytopes of cluster variables have also been studied, by Sherman and Zelevinsky in rank 2 [29], by Cerulli Irelli for A 2 [9], and by Kalman in type A n [21]. Note that, by [17,Corollary 6.3], the Newton polytope of a cluster variable is an affine transformation of the corresponding F -polynomial, so the two questions are quite close.
For (i, j) ∈ I + , let e ij denote the standard basis vector in R I + which has a 1 in position (i, j) and zeros elsewhere.
Subsequent to the appearance of the first version of the present paper, this result has been extended to the non-simply laced case [2], by using a folding argument to reduce to the simply laced case, for which they rely on this result. Before we prove the theorem, we will state and prove a key lemma, and then a proposition.  Looking at the second vertex listed, we see that T 2 consists of direct sums of all indecomposables except S 3 . The corresponding torsion part t 2 M c therefore has dimension (3, 4, 2) = π(q T2 ). (Recall that in the definition of π, the order in which the final slice of coordinates appear is determined by their corresponding g-vectors, which is why, in type A 3 , π is in effect reading the final slice of coordinates from bottom to top.) Proof. As before, let G be the n × |I| matrix whose (i, j)-th column consists of the g-vector g(i, j).
Let the summands of T be T 1 , . . . , T n . Let us write G| T for the n × n matrix formed by taking the columns of G corresponding to T 1 , . . . , T n . We can interpret G| T as the change-of-basis matrix from the basis dimT 1 , . . . , dimT n to the basis dimP 1 , . . . , dimP n . G| T is therefore invertible, with inverse given by the inverse change of basis. It follows that (G| T ) −1 G is a matrix whose restriction to the columns corresponding to summands of T is an identity matrix.
The point in E c which has zeros in the columns corresponding to the T i is therefore Note that since T is a tilting object, each indecomposable projective module P i admits a coresolution , where T i0 and T i1 are in add T . The entries in the i-th column of (G| T ) −1 encode the signed multiplicity of T 1 , . . . , T n in this coresolution of P i .
Note that the final n coordinates of v c are zero. The i-th coordinate of A c is therefore dim Hom(T i0 , M c ) − dim Hom(T i1 , M c ).
From the coresolution of P i , we obtain the following commutative diagram, with the rows exact: The zeros on the lefthand end follow from the fact that Hom(P i , M c [−1]) = 0 = Hom(P i , tM c [−1]). Further, Ext 1 (T i1 , tM c ) = 0 because T i1 is Ext-projective in T while tM c is in T . Therefore the map from Hom(T i0 , tM c ) to Hom(P i , tM c ) is surjective.
The first three vertical maps are injective because they are induced from the inclusion of tM c into M c .
Any map from a torsion module to M c necessarily lands in the torsion part of M c , so factors through tM c . This means that the first two vertical arrows are also surjective.
Our goal is to understand the image of Hom(T i0 , M c ) inside Hom(P i , M c ); by what we have already shown, it equals Hom(P i , tM c ); in other words, the dimension of this image is the dimension of tM c at vertex i, as desired.
From the previous lemma, the following proposition is almost immediate: Proof. Thanks to Corollary 1, we know that the vertices of U c are the set of points q T for T a tilting object in the additive hull of the W α , with α ∈ I. (Note that it is of course posible that q T = q T for two distinct tilting objects T and T .) By the previous lemma, π(q T ) is the dimension vector of the torsion part of M c with respect to the corresponding torsion class. All torsion classes are of this form, so A c is the convex hull of the dimension vectors of all possible torsion parts of M c . Lemma 4 now tells us that A c is therefore the submodule polytope of M c , as desired.
We can now prove Theorem 3.
Proof of Theorem 3. Since we are interested in A e ij , we set c = e ij , M c = W ij . The quiver Grassmannian Gr e (W ij ) is empty if e is not the dimension vector of a submodule. Thus, the Newton polytope of F (i+1,j) is contained in the convex hull of the dimension vectors of submodules of W ij , which we have established in Proposition 1 is A e ij . It remains to check that the vertices of A e ij correspond to quiver Grassmannians with non-zero Euler characteristics. Lemma 4 tells us that for each vertex of A e ij , there is a unique submodule of the appropriate dimension vector. The quiver Grassmannian is therefore a single point, and the Euler characteristic of a single point is 1.

The use of principal coefficients
Let Q ice be an ice quiver, whose unfrozen part is Q. We will explain, following [17], how the cluster variables of A(Q ice ) can be calculated directly from those of A prin (Q), rather than via mutation.
Let z 1 , . . . , z m be the coefficients corresponding to the frozen vertices of Q ice (equivalently, these correspond to rows n + 1 to n + m of the matrix B 0 .) Definẽ We write F trop (ỹ 1 , . . . ,ỹ n ) for the tropical evaluation of F atỹ 1 , . . . ,ỹ n . This is the monomial such that the power of z i that appears in it is the minimum over all terms of F of the power of z i in that term. (This is the gcd of the monomials that appear.) Then [17,Theorem 3.7] says that the cluster variable in A(Q) corresponding to f is equal to f (x 1 , . . . , x n ,ỹ 1 , . . . ,ỹ n )/F trop (ỹ 1 , . . . ,ỹ n ).
Example 8. Let us consider the following example, with vertex 4 frozen, and associated to the variable z. The corresponding B-matrix is:  Using the labeling of the vertices as in Example 1, the cluster variables as well as the cluster variables with coefficients associated to every vertex are as following: Cluster variables for Q ice Cluster variables with principal coefficients (0, 1) (1, 1) By the definition ofỹ i , in this example we haveỹ 1 = z, whileỹ 2 = 1 andỹ 3 = z −1 . Substituting them in the polynomials listed above, we obtain the polynomials f (x 1 , . . . , x n ,ỹ 1 , . . . ,ỹ n ). Moreover, the monomials F trop (ỹ 1 , . . . ,ỹ n ) associated to every polynomial f are as follows: Calculating f (x 1 , . . . , x n ,ỹ 1 , . . . ,ỹ n )/F trop (ỹ 1 , . . . ,ỹ n ) and comparing with the cluster variables above, we observe that the theorem holds in this case.

Universal coefficients
As mentioned in Section 4, there are two choices of coefficients which are particularly interesting. One is the principal coefficients which we discussed in the previous section. The other is universal coefficients.
In fact, there are two closely related notions: universal coefficients, introduced by Fomin and Zelevinsky [17], and universal geometric coefficients, introduced by Reading [27]. In both cases, the goal is a cluster algebra with sufficiently general coefficients that it will admit a ring homomorphism (with certain good properties) to the cluster algebra defined for any other choice of coefficients. In the finite type case, the two definitions yield the same system of coefficients, see [27]. We will not need any properties of universal coefficients, so we do not give the precise definitions.
Reading [27,Theorem 10.12] proves B 0 provides universal coefficients if the coefficient rows of the extended exchange matrix are the g-vectors B T 0 , where T indicates transposition.
Since, for us, B 0 is skew-symmetric, B T 0 = −B 0 . Thus, the desired coefficient rows are the g-vectors for Q op . Reading the quiver for I from right to left instead of left to right, we see that the g-vectors for Q op are simply the negatives of the g-vectors for Q. Thus, we shall be interested in the setting where we add a row to the exchange matrix B 0 for each element of I, with the row corresponding to (i, j) being given by −g(i, j). We will call the corresponding algebra A univ (Q).

Example 9. Consider again Example 1. The extended exchange matrix
The additional rows in B 0 define the behaviour of z 0,1 , . . . , z 2,3 , where z ij is the frozen variable corresponding to the row whose entries are −g(i, j).
The cluster variables with universal coefficients are computed below.
Example 10. Consider, for example, the universal F -polynomial F univ (1,1) . According to the above calculation, it is z 0,1 + z 1,1 z 1,2 z 2,3 . Thus, its Newton polytope is the line segment from (1, 0, 0, 0, 0, 0, 0, 0, 0) to (0, 0, 0, 1, 1, 0, 0, 0, 1). To find the vertices of A e (0,1) , we find solutions to the e (0,1) -deformed mesh relations which are all nonnegative and have at least three zeros. The results are the following, as expected. Proof. We will use the strategy explained in Section 7 to calculate F univ (i+1,j) on the basis of F (i+1,j) . Defineỹ 1 , . . . ,ỹ n as in Section 7 with respect to the matrix B 0 as defined above. Then . Let x be a vertex of A c . It corresponds to a term y x in F (i+1,j) . There is a unique elementx ∈ E 0 such that π(x) = x. This is precisely the exponent vector of the result of substitutingỹ i for y i in the monomial y x .
The element of E c which projects onto x is exactlyx + v c . This, then, is the vertex of U c corresponding to x. What remains to be verified is that v c equals negative the exponent of F trop (i+1,j) (ỹ 1 , . . . ,ỹ n ). By definition, F trop (i+1,j) (ỹ 1 , . . . ,ỹ n ) is the least common multiple of all the terms in F (i+1,j) (ỹ 1 , . . . ,ỹ n ). Subtracting it from the Newton polytope of F (i+1,j) (ỹ 1 , . . . ,ỹ n ) translates the polytope so that for each z ij , the minimal power that appears is zero. We know that U c also has the property that the minimum value which any coordinate takes on within U c is zero. Thus v c = −F trop (i+1,j) (ỹ 1 , . . . ,ỹ n ), as desired.
Example 11. We look at F univ (1,1) in our running example. y 1 = z 1,1 z 1,2 z 2,3 /z 0,1 y 2 = z 2,2 /z 0,2 z 1,1 z 1,2 z 1,3 and we indeed obtain 9. The nef cone of the toric variety associated to the g-vector fan is simplicial 9.1. Brief reminder on toric varieties. Our main reference for toric varieties is [11]. We do not give specific references for the basic facts which are to be found in the first few chapters of that book. Let N be a free abelian group of rank n. Write M = Hom(N, Z) for its dual. We write ·, · for the duality pairing from M × N to Z. We write N R for N ⊗ Z R, and in general use a subscript R to denote tensoring by R.
A cone in a real vector space is a semigroup closed under multiplication by nonnegative reals. A strongly convex, rational, polyhedral cone in N R is a cone which is generated by a finite collection of vectors from N , all lying in a proper half-space. A fan in N R is a collection of strongly convex, rational, polyhedral cones such that the intersection of any two cones is necessarily a face of each. (The examples to which we will apply this theory are the outer normal fan to the generalized associahedra A c which we have constructed, which are particular instances of g-vector fans of finite type cluster algebras. They are indeed fans in the above sense.) Let Σ be a fan in N R . For simplicity of exposition, we will assume that the n-dimensional cones of Σ cover R n , since this is true in particular for the outer normal fans which we are considering. We write Σ i for the i-dimensional cones of Σ.
Associated to a fan Σ, there is a normal toric variety X Σ . The field of rational functions on X Σ is the fraction field of the group ring C[M ]. For m ∈ M , we write χ m for the corresponding function field element.
There is a torus T (C * ) n acting on X Σ . There is a bijection between cones of Σ and T -orbits in X Σ . We write O σ for the orbit corresponding to the cone σ ∈ Σ. The dimension of O σ is n minus the dimension of the span of σ.
A divisor on a normal variety is a formal Z-linear combination of irreducible codimension one subvarieties. On a toric variety, we are particularly interested in those divisors which are torus invariant. For ρ ∈ Σ 1 , we define D ρ to be the closure of O ρ . This is an irreducible codimension one subvariety which is torus-invariant. The torus-invariant divisors of X Σ are Div( Given a normal variety X and an element of the function field of X, say f , there is an associated divisor, div(f ). Informally, it consists of the zero locus of f minus the locus where f blows up. We shall shortly define this notion precisely in the setting of toric varieties. A divisor D on a normal variety X is called a Cartier divisor if there exists an open cover {U i } of X such that D| Ui is principal for each i ∈ I, that is to say, there exists an element f i of the function field of U i such that div(f i ) equals the restriction of D to U i .
In the case of toric varieties, we are interested in torus-invariant Cartier divisors. It turns out that there is a canonical choice of open cover which works for any torusinvariant Cartier divisor. For each maximal cone σ, let X σ denote the toric variety associated to the fan consisting of σ and its faces. X σ is open in X Σ , and X Σ is covered by the varieties X σ . Any torus-invariant Cartier divisor on X Σ is given by a collection of functions, one on each X σ . In fact, we can take the function on X σ to be of the form χ m σ for some m σ ∈ M . We call the collection {m σ } σ∈Σ n the local data of the Cartier divisor. For each ray ρ ∈ Σ 1 , let u ρ be the first lattice point along the ray ρ. The multiplicity of D ρ in the divisor corresponding to the local data {m σ } σ∈Σ n is given by − m σ , u ρ for σ any cone of Σ containing ρ.
A collection {m σ } σ∈Σ n is not necessarily the local data of any Cartier divisor. The following condition provides a necessary and sufficient condition to verify that it is. As we have already remarked, − m σ , u ρ is the multiplicity of D ρ in the divisor; the condition for {m σ } to be local data amounts to saying that the formula for the multiplicity of D ρ does not depend on which σ is used, among those containing ρ.
We write CDiv(X Σ ) for the torus invariant Cartier divisors on X Σ . We write Div 0 (X Σ ) for the principal divisors on X Σ . These are the Cartier divisors for which all m σ are equal. The Picard group of X Σ , denoted Pic(X Σ ), is defined to be CDiv(X Σ )/ Div 0 (X Σ ). Under our assumptions on the fan Σ, CDiv(X Σ ) is a free abelian group whose rank is |Σ 1 | − n [11, Theorem 4.2.1].
Let D be a Cartier divisor on a normal variety X and let C a complete curve in X. We write D · C for the intersection product of D and C. We will not define it in full generality, but the following can be taken as a definition in the toric setting. Definition 1. On the toric variety X Σ , let D be a torus-invariant Cartier divisor, which is thus given by a collection of local data {m σ } σ∈Σ n . Let C be a complete, irreducible torus-invariant curve in X Σ ; it is therefore the closure of O τ for some codimension one τ in Σ. The cone τ separates two maximal cones σ and σ . Let u be an element of σ which maps to a generator of N/N τ . (Here N τ is the lattice generated by τ , so N/N τ is isomorphic to Z.) Then We take this as the definition of D · C; see also [11,Proposition 6.3.8], which shows that this definition agrees with the definition for general varieties.
A Cartier divisor on a normal variety X is called nef ("numerically effective") if D · C ≥ 0 for every irreducible complete curve C in X. In the toric case, D is nef if and only if D · C ≥ 0 for every irreducible torus-invariant complete curve, i.e., for those curves which are the closure of O τ for some τ ∈ Σ n−1 [11,Theorem 6.3.12]. Thus, the definition of D · C which we have given above is sufficient to determine, for any Cartier divisor on X Σ , whether or not it is nef.
A Cartier divisor D on a normal variety is said to be numerically equivalent to zero iff D · C = 0 for all irreducible complete curves C; two Cartier divisors are numerically equivalent if their difference is numerically equivalent to zero. The nef cone of a variety is defined in the vector space of its Cartier divisors modulo numerical equivalence, and tensored by R. For a toric variety, a torus-invariant Cartier divisor is numerically equivalent to zero if and only if it is principal [11,Proposition 6.3.15]. Thus, we can view Nef(X Σ ) as contained in Pic(X Σ ) R . Then, Nef(X Σ ) is the cone generated by the classes of the nef Cartier divisors in Pic(X Σ ) R .
Combining [11, Theorems 6.1.7 and 6.3.12], we obtain the following useful criterion for a Cartier divisor's being nef: Proposition 2. A Cartier divisor D on X Σ with local data {m σ } σ∈Σ n is nef if and only if m σ ∈ P D for all σ ∈ Σ n .
A cone in a real vector space is said to be simplicial if its number of generating rays is equal to the dimension of its span.
We can now state the theorem which we seek to prove about the toric variety associated to the g-vector fan.
Theorem 5. Let Σ be the outer normal fan of A c (or equivalently the g-vector fan corresponding to a Dynkin quiver). The nef cone of the toric variety X Σ is simplicial.

9.2.
Combinatorics of Pic(X Σ ). Let N = Z n be a rank n free abelian group with a fixed basis b 1 , . . . , b n , and let M = Hom(N, Z), equipped with the dual basis b * 1 , . . . , b * n . Let Σ be the g-vector fan realized in N R with respect to the basis {b i }, and let X Σ be the corresponding toric variety. The irreducible divisors correspond to rays of Σ, which themselves correspond to elements (i, j) ∈ I. We will write D ij for the divisor corresponding to the ray g(i, j).
We identify M R with the vector space R n which is the image of the projection π, identifying b * i with the standard basis of R n . We will now construct a bijection between Z |I + | and Pic(X Σ ), the Cartier divisors of X Σ up to linear equivalence.
Fix c = (c ij ) (i,j)∈I + , with all c ij ∈ Z. Unlike earlier in the paper, we do not assume that the entries of c are non-negative. Nonetheless, as earlier, we get a well-defined affine subspace E c of V . There is a section of π, which we denote i c , sending R n to E c .
The definition we gave of v c in Section 3 does not make sense any longer, since some of the c ij are negative. However, it can be extended to the more general setting in the following way.
Define a matrix A, whose rows are indexed by I and whose columns are indexed by I + , and such that the (i, j), (k, l) entry is dim Hom(M ij , M kl ). If we order the rows and columns of A in the same way, respecting the left-to-right order of the Auslander-Reiten quiver of Q and keeping the elements of I \I + for last, we obtain a matrix with 1's on the diagonal and zeros below it, with the last n rows consisting entirely of zeros.
We then define v = Ac. Clearly, this recovers the definition of Section 3 when the c ij are non-negative. The proof that v ∈ E c goes through without essential alterations.
For future use, let us write A for the submatrix of A consisting of the rows indexed by elements of I + . Since it is upper triangular with 1's on the diagonal, it is invertible in GL |I + | (Z).
Example 12. Consider the case that Q = 1 G G 2 .