Littlewood–Richardson coefficients via mirror symmetry for cluster varieties

I prove that the full Fock–Goncharov conjecture holds for Conf3×(Fℓ∼) — the configuration space of triples of decorated flags in generic position. As a key ingredient of this proof, I exhibit a maximal green sequence for the quiver of the initial seed. I compute the Landau–Ginzburg potential W on Conf3×(Fℓ∼)∨ associated to the partial minimal model Conf3×(Fℓ∼)⊂Conf3(Fℓ∼) . The integral points of the associated ‘cone’ Ξ:={WT⩾0}⊂Conf3×(Fℓ∼)∨(RT) parametrize a basis for O(Conf3(Fℓ∼))=⨁(Vα⊗Vβ⊗Vγ)G and encode the Littlewood–Richardson coefficients cαβγ . In the initial seed, the inequalities defining Ξ are exactly the tail positivity conditions of [18]. I exhibit a unimodular p∗ map that identifies W with the potential of Goncharov–Shen on Conf3×(Fℓ∼) [8] and Ξ with the Knutson–Tao hive cone [14].


Summary of results
In this paper I obtain polytopes whose number of integral points are the Littlewood-Richardson coefficients by a method that has essentially nothing to do with representation theory. The same method will, in theory, produce analogous polytopes whose integral points parametrize a canonical basis for the space of sections of any line bundle on any Fano variety with a choice anti-canonical divisor. I prove [12,Corollary 0.21], and recover Corollary 0.20 as well with very little additional work † . In particular I recover polytopes parametrizing canonical bases for each irreducible representation of GL n . Let G = GL n , let B be the Borel subgroup of upper triangular matrices in G, and let U be the unipotent radical of B -the subgroup of upper triangular matrices with diagonal entries all 1. The flag variety F is isomorphic to G/B, and the decorated flag variety F -whose points consist of a complete flag X • = (X 1 ⊂ · · · ⊂ X n ) together with a non-zero vector x i in each successive quotient X i /X i−1 -is isomorphic to G/U ‡ . Setting H := B/U , F is naturally a principal H-bundle over F . Next, following [5,8], define Conf 3 (F ) to be G\ F ×3 and Conf × 3 (F ) to be the affine subvariety where pairs of flags intersect generically. Define Conf 3 ( F ) and Conf Each irreducible representation of G can be described as the space of sections of a line bundle over F . Similarly, for L ∈ Pic(Conf 3 (F )), for some triple of weights (α, β, γ), where V λ denotes the irreducible representation of highest weight λ. The space and the dimension of each summand is a Littlewood-Richardson coefficient. See Section 2.1 for details. The first key result of this paper pertains to the open subvariety Conf × 3 ( F ).
The full Fock-Goncharov conjecture [12, Definition 0.6] will be described in detail in the next Section 1.2. For now, there is a piecewise linear manifold Conf × 3 ( F ) ∨ (R T ) encoding the logarithmic geometry of the mirror Conf × 3 ( F ) ∨ to Conf × 3 ( F ). Theorem 1 says that its integral points Conf × 3 ( F ) ∨ (Z T ) parametrize a canonical basis for O(Conf × 3 ( F )). Next observe that Conf 3 ( F ) is a partial compactification of Conf × 3 ( F ). This compactification corresponds to a Landau-Ginzburg potential W on Conf × 3 ( F ) ∨ . A piecewise linear analogue of W -its tropicalization W T -yields a subset Ξ := {W T 0} of Conf × 3 ( F ) ∨ (R T ), and so a subset of our basis for O(Conf × 3 ( F )). Now, both Conf × 3 ( F ) and Conf × 3 ( F ) ∨ have an atlas of algebraic tori, with tori in the two atlases coming in dual pairs. These tori are glued via birational maps, known as mutations. Choosing a pair of tori identifies Conf × 3 ( F ) ∨ (R T ) with a real vector space and Ξ with a rational polyhedral cone in this vector space. Under one particularly simple choice of tori, the inequalities defining Ξ are precisely the tail-positivity conditions of [18] † .
We need one more ingredient to identify Ξ with the Knutson-Tao hive cone of [14]. There is a class of maps (known as p * maps) that in this setting send the cocharacter lattice N of a torus in Conf × 3 ( F ) to the cocharacter lattice M of the dual torus in Conf × 3 ( F ) ∨ and commute with mutation. Such a map of lattices gives rise to a map of schemes p : Theorem 2. For a particular pair of dual tori § , the cone Ξ defined by the Landau-Ginzburg potential W on Conf × 3 ( F ) ∨ is precisely the tail-positive cone of [18]. Furthermore, a particular choice of the map p * : N → M identifies W with the potential of Goncharov-Shen on Conf × 3 ( F ) and Ξ with the Knutson-Tao hive cone.
Note that these identifications are not used to prove any of the combinatorial results in this paper. Instead, the identifications should be interpreted as placing the tail-positive cone and the Knutson-Tao hive cone within the broad framework of log Calabi-Yau mirror symmetry. It is an instance of the mirror construction described, for example, in [12].
Remark 3. Conf 3 ( F ) and F have very similar cluster structures. As a result, many of the proofs in [16] apply here as well. For completeness and convenience, I have provided them here. Since the current paper encompasses the main results in [16], I will seek publication of this paper and not [16].

Full Fock-Goncharov conjecture
This subsection provides a bit of background on the full Fock-Goncharov conjecture following [12].
The full Fock-Goncharov conjecture is a canonical basis conjecture for regular functions on cluster varieties. Cluster varieties are schemes built out of birationally glued algebraic tori. A torus (C * ) n comes with a canonical volume form dz1 z1 ∧ · · · ∧ dz1 z1 . The birational gluing maps are required to patch these volume forms on tori, giving a global volume form. In cluster theory, † Thanks to G. Fourier and an anonymous referee for pointing this out to me. ‡ A complete discussion of p * maps appears in [10,Section 2], and I give a brief review in Section 4.2.3. § For readers already familiar with the cluster structure of Conf × 3 ( F ), these are the tori of the initial seed. ¶ This condition allows us to associate a regular function on Conf × 3 ( F ) ∨ to each irreducible component of D := Conf 3 ( F ) \ Conf × 3 ( F ), and it ensures D interacts well with the canonical basis for O(Conf × 3 ( F )). This will be discussed in greater detail in Section 4.1.
there are two classes of such gluing maps (mutations), producing two types of cluster varieties -known as A-varieties and X -varieties. The A and X mutations are defined dually from the same data, and up to certain multipliers A and X are built from dual tori. There is a notion of Langlands dual data that accounts for these multipliers, giving rise to another pair of cluster varieties ( L A, L X ) built precisely out of the dual tori for (A, X ). The pairs (A, L X ) and similarly ( L A, X ) are said to be Fock-Goncharov dual. For a discussion of A and X varieties and their Fock-Goncharov duals, please see [6, Section 1.2; 10, Section 2; 12, Appendix A].
Let V be a cluster variety (of whichever type), and V ∨ its Fock-Goncharov dual. We think of V ∨ as mirror to V as, if V and V ∨ are sufficiently close to affine, the Fock-Goncharov canonical basis conjecture becomes a special case of the log Calabi-Yau mirror symmetry conjecture [11,Conjecture 0.6]. This is discussed in greater detail in the introduction of [12], where they additionally give a log Calabi-Yau mirror symmetry interpretation of the Fock-Goncharov canonical basis conjecture for arbitrary cluster varieties -not only those that satisfy certain affineness assumptions † .
In [12], several algebras are associated to V. First, there is the upper cluster algebra up(V) = O(V), originally defined for A-type cluster varieties in [1]. Its subalgebra generated by global monomials, that is, global regular functions restricting to a character on some torus in the atlas for V, is the ordinary cluster algebra ord(V). In the case of an A-type cluster variety, this corresponds to the usual notion of a cluster algebra. If V ∨ is the Fock-Goncharov dual of V, can(V) is a vector space with basis parametrized by V ∨ (Z T ) ‡ . Scattering diagrams and broken lines are used to associate to each q ∈ V ∨ (Z T ) a (possibly infinite) sum of characters on each torus in the atlas of V, the result denoted by ϑ q , and to define a multiplication rule for the ϑ q . A scattering diagram for V is a collection of walls, each decorated with a scattering function, living in a piecewise linear manifold V ∨ (R T ). Broken lines are lines drawn in this piecewise linear manifold which are allowed to bend in a prescribed fashion at the walls. Each linear piece is decorated with a monomial corresponding to the tangent direction, and the allowed bending at a wall is determined by the wall's scattering function. The ϑ-functions are expressed in patches by summing decorating monomials over broken lines with a given initial direction and end point. The details of this construction are beyond the scope of this paper. In situations where the full Fock-Goncharov conjecture holds, can(V) will be identified with up(V), and the ϑ q will form a canonical basis for up(V). More generally, can(V) has a subspace mid(V) parametrized by the subset Θ ⊂ V ∨ (Z T ) consisting of ϑ q which restrict to finite sums of characters, that is, Laurent polynomials, on tori from the atlas. Then each element of mid(V) naturally corresponds to an element of up(V), but there is no reason a priori that distinct elements of mid(V) must correspond to distinct elements of up(V). More formally, there is a canonical algebra homomorphism ν : mid(V) → up(V), and we have the following definition.
To recap, we have the following.
(2) ord(V) ⊂ up(V), the subalgebra generated by global monomials. † While the original Fock-Goncharov conjecture is false as stated, [10,12] build toric degenerations of Avarieties and show that the conjecture holds in a formal neighborhood of the central fiber of the degeneration. They interpret this as saying the Fock-Goncharov conjecture always holds in the large complex structure limit. ‡ This set -the integral tropical points of V ∨ -is essentially the divisors D that we can use to partially compactify V ∨ such that the canonical volume form on V ∨ has a pole along D. It will be discussed further in the following Section 1.3. For a more complete description of tropicalization, see [12,Section 2].
(3) can(V), the vector space spanned by ϑ q for q ∈ V ∨ (Z T ), endowed with a multiplication rule via scattering diagrams and broken lines. (4) mid(V) ⊂ can(V), the span of Θ ⊂ V ∨ (Z T ), the ϑ q which are Laurent polynomials on cluster tori.
Many conditions implying the full Fock-Goncharov conjecture holds for a given cluster variety are provided in [12]. I will use [12,Proposition 8.25 The cluster complex is a subset of the scattering diagram for A, where chambers correspond to tori in the atlas for A and rays spanning these chambers correspond to cluster variables † . Broken lines initiating within this region correspond to cluster monomials -ϑ-functions that restrict to characters on some torus of the atlas of A. Each choice of torus in the atlas of A gives an identification of A ∨ (R T ) with a real vector space. The statement that A has large cluster complex means that, for some choice of torus, the cluster complex is not contained in a half-space.
Before summarizing how I prove that Conf × 3 ( F ) has large cluster complex, I need to provide a bit more background. Cluster varieties are built recursively from a Z-basis for the character lattice of a torus and data that indicate how to replace this torus and basis with new tori and bases. The bases are known as seeds, and the data used to replace one seed with another are often encoded in a quiver -a directed graph. Mutation changes both the seed and the quiver.
To show that Conf × 3 ( F ) has large cluster complex, I exhibit a maximal green sequence [3, Definition 2.8] for the quiver of the initial seed. I review the notion of a maximal green sequence in Section 3.1. The existence of a maximal green sequence implies Conf × 3 ( F ) has large cluster complex by [12,Proposition 8.24] ‡ .
Theorem 5. The quiver for the initial seed of Conf × 3 ( F ) has a maximal green sequence, and therefore Conf × 3 ( F ) has large cluster complex.
Next, the convexity condition (7) of [12, Theorem 0.3] referenced in [12,Proposition 8.25] is the following: There is a seed s = (e 1 , . . . , e n ) for which all the covectors {e i , ·}, i ∈ I uf , lie in a strictly convex cone.

Partial compactifications and potentials
The space we are really interested in is Conf 3 But Section 1.2 describes a canonical basis for O(Conf × 3 ( F )). This situation is typical. Generally spaces we are interested in, say for representation theoretic reasons, will not be cluster varieties, or even log Calabi-Yau varieties † . However, many representation theoretically interesting spaces are partial compactifications of cluster varieties or log Calabi-Yau varieties in a nice way. For log Calabi-Yau varieties, the 'nice' type of partial compactification we are interested in is called a partial minimal model. Take U to be a log Calabi-Yau with canonical volume form Ω. Then an inclusion U ⊂ Y as an open subset is a partial minimal model if Ω has a simple pole along every irreducible divisor of Y \ U. In the special case that U is a cluster A-variety with frozen variables, there is a simple way these partial minimal models may arise -by taking Y to be the partial compactification given by allowing some frozen variables to vanish [12, Section 0.3]. Conf × 3 ( F ) and Conf 3 ( F ) are related in precisely this way.
In this situation, each irreducible divisor in D := Y \ U defines a divisorial discrete valuation whose evaluation at Ω is negative. So, each irreducible component gives a point in U trop (Z) by definition [10,Definition 1.7]. A divisorial discrete valuation (ddv) is a discrete valuation v : C(U ) \ {0} → Z on the field of rational functions of U given by order of vanishing along a divisor on some variety birational to U . The set U trop (Z) is defined to be This definition applies to all log Calabi-Yau varieties.
There is a related notion that specifically references the atlas of tori and mutation maps, known as Fock-Goncharov tropicalization. Fock-Goncharov tropicalization applies to positive schemes. Let U be a scheme where an open dense subset has an atlas of tori, glued birationally. Suppose additionally that the transition functions between tori have the form f g where f and g are linear combinations of characters (Laurent monomials) with coefficients in Z 0 . Such a U is said to be a positive scheme [6, Definition 1.1]. If P is a semifield, we can take the P -points of each torus-T (P ) := χ * (T ) ⊗ Z P , where χ * (T ) is the cocharacter lattice of T and we use the abelian group structure of P for the tensor product. The transition functions of a positive scheme induce isomorphisms of the P -points of the tori, giving a notion of the P -points of U . In particular, P can be taken to be a tropical semifield such as Z T -the set Z with multiplication a ⊗ b := a + b and addition a ⊕ b := max(a, b). This gives the notion of the Fock-Goncharov tropicalization U (Z T ).
Cluster varieties are both log Calabi-Yau varieties and positive schemes, so both notions of tropicalization make sense here. They are closely related as well. There is a different tropical semifield structure on Z, denoted Z t , given by replacing max with min. For a cluster variety U , the notions U trop (Z) and U (Z t ) coincide. We have an isomorphism of the tropical semifields Z T and Z t given by sign change x → −x. So, sign change also induces an identification † Cluster varieties, both Aand X -varieties, form a very special class of log Calabi-Yau varieties, which generalize Calabi-Yau varieties to the non-compact world. See [10, Definition 1.1]. I frame this discussion in terms of log Calabi-Yau varieties rather than just cluster varieties in this introductory section simply because it is the log Calabi-Yau structure, not the cluster structure, that is essential to the narrative. The cluster structure will be very useful in many of the proofs later on, but this is a case where framing the narrative in a more general context clarifies the key arguments rather than obscuring them.
i : U (Z T ) → U trop (Z). These different notions of tropicalization and the relations between them are carefully discussed in [12,Section 2]. With regard to the current paper, the reader is encouraged to ignore the distinction between the two forms of tropicalization, only paying it mind in the event that some sign confusion arises. The identification map i can be thought of as a minor detail that simply allows us to relate differing conventions in the literature.
From now on, take U to be a cluster variety. Evaluation gives a pairing between U trop (Z) and C(U ). Since each element of Θ is canonically identified with a regular function on U , we can restrict this pairing from C(U ) to Θ. We can similarly restrict from U trop (Z) to Θ ∨ . In this way, evaluation defines a pairing We could just as well start with the evaluation pairing between (U ∨ ) trop (Z) and C(U ∨ ), which would restrict to the pairing These two pairings are conjectured to agree in general, and they are known to agree when either v or w corresponds to a global monomial [12, Lemma 9.10, Remark 9.11].
Since the irreducible components D i of the divisor D in our partial minimal model Y ⊃ U define points v i in U trop (Z), it is natural to ask whether these v i further lie in Θ ∨ , canonically giving regular functions ϑ vi on U ∨ . If so, and if Assuming the full Fock-Goncharov conjecture holds for U , this gives a candidate basis for O(Y ), namely {ϑ w |W trop (w) 0}. This need not be a basis though. Poles can cancel when we add functions, so in principal we could have ϑ p + ϑ q regular on Y even if ϑ p and ϑ q have poles along D.
So, once we know that (U ∨ ) trop (Z) is canonically identified with a basis for O(U ), there are three issues we need to address in order to cut this down to a basis for O(Y ).
(3) If a linear combination of ϑ-functions is regular on Y , then so is each ϑ-function appearing in the sum.
If U is a cluster A-variety and Y is obtained by allowing frozen variables to vanish, all three hold if each frozen index has an optimized seed. By construction, if a seed s is optimized for the frozen index i, then ϑ vi restricts to a Laurent monomial on the cluster torus in X associated to s. I provide a brief review of optimized seeds in Section 4.1. A more complete discussion can be found in [12, Section 9.1].
In Section 4.1, I show that each frozen index for Conf × 3 ( F ) has an optimized seed.
, and let W be the Landau-Ginzburg potential on Conf × 3 ( F ) ∨ associated to the partial minimal model In general if we have some partial minimal model U ⊂ Y for a log Calabi-Yau, we cannot expect to get a canonical basis for O(Y ) itself. The basis will be determined by the geometry of the pair U ⊂ Y rather than the geometry of Y alone. However, in the particular case Y = Conf 3 ( F ), the log Calabi-Yau open subset U = Conf × 3 ( F ) is determined entirely by the geometry of Y itself. It is simply the locus where underlying flags intersect generically, described in more detail in Section 2. In this sense, Ξ(Z T ) can be viewed as a canonical basis for O(Conf 3 ( F )) itself -it is a basis determined entirely by the geometry of Conf 3 ( F ).

Representation theory background
Interest in Conf 3 ( F ) has its roots in representation theory. The starting point is the Peter-Weyl theorem. A group G acts on itself both by left and right multiplication, and this action gives O(G) the structure of a G × G-bimodule. The following statement of the Peter-Weyl theorem comes from [17].

Theorem 7 (Peter-Weyl). Let G be a linearly reductive group. Then as G × G-bimodules
where the sum is over isomorphism classes of irreducible representations of G.
For GL n , the span of the highest weight vector v λ in the irreducible representation V λ of highest weight λ is the one-dimensional subspace fixed by U -the subgroup of upper triangular matrices with 1's along the diagonal. So where u λ is the highest weight vector for V * λ . The weight of u λ is −w 0 (λ), where w 0 is the longest element of the Weyl group W of G † . So the copy of V λ appearing in equation (1) is a weight space for the right action of the maximal torus H in G, and its weight is −w 0 (λ). To stress this point, the left action of H splits V λ into weight spaces, the highest weight being λ, but under the right action V λ is the −w 0 (λ)-weight space. The next thing to observe is that functions on G that are fixed by U -so f (xu) = f (x) for all u ∈ U -are the same as functions on G/U ∼ = F . Then and this is a weight space decomposition for the right action of H. † The Weyl group for GLn is just Sn, and w 0 is the permutation sending 1, 2, . . . , n to n, n − 1, . . . , 1. Now if we were to take three copies of F instead of one, we would have . By Schur's lemma, this is just a copy of the trivial representation for every The term on the right is a Littlewood-Richardson coefficient. These are the structure constants giving the decomposition This is the connection between Conf 3 ( F ) and the Littlewood-Richardson coefficients.

Geometric background
As mentioned in Section 1.1, Conf 3 (F ) is Fano and π : are triples of complete flags, defined up to an overall G-action. Given two arbitrary flags X • = (X 1 ⊂ · · · ⊂ X n ) and is the subset consisting of such triples of flags. It is log Calabi-Yau -its complement is an anti-canonical divisor D in Conf 3 (F ). Furthermore, the canonical volume form on Conf × 3 (F ) has a pole along all of D. We could in principle use the log Calabi-Yau mirror symmetry machinery to study the pair Conf × 3 (F ) ⊂ Conf 3 (F ). After all, the vector spaces of interest (V α ⊗ V β ⊗ V γ ) G from Section 2.1 are spaces of sections of line bundles over Conf 3 (F ). However, lifting to Conf 3 ( F ) will allow us to tackle all of the line bundles, and so all of the vector spaces (

Cluster structure
Conf × 3 ( F ) is not just log Calabi-Yau. Fock and Goncharov described a cluster structure for it in [5]. The discussion here is based on [5,8].
analogously. It will also be handy later to define W and Ξ to be the Landau-Ginzburg potential on Conf 3 × ( F ) ∨ and the cone given by its tropicalization. I will describe the initial seed of Conf 3 × ( F ), viewed as a cluster A-variety, and we will view Conf × 3 ( F ) as a quotient of The quiver for the initial seed comes from the 'n-triangulation' of a triangle, illustrated below for n = 4. The vertices in the n-triangulation will be the vertices of our quiver. For the arrows, we need to orient the edges of the n-triangulation. First, the boundary of the original triangle is given a clockwise orientation. The edges of the n-triangulation inherit their orientation from this one in the manner illustrated below. The vertices on the boundary of the original triangle are frozen vertices of the quiver, and the vertices in the interior are unfrozen. We ignore arrows between frozen vertices as they do not affect mutation, so the quiver we are after is: . The vertices of the quiver can be indexed by triples of non-negative integers (a, b, c) satisfying The cluster variables in the initial seed of Conf 3 × ( F ) are defined as follows: Note that by definition a linear transformation T : V → V is in SL(V ) if and only if ∧ n T acts by the identity on ∧ n V . So A (a,b,c) is indeed a well-defined function on Conf 3 × ( F ) -it respects the quotient by the diagonal SL(V ) action. None of these cluster variables are invariant under the diagonal action of GL(V ) -they are not functions on Conf × 3 ( F ) -but rational functions in these variables can still be GL(V ) invariant. In fact, take a Laurent monomial in these variables: Then for g ∈ GL(V ), This will lead to a condition on g-vectors (Proposition 21), which will index our ϑ-functions. I review the notion of g-vectors in Section 4.2.3.
Remark 8. The initial data I have described is for Conf 3 × ( F ) rather than Conf × 3 ( F ). That said, it can easily be translated into initial data for Conf × 3 ( F ). That is, we can view Conf × 3 ( F ) as a cluster variety in its own right, rather than as a quotient of a cluster variety. One way to do this is to replace all cluster variables A (a,b,c) of the initial seed with a new collection of variables, say A (a,b,c) = A (a,b,c) /A (n,0,0) . Upon doing so, the proofs I give in the following sections using the cluster structure of Conf 3 × ( F ) translate immediately to Conf × 3 ( F ). However, I find it more natural to avoid such choices. In what follows, I will freely use the cluster structure of I will show that the full Fock-Goncharov conjecture holds for Conf × 3 ( F ) by proving the following two conditions.
(1) The quiver Q s0 for the initial seed of Conf × 3 ( F ) has a maximal green sequence. (2) In the initial seed s 0 = (e 1 , . . . , e n ), all of the covectors {e i , ·}, i ∈ I uf , lie in a strictly convex cone † .
Together, (1) and (2) imply that the full Fock-Goncharov conjecture holds for Conf × 3 ( F ) [12,Proposition 8.25]. We will begin with (2) as its proof is much shorter. Proof. This is implied by the existence of a unimodular p * map. In Section 4.2.3, I construct a particular p * map and prove its unimodularity in Proposition 24.

Maximal green sequence
Let us first review what maximal green sequences are. Recall that the A prin construction involves a 'doubled' quiver, where a new frozen vertex w i is introduced for each vertex v i of the original quiver for A, and for each unfrozen vertex v i we introduce an arrow v i → w i . See [10, Construction 2.11] for a more complete discussion ‡ . This quiver is called the framed quiver Q associated to Q in [3]. As an example, if we take principal coefficients at s 0 for Conf × 3 ( F ), we would replace the quiver Q s0 of Figure 3 with Only the faded orange vertices are unfrozen. † I uf is the unfrozen subset of the indexing set I -the subset corresponding to mutable vertices. ‡ The description I am giving here is for the skew-symmetric case, so at first glance it could look different from the more general construction in [10] Now let Q be an arbitrary quiver mutation-equivalent to Q. An unfrozen vertex v i of Q is said to be green if all arrows between v i and any w j are outgoing: v i → w j † . On the other hand, v i is red if all arrows between v i and any w j are incoming: v i ← w j . A major result in cluster theory -known as sign coherence of c-vectors [12,Corollary 5.5] -implies that all unfrozen vertices are either green or red ‡ . A sequence of mutations is called a green sequence if each mutation in the sequence is mutation at a green vertex. It is a maximal green sequence if every unfrozen vertex in the resulting quiver is red.
Let r be the top r rows of unfrozen vertices in Q s0 , and let i r be mutation at each of these vertices in order -left to right, top to bottom. For example, for n = 6, i 3 is the following sequence: . Figure 6 (colour online). The mutation sequence i 3 for n = 6. We mutate at the indicated vertices in the indicated order.
Note that r can be at most n − 2 (the number of rows of unfrozen vertices).
Remark 11. This maximal green sequence induces a simple involution on O(Conf 3 ( F )) that I think is worth mentioning. It sends Remark 12. This sequence is one of the pieces I have lifted from the earlier paper [16]. It has been brought to my attention that the sequence has since appeared in [9, Proposition 9.8; 15, Section 3.3]. Le discusses the sequence in the context of a cactus transformation, making observations similar to Corollary 32. Goncharov and Shen discuss the sequence in the context of Donaldson-Thomas transformations, and use it to show that an involution on Conf 3 (SL n /U ) is a cluster transformation.
Let us start by looking at i r . Define r := {w i |v i ∈ r } and let F be the frozen vertices of Q s0 . We will split up the effects of i r into three parts: (1) how it affects the full subquiver with vertices n−2 , † Note that by construction all unfrozen vertices of Q are green. ‡ In the skew-symmetric case, the c-vector c i;s at the seed s associated to Q is c i;s : The statement that c i;s is sign coherent means either the left or right sum is empty.
(2) how it affects the collection of arrows between F and n−2 , and (3) how it affects the collection of arrows between n−2 and n−2 .
Note that we can split up the analysis this way. Since we never mutate at frozen vertices, arrows between the vertices of n−2 are unaffected by the presence of the frozen vertices. There is never a composition with the center vertex frozen. Additionally, since we never introduce arrows between frozen vertices, we could in principle treat each frozen vertex separately if we wanted to.
Lemma 13. The mutation sequence i r sends the subquiver So Q r−1 -the full subquiver with vertex set r−1 -remains unchanged, Q r only has its bottom horizontal arrows deleted, Q r+1 additionally has its bottom horizontal arrows deleted and its bottom diagonal arrows reversed, and this accounts for all changes to Q n−2 .
Proof. It is immediate that the claim holds for r = 1 -there is only one mutation to perform. Suppose it holds for all q < r. Then the quiver after performing i r−1 is Q i r−1 . All that remains is mutation through row r. We start with the leftmost vertex: This is followed by Finally, mutation at v r yields Q i r . Now let us move on to how i r affects the collection of arrows between F and n−2 . This is not really necessary to prove Proposition 10 -these arrows are not considered when determining if an unfrozen vertex is red or green -but it is worth knowing in any case, and it will provide a nice sanity check later. See Remark 16.
Lemma 14. For each Q s mutation equivalent to Q s0 , let A s be the subquiver having all vertices of Q s but only those arrows for which either the head or tail is in F . Then (2) Reverse arrows involving the vertices now in the v (n−r−1,r+1,0) and v (n−r−1,0,r+1) positions.
I will illustrate the claim with an example before proving it. For n = 6, A s0 is , which we can rearrange as So the claim holds for A i 1 (s0) . Suppose it holds for all q < r. Then, using Lemma 13, It remains to mutate through row r. Mutating at the leftmost unfrozen vertex of row r gives . . . . . . . . .
The next vertex of mutation sharing an arrow with some frozen vertex is the rightmost vertex of row r -the final vertex in our sequence. The penultimate quiver in the sequence is The final mutation gives completing the proof. Now onto the arrows between n−2 and n−2 .
Lemma 15. For each Q s mutation equivalent to Q s0 , let R s be the subquiver with vertex set n−2 n−2 but only those arrows for which either the head or tail is in n−2 . Then R i r (s0) can be constructed from R s0 as follows. To illustrate the claim, if we take n = 7, then R s0 is  and we just have to mutate through row r. The unfrozen portion of the quiver for each of the remaining mutations is given in the proof of Lemma 13. Note that there is a cyan arrow emanating from v (n−r−1,b,r−b+1) corresponding to each magenta arrow terminating at v (n−r−2,b,r−b+2) , and there is one additional cyan arrow v (n−r−1,b,r−b+1) →w (n−r−1,b,r−b+1) . Now, v (n−r−1,b,r−b+1) is the bth vertex of mutation in this row, and each of the magenta arrows are killed by a composition  We now have all of the ingredients we need to tackle Proposition 10.
Proof. We start with i n−2 . From the proof of Lemma 15, we see that each time we mutate at a vertex v k in i n−2 , all arrows between v k and n−2 emanate from v k -so v k is green. Then i n−2 is a green sequence. Using Lemmas 13-15, performing i n−2 on Q s0 and rearranging vertices as indicated in the lemmas results in the quiver The vertices of the bottom unfrozen row are now red, while the remaining unfrozen vertices are all green. For consistency with the r notation, let us only consider unfrozen vertices when indexing the rows. So the bottom unfrozen row we will call row n − 2, the one above it row n − 3, and so forth. This quiver is very similar to the one we started with. Above row n − 2 the only relevant difference is the introduction of the magenta arrows from n−3 to row n − 2.
Referring to the proof of Lemma 13, we note that no vertex of mutation in the sequence i n−4 shares an arrow with row n − 2. As a result, no composition affecting these arrows can occur until we mutate at row n − 3. That is, the subsequence i n−4 of i n−3 proceeds exactly as before, with these magenta arrows tagging along for the ride. Then prior to mutation through row n − 3, there is a cyan arrow emanating from v (2,b,n−b−2) for all but one of the magenta arrows terminating at v (1,b,n−b−1) . These paired magenta arrows are canceled upon mutation at v (2,b,n−b−2) . So after performing i n−3 and rearranging frozen vertices, we have
Remark 16. With the indicated rearranging of frozen vertices, the final quiver we obtained is the same as the original framed quiver with every arrow reversed. Imagine each w (a,b,c) as lying above v (a,b,c) . Now ignore temporarily all vertices that are not attached to any arrows, and reflect the rest of the final quiver over the plane given by a = c. The quiver itself is obviously the same. We have just changed its embedding into R 3 and returned each w (a,b,c) to its original position. Note that this also gives an isomorphism of the full subquiver whose vertex set is all of the v (a,b,c) vertices with the quiver Q s0 . So there is an isomorphism of the final quiver with the coframed quiver † } Q s0 fixing the w (a,b,c) vertices. This is what we expect by [3, Proposition 2.10], and it is a sanity check for the work in this section.

Existence of optimized seeds
The main result of this subsection is that every frozen index for Conf × 3 ( F ) has an optimized seed. In the skew-symmetric case, a seed s is optimized for the frozen index i if the vertex v i is a sink of the associated quiver Q s . The frozen index i corresponds to a component D i of the divisor we have added to Conf × 3 ( F ), and the existence of a seed optimized for i ensures that the ϑ-function on Conf × 3 ( F ) ∨ associated to D i is a global monomial. See [12, Section 9.1] for a more complete discussion of optimized seeds.
So far we have a basis B × for O(Conf × 3 ( F )). What we really want is a basis B for O(Conf 3 ( F )). A natural candidate for B is the subset of B × that extends to the divisors we have added, that is, B × O(Conf 3 ( F )). But this candidate is not automatically a basis for O(Conf 3 ( F )). Maybe ϑ p , ϑ q ∈ B × both have a pole along some component D i , but these poles cancel in their sum ϑ p + ϑ q . Then we would have ϑ p + ϑ q ∈ O(Conf 3 ( F )), but . The existence of an optimized seed for each frozen index ensures that this does not happen -if a linear combination of ϑ-functions extends to D i , then each ϑ-function in the sum extends as well [12,Proposition 9.7]. This condition is needed to utilize [12, Theorem 0.19], which will be used in the coming subsection on the potential W and cone Ξ for Conf 3 ( F ).

Proposition 17. Every frozen index for Conf
The sequence of mutations v 1 , v 2 , . . . , v r yields the quiver making v f a sink. The initial seed quiver for Conf × 3 ( F ) is shown in Figure 3. Call it Q s0 . Since there are no arrows to or from the corner vertices v (n,0,0) , v (0,n,0) , and v (0,0,n) , every quiver mutation equivalent to Q s0 will trivially be optimized for these three vertices. Beyond that, Q s0 is optimized for v (n−1,1,0) , v (0,n−1,1) , and v (1,0,n−1) . For the remaining frozen vertices v f , there is a subquiver of Q s0 isomorphic to Q L . Performing these mutations on Q s0 only affects the subquiver whose vertices are either in Q L or connected to Q L by an arrow. As arrows between frozen vertices are deleted, any frozen vertices besides v f can be ignored when determining if v f becomes a sink. Then the relevant subquiver of Q s0 has the form · · · · · · · · · , · · · · · · · · · · · · · · · · · · · · · · · · · · · . . .

· · ·
· · · · · · · · · · · · · · · 4.2. The potential W and cone Ξ for Conf 3 ( F ) In this subsection I compute the Landau-Ginzburg potential W and corresponding cone Ξ := {W T 0} ⊂ Conf × 3 ( F ) ∨ (R T ). By [12, Theorem 0.19], the analogous cone for A prin gives a canonical basis for the finitely generated algebra mid(A prin ) = up(A prin ). However, the exchange matrix for Conf × 3 ( F ) is full rank over Z. This is immediate from the stronger result Proposition 24. Then, as explained in [12, Proofs of Corollaries 0.20 and 0.21, p. 581], the desired results for Conf 3 ( F ) are implied by the results for A prin . I will say a few more words about this in Subsection 4.3.
I give an explicit description of W and Ξ in the initial seed. In this seed, the inequalities defining Ξ are precisely the tail positivity conditions of [18]. I also exhibit a map p * : N → M that identifies W with the representation theoretically defined potential W GS of [8] on Conf × 3 ( F ) and identifies Ξ with the Knutson-Tao hive cone [14]. † Before doing this, let us recall what W GS and the Knutson-Tao hive cone are. (a, b, c) ∈ (Z 0 ) 3 with a + b + c = n, for some fixed n, just like in Figure 4. Let H be the set of † Here N is the cocharacter lattice for a torus in Conf × 3 ( F ) and M is its dual -the cocharacter lattice for a torus in Conf × 3 ( F ) ∨ . The map p * commutes with mutation and is closely related to the exchange matrix. See [10, Section 2] for a general discussion of p * maps, and Section 4.2.3 below for a brief review. these vertices. R H is the possible labelings of these vertices by real numbers. Now take any pair of neighboring triangles, together forming a rhombus. This rhombus defines a linear inequality in R H by requiring the sum of the labels on the obtuse vertices to be greater than or equal to the sum of the labels on the acute vertices. (1) The polyhedral cone in R H satisfying all rhombus inequalities.

Knutson-Tao hive cone. Consider a triangular array of vertices indexed by triples
(2) The slice of (1) having top entry 0.
(2) and (3) clearly have completely equivalent combinatorics, with each point in (2) giving a representative of one of the equivalence classes in (3). The points in the Knutson-Tao hive cone are called hives.
The Knutson-Tao hive cone encodes the Littlewood-Richardson coefficients in a really beautiful way. Suppose we want to know dim (V α ⊗ V β ⊗ V γ ) G . The choice of weights (α, β, γ) determines the border of a hive, which I will illustrate in terms of (3). If λ = (λ 1 , . . . , λ n ), define |λ| = λ 1 + · · · + λ n . Now take |α| + |β| + |γ| = 0 -otherwise dim (V α ⊗ V β ⊗ V γ ) G = 0. Then we label the border of the hive as follows: Note that the condition |α| + |β| + |γ| = 0 is exactly what we need to be able to fill in the border this way. Also note that, since we are working only up to translations in the 1 H direction, the border is completely determined by the choice of (α, β, γ). Furthermore, this picture is manifestly symmetric under cyclically permuting α, β, and γ. Knutson and Tao showed that the number of integral hives with this border is precisely dim (V α ⊗ V β ⊗ V γ ) G [4,14].
They then show that in the initial seed of the cluster variety, W T GS 0 gives exactly the rhombus inequalities cutting out the Knutson-Tao hive cone.
Remark 18. Using this exponential notation, the X -variables of a given seed s = (e 1 , · · · , e n ) are defined by X i := z ei and the A-variables by A i := z e * i . See [10, Section 2], keeping in mind that we are considering the skew-symmetric case here.
For each frozen index f , we have an explicit sequence of mutations from s 0 to a seed s f optimized for f from the proof of Proposition 17. We want to pullback ϑ f from T M ;s f to T M ;s0 , so we reverse this sequence. All of the mutations occur at vertices of the subquiver Q L , so, by the mutation formula given above, only indices of Q L will come into play in computing the pullback of ϑ f .

Proof. The quiver for the first mutation is
The next quiver is This pattern continues with the ith mutation yielding ). § Generally there would be multipliers d j modifying this expression, but in the case under consideration all multipliers d j are 1. This is known as the skew-symmetric case. Additionally, the reader who wishes to compare results here with work following Fomin and Zelevinsky's conventions should set the Fomin-Zelevisky exchange matrix B equal to T .

The result after all n mutations is
as claimed.
Using Proposition 19, we can immediately express W on the torus T M ;s0 of the initial seed Corollary 20. Take a, b, c ∈ Z >0 . The restriction of W to T M ;s0 is Note that we now have the basis B of O(Conf 3 ( F )) that we were after -it is canonically identified with

Explicitly, the ϑ-functions ϑ p is in B if and only if p is in Ξ(Z T ). This basis is canonically determined by the pair Conf
, so B must be preserved by this action. Since B is a discrete set and H ×3 acts continuously, the only possibility is that each element of B is fixed by H ×3 . The elements of B are defined up to scaling, so this means that every element of B is an H ×3 -eigenfunction. The H ×3 action and the weights of basis elements under this action are discussed further in Section 4.3.
At this point we would like to see if W to pulls back to the Goncharov-Shen potential W GS on Conf × 3 ( F ) for some carefully chosen p * . The guideline for writing down this map will be the representation theoretic interpretation of the cones on both sides. For this, we will compare version (3) of the Knutson-Tao hive cone to Ξ. To have a nice representation theoretic interpretation of Ξ, we need to relate the g-vector of a ϑ-function to its weight under the H ×3 action.
First let us review what g-vectors are and fix some notation. Given any cluster A-variety, Gross-Hacking-Keel-Kontsevich build a family of deformations of A, denoted A prin , that itself has the structure of an A-variety. I recall how to build a quiver for A prin at the beginning of Section 3.1. A choice of seed s determines a cluster torus T N ;s in A, an action of T N ;s on A prin , and a canonical extension of each cluster monomial to A prin (simply by performing mutations in A prin rather than A) † . The T N ;s action is induced by the inclusion The extended cluster monomials are all eigenfunctions of this T N ;s action, and the weight under this action is known as the g-vector at s for the cluster monomial. The choice of seed s also determines an identification of X (Z T ) with M , and this weight (an element of M ) is how we explicitly associate a ϑ-function to a point in X (Z T ).
Note that . A (a,b,c) .
Decompose λ ∈ χ * (H) by λ(h) = h 1 λ1 · · · h n λn . Then the following picture lets us read off the H ×3 weight (α, β, γ) of z gs 0 (ϑp) , and in turn ϑ p (denoted w(ϑ p )). † I am restricting to the skew-symmetric case here to avoid a discussion of multipliers d i and to simplify notation. ‡ The partial ordering on terms comes from the monoid of bending parameters. See [12,Section 3].
Before finding the map p * : N → M , let us recall briefly the properties it must satisfy: Here N uf is the unfrozen sublattice of N -the span of {e i } i∈I uf . Up to some map N/N uf → N ⊥ uf , p * is essentially just the exchange matrix . The ambiguity we can exploit -the map 4 . Figure 14 (colour online). Hyperplanes defining P α,β,γ for n = 4, with obvious generalization to arbitrary n. All entries on the indicated side of a line sum to the given value, so each line corresponds to one hyperplane. Figure 15 (colour online). Pictorial representation of the inequalities cutting out Ξ in the initial seed for n = 4. The boxes indicate that the contained entry must be non-negative. Arrows indicate that the sum of the entries along the line of the arrow, starting with the boxed entry and ending with the entry at the tip of the arrow, must be non-negative. Each inequality corresponds to an exponent appearing in Corollary 20. These are exactly the tail-positivity inequalities of [18].
N/N uf → N ⊥ uf -amounts to choosing pairings between frozen indices, without considering any skew-symmetry requirements for this portion of our exchange matrix.
So we know what properties p * must satisfy, and we can compare Figures 11 and 14 to further guide our efforts to write down a candidate p * map. First, take e (a,b,c) ∈ N uf . Using (1) we can immediately write down p * (e  Figure 17. p * (e (2,1,1) ).
Note that the sum of the entries of p * (e (2,1,1) ) is 0, as are all sums indicated in Figure 14. So things look good so far.
We can do the same procedure for every e (a,b,c) . The resulting map is given by and rotations of these. Every entry not explicitly given is 0.
So this is our proposed map p * . It certainly satisfies (1) and (2). What we want to show now is that p * gives a unimodular equivalence between version (3) of the Knutson-Tao hive cone and Ξ in the initial seed, and furthermore that p * W = W GS -so the representation theoretic Goncharov-Shen potential has a purely geometric description.
The last line above is expressed in [8] as Summing over all ϑ-functions in W yields the potential W in [8, Section 3.1], with each monomial summand corresponding to a different rhombus inequality.
Pictorially, p * identifies the inequalities defining Ξ in the initial seed with those defining the Knutson-Tao hive cone in the following way: Proposition 24. The map p * is unimodular, so in the initial seed Ξ is unimodularly equivalent to the Knutson-Tao hive cone.
Remark 25. Keep in mind that upon identifying our tropical spaces with real vector spaces, the domain of p * will look like R H /R · 1 H and the codomain will look like the subspace V of R H in which the sum of all entries is 0. Note that if we view the two copies of R H as dual spaces in the obvious way, then 1 H ⊥ is exactly V , and so the domain and codomain of p * are also dual spaces.
Proof. First note that (a,b,c)∈H p * e (a,b,c) = p * (1 H ) = 0. (2) Next, I claim that On account of (2), an immediate corollary of this claim would be that so is e * (n−1,0,1) . Next we use the other vector associated to v (n−1,0,1) (the horizontally oriented one e * (n−1,1,0) − e * (n−1,0,1) this time) to see that e * (n−1,1,0) is also in Λ. We then go one step southeast to the vertex v (n−2,1,1) and repeat this process, starting with the diagonally oriented vector and following up with the horizontally oriented vector, to find that e * (n−2,1,1) and e * (n−2,2,0) are in Λ as well. Continuing southeast gives every e * (a,b,c) with c = 0 or 1. Now we will push toward the southwest corner using rhombi of the only remaining orientation. Since p * commutes with mutation, This is discussed in greater detail in [10,Section 2]. Here I will identify the action of T K on Conf × 3 ( F ) with the H ×3 action, and the tropicalization of (3) with the map w sending p ∈ Conf × 3 ( F ) ∨ (Z T ) to the H ×3 -weight of the corresponding ϑ-function w(ϑ p ).
The  To see that (3) tropicalizes to the weight map, first restrict to tori for a fixed seed s. Then ϑ p is a finite sum of characters on T N ;s , and since ϑ p is an H ×3 eigenfunction, each of these characters has the same H ×3 = T K weight. Let one of the characters be z m . Then the weight of ϑ p under the T K action is the map z k → z k,m for z k ∈ T K . In other words, the T K weight of z m is (m mod K ⊥ ) ∈ K * † . The map m → m mod K ⊥ dualizes the inclusion K → N , so for each seed s the weight of ϑ p is the tropicalization of (3). Since this holds when we restrict to every torus, it holds for all of Conf × 3 ( F ) ∨ . As alluded to previously, there is a related action on A prin . Let K be the kernel of The fact that the exchange matrix is full rank implies that π is isomorphic to the trivial bundle Conf × 3 ( F ) × T M [12,Lemma B.7]. This † As a sanity check, note that in the scattering diagram approach of [12], all scattering functions have the form (1 + z p * (n) ) c for some n ∈ N uf and c ∈ Z >0 . As a result, the exponents of the summands of ϑp differ by elements p * (N uf ). It is straightforward to check that p * (N uf ) ⊂ K ⊥ , so all of the summands of ϑp do indeed have the same weight.
is used in [12, Proof of Corollaries 0.20 and 0.21, p. 581] to translate basis results for A prin to Conf 3 ( F ).

Ray representation of Ξ
In [16], I showed that for the decorated flag variety, the cone Ξ F is generated by the g-vectors for Plücker coordinates. This is probably the most natural generating set for the homogeneous coordinate ring of the flag variety, and I wonder if the generators of Ξ can fill this role for Conf 3 (F ). This is necessarily vague and subjective. I am primarily asking if the ϑ-functions corresponding to generators of Ξ in the initial seed have a simple explicit description. In this subsection, I describe the rays generating Ξ and give partial results relating these rays to functions. Since Ξ is not strictly convex for G = GL n , we will temporarily restrict to SL n . The line spanned by such a g-vector is the intersection of the hyperplanes defined by the boxes for the remaining frozen variables and the arrows parallel to the boundary edge containing v (a,b,c) . So for A (2,0,2) , we get the line described by the following picture: The ray R 0 · g(A (a,b,c) ) contained in this line satisfies the remaining inequalities and is an edge of Ξ.
Proposition 28. The g-vectors of all initial seed variables generate edges of Ξ.
Proof. Proposition 27 took care of the frozen variables. For the unfrozen variable A (a,b,c) , there is a 1 in the interior of our triangle with every other entry 0. For starters, take the hyperplanes defined by each box. Then fix a box and add as many consecutive arrows in the string emanating from it as possible without hitting v (a,b,c) . Doing this for all boxes gives enough hyperplanes to determine the line given by a free parameter in position v (a,b,c) and 0 elsewhere. It really gives more hyperplanes than needed, but that is not a problem. For example, for A (2,1,1) we would have the following picture: The ray R 0 · g(A (a,b,c) ) contained in this line satisfies the remaining inequalities and is an edge of Ξ.
The functions associated to the remaining edges take more effort to describe. We will figure out what these edges actually are before worrying what functions they might correspond to. Consider for a moment the inequalities represented by arrows. These are described in Figure 15. Staring at this for a little while suggests the following picture for some of the remaining rays of Ξ. Take an interior vertex v (a,b,c) . There are three subquivers Q L (from Proposition 17) starting at a frozen vertex and ending at v (a,b,c) . Draw a line segment through the vertices of each of these subquivers. Here is an example: .  (a, b, c) position, a single '1' along each of the three segments, and '0' elsewhere. The idea is that each of the three arrows pointing to v (a,b,c) indicate that the sum of the entries along one of the three segments should be non-negative. If we make any entry negative, we will get something outside of the cone generated by vectors in Propositions 27 and 28. The construction described is the simplest way to achieve this without violating any inequalities.
Let us call such a picture a trivalent vertex.
Proposition 29. The vectors associated to trivalent vertices generate edges of Ξ. Proof. Consider all of the hyperplanes defining faces of Ξ. It is easiest to say which to exclude from our intersection. Essentially, we want to intersect all of the hyperplanes associated to inequalities that should reduce to equalities for the vector in question, and only these hyperplanes. So in Figure 27, we would remove exactly the blue boxes and arrows corresponding to strict inequalities, and the intersection of the remaining hyperplanes is a line containing the given vector. To do this for an arbitrary vector associated to a trivalent vertex, we start by going to the position of the entry 1 along each line segment. If this is not on the boundary, then the box at the end of the line segment and all arrows leading to this entry apart from the last one give equalities. But the arrow whose tip hits this entry gives a strict inequality, as do the arrows coming after it, until we get to the arrow whose tip hits the −1 entry. So given the segment we would not include the following hyperplanes in the intersection: −1 1 . Figure 29 (colour online). The blue arrows give strict inequalities, so we exclude the corresponding hyperplanes from the intersection that will yield the span of our vector.
Do this for all three line segments and then intersect all remaining hyperplanes. The result is the real span of the given vector, and its R 0 span also satisfies the inequalities that have been omitted from the intersection.
The next thing to notice is that we can overlay two trivalent vertices, and as long as none of the line segments are colinear we will have two line segments intersecting at a vertex. If we make the entry of this vertex 1, we will get a vector outside the span of the vectors previously described. For example, take which gives the vectors and Consider either of these two vectors. If it were the sum of vectors described previously, we would have to take at least two vectors associated to trivalent vertices to account for the two minus signs. Then the sum of all entries must be at least 4, but it is in fact only 3. So it is indeed outside of the span of the vectors described previously, and it clearly lies in our cone. The proof that it generates an edge of Ξ is basically identical to the proof of Proposition 29.
Next, there is no reason to limit ourselves to only overlaying two trivalent vertices. We can overlay as many as we want. Say we overlay k of them. Then we are describing vectors for which k entries are −1. As long as we place our 1's in such a way that our vector cannot be a positive combination of the vectors associated to k − 1 or fewer trivalent vertices, we will get an edge of Ξ by the argument used above. In particular, if we ensure that the sum of the entries is less than could be achieved with such positive combinations, our vector must be an edge of Ξ.
Proposition 30. Every edge of Ξ is generated by an initial seed g-vector, a trivalent vertex, or overlaid trivalent vertices.
Proof. The initial seed g-vectors already generate the entire positive orthant, so any additional edge of Ξ must have some negative entry. Negative entries must be at interior vertices on account of the box inequalities. Say the entry at (a, b, c) is negative, with value −x. Then the three incoming arrows at v (a,b,c) indicate that the sum of the remaining entries along each of the three line segments leading to v (a,b,c) must be at least x. However, if the sum is more than x, we would be able to realize this vector as a sum of vectors from trivalent vertices and vectors in the positive orthant. You can see this by restricting to a line segment first. It is easy to see for this restriction, and the result transfers over directly as the trivalent vertices are made up of three line segments, and the position of a 1 along one line segment is completely independent of the other two line segments. It follows that if some edge of Ξ lies outside of the positive orthant, it must be generated by a trivalent vertex or a collection of overlaid trivalent vertices. So this is indeed all of the edges of Ξ.
Proposition 31. Consider the trivalent vertex p having 1's in positions (a 1 , b 1 , c 1 ),  (a 2 , b 2 , c 2 ), and (a 3 , b 3 , c 3 ), labeled such that the top left 1 is in position (a 1 , b 1 , c 1 ), the rightmost 1 is at (a 2 , b 2 , c 2 ), and the bottom left 1 is (a 3 , b 3 , c 3 ). Now take the triangle with sides b = b 1 + 1, c = c 2 + 1, and a = a 3 + 1. Then ϑ p is obtained by performing the maximal green sequence of Proposition 10 on the subquiver with vertices .
This proves the claim † .
It is not clear yet whether the rays of Ξ correspond to a 'simple' collection of functions, but I hope that observations in this subsection provide a foundation for addressing this question. It would be philosophically more appealing to have a description that avoids such choices. So, instead of choosing a single pair, we will choose all pairs at once, and later we will mod out to identify all of these choices. First note that G = GL(V ) acts freely and transitively on the generic locus ( F × F ) × of F × F . It is clear that G acts freely and transitively on ordered bases for V , and the correspondence between ordered bases and generic pairs ((X • , x • ), Y • ) is straightforward. Given an ordered basis (v 1 , . . . , v n ), set X i := span{v 1 , . . . , v i }, Y n−i := span{v i+1 , . . . , v n }, and x i := v i mod X i−1 . On the other hand, given a generic pair ((X • , x • ), Y • ), note that x i+1 can be identified with an i-dimensional affine subspace of X i+1 : x i+1 '='{v ∈ X i+1 |v mod X i = x i+1 }. Then x i+1 and Y n−i intersect in a point since X i and Y n−i do, and x i+1 is just a translation of X i . Set v i+1 := x i+1 Y n−i . These two maps are clearly inverses of each other. Now we can view F as the subset of Conf( F , F , F ) with (A 1 , B 3 ) generic, and G e,w0 as Conf( F , F , F ) × . The above description is based on [8].

Recovering
So we start with Conf × 3 ( F ) ∨ (Z T ) and then take the slice whose H z weight γ is 0 † . This gives us a basis for O(G e,w0 ). When we partially compactify to F , we still ask for the first and third flags to intersect generically, so we are leaving off the divisors D (i,0,n−i) . Then in the initial seed, the Landau-Ginzburg potential in this case is where a, b, c ∈ Z >0 , ϑ (a,b,0) = n−a−1 i=0 z − i j=0 e (a,b−j,j) , † Recall that G = SLn here. † As in Remark 8, using this condition we could recover the usual cluster structure for this space by defining a new collection of variables, say A (a,b,c) := A (a,b,c) /A (0,n−c,c) . But the point here is to avoid making choices, so we will not do that. It is immediate from Proposition 23 that W F pulls back to the potential of [8] on Conf ( F , F , F ). Furthermore, Proposition 24 and [8, Theorem 3.2] immediately imply that Ξ F is unimodularly equivalent to the Gelfand-Tsetlin cone. The polytope in Ξ F where β = −w 0 (λ) parametrizes a canonical basis for the irreducible representation V λ .
Remark 33. There is one semantic caveat worth mentioning here. I have generally seen the term 'Gelfand-Tsetlin cone' applied to a strictly convex cone encoding polynomial GL n representations. O( F ) decomposes as a sum of rational GL n representations, which include duals of polynomial representations. Essentially, det is an invertible function on GL n and this gives Ξ F a 1-dimensional linear subspace. That said, the cone defined by Gelfand and Tsetlin in [7,Equation 3] encodes rational representations and is the cone identified with Ξ F by p * .
Using [8, Figure 31], the inequalities defining Ξ F and the Gelfand-Tsetlin cone are identified via p * as follows:

⇐⇒
. Figure 33 (colour online). Correspondence between inequalities defining Ξ F in the initial seed and the Gelfand-Tsetlin cone for n = 4. For the Gelfand-Tsetlin cone, an arrow indicates that the entry at the tail is at least as large as the entry at the tip.
Goncharov and Shen describe their potential as part of a 6-tuple defining a positive decorated geometric crystal. In this setting, p * W = W GS plays the role of Berenstein and Kazhdan's potential f from [2]. See the appendix in the arXiv version of [8] for details.
To describe U in this way, we view F as the subset of Conf( F , F , F ) with (A 1 , B 3 ) generic, U as the subset where (A 1 , B 2 ) is also generic, and the cluster varietyŮ in U as Conf × ( F , F , F ). UsuallyŮ is described as the subset of U (upper triangular unipotent matrices) where the minors Δ 1,...,i n−i+1,...,n are non-vanishing. Here we are getting to U from F by requiring (A 1 , B 2 ) to be generic, and we get toŮ from U by requiring (B 2 , B 3 ) to be generic.
A basis for O(Ů ) is given by taking the slice of Conf × 3 ( F )(Z T ) with H y × H z weight (β, γ) = 0. When we partially compactify to U , the divisors that we add are D (0,i,n−i) . The corresponding inequalities are the solid (as opposed to dashed) boxes and arrows of Figure 33. Then Ξ U is a simplicial cone of dimension n 2 .
and L. Shen cleared up a variety of confusions that arose in the course of this work. I thank L. Bossinger for carefully reading through a preliminary version of this paper, and providing numerous edits and suggestions. I would also like to thank the two anonymous referees. Both provided amazingly thorough and knowledgeable reports, and I feel their suggestions have greatly improved the exposition here.