Stasheff polytopes and the coordinate ring of the cluster X-variety of type A_n
SStasheff polytopes and the coordinate ring of the cluster X -variety of type A n Linhui Shen
Abstract
We define Stasheff polytopes in the spaces of tropical points of cluster A -varieties.We study the supports of products of elements of canonical bases for cluster X -varieties.We prove that, for the cluster X -variety of type A n , such supports are Stasheff poly-topes. Contents A -varieties of type A A k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 The set of A -laminations on a convex polygon . . . . . . . . . . . . . . . . . 253.4 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Further discussion and conjecture . . . . . . . . . . . . . . . . . . . . . . . . 29 A cluster ensemble is a pair ( X , A ) of positive spaces defined in [FG]. Cluster A -varietiesare closely related to cluster algebras introduced in [FZ] – the ring of regular functions onthe space A coincides with the upper cluster algebra of [BFZ]. In this paper, we focus onthe cluster X -variety X A n of type A n , which is closely related to the moduli space M ,n +3 .1 a r X i v : . [ m a t h . AG ] S e p n detail, let ( A Φ , X Φ ) be the cluster ensemble assigned to a root system Φ of finite type.Denote by A Φ ∨ ( Z t ) the space of Z - tropical points of the Langlands dual cluster A -variety A Φ ∨ . According to the Duality Conjectures from [FG], one should have an isomorphism I A : A Φ ∨ ( Z t ) −→ E ( X Φ ) (1.1)where E ( X Φ ) is the set of indecomposable functions on X Φ , providing a canonical basisof the ring of regular functions on X Φ . In the case of type A n , such a map I A has beenconstructed in [FG3]. Denote by P ( X A n ) the set of finite products of elements of E ( X A n ).The main goal of this paper is to study P ( X A n ).The basis E ( X A n ) has remarkable combinatorial properties. One of them is that, giventropical points l , ..., l m ∈ A A n ( Z t ), their corresponding product in P ( X A n ) can be uniquelydecomposed into a finite sum of elements of E ( X A n ) with non-negative coefficients: m (cid:89) i =1 I A ( l i ) = (cid:88) l ∈A An ( Z t ) c ( l , ..., l m ; l ) I A ( l ) . (1.2)The support of the product is the set of l ∈ A A n ( Z t ) such that c ( l , ..., l m ; l ) (cid:54) = 0. TheConvexity Conjecture from [FG] suggests that the support is universally convex in A A n ( Z t ).We prove that it is not only convex but also a Stasheff polytope.The Stasheff polytope is a remarkable convex polytope first described combinatoriallyby J. Stasheff in 1963. In Section 1.3, we define Stasheff polytopes in tropical positive spaces .The main idea of the definition is given below.Denote by A ( R t ) the space of R -tropical points of a positive space A . It has a piecewise-linear structure, isomorphic to R dim A in many different ways. We define a convex polytope in A ( R t ) as the intersection of “half-spaces” given by inequalities of tropical indecompos-able functions (see Section 1.1). The tropical indecomposable functions are usually notlinear in all coordinate systems, but are convex and piecewise linear.Let Φ ≥− be the set of positive and simple negative roots of a root system Φ of finitetype. In [CFZ], polytopal realizations of Stasheff polytopes (or generalized associahedra)in R n were constructed by a set of linear inequalities indexed by Φ ≥− . For the cluster A -variety A Φ , there are cluster variables A α indexed by Φ ≥− ([FZ1, Theorem 5.7]). WhenΦ is of classical Cartan-Killing type, the cluster variables are indecomposable functions([ loc.cit. , Theorem 4.27]). Their tropicalizations A tα are functions on the space A Φ ( Z t ).Our Stasheff polytopes in A Φ ( Z t ) are defined via tropical cluster variables (Definition 1.9).One of our main results is as follows. Theorem 1.1.
The support of the product (1.2) is the Minkowski sum of the points l , ..., l m : S ( l , ..., l m ) = { x ∈ A A n ( Z t ) | A tα ( x ) ≤ m (cid:88) i =1 A tα ( l i ) for any α ∈ Φ ≥− } . (1.3) In the case of type A n , the Langlands dual A A n ∨ coincides with A A n . t is a Stasheff polytope in the space A A n ( Z t ) . We conjecture the same result for any root system Φ of finite type.In R n the Minkowski sum of finitely many points is the sum of these points as vectors.It is still a point.In our case, A A n has many coordinate systems which are called clusters . Each clusterprovides a way of taking the sum of the tropical points as vectors. We thus get a set ofvertices parametrized by the set of clusters. The Minkowski sum turns out to be the convexhull of these vertices. It is a Stasheff polytope.An example of Stasheff polytopes in A A is shown on Fig. 1-5. These figures show thesame polytope in 5 different coordinate systems (clusters). Notice that it is a “tropical”pentagon, but looks like a heptagon from the usual point of view: the reason is that someof its sides are given by piecewise linear functions, and thus they are not line segments.But they become line segments in other coordinate systems. In general, the Minkowskisum gives rise to a new family of convex polytopes in tropical positive spaces (Definition3.15).Section 2 focuses on the cluster A -variety (cid:101) A A n . It is a cluster A -variety with coeffi-cients (see Definition 2.2). Theorems 2.4, 2.7 provide two criteria for recognizing Stasheffpolytopes. Theorem 1.10 is a corollary of Theorem 2.4. The variety (cid:101) A A n also has a canon-ical basis E ( (cid:101) A A n ) ([FZ1], [SZ],[C]). Similarly, we define the set P ( (cid:101) A A n ) of finite productsof elements of E ( (cid:101) A A n ). It has a natural partial order structure. Theorem 2.13 determinesthe partial order on P ( (cid:101) A A n ). Theorem 2.10 is the main technical tool for proving Theorem1.1.Section 3 focuses on the cluster X -variety X A n . There is a surjective map k : (cid:101) A A n →X A n . The induced map k ∗ on their coordinate rings takes E ( X A n ) (respectively P ( X A n ))into E ( (cid:101) A A n ) (respectively P ( (cid:101) A A n )). By using Theorem 2.10 and the map k , we provethe first part of Theorem 1.1. The second part is a direct consequence of Theorem 1.10.Theorem 3.14 provides a bijection between P ∗ ( X A n ) and the set of Stasheff polytopes in A A n ( Z t ). Conjecture 3.16 is a generalization of Theorem 1.10 to all cluster X -varieties. Acknowledgements . I wish to thank K. Peng, R. Raj for helpful conversations. Iam especially grateful to my advisor A. Goncharov for suggesting the problem and forhis enlightening suggestions and encouragement. In particular, the main definitions ofthis paper concerning convex polytopes in tropical positive spaces follow the idea in [FG3,Section 2.4]. Finally, I thank the referee for very careful reading of this paper and for manyuseful suggestions.
For the convenience of the reader, let us briefly recall some basic definitions from [FG].3 positive space is a variety A equipped with a positive atlas C A . Namely, the transitionmaps between coordinate systems in C A are given by rational functions presented as a ratioof two polynomials with positive integral coefficients. We denote such a space by ( A , C A ).A universally positive Laurent polynomial on A is a regular function on A which isa Laurent polynomial with non-negative integral coefficients in every coordinate systemin C A . Denote by L + ( A ) the set of universally positive Laurent polynomials. Given F , F ∈ L + ( A ), if F − F ∈ L + ( A ), then we say F ≥ F . We say that F ∈ L + ( A ) is indecomposable if it cannot be decomposed into a sum of two nonzero universally positiveLaurent polynomials. Denote by E ( A ) the set of indecomposable functions. Let P ( A ) bethe set of finite products of indecomposable functions. Clearly, P ( A ) is a subset of L + ( A ),and is a semigroup under multiplication.A semifield is a set P equipped with operations of addition and multiplication, so thataddition is commutative and associative, multiplication makes P into an abelian group,and they are compatible in a natural way: ( a + b ) c = ac + bc for a, b, c ∈ P . For anypositive space A , the transition maps are subtraction free. Thus one can consider the set A ( P ) of P -points of A . The transition maps are bijective on A ( P ) because they are welldefined on every P -point. Therefore A ( P ) (cid:39) P n . For example, the set R > of positive realnumbers with usual operations is a semifield. The set A ( R > ) (cid:39) R n> of positive points of A is well defined.The tropical semifield R t is a set of real numbers R but with the multiplication · t andaddition + t given by a · t b := a + b, a + t b := max { a, b } . (1.4)The semifields Z t , Q t are defined in the same way. Let A ( A t ) be the set of A t -points, here A can be Z , Q , R . One can tropicalize F ∈ L + ( A ) via evaluating it on A ( A t ). It is easyto see that the tropicalization F t is a convex piecewise linear function in each positivecoordinate system. Example.
Let F = 2 x x + x − + 1 = x x + x x + x − + 1. When taking themaximum, the coefficients of monomials in F do not matter. One can drop them first.Hence F t = max { x + x , − x , } . Given a subset S of A ( A t ), define c F,S = sup x ∈ S F t ( x ) , c F,S ∈ A (cid:91) { + ∞} . (1.5)Following [FG3], the convex hull of S and the Minkowski sum of two convex subsets are asfollows. Definition 1.2.
The convex hull of a subset S of A ( A t ) is C ( S ) = { x ∈ A ( A t ) | F t ( x ) ≤ c F,S for all F ∈ E ( A ) } . (1.6) Clearly S ⊆ C ( S ) . We say S is convex if S = C ( S ) . efinition 1.3. The Minkowski sum of two convex subsets S , S of A ( A t ) is S + S = { x ∈ A ( A t ) | F t ( x ) ≤ c F,S + c F,S for all F ∈ E ( A ) } . (1.7) Remark . Clearly S + S is also convex, but the number c F,S + S = sup x ∈ S + S F t ( x ) (1.8)is not necessarily equal to c F,S + c F,S . Furthermore, the Minkowski sum may not beassociative. This is because in our definition, the defining functions may not be linear. Forexample, let L = x , L = y , L = max { x + y, x } be functions on R . Given three sets S = { L ≤ , L ≤ , L ≤ } , S = { L ≤ , L ≤ − , L ≤ } ,S = { L ≤ , L ≤ , L ≤ } . Consider S + S = { L ≤ , L ≤ , L ≤ } , then c L ,S + S = 1 < c L ,S + c L ,S and( S + S ) + S (cid:54) = S + ( S + S ) . In the cases of cluster A -varieties of classical Cartan-Killing type (see [FZ1], Section4), the Minkowski sum behaves well in the following sense. Proposition 1.4.
Given a cluster A -variety A of classical type, let S , S , S be convexsubsets of A ( A t ) . For any F ∈ E ( A ) , we have c F,S + S = c F,S + c F,S . (1.9) The associativity holds: ( S + S ) + S = S + ( S + S ) . (1.10) Proof.
For any F ∈ E ( A ), it becomes a monomial in a certain coordinate system ([ loc.cit. ,Theorem 4.27], [C, Theorem 1.1]). Its tropicalization F t is linear in this coordinate system.Let x ∈ S , x ∈ S be such that F t ( x ) = c F,S , F t ( x ) = c F,S . Let x + x be the usualsum of two vectors in the same coordinate system. Then F t ( x + x ) = c F,S + c F,S .On the other hand, for any G ∈ E ( A ), the convexity of G t implies that G t ( x + x ) ≤ G t ( x ) + G t ( x ) ≤ c G,S + c G,S . Therefore, by definition, x + x ∈ S + S . The number c F,S + S ≥ F t ( x + x ) = c F,S + c F,S . The other direction that c F,S + S ≤ c F,S + c F,S follows from (1.7), (1.8). The first part is proved. The associativity follows directly fromthe first part. Conjecture 1.5.
The formula (1.9) holds for all cluster A -varieties. For cases when (1.9) holds, given finitely many subsets S , ..., S m of A ( A t ), theirMinkowski sum is m (cid:88) i =1 S i = { x ∈ A ( A t ) | F t ( x ) ≤ m (cid:88) i =1 c F,S i for all F ∈ E ( A ) } . (1.11)5 .2 Supports of products of elements of a canonical basis Let ( X , A ) be a cluster ensemble. We briefly recall its definition in Section 3.1. Let( X ∨ , A ∨ ) be the pair of their Langlands dual spaces ([FG]). Fock and Goncharov’s DualityConjecture asserts that the set of Z t -points of A ∨ -(or X ∨ -) space parametrizes a canonicalbasis of the coordinate ring of X -(or A -) space. For our purposes, we need only onedirection of this conjecture. Conjecture 1.6 ([ loc.cit ], Section 4) . There is a canonical isomorphism I A : A ∨ ( Z t ) ∼ −→ E ( X ) . (1.12) The set E ( X ) provides a Z -basis of the coordinate ring of X . For the cluster X -variety X A n , the Duality Conjecture is proved in [FG3]. We sketchthe proof in Section 3. In this case, A ∨ A n coincides with A A n . Let f be a regular functionon X A n . It can be uniquely decomposed as a finite sum f = (cid:88) l ∈A An ( Z t ) c ( f ; l ) I A ( l ) . (1.13)The numbers c ( f ; ∗ ) are called the structure coefficients of f . Thanks to the followingLemma, we have c ( f ; ∗ ) ∈ Z ≥ for each f ∈ L + ( X A n ). Lemma 1.7.
If the set E ( X ) of indecomposable functions is a Z -basis of the coordinatering of X , then it provides a Z ≥ -basis of L + ( X ) .Proof. Let α = { X i } be a local coordinate system in C X . By definition, each f ∈ L + ( X )can be uniquely expressed as f = (cid:88) a =( a ,...,a n ) ∈ Z n c a X a . . . X a n n , c a ∈ Z ≥ . Define l α ( f ) := f (1 , . . . ,
1) = (cid:80) a ∈ Z n c a . Clearly here if f (cid:54) = 0, then l α ( f ) ≥ f ∈ L + ( X ) − { } . Let f = (cid:80) mk =1 g k be a decomposition of f such that g , . . . , g m ∈ L + ( X ) − { } . Then l α ( f ) = (cid:80) mk =1 l α ( g k ) ≥ m . Therefore m is bounded. Let N be themaximal number such that f = g + . . . + g N , g , . . . , g N ∈ L + ( X ) − { } . (1.14)Here g , . . . , g N ∈ E ( X ). Otherwise, we may assume that g N / ∈ E ( X ). By definition,we have g N = g (cid:48) N + g (cid:48) N +1 , where g (cid:48) N , g (cid:48) N +1 ∈ L + ( X ) − { } . Therefore we get a newdecomposition f = g + . . . + g (cid:48) N + g (cid:48) N +1 , which contradicts the assumption that N is themaximal number. 6et us combine the same terms appearing in the right hand side of (1.14). It gives usa decomposition f = (cid:88) f i ∈ E ( X ) c i f i , c i ∈ Z ≥ . (1.15)If E ( X ) is a Z -basis of the coordinate ring, then the decomposition (1.15) is unique. Con-versely, if f can be decomposed as in (1.15), then f ∈ L + ( X ). The Lemma is proved. Definition 1.8.
The support of f is a finite set S f := { l ∈ A A n ( Z t ) | c ( f ; l ) (cid:54) = 0 } . (1.16) Example.
We consider the case of type A . Let l = ( − , , l = (0 , , l = (1 , , l = (1 , , l = (0 , − . (1.17)In a certain coordinate system, the set A A ( Z t ) (cid:39) Z = { bl i + cl i +1 | b, c ∈ Z ≥ , i ∈ Z / } . (1.18)We have I A ( l ) = X − , I A ( l ) = X , I A ( l ) = X X + X , I A ( l ) = X + X X − + X − , I A ( l ) = X − + X − X − . (1.19)In general I A ( bl i + cl i +1 ) = (cid:0) I A ( l i ) (cid:1) b · (cid:0) I A ( l i +1 ) (cid:1) c . (1.20)Every f ∈ P ( X A ) can be expressed as f = (cid:89) i =1 (cid:0) I A ( l i ) (cid:1) d i = (cid:88) l ∈A A ( Z t ) c ( f ; l ) I A ( l ) . (1.21)Here d , . . . , d ∈ Z ≥ . The coefficients are c ( f ; bl i + cl i +1 ) = (cid:88) k (cid:18) d i +3 k (cid:19)(cid:18) d i +4 + kd i +2 + d i +3 − d i + b (cid:19)(cid:18) d i +2 d i +4 − d i +1 + c + k (cid:19) . (1.22)For example, if d = 10 , d = 30 , d = 10 , d = 20 , d = 30, the support S f is shown inFig.1. 7 .3 Stasheff polytopes We define Stasheff polytopes in A Φ ( A t ) by using tropical cluster variables. We borrow thefollowing notations from [FZ2].Recall the set Φ ≥− of almost positive roots (i.e. positive and simple negative roots).There is a compatibility degree map ([ loc.cit ]):Φ ≥− × Φ ≥− −→ Z ≥ , ( α, β ) (cid:55)−→ ( α || β ) . (1.23)In particular, ( α || β ) = 0 implies ( β || α ) = 0. In this case, α and β are called compatible .A compatible set T is a subset of Φ ≥− whose elements are mutually compatible. We setS( T ) = { α ∈ Φ ≥− | α / ∈ T and T ∪ { α } is still a compatible set } (1.24)and call S( T ) the supplement of T . A compatible set is called the cluster associated to Φ ifits supplement is empty. From [ loc.cit , Theorem 1.8], each cluster is a Z -basis of the rootlattice.Recall the cluster A -variety A Φ of classical type. Its cluster variables A α are indecom-posable functions indexed by Φ ≥− ([FZ1, Theorem 4.27, 5.7]). Let c = { c α } be a set ofreal numbers. For each compatible set T , define the T -face F Tc = { x ∈ A ( R t ) | A tα ( x ) = c α for all α ∈ T, and A tα ( x ) ≤ c α for all α ∈ S( T ) } . (1.25)If T is empty, then by definition F ∅ c = { x ∈ A ( R t ) | A tα ( x ) ≤ c α for all α ∈ Φ ≥− } . (1.26) Definition 1.9.
The polytope F ∅ c defined in (1.26) is called a Stasheff polytope ( or a generalized associahedron ) if T ⊆ T = ⇒ F T c ⊆ F T c , ∀ T , T . (1.27) Remark.
For simplicity, we will only consider the cases when the seed is reduced(see Section 3.1). For each T , there is a coordinate system in which the tropical clustervariables of T are simultaneously linear. Then F Tc becomes a convex polytope and thushomeomophic to a unit ball D k . Clearly its dimension k ≤ n − T , where n is the rank ofΦ. We say F ∅ c is non-degenerate if k = n − T for each T .The set A Φ ( A t ) is a subset of A Φ ( R t ). With an abuse of notation, the intersection A Φ ( A t ) (cid:84) F ∅ c is called a Stasheff polytope in A Φ ( A t ). Examples . We consider the case of type A . Then Φ ≥− = {− α, − β, α, β, α + β } . Thecluster variables are A − α = A , A − β = A , A α = A − + A − A ,A β = A − + A − A , A α + β = A − A − + A − + A − . (1.28)8heir tropicalization and a set c = { c i } of real numbers are given as follows A t − α = a , c − α = 20; A t − β = a , c − β = 10; A tα = max {− a , − a + a } , c α = 20; A tβ = max {− a , a − a } , c β = 20; A tα + β = max {− a − a , − a , − a } , c α + β = 30 . (1.29)Fig.1 shows the Stasheff polytope F ∅ c in A A ( R t ). Fig.2-5 show the same Stasheff polytopein A A ( R t ) in the other four coordinate systems. -30 -20 -10 0 10 20 30 40-30-20-101020 F - β =10 F -a =20F β =20F a =20F a + β =30 Figure 1: A Stasheff polytope in the tropical cluster A -variety of type A .Fig.6 is a generalized associahedron of type B .Fig.7 shows a Stasheff polytope of type A . There are three kinds of lines: visible lineswhich form part of the faces, lines on the back of the polytope which form part of faces(which are dashed), and “creases” (which are dotted). Although the three creases are partof the boundary, they are not faces of the Stasheff polytope in the sense of this paper.They appear because the defining functions A tα ≤ c α are not linear in general. The creaseson F Tc will disappear once the tropical cluster variables of T are simultaneously linear. Forthe same reason, the white vertex is not a face either.Let Π = {− α , . . . , − α n } be the set of simple negative roots. It is a cluster associatedto Φ ≥− . The set { A − α , . . . , A − α n } is a positive coordinate system of A Φ . For any root α = c α + . . . + c n α n ∈ Φ ≥− , [FZ1, Theorem 5.8] shows that the cluster variable A α = P α ( A − α , . . . , A − α n ) A c − α . . . A c n − α n , (1.30)9
32 -24 -16 -8 0 8 16 24 32 40-16-881624 F - β =10F -a =20F β =20 F a =20F a + β =30 Figure 2: A Stasheff polytope in the tropical cluster A -variety of type A .where P α is a polynomial in A − α , . . . , A − α n with nonzero constant term. Clearly every A tα becomes linear in the negative part { x ∈ A Φ ( R t ) | A t − α i ( x ) ≤ , i = 1 , . . . , n } (cid:39) R n ≤ . (1.31)If a Stasheff polytope F ∅ c is contained in (1.31), then all faces of F ∅ c become flat. It coincideswith the usual Stasheff polytope in R n . When Φ is of type A n , such a Stasheff polytopecan be easily constructed. For example, given any Stasheff polytope F ∅ c , one can choose apoint x = ( − x , . . . , − x n ) ∈ (1.31) with x i ≥ { x } + F ∅ c is contained in (1.31). We can show that { x } + F ∅ c is still a Stasheff polytope. Itlooks like the usual Stasheff polytope in R n . Fig.8 shows an example of such a polytope.We will focus on the cluster A -variety A A n (see Definition 2.3). Our basic result is asfollows. Theorem 1.10.
Every single point set { l } ⊂ A A n ( A t ) is a convex set. Given finitely manypoints l , ..., l m ∈ A A n ( A t ) , their Minkowski sum m (cid:88) k =1 l k := { x ∈ A A n ( A t ) | F t ( x ) ≤ m (cid:88) i =1 F t ( l k ) for all F ∈ E ( A A n ) } (1.32) is a Stasheff polytope F ∅ c with c α = m (cid:88) k =1 A tα ( l k ) , ∀ α ∈ Φ ≥− . (1.33)10
32 -24 -16 -8 0 8 16 24 32 40-16-88162432 F - β =10F -a =20 F β =20 F a =20F a + β =30 Figure 3: A Stasheff polytope in the tropical cluster A -variety of type A . A -varieties of type A In this section, we give two realizations of cluster A -varieties of type A n (Definition 2.2,2.3), which are the models used in the proof of Theorem 1.10.Let S be an ( n +3)-gon with vertices labeled by 1 through ( n +3) clockwise. We considerthe line segments connecting two different vertices. Segments connecting two adjacentvertices are called edges . Segments that are not edges are called diagonals .With an abuse of notation, we will identify a triangulation T of the ( n +3)-gon withthe set of diagonals contained in T . Denote by (cid:101) T the union of T and the set of edges.Recall the notion of compatible sets and clusters from Section 1.3. In the case of type A n , they have the following description. Proposition 2.1 ([FZ2]) . Let Φ be a root system of type A n . Its almost positive roots areone-to-one corresponding to the diagonals of an ( n +3) -gon. Two almost positive roots arecompatible if and only if they correspond to two diagonals which have no intersection inside.Every compatible subset thus corresponds to a ( partial ) triangulation of the ( n +3) -gon. Itis a cluster if and only if the triangulation is complete. Denote by { ij } the segment connecting the vertices labeled by i, j . Assign to eachsegment { ij } a variable A ij . When i, j, k, l are seated clockwise, the following is called thePl¨ucker relation: A ik A jl = A ij A kl + A il A jk . (2.1) Definition 2.2.
Assign to each complete triangulation T a coordinate system (cid:101) α T = { A ij | { ij } ∈ (cid:101) T } . (2.2)11
32 -24 -16 -8 0 8 16 24 32 40-16-88162432 F - β =10F -a =20 F β =20F a =20 F a + β =30 Figure 4: A Stasheff polytope in the tropical cluster A -variety of type A . Each (cid:101) α T determines a torus Spec( C [ A ± ij ]) , { ij } ∈ (cid:101) T . Denote by (cid:101) A A n the space obtained bygluing the tori via the transition maps generated by the Pl¨ucker relation (2.1) . Definition 2.3.
Assign to each complete triangulation T a coordinate system α T = { A ij | { ij } ∈ T } . (2.3) Each α T determines a torus Spec( C [ A ± ij ]) , { ij } ∈ T . Denote by A A n the space obtainedby gluing the tori via the transition maps generated by the Pl ¨ u cker relation (2.1) and thefollowing condition: A ij = 1 for all edges { ij } . (2.4) Remark . The transition maps defined above are subtraction free. Thus both (cid:101) A A n and A A n are positive spaces. They are cluster A -varieties of type A n with different coefficients.By [FZ1] [C], indecomposable functions on these spaces are monomials in the A ij ’s fromthe same coordinate systems.In particular, we are interested in A A n which corresponds to the case with reducedseed. Given a set of real numbers c = { c ij } indexed by the set of diagonals, every partialtriangulation T gives rise to a face: F Tc = { x ∈ A A n ( R t ) | A tij ( x ) = c ij for all diagonals { ij } ∈ T, and A tij ( x ) ≤ c ij for all diagonals { ij } ∈ S( T ) } . (2.5)When T contains no diagonals, we obtain the ∅ -face: F ∅ c = { x ∈ A A n ( R t ) | A tij ( x ) ≤ c ij for all diagonals { ij }} . (2.6)12
32 -24 -16 -8 0 8 16 24 32-24-16-881624 F - β =10 F -a =20 F β =20F a =20 F a + β =30 Figure 5: A Stasheff polytope in the tropical cluster A -variety of type A .If T is complete, then F Tc contains only one point. In this case, F Tc is called a vertex. Ournext theorem provides a criterion for recognizing Stasheff polytopes in A A n ( R t ). Theorem 2.4.
The polytope F ∅ c defined by (2.6) is a Stasheff polytope if and only if everyvertex F Tc is contained in F ∅ c .Proof. The “only if” part is by definition.We tropicalize the Pl¨ucker relation and the condition (2.4): A tpr + A tqs = max { A tpq + A trs , A tqr + A tps } if p, q, r, s are seated clockwise; (2.7) A tpq = 0 if p, q are adjacent. (2.8)For the “if” part, given a partial triangulation T , let α = { ij } ∈ S( T ). Then T = T (cid:83) { α } is still a triangulation. By induction, we only need to show that F T c ⊆ F T c . It isenough to show that ∀{ kl } ∈ S( T ) , ∀ y ∈ F T c ,A tkl ( y ) ≤ c kl . (2.9)If { kl } ∈ S( T ) or { kl } = { ij } , then (2.9) follows by definition. Otherwise { kl } intersects { ij } . Thus the vertices i, k, j, l are seated clockwise. Consider the segments { ik } , { kj } , { jl } , { il } . They are either in S( T ) ∪ T or they are edges. Meanwhile, thesefour segments are mutually compatible. Therefore, there exists a cluster T that contains T and all the diagonals among { ik } , { kj } , { jl } , { il } . Such a cluster T gives rise to a vertex F Tc := { x } . Now by assumption, one has x ∈ F ∅ c . Therefore c kl ≥ A tkl ( x ) = max { A tik ( x )+ A tjl ( x ) , A til ( x )+ A tkj ( x ) }− A tij ( x ) = max { c ik + c jl , c il + c kj }− c ij . (2.10)13
15 -12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5-5-2.52.557.510 F a + β =8 F a +2 β = F - β =10 F - a =11 F β =9 F a =15 Figure 6: A generalized associahedron of type B .Here c pq = 0 if the segment { pq } is an edge. For any y ∈ F T c , by definition, A tik ( y ) ≤ c ik , A tjl ( y ) ≤ c jl , A til ( y ) ≤ c il , A tkj ( y ) ≤ c kj , A tij ( y ) = c ij . (2.11)By (2.7), (2.10) and (2.11), A tkl ( y ) = max { A tik ( y ) + A tjl ( y ) , A til ( y ) + A tkj ( y ) } − A tij ( y ) ≤ c kl . (2.12)The Theorem is proved. Theorem 1.10 is a consequence of the following Lemmas.
Lemma 2.5.
Given finitely many points l , ..., l m ∈ A A n ( R t ) and a set c = { c ij } of realnumbers: c ij = m (cid:88) k =1 A tij ( l k ) , (2.13) the polytope F ∅ c defined by (2.6) is a Stasheff polytope.Proof. By Theorem 2.4, it is enough to show that all vertices are contained in F ∅ c . Eachcluster T gives rise to a coordinate system α T = { A ij | { ij } ∈ T } , that maps A A n ( R t )isomorphically to R n . Let l be the sum of l k ’s as vectors in R n . Clearly A tij ( l ) = c ij , ∀{ ij } ∈ T . In other words, F Tc = { l } . Meanwhile, all tropical cluster variables are convex piecewiselinear functions in this coordinate system α T . The convexity shows that A tij ( l ) ≤ m (cid:88) k =1 A tij ( l k ) = c ij , for all diagonals { ij } . (2.14)Therefore F Tc ⊆ F ∅ c . The Lemma is proved.14igure 7: A Stasheff polytope of type A . Lemma 2.6.
The polytope F ∅ c in Lemma 2.5 is the Minkowski sum of the defining l i ’s.Proof. By definition, the Minkowski sum of the defining l i ’s is contained in F ∅ c . It remainsto show that F ∅ c is contained in the Minkowski sum. For cluster A -varieties of classicaltype, all indecomposable functions are cluster monomials. Namely, ∀ F ∈ E ( A ), thereexists a cluster T such that F t = (cid:80) { ij }∈ T a ij A tij , where the numbers a ij are all nonnegativeintegers. Thus ∀ x ∈ F ∅ c , F t ( x ) = (cid:88) { ij }∈ T a ij A tij ( x ) ≤ (cid:88) { ij }∈ T a ij c ij = (cid:88) k F t ( l k ) . (2.15)The Lemma is proved. Proof. (Of Theorem 1.10) By Lemma 2.5 and 2.6, the second part is proved. It remains toshow that every single point set { l } ⊂ A A n ( R t ) is convex. Notice that its convex hull C ( { l } )gives rise to a Stasheff polytope whose only vertex is l . By induction on the dimensions ofthe faces, one can easily show that all faces of this polytope contain only l . The first partis proved. Remark.
In fact, suppose { l } is not convex, by the same argument used in the proof ofTheorem 2.10, one can show that there exist cyclic ordered k, s, m, t such that the tropicalPl¨ucker relation fails on l . This will give another proof of the convexity of { l } .15
30 −20 −10 0−60−40−200−30−25−20−15−10−50
Figure 8: A Stasheff polytope of type A .The proof of Theorem 2.4 crucially uses (2.10). It provides another criterion for recog-nizing Stasheff polytopes. Theorem 2.7.
The polytope F ∅ c defined in (2.6) is a Stasheff polytope if and only if c ij + c kl ≥ max { c ik + c jl , c kj + c il } for all i, k, j, l seated clockwise. (2.16) Furthermore, F ∅ c is non-degenerate if and only if these inequalities are strict for all i, k, j, l seated clockwise.Proof. Let T be a cluster containing all diagonals among { ik } , { jl } , { kj } , { il } , { ij } . Let F Tc = { x } . If { x } ∈ F ∅ c , then by (2.10), the condition (2.16) follows. The “only if” partof the first statement is proved. The “if” part follows from the same argument used in theproof of Theorem 2.4. By induction on the dimensions of the faces, the second statementfollows. Let (cid:101) A A n be as in Definition 2.2. The set E ( (cid:101) A A n ) provides a canonical basis for the coordi-nate ring of (cid:101) A A n ([C], Theorem 1.1). In this section, we study the partial order structureon P ( (cid:101) A A n ).Label the vertices of an ( n +3)-gon by 1 through ( n +3) clockwise as before. Every F ∈ P ( (cid:101) A A n ) can be expressed as a product of the variables A ij corresponding to segments { ij } of the ( n +3)-gon. Thus F can be represented by a weighted graph . Definition 2.8.
A weighted graph is a collection of segments of an ( n +3) -gon with integralweights such that the weights of diagonals are non-negative.
16e present a weighted graph by a symmetric ( n +3) × ( n +3) matrix G = ( w ij ), suchthat w ij is the weight of { ij } if i (cid:54) = j and w ii = 0 otherwise. The matrix G is called trivialif all its entries are zero.The map I ( G ) = (cid:89) ≤ i For any matrix G = ( w ij ) , we set Γ kl ( G ) := 12 (cid:88) k ≤ i,j ≤ l w ij , ∀ ≤ k ≤ l ≤ n + 3; R p ( G ) := (cid:88) ≤ j ≤ n +3 w pj , ∀ ≤ p ≤ n + 3 . (2.18) Theorem 2.10. Given two weighted graphs presented by matrices G , G , if1. Γ kl ( G ) ≥ Γ kl ( G ) , ∀ ≤ k ≤ l ≤ n + 3 ,2. R p ( G ) = R p ( G ) , ∀ ≤ p ≤ n + 3 ,then I ( G ) ≤ I ( G ) . Remark. Let [ p, q ] := { p, . . . , q } . For any G = ( w ij ), define I kl ( G ) = (cid:88) k +1 ≤ i ≤ l R i ( G ) − k +1 ,l ( G ) = (cid:88) i ∈ I,j ∈ J w ij , (2.19)where I = [ k +1 , l ], J = [1 , n +3] − I are partitions of the set of vertices obtained by cuttingthe boundary of the ( n +3)-gon into two connected parts. Geometrically, I kl is the sumof weighted segments that connect I and J , and R p ( G ) is the sum of weighted segmentsemanating from the vertex labeled p . The two conditions in Theorem 2.10 is equivalent tothe condition that R p ( G ) = R p ( G ) , ∀ vertices p,I kl ( G ) ≤ I kl ( G ) , ∀ diagonals { kl } . (2.20)The condition (2.20) is more geometric and does not depend on the labeling of the vertices.Because of this, if necessary, one can relabel the vertices so that the cyclic order is preserved.The conditions in the Theorem are easier for calculation.17et G = ( u ij ) , G = ( v ij ) be two weighted graphs satisfying condition (2.20). Let G = ( s ij ) such that s ij = min { u ij , v ij } . Then G, G − G, G − G are all weighted graphsand I ( G i ) = I ( G i − G ) I ( G ) , for i = 1 , . (2.21)Clearly I ( G − G ) ≤ I ( G − G ) implies I ( G ) ≤ I ( G ). Notice that G − G , G − G stillsatisfy condition (2.20). Theorem 2.10 can be reduced to the case when G = ( u ij ) , G = ( v ij ) such that min { u ij , v ij } = 0 , ∀{ ij } . (2.22) Definition 2.11. The length of the segment γ = { ij } is l ( γ ) = min {| i − j | , n + 3 − | i − j |} . (2.23) For each nontrivial weighted graph G , the depth of G is dep( G ) := min γ | w γ (cid:54) =0 { l ( γ ) } . (2.24)Clearly both are well defined as long as the cyclic order of the labeling is preserved. Lemma 2.12. If G and G are two nontrivial weighted graphs satisfying both conditions (2.20) and (2.22) , then ≤ dep( G ) < dep( G ) ≤ n + 32 . (2.25) Proof. Here 1 ≤ dep( G ) and dep( G ) ≤ ( n + 3) / G ) < dep( G ).Let α be the shortest segment such that its weight v α in G is strictly positive. Let k := l ( α ) = dep( G ). Relabel the vertices such that α = { , k + 1 } . By definition and thefirst condition in Theorem 2.10, we get0 < v ,k +1 = (cid:88) ≤ i 0. Thus R k ( G ) = R k ( G ) > . By Lemma 2.12, dep( G ) > k .Therefore v kj = 0 for all j ∈ [1 , k ] ∪ { n + 3 } . There is at least one j ∈ [2 k + 1 , n + 2] suchthat v kj > R k ( G ) > 0. Let m be the largest one in [2 k + 1 , n + 2] such that v km > 0. Notice thatΓ km ( G ) + Γ m,n +3 ( G ) ≤ Γ km ( G ) + Γ m,n +3 ( G ) ≤ Γ k,n +3 ( G ) − u k,n +3 < Γ k,n +3 ( G ) = Γ k,n +3 ( G ) . (2.28)By the choice of m , we have (cid:88) m The map (2.17) is a bijection from the set of weighted graphs to P ( (cid:101) A A n ) .Furthermore, I ( G ) ≤ I ( G ) if and only if the condition (2.20) holds.Proof. The “if” part of the second statement follows directly from Theorem 2.10. We provethe “only if” part.Let G = ( u ij ) be a weighted graph. If I ( G ) is not indecomposable, then there exist1 ≤ r < s < m < t ≤ n + 3 such that u rm > u st > 0. Let G (cid:48) = ( v ij ), G (cid:48)(cid:48) = ( w ij ) be twonew weighted graphs such that v ij = w ij = u ij − , if { ij } = { rm } or { st } ; v ij = u ij + 1 , w ij = u ij if { ij } = { rt } or { sm } ; v ij = u ij , w ij = u ij + 1 if { ij } = { rs } or { tm } ; v ij = w ij = u ij , otherwise . (2.34)Then by the Pl¨ucker relation (2.1), I ( G ) = I ( G (cid:48) ) + I ( G (cid:48)(cid:48) ). For any { kl } , I kl ( G ) = max { I kl ( G (cid:48) ) , I kl ( G (cid:48)(cid:48) ) } . (2.35)If I ( G (cid:48) ), I ( G (cid:48)(cid:48) ) are not indecomposable, repeat the above process. The product I ( G ) canbe uniquely decomposed into a finite sum: I ( G ) = (cid:88) i ∈ I I ( L i ) , (2.36)where every I ( L i ) is indecomposable, and I kl ( G ) = max i ∈ I { I kl ( L i ) } , ∀{ kl } . (2.37)Let G , G be two weighted graphs such that I ( G ) ≤ I ( G ). The decomposition of I ( G ) contains all indecomposable functions appearing in the decomposition of I ( G ). By(2.37), I kl ( G ) ≤ I kl ( G ). The condition R i ( G ) = R i ( G ) follows by the same argument.Therefore the condition (2.20) is necessary.It is clear that the map (2.17) is surjective. It remains to show that it is injective.Notice that for each G = ( u ij ), we have u ij = 12 (cid:0) I ij ( G ) + I i − ,j − ( G ) − I i,j − ( G ) − I i − ,j ( G ) (cid:1) . (2.38)If I ( G ) = I ( G ), then by condition (2.20), I kl ( G ) = I kl ( G ) , ∀{ kl } . Therefore by (2.38), G = G . The Theorem is proved. 20 Cluster ensembles of type A For the convenience of the reader, we briefly recall the definition of cluster ensembles from[FG]. Definition 3.1. A seed is a datum i = ( I, I , ε, d ) , where I is a finite set, I is a subset of I , ε = { ε ij } is a Z -valued function on I × I , and d = { d i } i ∈ I is a set of positive rationalnumbers such that ε ij d − j = − ε ji d − i . Given a seed i , any element k ∈ I − I provides a new seed µ k ( i ) = i (cid:48) = { I (cid:48) , I (cid:48) , ε (cid:48) , d (cid:48) } such that I (cid:48) := I , I (cid:48) := I (cid:48) , d (cid:48) := d and ε (cid:48) ij = − ε ij , if k ∈ { i, j } ε ij , if ε ik ε kj ≤ , k / ∈ { i, j } ε ij + | ε ik | · ε kj , if ε ik ε kj > , k / ∈ { i, j } (3.1)Here µ k is called the seed mutation in the direction k . This mutation is involutive: µ k ( i ) = i . Repeating the process in every direction for each new seed got via seed mutations, weobtain an n -regular tree such that each of its vertices corresponds to a seed. Here n is thecardinality of the set I − I .Now assign to each seed i two coordinate systems: X i = { X i | i ∈ I } and A i = { A i | i ∈ I } . There is a homomorphism p relating X i and A i : p ∗ X i = (cid:89) j ∈ I A ε ij j . (3.2)The transition maps between the coordinate systems assigned to i and i (cid:48) = µ k ( i ) are asfollows: µ ∗ k X (cid:48) i = (cid:40) X − k , if i = kX i (1 + X − sgn( ε ij ) k ) − ε ij , if i (cid:54) = k, (3.3) µ ∗ k A (cid:48) i = (cid:40) A − k ( (cid:81) j | ε kj > A ε kj j + (cid:81) j | ε kj < A − ε kj j ) , if i = kA i , if i (cid:54) = k, (3.4)Clearly the transition maps are subtraction free and thus give rise to a pair of positivespaces. Denote it by ( X | i | , A | i | ) and call it a cluster ensemble.Given a seed i = ( I, I , ε, d ), let i = ( I − I , ∅ , ε, d ), where ε in i is the restrictionof ε in i to ( I − I ) × ( I − I ). Call i the reduced seed of i . Let ( X | i | , A | i | ) be the21luster ensemble corresponding to the reduced seed i . We have the following commutativediagram: A | i | p −→ X | i | ↑ i (cid:38) k ↓ j A | i | p −→ X | i | . (3.5)The maps p, p are natural maps defined by (3.2).The map i is an injective map such that i ∗ A i := A i if i ∈ I − I , otherwise i ∗ A i := 1.The map j is a surjective map such that j ∗ X i := X i for all i ∈ I − I .The map k is the composition of p and j . Here k is surjective if and only if the submatrix ε I − I ,I = ( ε ij ) is of full rank, where ( i, j ) runs through ( I − I ) × I . k In this section, we assume that the map k is surjective. It induces an injective linear map τ i : Z n −→ Z m , ( b , . . . , b n ) (cid:55)−→ ( a , . . . , a m ) . (3.6)Here n = I − I ), m = I , and a j = (cid:80) ni =1 b i ε ij . For each b = ( b , . . . , b n ), let X b := (cid:81) ni =1 X b i i . By (3.2), we have k ∗ ( X b ) = A τ i ( b ) . Let Q ( X | i | ) be the field of rational functions on X | i | . Lemma 3.2. Let f ∈ Q ( X | i | ) . Then f ∈ L + ( X | i | ) if and only if k ∗ ( f ) ∈ L + ( A | i | ) .Proof. For each seed i , let X i = { X i | i ∈ I − I } , A i = { A j | j ∈ I } be correspondingcoordinate systems. Let Z ≥ [ X ± i ], Z ≥ [ A ± j ] be the semirings of Laurent polynomials withnon-negative integral coefficients. Since k ∗ is injective and it maps Laurent monomials toLaurent monomials, we have f ∈ Z ≥ [ X ± i ] ⇐⇒ k ∗ ( f ) ∈ Z ≥ [ A ± j ] . (3.7)By definition, we have L + ( X | i | ) = (cid:84) i Z ≥ [ X ± i ], and L + ( A | i | ) = (cid:84) i Z ≥ [ A ± j ]. The Lemmafollows directly. Lemma 3.3. Let f ∈ L + ( X | i | ) , g ∈ L + ( A | i | ) be such that k ∗ ( f ) ≥ g . Then there exists aunique g (cid:48) ∈ L + ( X | i | ) such that k ∗ ( g (cid:48) ) = g. Proof. Fix a seed i . Let f = (cid:81) b ∈ Z n c b X b . Then k ∗ ( f ) = (cid:89) b ∈ Z n c b A τ i ( b ) := (cid:89) a ∈ Z m c (cid:48) a A a . (3.8)22ere the number c (cid:48) a > a ∈ τ i ( Z n ).Let g = (cid:81) a ∈ Z m d a A a . Since k ∗ ( f ) ≥ g , then c (cid:48) a ≥ d a . If d a > 0, then c (cid:48) a > a ∈ τ i ( Z n ). In other words, there exists a unique g (cid:48) = (cid:89) b ∈ Z n d τ i ( b ) X b ∈ Z ≥ [ X ± j ] (3.9)such that k ∗ ( g (cid:48) ) = g .By Lemma 3.2, g (cid:48) ∈ L + ( X | i | ). The Lemma is proved. Lemma 3.4. Let f ∈ Q ( X | i | ) . Then f ∈ E ( X | i | ) if and only if k ∗ ( f ) ∈ E ( A | i | ) .Proof. If f ∈ E ( X | i | ), then by Lemma 3.2, k ∗ ( f ) ∈ L + ( A | i | ). Here k ∗ ( f ) must be indecom-posable. Otherwise assume k ∗ ( f ) = g + h , where g, h ∈ L + ( A | i | ) and g, h (cid:54) = 0. By Lemma3.3, there exist g (cid:48) , h (cid:48) ∈ L + ( X | i | ) such that g = k ∗ ( g (cid:48) ) , h = k ∗ ( h (cid:48) ). Therefore f = g (cid:48) + h (cid:48) ,which contradicts the assumption that f is indecomposable. The other direction followssimilarly. The Lemma is proved. Lemma 3.5. For each f ∈ P ( X | i | ) , we have k ∗ ( f ) ∈ P ( A | i | ) . Moreover, it preserves thepartial order structure: ∀ f, g ∈ P ( X | i | ) , f ≤ g ⇐⇒ k ∗ ( f ) ≤ k ∗ ( g ) . (3.10) Proof. The first part follows from Lemma 3.4. The second part follows from Lemma3.2.Let T be a split algebraic torus. We define the group of characters X (T) := Hom(T , G m ) . (3.11)In particular, each seed i gives rise to a pair of toriT i , A := Spec( C [ A ± j ]) , T i , X := Spec( C [ X ± i ]); j ∈ I, i ∈ I − I . (3.12)Since k is a surjective morphism from T i , A to T i , X , the group X (T i , X ) can be viewed as asub lattice of X (T i , A ). Let Q i := X (T i , A ) /X (T i , X ) be the corresponding quotient group.Let L ( A | i | ) be the coordinate ring of A | i | . Each f ∈ L ( A | i | ) can be uniquely expandedas f = (cid:88) [ a ] ∈ Q i f i [ a ] (3.13)where f i [ a ] is a Laurent polynomial consisting of characters belong to the coset [ a ]. Namely,if a ∈ X (T i , A ) is a representative of [ a ] ∈ Q i , then f i [ a ] = A a · k ∗ ( g a ), where g a ∈ Q ( X | i | ).23 emma 3.6. For each pair i , i ∈ | i | , there is a canonical isomorphism η : Q i ∼ −→ Q i suchthat f i [ a ] = f i η ([ a ]) , ∀ [ a ] ∈ Q i , ∀ f ∈ L ( A | i | ) . (3.14) Proof. First assume i = µ k ( i ). By definition, we have A k = (cid:0) A − k (cid:89) j | ε kj < A − ε kj j (cid:1) · k ∗ (1 + X k ) A j = A j , ∀ j (cid:54) = k. (3.15)We consider the following bijective map η : X (T i , A ) −→ X (T i , A ) , ( a , . . . , a m ) (cid:55)−→ ( b , . . . , b m ) , such that b j = − a k , if j = k,a j − ε kj · a k , if ε kj < ,a j , otherwise. (3.16)We pick a representative a ∈ X (T i , A ) for each [ a ] ∈ Q i . Then by (3.15), we have f i [ a ] = A a · k ∗ ( g a ) = A η ( a ) · k ∗ (cid:0) (1 + X k ) a k · g a (cid:1) , where (1 + X k ) a k · g a ∈ Q ( X | i | ).Notice that η maps the sub lattice X (T i , X ) onto X (T i , X ). It descends to a bijectivemap η : Q i −→ Q i . We consider the expansion of f under the seed i : f = (cid:88) η ([ a ]) ∈ Q i f i η ([ a ]) = (cid:88) η ([ a ]) ∈ Q i A η ( a ) · k ∗ (cid:0) (1 + X k ) a k · g a (cid:1) . Since this expansion is unique, we have f i η ([ a ]) = A η ( a ) · k ∗ (cid:0) (1 + X k ) a k · g a (cid:1) = f i [ a ] . The Lemma follows. For arbitrary seed i , it is a composition of mutations. Thus theLemma is proved. Lemma 3.7. Let f be as in (3.13). If f ∈ L + ( A | i | ) , then f i [ a ] ∈ L + ( A | i | ) , ∀ [ a ] ∈ Q i . (3.17) Proof. It follows immediately from Lemma 3.6. Lemma 3.8. If E ( A | i | ) is a basis of L ( A | i | ) , then E ( X | i | ) is a basis of L ( X | i | ) . roof. If f ∈ E ( A | i | ), then there exists a unique [ a ] ∈ Q i such that f = f i [ a ] , otherwise it isdecomposable due to Lemma 3.7.Let g ∈ L ( X | i | ). Let h = k ∗ ( g ) ∈ L ( A | i | ). Then by definition h = h i [0] . Since E ( A | i | ) isa basis of L ( A | i | ), one has a unique decomposition h = (cid:88) i c i h i , h i ∈ E ( A | i | ) . (3.18)Clearly here if c i (cid:54) = 0, then h i = h i i, [0] . Namely h i ∈ k ∗ ( Q ( X | i | )). Let h i = k ∗ ( g i ). ByLemma 3.4, g i ∈ E ( X | i | ). We thus get a unique decomposition: g = (cid:88) i c i g i , g i ∈ E ( X | i | ) . (3.19)The Lemma is proved. A -laminations on a convex polygon Let us recall the definition of A -laminations from [FG1]. Most of the results of this sectionare from [FG3] Section 3. Definition 3.9 ([ loc.cit, Definition 3.2]) . An A -lamination on a convex polygon is a collec-tion of edges and mutually non-intersecting diagonals of the polygon with A -valued weights,subjecting to the following conditions:1. The weights of the diagonals are non-negative.2. The sum of the weights of the diagonals and edges incident to a given vertex is zero. Denote by A L ( n, A ) the set of A -laminations on a convex ( n +3)-gon. Recall that A can be Z , Q , R . Clearly A L ( n, A ) is a subset of weighted graphs. Let G ∈ A L ( n, A ). Set a kl := 12 I kl ( G ) . (3.20)Notice that R i ( G ) = 0 , ∀ i ∈ [1 , n + 3]. By (2.19), we have a kl = − Γ k +1 ,l ( G ) ∈ A . It iseasy to show that a ik + a jl = max { a ij + a kl , a il + a jk } , if i, j, k, l seated clockwise ,a ij = 0 , if i, j are adjacent. (3.21)Let T be a triangulation of the ( n +3)-gon. Lemma 3.10. There is a bijection φ T : A L ( n, A ) ∼ −→ A { diagonals of T } , G (cid:55)−→ { a ij ( G ) } , { ij } ∈ T. (3.22)25 roof. We prove the Lemma by constructing the inverse map of φ T .Let { a ij } ∈ A { diagonals of T } . It can be uniquely extended to { a pq } , p, q ∈ [1 , n + 3] suchthat they satisfy (3.21) and that a pp = 0 , ∀ p ∈ [1 , n + 3]. Let u pq = a q − ,p − + a pq − a p,q − − a p − ,q (3.23)We prove that G = ( u pq ) is the pre-image of { a ij } . We first prove that G is an A -lamination.By the tropical Pl¨ucker relation, the weight u pq ≥ { pq } . For eachdiagonal { pq } such that u pq > 0, let I = [ p + 1 , q − , J = [ q + 1 , n + 3] ∪ [1 , p − { pq } is (cid:88) i ∈ I, j ∈ J u ij = (cid:88) j ∈ J (cid:0) (cid:88) i ∈ I ( a ij − a i − ,j ) + (cid:88) i ∈ I ( a i − ,j − − a i,j − ) (cid:1) = (cid:88) j ∈ J (cid:0) a q − ,j − a pj + a p,j − − a q − ,j − (cid:1) = (cid:88) j ∈ J ( a q − ,j − a q − ,j − ) + (cid:88) j ∈ J ( a p,j − − a p,j )= a q − ,p − + a p,q − a q − ,q − a p,p − . (3.24)The tropical Pl¨ucker relation tells us thatmin { u pq , (cid:88) i ∈ I, j ∈ J u ij } = min { a q − ,p − + a p,q − a q − ,q − a p,p − , a q − ,p − + a pq − a p,q − − a p − ,q } = 0 . (3.25)Since u pq > 0, one must have (cid:88) i ∈ I, j ∈ J u ij = 0 . (3.26)Each of u ij is non-negative. Therefore u ij = 0 , ∀ i ∈ I, j ∈ J . Therefore G is a collectionof edges and mutually non-intersecting diagonals of the polygon.Let I = [ k + 1 , l ] , J = [ l + 1 , n + 3] ∪ [1 , k ]. Similarly, we have a kl ( G ) = 12 I kl ( G ) = 12 (cid:88) i ∈ I, j ∈ J u ij = 12 ( a lk − a ll + a kl − a kk ) = a kl . (3.27)In particular R k ( G ) = I k,k +1 ( G ) = 2 a k,k +1 = 0, ∀ k ∈ [1 , n + 3]. Therefore G is an A -lamination. The map φ T takes G to { a ij } . Hence φ T is surjective.It is injective because every G = ( u pq ) is uniquely determined by (3.23).By Lemma 3.10 and the relations (3.21), the following proposition is clear. Proposition 3.11. ([FG3]) There is a canonical isomorphism of sets A A n ( A t ) = A L ( n, A ) (3.28) such that A tij ( l ) = a ij ( l ) , ∀ diagonals { ij } . (3.29)26 .4 Proof of Theorem 1.1 Given a Cartan matrix of type A n , we obtain a skewsymmetric matrix ε by killing the 2’son the diagonal and changing signs under the diagonal. Let i = { I, ∅ , ε, d } be such that | I | = n and every d i ∈ d is 1. Denote by X A n the cluster X -variety corresponding to thisseed. Recall the cluster A -variety (cid:101) A A n from Definition 2.2. As shown in Section 3.1, thereis a canonical surjective map k : (cid:101) A A n → X A n . The space X A n is a partial completion ofthe moduli space M ,n +3 . The next Lemma follows from [FG3], Section 3. Lemma 3.12. For each weighted graph G, I ( G ) ∈ k ∗ ( Q ( X A n )) if and only if R i ( G ) = 0 , ∀ i ∈ [1 , n + 3] . (3.30) Theorem 3.13. There is a canonical isomorphism: I A : A A n ( Z t ) ∼ −→ E ( X A n ) . The set E ( X A n ) is a basis of L ( X A n ) .Proof. The set E ( (cid:101) A A n ) is a canonical basis of L ( (cid:101) A A n ) ([FZ1, Section 4], [C]). Thus thesecond part of our theorem follows from Lemma 3.8.Moreover E ( (cid:101) A A n ) can be parametrized by the set of weighted graphs with mutuallynon-intersecting diagonals. By Lemma 3.4 and Lemma 3.12, the set E ( X A n ) is isomorphicto A L ( n, Z ). By Proposition 3.11, we construct a canonical isomorphism between E ( X A n )and A A n ( Z t ). The first part is proved.Now we prove Theorem 1.1. Proof. By Proposition 3.11, we may replace A A n ( Z t ) by A L ( n, Z ). For each f = m (cid:89) i =1 I A ( l i ) = (cid:88) l ∈A L ( n, Z ) c ( f ; l ) I A ( l ) , (3.31)we have l ∈ S f ⇐⇒ c ( f ; l ) > ⇐⇒ f ≥ I A ( l ) ⇐⇒ k ∗ ( f ) ≥ k ∗ (cid:0) I A ( l ) (cid:1) . (3.32)The first two equivalences are by definition. The third one is by Lemma 3.5. By theconstruction of I A , we have k ∗ ( I A ( l )) = I ( l ) , k ∗ ( f ) = I ( m (cid:88) i =1 l i ) . (3.33)Here (cid:80) mi =1 l i is the sum of the laminations l i as matrices. Notice that l, l , . . . , l m are A -laminations. Then R p ( l ) = 0 , R p ( m (cid:88) i =1 l i ) = m (cid:88) R p ( l i ) = 0 , ∀ p. I ( l ) ≤ I ( m (cid:88) i =1 l i ) ⇐⇒ I jk ( l ) ≤ m (cid:88) i =1 I jk ( l i ) , ∀{ jk } . (3.34)By (3.20) and Proposition 3.11, we have A tjk ( l ) = I jk ( l ). Therefore l ∈ S f ⇐⇒ I ( l ) ≤ I ( m (cid:88) i =1 l i ) ⇐⇒ A tjk ( l ) ≤ m (cid:88) i =1 A tjk ( l i ) , ∀{ jk } . (3.35)Then the support is S f = { l ∈ A A n ( Z t ) | A tjk ( l ) ≤ m (cid:88) i =1 A tjk ( l i ) , ∀{ jk }} . (3.36)The first part of Theorem 1.1 is proved. The second part follows from Theorem 1.10.A Stasheff polytope F ∅ c is called regular if the defining set c is a set of integers. Denoteby St( A A n ) the set of regular Stasheff polytopes in A A n ( Z t ). Define the set P ∗ ( X A n ) := { f ∈ L + ( X A n ) | k ∗ ( f ) ∈ P ( (cid:101) A A n ) } . (3.37) Theorem 3.14. The following map is a bijection: P ∗ ( X A n ) −→ St( A A n ) , f (cid:55)−→ S f . (3.38) Furthermore f ≤ f if and only if S f ⊆ S f .Proof. For each f ∈ P ∗ ( X A n ), by the above discussion, we have k ∗ ( f ) = I ( G ) such that S f = { l ∈ A A n ( Z t ) | A tjk ( l ) ≤ c jk ( G ) , ∀{ jk }} , (3.39)where c jk ( G ) := 12 I jk ( G ) = − Γ k +1 ,l ( G ) ∈ Z , ∀{ jk } . (3.40)Furthermore, the set { c jk ( G ) } satisfies (2.16). By Theorem 2.7, S f ∈ St( A A n ).For f = I ( G ) , f = I ( G ) ∈ P ∗ ( X A n ), following the same argument of the last proof, f ≤ f ⇐⇒ I jk ( G ) ≤ I jk ( G ) , ∀{ jk } . (3.41)The right hand side is equivalent to S f ⊆ S f . The second part is proved.Therefore if S f = S f , then f = f , so the map (3.41) is injective.Given a set c = { c ij } of integers satisfying (2.16), let u ij = c ij + c i − ,j − − c i,j − − c i − ,j ,where c ij = 0 if i, j are adjacent or i = j . Let G = ( u ij ) be the corresponding weightedgraph. Let I = [ k + 1 , l ] and let J = [ l + 1 , n + 3] ∪ [1 , k ]. By (3.27), c kl ( G ) = 12 I kl ( G ) = c kl , ∀{ kl } . (3.42)In particular R i ( G ) = I i − ,i ( G ) = 0 for all i . By Lemma 3.12, there is f ∈ P ∗ ( X A n ) suchthat k ∗ ( f ) = I ( G ), and S f = F ∅ c . Thus the map is surjective.28 .5 Further discussion and conjecture The Minkowski sum gives rise to a new family of convex polytopes in tropical positivespaces. Definition 3.15. A subset S of a tropical positive space A ( A t ) is called a Minkowskipolytope if there exist finitely many points l , ...l m ∈ A ( A t ) such that S is the Minkowskisum of these l i ’s: S = { x | F t ( x ) ≤ m (cid:88) i =1 F t ( l i ) for all F ∈ E ( A ) } . (3.43) Given any set T ⊂ E ( A ) , the T -face of S is the following set F T = S (cid:92) { x | F t ( x ) = (cid:88) i ∈ I F t ( l i ) for all F ∈ T } . (3.44) Remark . Not all single point sets in A ( A t ) are convex. For example, it follows from [SZ]that for rank 2 cluster A -varieties of affine type, point sets corresponding to imaginary rootsare not convex. Therefore they are not Minkowski polytopes. In this case, the supportsof products of indecomposable functions are not necessarily convex either. For example,let δ be the imaginary root. For each n ≥ 1, there is an indecomposable function z n corresponding to nδ . By [ loc.cit. , Proposition 5.4], for all p ≥ n ≥ z p z n = (cid:26) z p − n + z p + n , if p > n ;2 + z n , if p = n. (3.45)The support of z p z n is clearly not convex.Assuming the Duality Conjecture, a generalization of Theorem 1.1 is as follows. Conjecture 3.16. Let f, g ∈ L + ( X ) be two universally positive Laurent polynomials on acluster X -variety. If both supports S f and S g are Minkowski polytopes in A ∨ ( Z t ) , then thesupport S fg is the Minkowski sum of S f and S g . References [BFZ] Berenstein A., Fomin S., Zelevinsky A.: Cluster algebras. III. Upper bounds anddouble Bruhat cells. Duke Math. J. 126 (2005), no. 1, 1-52.[C] Cerulli Irrelli G.: Positivity in skew-symmetric cluster algebras of finite type, arXivmath.RA/1102.3050.[CFZ] Chapoton F., Fomin S., Zelevinsky A.: Polytopal Realizations of generalized asso-ciahedra, Canad. Math. Bull. Ann.Sci. ´Ec. Norm. Sup´er. (4) 42 (2009), no. 6, 865-930.[FG1] Fock V., Goncharov A.: Dual Teichm¨uller and and lamination spaces, in Handbookof Teichm¨uller theory. Vol.I, Eur.Math. Soc., Z¨urich, Publ. Math. Inst. Hautes ´Etudes Sci. X -varieties at infinity, arXiv math.AG/1104.0407.[FZ] Fomin S., Zelevinsky A.: Cluster algebras I: Foundations, J. Amer. Math. Soc. Current developments in mathematics, Ann. of Math. Mosc. Math. J. E-mail address : [email protected]@yale.edu