Positivity for cluster algebras from surfaces
We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.
2000 Mathematics Subject Classification:16S99, 05C70, 05E15
- 1 Introduction
- 2 Cluster algebras
- 3 Cluster algebras arising from surfaces
- 4 Main results: cluster expansion formulas
- 5 Examples of results, and identities in the coefficient-free case
- 6 Outline of the proof of the cluster expansion formulas
- 7 Construction of a triangulated polygon and a lifted arc
- 8 Construction of and the map
- 9 Quadrilateral lemma
- 10 The proof of the expansion formula for ordinary arcs
- 11 Positivity for notched arcs in the coefficient-free case
- 12 Proofs of Laurent expansions for notched arcs
- 13 Applications to F-polynomials, g-vectors, Euler characteristics
Since their introduction by Fomin and Zelevinsky in [FZ1], cluster algebras have been shown to be related to diverse areas of mathematics such as total positivity, quiver representations, Teichmüller theory, tropical geometry, Lie theory, and Poisson geometry. One of the main outstanding conjectures about cluster algebras is the positivity conjecture, which says that if one fixes a cluster algebra and an arbitrary cluster , one can express each cluster variable as a Laurent polynomial with positive coefficients in the variables of .
There is a class of cluster algebras arising from surfaces with marked points, introduced by Fomin, Shapiro, and Thurston in [FST] (generalizing work of Fock and Goncharov [FG1, FG2] and Gekhtman, Shapiro, and Vainshtein [GSV]), and further developed in [FT]. This class is quite large: (assuming rank at least three) it has been shown [FeShTu] that all but finitely many skew-symmetric cluster algebras of finite mutation type come from this construction. Note that the class of cluster algebras of finite mutation type in particular contains those of finite type.
The goal of this paper is to give a combinatorial expression for the Laurent polynomial which expresses any cluster variable in terms of any cluster, for any cluster algebra arising from a surface. An immediate corollary is the proof of the positivity conjecture for all cluster algebras arising from surfaces.
A cluster algebra of rank is a subalgebra of an ambient field isomorphic to a field of rational functions in variables. Each cluster algebra has a distinguished set of generators called cluster variables; this set is a union of overlapping algebraically independent -subsets of called clusters, which together have the structure of a simplicial complex called the cluster complex. See Definition 2.5 for precise details. The clusters are related to each other by birational transformations of the following kind: for every cluster and every cluster variable , there is another cluster , with the new cluster variable determined by an exchange relation of the form
Here and belong to a coefficient semifield , while and are two monomials in the elements of . There are two dynamics at play in the exchange relations: that of the monomials and , which is encoded in the exchange matrix, and that of the coefficients.
A classification of finite type cluster algebras – those with finitely many clusters – was given by Fomin and Zelevinksy in [FZ2]. They showed that this classification is parallel to the famous Cartan-Killing classification of complex simple Lie algebras, i.e. finite type cluster algebras either fall into one of the infinite families , , , , or are of one of the exceptional types , or . Furthermore, the type of a finite type cluster algebra depends only on the dynamics of the corresponding exchange matrices, and not on the coefficients. However, there are many examples of cluster algebras of geometric origin which – despite having the same type – have totally different systems of coefficients. This motivated Fomin and Zelevinsky’s work in [FZ4], in which they studied the dependence of a cluster algebra structure on the choice of coefficients. One surprising result of [FZ4] was that there is a special choice of coefficients, the principal coefficients, which have the property that computation of explicit expansion formulas for the cluster variables in arbitrary cluster algebras can be reduced to computation of explicit expansion formulas in cluster algebras with principal coefficients. A corollary of this work is that to prove the positivity conjecture in geometric type, it suffices to prove the positivity conjecture for the class of principal coefficients.
This takes us to the topic of the present work. Our main results are combinatorial formulas for cluster expansions of cluster variables with respect to any seed, in any cluster algebra coming from a surface. Our formulas are manifestly positive, so as a consequence we obtain the following result.
Let be any cluster algebra arising from a surface, where the coefficient system is of geometric type, and let be any choice of initial cluster. Then the Laurent expansion of every cluster variable with respect to the cluster has non-negative coefficients.
Our results generalize those obtained in [S2], where cluster algebras from the (much more restrictive) case of surfaces without punctures were considered. This work in turn generalized [ST], which treated cluster algebras from unpunctured surfaces with a very limited coefficient system that was associated to the boundary of the surface. The very special case where the surface is a polygon and coefficients arise from the boundary was covered in [S], and also in unpublished work [CP, FZ3]. See also [Pr2]. The recent paper [MS] gave an alternative formulation of the results of [S2], using the language of perfect matchings as opposed to -paths.
Many other people have worked on the problem of finding Laurent expansions of cluster variables, and on the positivity conjecture. However, most of the results so far obtained have strong restrictions on the cluster algebra, the choice of initial cluster (equivalently, initial seed), or on the system of coefficients.
In the case of rank 2 cluster algebras, the papers [SZ, Z, MP] gave cluster expansion formulas in affine types. Positivity in these cases was subsequently generalized to the coefficient-free rank 2 case in [Dup], using results of [CR]. In the case of finite type cluster algebras, the positivity conjecture with respect to a bipartite seed follows from [FZ4, Corollary 11.7]. Other work [M] gave cluster expansions for coefficient-free cluster algebras of finite classical types with respect to a bipartite seed.
A recent tool in understanding Laurent expansions of cluster variables is the connection to quiver representations and the introduction of the cluster category [BMRRT] (see also [CCS1] in type ). More specifically, there is a geometric interpretation (found in [CC] and generalized in [CK]) of coefficients in Laurent expansions as Euler-Poincaré characteristics of appropriate Grassmannians of quiver representations. Using this approach, the works [CC, CK, CK2] gave an expansion formula in the case where the cluster algebra is acyclic and the initial cluster lies in an acyclic seed (see also [CZ] in rank 2); this was subsequenty generalized to arbitrary clusters in an acyclic cluster algebra [Pa]. Note that these formulas do not give information about the coefficients. Later, [FK] generalized these results to cluster algebras with principal coefficients that admit a categorification by a 2-Calabi-Yau category [FK]; by [A] and [ABCP, LF], such a categorification exists in the case of cluster algebras associated to surfaces with non-empty boundary. Recently [DWZ] gave expressions for the -polynomials in any skew-symmetric cluster algebra. However, since all of the above formulas are in terms of Euler-Poincaré characteristics (which can be negative), they do not immediately imply the positivity conjecture.
The work [CR] used the above approach to prove the positivity conjecture for coefficient-free acyclic cluster algebras, with respect to an acyclic seed. Building on results of [HL], Nakajima recently used quiver varieties to prove the positivity conjecture for cluster algebras that have at least one bipartite seed, with respect to any cluster [N]. This is a very strong result, but it does not overlap very much with our Theorem 1.1. Note that a bipartite seed is in particular acyclic, but not every acyclic type has a bipartite seed; the affine type , for example, does not. And the only surfaces that give rise to acyclic cluster algebras are the polygon, the polygon with one puncture, the annulus, and the polygon with two punctures (corresponding to the finite types and , and the affine types and ). All other surfaces yield non-acyclic cluster algebras, see [FST, Corollary 12.4].
The paper is organized as follows. We give background on cluster algebras and cluster algebras from surfaces in Sections 2 and 3. In Section 4 we present our formulas for Laurent expansions of cluster variables, and in Section 5 we give examples, as well as identities in the coefficient-free case. As the proofs of our main results are rather involved, we give a detailed outline of the main argument in Section 6, before giving the proofs themselves in Sections 7 to 10 and 12. In Section 13, we combine our results with those of [FK] and [DWZ] to give formulas for F-polynomials, g-vectors, and Euler-Poincaré characteristics of quiver Grassmannians.
Recall that cluster variables in cluster algebras from surfaces correspond to ordinary arcs as well as arcs with notches at one or two ends. We remark that working in the generality of principal coefficients is much more difficult than working in the coefficient-free case. Indeed, once we have proved positivity for cluster variables corresponding to ordinary arcs, the proof of positivity for cluster variables corresponding to tagged arcs in the coefficient-free case follows easily, see Proposition 5.3 and Section 11. Putting back principal coefficients requires much more elaborate arguments, see Section 12. A crucial tool here is the connection to laminations [FT].
Note that all cluster algebras coming from surfaces are skew-symmetric. In a sequel to this paper we will explain how to use folding arguments to give positivity results for cluster algebras with principal coefficients which are not skew-symmetric.
Acknowledgements: We are grateful to the organizers of the workshop on cluster algebras in Morelia, Mexico, where we benefited from Dylan Thurston’s lectures. We would also like to thank Sergey Fomin and Bernard Leclerc for useful discussions.
2. Cluster algebras
2.1. What is a cluster algebra?
To define a cluster algebra we must first fix its ground ring. Let be a semifield, i.e., an abelian multiplicative group endowed with a binary operation of (auxiliary) addition which is commutative, associative, and distributive with respect to the multiplication in . The group ring will be used as a ground ring for . One important choice for is the tropical semifield; in this case we say that the corresponding cluster algebra is of geometric type.
Definition 2.1 (Tropical semifield).
Let be an abelian group (written multiplicatively) freely generated by the . We define in by
and call a tropical semifield. Note that the group ring of is the ring of Laurent polynomials in the variables .
As an ambient field for , we take a field isomorphic to the field of rational functions in independent variables (here is the rank of ), with coefficients in . Note that the definition of does not involve the auxiliary addition in .
Definition 2.2 (Labeled seeds).
A labeled seed in is a triple , where
is an -tuple of elements of forming a free generating set over ,
is an -tuple of elements of , and
is an integer matrix which is skew-symmetrizable.
That is, are algebraically independent over , and . We refer to as the (labeled) cluster of a labeled seed , to the tuple as the coefficient tuple, and to the matrix as the exchange matrix.
We obtain (unlabeled) seeds from labeled seeds by identifying labeled seeds that differ from each other by simultaneous permutations of the components in and , and of the rows and columns of .
We use the notation , , and
Definition 2.3 (Seed mutations).
Let be a labeled seed in , and let . The seed mutation in direction transforms into the labeled seed defined as follows:
The entries of are given by
The coefficient tuple is given by
The cluster is given by for , whereas is determined by the exchange relation
We say that two exchange matrices and are mutation-equivalent if one can get from to by a sequence of mutations.
Definition 2.4 (Patterns).
Consider the -regular tree whose edges are labeled by the numbers , so that the edges emanating from each vertex receive different labels. A cluster pattern is an assignment of a labeled seed to every vertex , such that the seeds assigned to the endpoints of any edge are obtained from each other by the seed mutation in direction . The components of are written as:
Clearly, a cluster pattern is uniquely determined by an arbitrary seed.
Definition 2.5 (Cluster algebra).
Given a cluster pattern, we denote
the union of clusters of all the seeds in the pattern. The elements are called cluster variables. The cluster algebra associated with a given pattern is the -subalgebra of the ambient field generated by all cluster variables: . We denote , where is any seed in the underlying cluster pattern.
The remarkable Laurent phenomenon asserts the following.
[FZ1, Theorem 3.1] The cluster algebra associated with a seed is contained in the Laurent polynomial ring , i.e. every element of is a Laurent polynomial over in the cluster variables from .
Let be a cluster algebra, be a seed, and be a cluster variable of . We denote by the Laurent polynomial given by Theorem 2.6 which expresses in terms of the cluster variables from . We refer to this as the cluster expansion of in terms of .
The longstanding positivity conjecture [FZ1] says that even more is true.
(Positivity Conjecture) For any cluster algebra , any seed , and any cluster variable , the Laurent polynomial has coefficients which are non-negative integer linear combinations of elements in .
In cluster algebras whose ground ring is (the tropical semifield), it is convenient to replace the matrix by an matrix whose upper part is the matrix and whose lower part is an matrix that encodes the coefficient tuple via
2.2. Finite type and finite mutation type classification
We say that a cluster algebra is of finite type if it has finitely many seeds. Fomin and Zelevinsky [FZ2] showed that the classification of finite type cluster algebras is parallel to the Cartan-Killing classification of complex simple Lie algebras.
More specifically, define the diagram associated to an exchange matrix to be a weighted directed graph on nodes , where is directed towards if and only if . In that case, we label this edge by the weight It was shown in [FZ2] that a cluster algebra is of finite type if and only is mutation-equivalent to an orientation of a finite type Dynkin diagram. In this case, we say that and are of finite type.
We say that a matrix (and the corresponding cluster algebra) has finite mutation type if its mutation equivalence class is finite, i.e. only finitely many matrices can be obtained from by repeated matrix mutations. A classification of all cluster algebras of finite mutation type with skew-symmetric exchange matrices was recently given by Felikson, Shapiro, and Tumarkin [FeShTu]. In particular, all but 11 of them come from either cluster algebras of rank 2 or cluster algebras associated with triangulations of surfaces (see Section 3).
2.3. Cluster algebras with principal coefficients
Fomin and Zelevinsky introduced in [FZ4] a special type of coefficients, called principal coefficients.
Definition 2.10 (Principal coefficients).
We say that a cluster pattern on (or the corresponding cluster algebra ) has principal coefficients at a vertex if and . In this case, we denote .
Definition 2.12 (The functions and ).
Let be the cluster algebra with principal coefficients at , defined by the initial seed with
By the Laurent phenomenon, we can express every cluster variable as a (unique) Laurent polynomial in ; we denote this by
Let denote the Laurent polynomial obtained from by
turns out to be a polynomial [FZ4] and is called an F-polynomial.
Knowing the cluster expansions for a cluster algebra with principal coefficients allows one to compute the cluster expansions for the “same” cluster algebra with an arbitrary coefficient system. To explain this, we need an additional notation. If is a subtraction-free rational expression over in several variables, a semifield, and some elements of , then we denote by the evaluation of at .
[FZ4, Theorem 3.7] Let be a cluster algebra over an arbitrary semifield and contained in the ambient field , with a seed at an initial vertex given by
Then the cluster variables in can be expressed as follows:
When is a tropical semifield, the denominator of equation (2.12) is a monomial. Therefore if the Laurent polynomial has positive coefficients, so does .
Let be the cluster algebra with principal coefficients at a vertex , defined by the initial seed . Let be any cluster algebra of geometric type defined by the same exchange matrix . If the positivity conjecture holds for , then it also holds for .
3. Cluster algebras arising from surfaces
Building on work of Fock and Goncharov [FG1, FG2], and of Gekhtman, Shapiro and Vainshtein [GSV], Fomin, Shapiro and Thurston [FST] associated a cluster algebra to any bordered surface with marked points. In this section we will recall that construction, as well as further results of Fomin and Thurston [FT].
Definition 3.1 (Bordered surface with marked points).
Let be a connected oriented 2-dimensional Riemann surface with (possibly empty) boundary. Fix a nonempty set of marked points in the closure of with at least one marked point on each boundary component. The pair is called a bordered surface with marked points. Marked points in the interior of are called punctures.
Table 1 gives examples of surfaces (using notation of Lemma 3.5). For technical reasons, we require that is not a sphere with one, two or three punctures; a monogon with zero or one puncture; or a bigon or triangle without punctures.
|0||0||0||4||6||sphere with 4 punctures|
|1||0||n||1||c||once punctured polygon|
|1||0||n-3||2||c+3||twice punctured polygon|
|1||1||n-3||0||c+3||torus with disk removed|
|2||1||n-6||0||c+6||torus with 2 disks removed|
|3||0||n-3||0||c+3||pair of pants|
3.1. Ideal triangulations and tagged triangulations
Definition 3.2 (Ordinary arcs).
An arc in is a curve in , considered up to isotopy, such that
the endpoints of are in ;
does not cross itself, except that its endpoints may coincide;
except for the endpoints, is disjoint from and from the boundary of ,
does not cut out an unpunctured monogon or an unpunctured bigon.
An arc whose endpoints coincide is called a loop. Curves that connect two marked points and lie entirely on the boundary of without passing through a third marked point are boundary segments. By (c), boundary segments are not ordinary arcs.
Definition 3.3 (Crossing numbers and compatibility of ordinary arcs).
For any two arcs in , let be the minimal number of crossings of arcs and , where and range over all arcs isotopic to and , respectively. We say that arcs and are compatible if .
Definition 3.4 (Ideal triangulations).
An ideal triangulation is a maximal collection of pairwise compatible arcs (together with all boundary segments). The arcs of a triangulation cut the surface into ideal triangles.
[FG3, Section 2] Each ideal triangulation consists of arcs, where is the genus of , is the number of boundary components, is the number of punctures and is the number of marked points on the boundary of . The number is called the rank of . The number of triangles in any triangulation is
There are two types of ideal triangles: triangles that have three distinct sides and triangles that have only two. The latter are called self-folded triangles. Note that a self-folded triangle consists of a loop , together with an arc to an enclosed puncture which we dub a radius, see the left side of Figure 1.
Definition 3.6 (Ordinary flips).
Ideal triangulations are connected to each other by sequences of flips. Each flip replaces a single arc in a triangulation by a (unique) arc that, together with the remaining arcs in , forms a new ideal triangulation.
Note that an arc that lies inside a self-folded triangle in cannot be flipped.
In [FST], the authors associated a cluster algebra to any bordered surface with marked points. Roughly speaking, the cluster variables correspond to arcs, the clusters to triangulations, and the mutations to flips. However, because arcs inside self-folded triangles cannot be flipped, the authors were led to introduce the slightly more general notion of tagged arcs. They showed that ordinary arcs can all be represented by tagged arcs and gave a notion of flip that applies to all tagged arcs.
Definition 3.7 (Tagged arcs).
A tagged arc is obtained by taking an arc that does not cut out a once-punctured monogon and marking (“tagging”) each of its ends in one of two ways, plain or notched, so that the following conditions are satisfied:
an endpoint lying on the boundary of must be tagged plain
both ends of a loop must be tagged in the same way.
Definition 3.8 (Representing ordinary arcs by tagged arcs).
One can represent an ordinary arc by a tagged arc as follows. If does not cut out a once-punctured monogon, then is simply with both ends tagged plain. Otherwise, is a loop based at some marked point and cutting out a punctured monogon with the sole puncture inside it. Let be the unique arc connecting and and compatible with . Then is obtained by tagging plain at and notched at .
Definition 3.9 (Compatibility of tagged arcs).
Tagged arcs and are called compatible if and only if the following properties hold:
the arcs and obtained from and by forgetting the taggings are compatible;
if then at least one end of must be tagged in the same way as the corresponding end of ;
but they share an endpoint , then the ends of and connecting to must be tagged in the same way.
Definition 3.10 (Tagged triangulations).
A maximal (by inclusion) collection of pairwise compatible tagged arcs is called a tagged triangulation.
Figure 1 gives an example of an ideal triangulation and the corresponding tagged triangulation . The notching is indicated by a bow tie.
3.2. From surfaces to cluster algebras
One can associate an exchange matrix and hence a cluster algebra to any bordered surface [FST].
Definition 3.11 (Signed adjacency matrix of an ideal triangulation).
Choose any ideal triangulation , and let be the arcs of . For any triangle in which is not self-folded, we define a matrix as follows.
and in the following cases:
and are sides of with following in the clockwise order;
is a radius in a self-folded triangle enclosed by a loop , and and are sides of with following in the clockwise order;
is a radius in a self-folded triangle enclosed by a loop , and and are sides of with following in the clockwise order;
Then define the matrix by , where the sum is taken over all triangles in that are not self-folded.
Note that is skew-symmetric and each entry is either , or , since every arc is in at most two triangles.
As noted in [FST, Definition 9.2], compatibility of tagged arcs is invariant with respect to a simultaneous change of all tags at a given puncture. So given a tagged triangulation , let us perform such changes at every puncture where all ends of are notched. The resulting tagged triangulation represents an ideal triangulation (possibly containing self-folded triangles): . This is because the only way for a puncture to have two incident arcs with two different taggings at is for those two arcs to be homotopic, see Definition 3.9. But then for this to lie in some tagged triangulation, it follows that must be a puncture in the interior of a bigon. See Figure 1.
Definition 3.13 (Signed adjacency matrix of a tagged triangulation).
The signed adjacency matrix of a tagged triangulation is defined to be the signed adjacency matrix , where is obtained from as in Remark 3.12. The index sets of the matrices (which correspond to tagged and ideal arcs, respectively) are identified in the obvious way.
[FST, Theorem 7.11] and [FT, Theorem 5.1] Fix a bordered surface and let be the cluster algebra associated to the signed adjacency matrix of a tagged triangulation as in Definition 3.13. Then the (unlabeled) seeds of are in bijection with tagged triangulations of , and the cluster variables are in bijection with the tagged arcs of (so we can denote each by , where is a tagged arc). Furthermore, if a tagged triangulation is obtained from another tagged triangulation by flipping a tagged arc and obtaining , then is obtained from by the seed mutation replacing by .
By a slight abuse of notation, if is an ordinary arc which does not cut out a once-punctured monogon (so that the tagged arc is obtained from by tagging both ends plain), we will often write instead of .
Given a surface with a puncture and a tagged arc , we let both and denote the arc obtained from by changing its notching at . (So if is not incident to , .) If and are two punctures, we let denote the arc obtained from by changing its notching at both and . Given a tagged triangulation of , we let denote the tagged triangulation obtained from by replacing each by .
Besides labeling cluster variables of by , where is a tagged arc of , we will also make the following conventions:
If is an unnotched loop with endpoints at cutting out a once-punctured monogon containing puncture and radius , then we set .111There is a corresponding statement on the level of lambda lengths of these three arcs, see [FT, Lemma 7.2]; these conventions are compatible with both the Ptolemy relations and the exchange relations among cluster variables [FT, Theorem 7.5].
If is a boundary segment, we set .
To prove the positivity conjecture for a cluster algebra associated to a surface, we must show that the Laurent expansion of each cluster variable with respect to any cluster is positive. The next result will allow us to restrict our attention to clusters that correspond to ideal triangulations.
Fix , , , , and as above. Let , and be the cluster algebras with principal coefficients at the vertices and , where , , , and Then
That is, the cluster expansion of with respect to in is obtained from the cluster expansion of with respect to in by substituting and
By Definition 3.13, the rectangular exchange matrix is equal to . The columns of are indexed by and the columns of are indexed by ; the rows of are indexed by and the rows of are indexed by .
To compute the -expansion of , we write down a sequence of flips (here ) which transforms into a tagged triangulation containing . Applying the corresponding exchange relations then gives the -expansion of in . By the description of tagged flips ([FT, Remark 4.13]), performing the same sequence of flips on transforms into the tagged triangulation , which in particular contains . Therefore applying the corresponding exchange relations gives the -expansion of in .
Since in both cases we start from the same exchange matrix and apply the same sequence of mutations, the -expansion of in will be equal to the -expansion of in after relabeling variables, i.e. after substituting and
Fix a bordered surface and let be the corresponding cluster algebra. Let be an arbitrary tagged triangulation. To prove the positivity conjecture for with respect to , it suffices to prove positivity with respect to clusters of the form , where is an ideal triangulation.
As in Remark 3.12, we can perform simultaneous tag-changes at punctures to pass from an arbitrary tagged triangulation to a tagged triangulation representing an ideal triangulation. By a repeated application of Proposition 3.16 – which preserves positivity because it just involves a substitution of variables – we can then express cluster expansions with respect to in terms of cluster expansions with respect to . ∎
The exchange relation corresponding to a flip in an ideal triangulation is called a generalized Ptolemy relation. It can be described as follows.
This follows from the interpretation of cluster variables as lambda lengths and the Ptolemy relations for lambda lengths [FT, Theorem 7.5 and Proposition 6.5]. ∎
3.3. Keeping track of coefficients using laminations
So far we have not addressed the topic of coefficients for cluster algebras arising from bordered surfaces. It turns out that W. Thurston’s theory of measured laminations gives a concrete way to think about coefficients, as described in [FT] (see also [FG3]).
Definition 3.19 (Laminations).
A lamination on a bordered surface is a finite collection of non-self-intersecting and pairwise non-intersecting curves in , modulo isotopy relative to , subject to the following restrictions. Each curve must be one of the following:
a closed curve;
a curve connecting two unmarked points on the boundary of ;
a curve starting at an unmarked point on the boundary and, at its other end, spiraling into a puncture (either clockwise or counterclockwise);
a curve whose ends both spiral into punctures (not necessarily distinct).
Also, we forbid curves that bound an unpunctured or once-punctured disk, and curves with two endpoints on the boundary of which are isotopic to a piece of boundary containing zero or one marked point.
In [FT, Definitions 12.1 and 12.3], the authors define shear coordinates and extended exchange matrices, with respect to a tagged triangulation. For our purposes, it will be enough to make these definitions with respect to an ideal triangulation.
Definition 3.20 (Shear coordinates).
Let be a lamination, and let be an ideal triangulation. For each arc , the corresponding shear coordinate of with respect to , denoted by , is defined as a sum of contributions from all intersections of curves in with . Specifically, such an intersection contributes (resp., ) to if the corresponding segment of a curve in cuts through the quadrilateral surrounding as shown in Figure 2 in the middle (resp., right).
Definition 3.21 (Multi-laminations and associated extended exchange matrices).
A multi-lamination is a finite family of laminations. Fix a multi-lamination . For an ideal triangulation of , define the matrix as follows. The top part of is the signed adjacency matrix , with rows and columns indexed by arcs (or equivalently, by the tagged arcs ). The bottom rows are formed by the shear coordinates of the laminations with respect to :
By [FT, Theorem 11.6], the matrices transform compatibly with mutation.
Definition 3.22 (Elementary lamination associated with a tagged arc).
Let be a tagged arc in . Denote by a lamination consisting of a single curve defined as follows. The curve runs along within a small neighborhood of it. If has an endpoint on a (circular) component of the boundary of , then begins at a point located near in the counterclockwise direction, and proceeds along as shown in Figure 3 on the left. If has an endpoint at a puncture, then spirals into : counterclockwise if is tagged plain at , and clockwise if it is notched.
The following result comes from [FT, Proposition 16.3].
Let be an ideal triangulation with a signed adjacency matrix . Recall that we can view as a tagged triangulation . Let be the multi-lamination consisting of elementary laminations associated with the tagged arcs in . Then the cluster algebra with principal coefficients is isomorphic to .
4. Main results: cluster expansion formulas
In this section we present cluster expansion formulas for all cluster variables in a cluster algebra associated to a bordered surface, with respect to a seed corresponding to an ideal triangulation; by Proposition 3.16 and Corollary 3.17, one can use these formulas to compute cluster expansion formulas with respect to an arbitrary seed by an appropriate substitution of variables. All of our formulas are manifestly positive, so this proves the positivity conjecture for any cluster algebras associated to a bordered surface. Moreover, since our formulas involve the system of principal coefficients, this proves positivity for any such cluster algebra of geometric type.
We present three slightly different formulas, based on whether the cluster variable corresponds to a tagged arc with , , or notched ends. More specifically, fix an ordinary arc and a tagged triangulation of , where is an ideal triangulation. We recursively construct an edge-weighted graph by glueing together tiles based on the local configuration of the intersections between and . Our formula (Theorem 4.9) for with respect to is given in terms of perfect matchings of . This formula also holds for the cluster algebra element , where is a loop cutting out a once-punctured monogon enclosing the puncture and radius . In the case of , an arc between points and with a single notch at , we build the graph associated to the loop such that . Our combinatorial formula for is then in terms of the so-called -symmetric matchings of . In the case of , an arc between points and which is notched at both and , we build the two graphs and associated to and . Our combinatorial formula for is then in terms of the -compatible pairs of matchings of . and .
Let be an ideal triangulation of a bordered surface and let be an ordinary arc in which is not in . Choose an orientation on , let be its starting point, and let be its endpoint. We denote by
the points of intersection of and in order. Let be the arc of containing , and let and be the two ideal triangles in on either side of .
To each we associate a tile , an edge-labeled triangulated quadrilateral (see the right-hand-side of Figure 4), which is defined to be the union of two edge-labeled triangles and glued at an edge labeled . The triangles and are determined by and as follows.
If neither nor is self-folded, then they each have three distinct sides (though possibly fewer than three vertices), and we define and to be the ordinary triangles with edges labeled as in and . We glue and at the edge labeled , so that the orientations of and both either agree or disagree with those of and ; this gives two possible planar embeddings of a graph which we call an ordinary tile.
If one of or is self-folded, then in fact must have a local configuration of a bigon (with sides and ) containing a radius incident to a puncture inscribed inside a loop , see Figure 5. Moreover, must either
intersect the loop and terminate at puncture , or
intersect the loop , radius and then again.
In case (1), we associate to (the intersection point with ) an ordinary tile consisting of a triangle with sides which is glued along diagonal to a triangle with sides . As before there are two possible planar embeddings of .
In case (2), we associate to the triple of intersection points a union of tiles , which we call a triple tile, based on whether enters and exits through different sides of the bigon or through the same side. These graphs are defined by Figure 5 (each possibility is denoted in boldface within a concatenation of five tiles). Note that in each case there are two possible planar embeddings of the triple tile. We call the tiles and within the triple tile ordinary tiles.
Definition 4.1 (Relative orientation).
Given a planar embedding of an ordinary tile , we define the relative orientation of with respect to to be , based on whether its triangles agree or disagree in orientation with those of .
Note that in Figure 5, the lowest tile in each of the three graphs in the middle (respectively, rightmost) column has relative orientation (respectively, ). Also note that by construction, the planar embedding of a triple tile satisfies .
Using the notation above, the arcs and form two edges of a triangle in . Define to be the third arc in this triangle if is not self-folded, and to be the radius in otherwise.
4.2. The graph
We now recursively glue together the tiles in order from to , subject to the following conditions.
After glueing together the tiles, we obtain a graph (embedded in the plane), which we denote . Let be the graph obtained from by removing the diagonal in each tile. Figure 5 gives examples of a dotted arc and the corresponding graph . Each intersects five times, so each has five tiles.
Even if is a curve with self-intersections, our definition of makes sense. This is relevant to our formula for the doubly-notched loop, see Remark 4.22.
4.3. Cluster expansion formula for ordinary arcs
Recall that if is a boundary segment then , and if is a loop cutting out a once-punctured monogon with radius and puncture , then . Also see Remark 3.15. Before giving the next result, we need to introduce some notation.
Definition 4.4 (Crossing Monomial).
If is an ordinary arc and is the sequence of arcs in which crosses, we define the crossing monomial of with respect to to be
Definition 4.5 (Perfect matchings and weights).
A perfect matching of a graph is a subset of the edges of such that each vertex of is incident to exactly one edge of . If the edges of a perfect matching of are labeled , then we define the weight of to be .
Definition 4.6 (Minimal and Maximal Matchings).
By induction on the number of tiles it is easy to see that