Cluster algebras and triangulated orbifolds

Cluster algebras and triangulated orbifolds

Anna Felikson Independent University of Moscow, B. Vlassievskii 11, 119002 Moscow, Russia School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, D-28759, Germany a.felikson@jacobs-university.de Michael Shapiro Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA mshapiro@math.msu.edu  and  Pavel Tumarkin Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, UK pavel.tumarkin@durham.ac.uk
Abstract.

We construct geometric realizations for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston [FT] to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds. We also compute the growth rate of these cluster algebras, provide the positivity of Laurent expansions of cluster variables, and prove the sign-coherence of -vectors.

Research was supported in part by grants DFG FE-1241/2 (A.F.), DNS 0800671 (M.S.) and RFBR 11-01-00289-a (P.T.)

1. Introduction

We continue investigation of cluster algebras of finite mutation type started in [FeSTu1] and [FeSTu2].

In [FeSTu1], we classified all the skew-symmetric exchange matrices with finite mutation class. It occurs that all but eleven exceptional mutation classes of skew-symmetric exchange matrices of rank at least can be obtained from triangulated marked bordered surfaces via construction provided by Fomin, Shapiro and Thurston [FST].

In [FeSTu2], we completed classification of finite mutation classes of exchange matrices by extending the combinatorial technique of [FST] to general (i.e., skew-symmetrizable) case. All but several exceptional finite mutation classes consist of so called s-decomposable exchange matrices (the precise definitions will be given below).

In this paper, we relate non-exceptional mutation-finite cluster algebras to triangulated orbifolds. Extending the technique of Fomin and Thurston [FT] to skew-symmetrizable case, we construct geometric realizations for algebras with s-decomposable exchange matrices. In these realizations, (tagged) triangulations of certain orbifolds form clusters with (modified) lambda lengths of arcs serving as cluster variables. The geometric realization provides various structural results, for example, we prove that the exchange graph in a cluster algebra with s-decomposable exchange matrices does not depend on coefficients.

One of the tools of [FeSTu2] was a notion of unfolding introduced by Zelevinsky (it can be understood as a counterpart of the unfolding procedure introduced by Lusztig in [L] for generalized Cartan matrices). In particular, we construct unfoldings for a class of mutation-finite matrices. In the current paper we provide a geometric version of unfolding, and construct unfoldings for almost all mutation-finite matrices. We then use unfoldings to compute the growth rate of all cluster algebras originating from orbifolds, and for generalization of positivity results by Musiker, Schiffler and Williams [MSW] to Laurent expansions of corresponding cluster variables.

Another application of the construction is a proof of the sign-coherence for -vectors. In [FZ3], Fomin and Zelevinsky conjectured that all the entries of -vectors are either nonnegative or nonpositive. This conjecture was proved for skew-symmetric cluster algebras by Derksen, Weyman and Zelevinsky [DWZ], and for a large class of skew-symmetrizable algebras by Demonet [D]. We extend the list of algebras for which the conjecture holds by proving the sign-coherence for -vectors for all algebras originating from orbifolds.

The paper is organized as follows.

In Section 2, we recall necessary definitions and basic facts on cluster algebras, exchange matrices, and their diagrams.

Section 3 is devoted to the technique of s-decomposable diagrams. We recall the basic facts and results from [FST] and [FeSTu2], and introduce block decompositions of matrices.

In Section 4, we construct a triangulated orbifold for any s-decomposable diagram. The simplicial complex of triangulations of this orbifold coincides with exchange graph of corresponding cluster algebra. The construction is close to the similar construction of Chekhov and Mazzocco [ChM].

In Section 5, a geometric realization of cluster algebras with s-decomposable exchange matrices is constructed. To do this, we proceed in a way similar to [FT], where cluster variables were represented by modified lambda lengths of arcs of triangulations of marked bordered surfaces. However, unlike [FT], we need to consider arcs of triangulations not of the given orbifold but of some its modification. We call this modified orbifold an associated orbifold. This associated orbifold can be constructed for any specific s-decomposable matrix. In some special cases an associated orbifold occurs to be a regular surface.

In Section 6, we generalize the notion of laminations and shear coordinates to the orbifold case. Sections 7 and 8 are devoted to a construction of a geometric realization of cluster algebras with arbitrary coefficients. Main results are contained in Section 9.

In Section 10, we investigate the growth of cluster algebras with s-decomposable exchange matrices. We use orbifolds to show that the exchange graph of a cluster algebra with an s-decomposable skew-symmetrizable exchange matrix is quasi-isometric to the exchange graph of a cluster algebra with some block-decomposable skew-symmetric exchange matrix. In this way we classify the growth rate of all cluster algebras with s-decomposable exchange matrices. This gives rise to a classification of the growth of all cluster algebras [FeSTTu].

In Section 11, we recall the definition of an unfolding of skew-symmetrizable matrices introduced by A. Zelevinsky (personal communication), extend it to a notion of unfolding of a diagram, and recall the results of [FeSTu2]. Section 12 is devoted to a construction of unfoldings of almost all mutation-finite skew-symmetrizable matrices.

In Section 13, we prove the positivity conjecture for (almost all) cluster algebras originating from orbifolds, namely, for ones with s-decomposable exchange matrices admitting unfolding. This is done by extending the results of [MSW] to the orbifold case.

Finally, in Section 14 we prove the sign-coherence of -vectors for all cluster algebras originating from orbifolds.

We would like to thank L. Chekhov and S. Fomin for fruitful discussions, and the anonymous referee for valuable comments.

2. Basics on cluster algebras

We briefly remind the definition of a cluster algebra.

An integer matrix is called skew-symmetrizable if there exists an integer diagonal matrix , such that the product is a skew-symmetric matrix, i.e., .

Let be a tropical semifield equipped with commutative multiplication and addition . The multiplicative group of is a coefficient group of cluster algebra, i.e, it is a free abelian group. is the integer group ring, is a field of rational functions in independent variables with coefficients in the field of fractions of . is called an ambient field.

Definition 2.1.

A seed is a triple , where

  • , a -tuple of elements of is a coefficient tuple of cluster ;

  • is a collection of algebraically independent rational functions of variables which generates over the field of fractions of ;

  • is a skew-symmetrizable integer matrix (exchange matrix).

The part of seed is called cluster, elements are called cluster variables.

Definition 2.2 (seed mutation).

For any , we define the mutation of seed in direction as a new seed in the following way:

(2.1)
(2.2)
(2.3)
(2.4)

We write . Notice that . We say that two seeds are mutation-equivalent if one is obtained from the other by a sequence of seed mutations. Similarly we say that two clusters or two exchange matrices are mutation-equivalent.

Notice that exchange matrix mutation (2.1) depends only on the exchange matrix itself. The collection of all matrices mutation-equivalent to a given matrix is called the mutation class of .

For any skew-symmetrizable matrix we define initial seed as a collection , where is the initial exchange matrix, is the initial cluster, is the initial coefficient tuple.

Cluster algebra associated with the skew-symmetrizable matrix is a subalgebra of generated by all cluster variables of the clusters mutation-equivalent to the initial seed .

Cluster algebra is called of finite type if it contains only finitely many cluster variables. In other words, all clusters mutation-equivalent to initial cluster contain only finitely many distinct cluster variables in total.

Definition 2.3.

A cluster algebra with only finitely many exchange matrices is called of finite mutation type.

Remark 2.4.

Since the orbit of an exchange matrix depends on the exchange matrix only, we may speak about skew-symmetrizable matrices of finite mutation type.

Following [FZ2], we encode an skew-symmetrizable integer matrix by a finite simplicial -complex with oriented weighted edges called diagram. The weights of a diagram are positive integers.

Vertices of are labeled by . If , we join vertices and by an edge directed from to and assign to this edge weight . Not every diagram corresponds to a skew-symmetrizable integer matrix: given a diagram of a skew-symmetrizable integer matrix , a product of weights along any chordless cycle of is a perfect square (cf. [Kac, Exercise 2.1]).

Distinct matrices may have the same diagram. At the same time, it is easy to see that only finitely many matrices may correspond to the same diagram. All weights of a diagram of a skew-symmetric matrix are perfect squares. Conversely, if all weights of a diagram are perfect squares, then there exists a skew-symmetric matrix with diagram .

As it is shown in [FZ2], mutations of exchange matrices induce mutations of diagrams. If is the diagram corresponding to matrix , and is a mutation of in direction , then we call the diagram associated to a mutation of in direction and denote it by . A mutation in direction changes weights of diagram in the way described in Figure 2.1 (see [FZ2]).

\psfrag

a \psfragb \psfragc \psfragd \psfragk \psfragmu

Figure 2.1. Mutations of diagrams. The sign before (resp., ) is positive if the three vertices form an oriented cycle, and negative otherwise. Either or may vanish. If is equal to zero then neither value of nor orientation of the corresponding edge does change.

Hence, for a given diagram, the notion of its mutation class is well-defined. We call a diagram (resp., matrix) mutation-finite if its mutation class is finite.

3. Block decompositions of diagrams and matrices

First, we remind the definitions from [FST] and [FeSTu2].

Definition 3.1.

In [FST], a block is a diagram isomorphic to one of the diagrams with black/white colored vertices shown in Fig. 3.1, or to a single vertex. Vertices marked in white are called outlets, we call the black ones dead ends. A connected skew-symmetric diagram is called block-decomposable if it can be obtained from a collection of blocks by identifying outlets of different blocks along some partial matching (matching of outlets of the same block is not allowed), where two single edges with same endpoints and opposite directions cancel out, and two single edges with same endpoints and same directions form an edge of weight . A non-connected diagram is called block-decomposable either if satisfies the definition above, or if is a disjoint union of several diagrams (without any edge joining one to another) satisfying the definition above. If a skew-symmetric diagram is not block-decomposable then we call non-decomposable. Depending on a block, we call it a block of type , , , , , or simply a block of -th type.

\psfrag

1 \psfrag2 \psfrag3a \psfrag3b \psfrag4 \psfrag5

Figure 3.1. Skew-symmetric blocks. Outlets are colored in white, dead ends are black.

Block-decomposable diagrams are in one-to-one correspondence with adjacency matrices of arcs of ideal (tagged) triangulations of bordered two-dimensional surfaces with marked points (see [FST, Section 13] for the detailed explanations). Mutations of block-decomposable diagrams correspond to flips of (tagged) triangulations. In particular, this implies that mutation class of any block-decomposable diagram is finite, and any subdiagram of a block-decomposable one is block-decomposable too.

It is proved in [FeSTu1] that block-decomposable diagrams almost exhaust mutation-finite ones. Namely, any mutation-finite non-decomposable skew-symmetric diagram of order at least is mutation-equivalent to one of exceptional diagrams, see [FeSTu1, Theorem 6.1].

Definition 3.2.

To adopt the technique of blocks to general (skew-symmetrizable) case, we introduce new blocks called s-blocks of types , , , , , and shown in Table 3.1, and exceptional blocks shown in Table 3.2.

s-blocks
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\psfrag2- \psfrag4 \psfrag512t
Unfoldings
Table 3.1. s-blocks and their local unfoldings (see Sections 1112). Vertex and the set are marked in the same way. Outlets are colored white.
s-blocks
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Unfoldings
Table 3.2. Exceptional s-blocks and their unfoldings. s-blocks shown in the table have no outlets, so they cannot be used to construct other s-decomposable diagrams
Definition 3.3.

A diagram is called s-decomposable if it can be obtained from a collection of blocks and s-blocks according to the same rules as block-decomposable diagram (the way of identification remains well-defined since any edge with two white ends has weight one). We keep the term “block-decomposable” for s-decomposable diagrams corresponding to skew-symmetric matrices. A diagram called non-decomposable if it is not s-decomposable.

It is proved in [FeSTu2] that any mutation-finite non-decomposable diagram of order at least is either skew-symmetric or mutation-equivalent to one of exceptional diagrams, see [FeSTu2, Theorem 5.13].

Remark 3.4.

The exceptional s-blocks shown in Table 3.2 have no outlets, so they cannot be used in constructing other s-decomposable diagrams. However, they are mutation-finite, cannot be decomposed into other blocks and s-blocks, and can be constructed as diagrams of triangulations of some orbifolds (see Table 3.5), so we call them s-blocks for completeness of the theory.

Now we can define s-decomposable matrices.

Definition 3.5.

A skew-symmetrizable matrix is s-decomposable (respectively, block-decomposable) if its diagram is s-decomposable (respectively, block-decomposable)

Block-decomposable (or s-decomposable) matrices can be indeed decomposed into blocks in the following way. Let be an s-decomposable matrix with diagram . For every block in spanned by vertices consider the following matrix : the matrix corresponding to the block (see Tables 3.3 and 3.4) is located on -places of , and the other entries are zeros. Then is the sum of all matrices for all blocks .

Block Diagram Matrix Surface
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\psfrag4 \psfragw \psfragu \psfragv1 \psfragv2 \psfragp1 \psfragp2
\psfrag5 \psfragu \psfragw1 \psfragw2 \psfragp1 \psfragp2
Table 3.3. Skew-symmetric blocks and the corresponding surfaces
s-Block Diagram Orbifold Matrix Associated orbifold
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Table 3.4. s-blocks, their matrices, orbifolds, and associated orbifolds, see Sections 5.3 and 5.4
Diagram Orbifold Matrix Associated orbifold
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Table 3.5. Exceptional s-blocks, their matrices, orbifolds, and associated orbifolds, see Sections 5.3 and 5.4

4. Triangulations of orbifolds

Let be a connected oriented 2-dimensional surface with (possibly empty) boundary .

By an orbifold we mean a triple , where is a bordered surface with a finite set of marked points , and is a finite (non-empty) set of special points called orbifold points, . Some marked points may belong to (moreover, every boundary component must contain at least one marked point; the interior marked points are also called punctures), while all orbifold points are interior points of (later on, as we will supply the orbifold with a metric, the orbifold points will have angle ). By boundary we mean .

An arc in is a curve in considered up to relative isotopy (of ) modulo endpoints such that

  • one of the following holds:

    • either both endpoints of belong to (and then is an ordinary arc)

    • or one endpoint belongs to and another belongs to (then is called pending arc);

  • has no self-intersections, except that its endpoints may coincide;

  • except for the endpoints, and are disjoint;

  • if cuts out a monogon then this monogon contains either a point of or at least two points of ;

  • is not homotopic to a boundary segment.

Note that we do not allow both endpoints of to be in .

Two arcs and are compatible if the following two conditions hold:

  • they do not intersect in the interior of ;

  • if both and are pending arcs, then the ends of and that are orbifold points do not coincide (i.e., two pending arcs may share a marked point, but neither an ordinary point nor a orbifold point).

A triangulation of is a maximal collection of distinct pairwise compatible arcs. The arcs of a triangulation cut into triangles. We allow self-folded triangles as well as triangles one or two of whose edges are pending arcs. See Fig. 4.1 for the list of possible triangles.

The following lemma is evident.

Lemma 4.1.

Any set of compatible arcs on an orbifold is contained in some triangulation.

Figure 4.1. Types of triangles admissible for triangulations of orbifolds (vertices marked by a cross denote orbifold points, bold edges denote pending arcs)

A flip of an arc of a triangulation replaces by a unique arc such that forms a new triangulation of . In Fig. 4.2 we show flips involving pending arcs.

Figure 4.2. Flips of pending arcs

4.1. Transitivity of flips on triangulations of orbifolds

In this section we prove the following theorem.

Theorem 4.2.

For any orbifold flips act transitively on triangulations of .

By a system of pending arcs of a triangulation we mean the union of all pending arcs of .

Lemma 4.3.

Flips act transitively on triangulations with the same system of pending arcs.

Proof.

Choose a system of pending arcs on . To prove the lemma we cut the orbifold along all pending arcs (i.e., we replace any pending arc by a hole with one marked point on the boundary) and denote by the obtained surface. The marked points of are the same as of , the orbifold points disappear. Every pending arc of produces a boundary component of with exactly one marked point.

The triangulations of containing the chosen system of pending arcs are in one-to-one correspondence with the triangulations of (and if two triangulations of are related by a flip in some arc, then the corresponding triangulations of are related by a flip in the corresponding arc). So, the lemma follows from transitivity of flips on triangulations of (see [H] and [FST]).

A set of pending arcs on is compatible if and are compatible for every .

Every maximal compatible set of pending arcs is a system of pending arcs for some triangulation of : to see this, we cut along all pending arcs and triangulate the surface. In the sequel we will use the notion of system of pending arcs as a maximal compatible set of pending arcs not related to any triangulation.

An elementary transformation of a system of pending arcs is a substitution of a pending arc by any other pending arc compatible with the set and not intersecting interior of .

Lemma 4.4.

Let be a system of pending arcs on . Let be an elementary transformation of a substituting by . Then there exist triangulations and containing the systems and respectively, such that where is a flip in the pending arc .

Proof.

Let be the common orbifold point of and . Let and be the marked (i.e., non-orbifold) ends of and (possibly, ). Consider a path from through to . Denote by and the paths in built as in Fig. 4.3: goes from to following the path and shifted to the left, while is the similar path shifted to the right from . Since the disc bounded by and contains no singularities except , the curves , and are sides of an admissible triangle on (in fact, may coincide with if ). Similarly, , and are sides of a triangle .

Now, delete the triangle with sides , , from the orbifold and choose any triangulation on compatible with the set of pending arcs . Then is a triangulation of compatible with the system of pending arcs . Similarly, is compatible with . It is left to note that the triangulation can be obtained from by a flip in the pending arc .

\psfrag

x \psfragx’ \psfragpl \psfragpr \psfragz’ \psfragz \psfragc

Figure 4.3. Proof of Lemma 4.4: paths and

Lemmas 4.3 and 4.4 show that to prove Theorem 4.2it is sufficient to prove the transitivity of action of elementary transformations on the set of systems of pending arcs of .

A system of pending arcs is centered at a marked point if is an endpoint of every pending arc of .

Lemma 4.5.

For any system of pending arcs on and any marked point one can find a sequence of at most elementary transformations which takes to a system centered at .

Proof.

For each orbifold point there exists a unique pending arc containing , so is connected. This implies that we can perform an elementary transformation that replaces by a pending arc connecting with a chosen fixed marked point .

Lemma 4.6.

Let and be two systems of pending arcs, both centered at the same marked point . Then there exists a sequence of elementary transformations taking to .

Proof.

First, we will show that using elementary transformations we can create a system containing a given pending arc.

Claim 1. Let be a system of pending arcs centered at and let be a path from to an orbifold point . Then there exists a system of pending arcs centered at , , and a sequence of elementary transformations taking to .

To prove the statement we perturb so that it intersects transversely. If then there is nothing to prove: an elementary transformation substituting by turns into required system . So, we assume that . Denote by the number of points of intersection. We will show that there exists an elementary transformation which takes to a system such that .

Let be the first intersection point of the path (from to ) with . Consider a path composed of the segment of the path and a segment of the path (and then shift to minimize the number of intersections with and , see Fig. 4.4). Notice that , so there exists an elementary transformation of substituting by . On the other hand, the path intersects system in all the intersection points of except , so .

\psfrag

x \psfragp \psfragt \psfragz’ \psfragz

Figure 4.4. Elementary transformation decreasing the number of intersections .

Thus, elementary transformations of the system of pending arcs allow us to decrease the number of intersection points of with by , which implies that after several elementary transformations we come to the case which was treated above. This proves Claim 1.

Now we will use elementary transformations to include a given pending arc in a system of pending arcs preserving some subset of the system.

Claim 2. Let be a system of pending arcs centered at and let be a path from to an orbifold point . Suppose that does not intersect the curves . Then there exists a system of pending arcs containing and a sequence of elementary transformations taking to .

Indeed, since does not intersect , elementary transformations described in the proof of Claim 1 never affect the curves , so, these pending arcs also belong to the resulting collection .

Now, to prove the lemma, it is sufficient to apply Claim 2 several times. Namely, given two systems and , we choose and apply Claim 2. As we obtain a system containing , choose and apply Claim 2 again to obtain a system containing both and (Claim 2 applies since does not intersect ). Applying Claim 2 times (and choosing at -th iteration) we obtain the required system . ∎

Proof of Theorem 4.2.

We will show that every triangulation can be transformed by flips into one fixed triangulation , the system of pending arcs of which is centered at randomly chosen marked point .

By Lemma 4.5, we can take system of pending arcs of to a system centered at . Applying Lemma 4.6, we can obtain system of pending arcs of . By Lemmas 4.3 and 4.4, all the elementary transformations above can be realized by sequences of flips. Now, applying Lemma 4.3 another one time, we perform a sequence of flips to obtain triangulation .

Later we will also need the following two easy statements concerning triangulations of orbifolds.

Lemma 4.7.

Let be an orbifold with even number of orbifold points. Then there exists a triangulation of such that every orbifold point of is contained in a monogon with two pending arcs.

Proof.

To find the required triangulation, first we connect all orbifold points with the same marked point (Lemma 4.5). Then we group the pending arcs in disjoint pairs of neighboring arcs, and for each pair we draw a curve which starts at , goes along , then goes along and returns back to (we assume that is close enough to and so that and compose a triangle). Notice, that the curves obtained for different pairs of adjacent pending arcs are distinct (excluding the case of a sphere with exactly one puncture and four orbifold points; in this case we just consider one of the two homotopy equivalent curves). So, if there are orbifold points in then we build a compatible set of pending arcs and curves enclosing the pairs of pending arcs in disks. Any triangulation containing this set of arcs satisfies the conditions of the lemma (such a triangulation does exist in view of Lemma 4.1).

Lemma 4.8.

Let be an orbifold with odd number of orbifold points, . Then there exists a triangulation of such that

  • contains triangles with two pending arcs and one triangle with one pending arc (and several triangles without pending arcs);

  • has a common edge with .

Proof.

The proof is similar to the proof of the previous lemma. First, we find a compatible system of pending arcs connecting all orbifold points with the same marked point . Then we enclose pairs of adjacent pending arcs by curves and add one extra curve enclosing the extra pending arc together with one of the discs above (as in Fig. 4.5). Similarly to the case of even number of orbifold points, there are several exclusions: namely, if is a sphere with only one puncture and at most orbifold points, then the curve coincides with one of the . Now, we are left to take any triangulation containing all the curves described above.

\psfrag

D3 \psfragD1 \psfragD2 \psfragc0 \psfragc1 \psfragc2 \psfragx \psfragc’

Figure 4.5. To the proof of Lemma 4.8.

4.2. Mutations of diagrams and tagged triangulations of orbifolds

To every triangulation of an orbifold we associate the following diagram :

  • vertices of correspond to arcs of (we denote by a vertex corresponding to an arc );

  • for every non-self-folded triangle and every pair of sides of we draw an arrow in from to if follows in in clockwise order;

  • for every self-folded triangle with sides and for every arrow from to (from to ) we draw an arrow from to (respectively, from to ).

  • arrows between a pending arc and a non-pending arc are labeled by ;

  • arrows between two pending arcs are labeled by ;

  • if and are connected by two arrows in opposite directions, then these arrows cancel out;

  • if and are connected by two arrows in the same direction, then these two arrows are substituted by one arrow labeled by .

Remark 4.9.

If there are no orbifold points (i.e., the orbifold is just a bordered surface with marked points), the construction above coincides with the one from [FST] leading to a quiver associated to a triangulation of a surface.

As it was shown in [FST], block-decomposable quivers are precisely ones corresponding to triangulations of surfaces. We will now generalize this result by establishing similar correspondence between s-decomposable diagrams and triangulations of orbifolds.

Lemma 4.10.

Any diagram obtained from triangulation of an orbifold is s-decomposable.

Proof.

Skew-symmetric blocks together with s-blocks represent all possible triangles which may appear in the triangulation. So, we take the blocks corresponding to the triangles and attach their outlets in accordance with the gluings in the triangulation. This results in the s-decomposition of the diagram corresponding to given triangulation.

Lemma 4.11.

Any s-decomposable diagram can be obtained from a triangulation of some orbifold.

Proof.

For each of the blocks we take the corresponding triangulated surface (or orbifold, see Tables 3.33.4 and 3.5) and attach them along the boundary edges in accordance with the gluing of the blocks in the diagram.

So, block decompositions of s-decomposable diagrams are in one-to-one correspondence with triangulations of orbifolds.

Remark 4.12.

One can see from Table 3.5 that all the exceptional blocks arise from triangulations of a sphere without boundary with four punctures and orbifold points in total, from which one, two, or three are punctures, and the remaining ones are orbifold points.

Now we will show that, as in the case of surfaces, the construction of a diagram is consistent with action of flips on triangulations. If is the diagram built by a triangulation and is a flip of an arc of , we denote by the mutation of in vertex .

Lemma 4.13.

For each flip one has .

Proof.

The proof is a straightforward exhaustion of finitely many possibilities for adjacent to triangles. Notice that one need to verify only the cases when one of the adjacent triangles contains a pending arc (in view of the similar fact known for the triangulated surfaces, see [FST]).

As in the case of surfaces, one can note that not every edge of triangulation can be flipped. More precisely, there is no flip in an interior edge of a self-folded triangle. This difficulty can be resolved by the same trick as in the case of surfaces, namely, by introducing tagged triangulations (see [FST][FT]). The construction is exactly the same as in the surface case, so we do not stop here for the details. We only mention that the ends of pending arcs being orbifold points are always tagged plain. It is easy to see that Lemma 4.13 holds for tagged triangulations as well.

We summarize the discussion above in the following lemma.

Lemma 4.14.

Let be an s-decomposable diagram, and let be an orbifold with tagged triangulation such that . Then a diagram is mutation-equivalent to if and only if can be obtained as for some tagged triangulation of . Moreover, if and only if .

4.3. Weighted orbifolds and matrix mutations

In the previous section we established a correspondence between s-decomposable diagrams and triangulated orbifolds. Every s-decomposable diagram can be considered as a diagram of some mutation-finite s-decomposable skew-symmetrizable matrix . In contrast to the skew-symmetric case, such matrix is not uniquely defined: every s-decomposable diagram with at least one edge labeled by can be constructed by several matrices. In this section, we associate with every s-decomposable matrix a triangulated orbifold with additional structure.

Given an s-decomposable skew-symmetrizable matrix , denote by a unique diagonal matrix with positive integer entries such that is skew-symmetric, and the greatest common divisor of the entries of is one. Given an s-decomposable diagram constructed by s-decomposable matrix , we call a weight of vertex . The matrix is the same for every matrix mutation equivalent to , hence the weights of vertices of a diagram do not change under mutations.

Definition 4.15.

An s-decomposable diagram with a collection of weights as above is called a weighted diagram and is denoted by . Weighted s-decomposable diagrams carry exactly the same information as s-decomposable matrices.

In other words, weighted s-decomposable diagrams are in one-to-one correspondence with s-decomposable matrices.

It was shown in [FeSTu2, Lemma 6.3] that the weights of outlets of all blocks of any weighted s-decomposable connected diagram are equal. The weight of any outlet is called weight of the regular part of and is denoted by .

Now fix an s-decomposable matrix and corresponding weighted diagram . According to Lemma 4.11, vertices of the diagram correspond to arcs of a triangulation of some orbifold . In this way we assign weights to every arc of . In particular, we assign weight to every pending arc.

Definition 4.16.

Given a weighted diagram and corresponding triangulation of orbifold , a weighted orbifold is the orbifold with weights assigned to all its orbifold points according to the following rule: the weight of an orbifold point is the weight of the pending arc of incident to divided by .

The definition of weighted orbifold is consistent: a flip in any arc of triangulation does not change weight of a pending arc incident to a given orbifold point.

All possible weights of orbifolds points of a weighted orbifold are easy to describe. It was proved in [FeSTu2] that the only weights that can appear in weighted diagram are , , and . In [FeSTu2, Lemma 6.3] we also proved that either or , and if then there is no vertex of weight . In terms of weights of orbifold points, every point has weight either or .

Conversely, given an orbifold with orbifold points, we can construct weighted orbifolds by assigning weights and . Every triangulation of each of these orbifolds can be constructed by some s-decomposable skew-symmetrizable matrix.

Summarizing the definitions above, for every s-decomposable skew-symmetrizable matrix we constructed a weighted orbifold with a triangulation via a weighted diagram . Now we are able to give a definition of signed adjacency matrix of a (tagged) triangulation of weighted orbifold.

Definition 4.17.

If is a weighted orbifold and its tagged triangulation corresponds to an s-decomposable skew-symmetrizable matrix , the matrix is called signed adjacency matrix of .

Remark 4.18.

It is easy to see that the actual number of distinct mutation classes of signed adjacency matrices constructed by one orbifold with orbifold points is usually much less than . Namely, every two weighted orbifolds with the same number of orbifold points of weight (and thus with the same number of orbifold points of weight ) give rise to mutation-equivalent matrices (after some permutation of rows and columns). More precisely, given two orbifold points, the transposition of them can be realized by a transposition of corresponding rows (and columns) of the signed adjacency matrix (with a mutation applied first in case of different weights).

At the same time, two weighted orbifolds with distinct number of orbifold points of weight (and thus as well) give rise to signed adjacency matrices with essentially different skew-symmetrizing matrices , and thus these signed adjacency matrices are not mutation-equivalent. In other words, an orbifold with orbifold points provides distinct mutation classes of signed adjacency matrices indexed by the number of orbifold points of weight (or ).

Theorem 4.19.

Let be a skew-symmetrizable matrix with weighted s-decomposable diagram . Let be a weighted orbifold with a triangulation built by an s-decomposition of . Then a skew-symmetrizable matrix is mutation-equivalent to if and only if is a signed adjacency matrix of a tagged triangulation of . Moreover, if and only if .

Proof.

A straightforward verification shows that the matrix analogue of Lemma 4.13 holds, i.e. for each flip one has , where is a signed adjacency matrix of the tagged triangulation of weighted orbifold . Now, the theorem follows from Lemma 4.14 combined with the fact that knowing a diagram together with the weights of the orbifold points of one can recover the matrix (via weighted diagram ).

5. Geometric realization of cluster algebras

5.1. Lambda lengths as cluster variables

In [FT] Fomin and Thurston show that the notion of lambda length introduced by Penner [P] works well for obtaining a geometric realization of some cluster algebras. More precisely, for every skew-symmetric block-decomposable matrix there exists a bordered hyperbolic surface with marked points such that lambda lengths of arcs of (tagged) triangulations on serve as cluster variables of some cluster algebra with exchange matrix in some seed.

In this section, we adjust the basic construction to the case of cluster algebras with s-decomposable skew-symmetrizable exchange matrices.

Let be a skew-symmetrizable matrix with weighted s-decomposable diagram . Let be a triangulated weighted orbifold built by an s-decomposition of . First, we will consider the case when all orbifold points on are of weight . In this case the role of cluster variables will be played by lambda lengths of the arcs of tagged triangulations of . Next, we will treat the case of orbifolds with all orbifold points of weight . In this case, we will introduce a surface “associated” to the orbifold (constructed from by a simple procedure). The lambda lengths of arcs of tagged triangulations of will serve as cluster variables. Finally, we consider the general case, when both orbifold points of weight and may appear. Then the cluster algebra will be modeled by lambda lengths of arcs of tagged triangulations of an “associated” orbifold (an orbifold constructed from by the same procedure as in the previous case). Notice, that in case of absence of the orbifold points (i.e. in case of skew-symmetric matrix ) we obtain exactly the initial construction described in [FT].

5.2. Orbifolds with orbifold points of weight 1/2

Let be a matrix with s-decomposable weighted diagram , let be a corresponding weighted orbifold. Suppose that all orbifold points in are of weight (in terms of matrix/weighted diagram this means that all the outlets have weight , and there are no vertices of weight ).

We endow with a hyperbolic structure with cusps in all marked points and with angles in orbifold points. To choose such a structure, one may take any ideal triangulation of and assume that each triangle of is an ideal hyperbolic triangle (there is a -parameter freedom in attaching ideal triangles along a given edge). Suppose also that for each marked point on we have chosen a horocycle centered at .

Such structures on form a decorated Teichmüller space (cf. [P]): a point of a decorated Teichmüller space is a hyperbolic structure as above with a collection of horocycles, one around each marked point. It is shown by Chekhov and Mazzocco [Ch][ChM] that is parametrized by the set of functions (lambda lengths) assigned to arcs of given triangulation of (including boundary segments), defined in the following way.

Definition 5.1 (Lambda length).

For an arc with both ends in marked points we define a lambda length as usual (see [P]):

where is the signed distance along between the horocycles (positive, if the horoballs bounded by the horocycles do not intersect, and negative otherwise).

If is a pending arc, we define

where is the signed distance from the orbifold point to the horocycle (negative, if the orbifold point is contained inside the horoball, and positive otherwise).

In the definition above can be understood as the length of the “round trip” from the horocycle to the orbifold point and back.

\psfrag

alpha \psfragbeta \psfraggamma \psfragdelta \psfragsigma \psfragtheta \psfragxi \psfragmu \psfragnu \psfragzeta \psfrageta \psfragphi \psfragpsi \psfraghi \psfraga(a) \psfragb(b) \psfragc(c) \psfragd(d)

Figure 5.1. Notation for Ptolemy relations (Lemma 5.2)
Lemma 5.2.

In the notation of Fig. 5.1 the following Ptolemy relations hold:

  • ;

  • ;

  • ;

  • .

Proof.

First, we prove the relation (a). We cut the digon shown in Fig 5.1.a along the pending arc , then glue together two copies of the obtained triangle as in Fig 5.2.a (together with the triangle we copy the chosen horocycles). Since all orbifold points are points with angle , we obtain a piece of hyperbolic surface. The relation (a) now follows from the Ptolemy relation for triangulations of surfaces (together with definitions of lambda lengths on surface and on orbifolds).

The relations (b)–(d) are proved similarly. All of these relations describe some flips of a triangulation of , so, we consider a quadrilateral (i.e.a union of two triangles) of containing the arcs included in the relation (or a unique triangle in case of a flip in a pending arc). We cut along all pending arcs of and along the boundary of the quadrilateral (respectively, along the boundary of the triangle ), so that we obtain a quadrilateral or a triangle on hyperbolic plane. In case of a flip in a pending arc, we attach two copies of the triangle along the image of the pending arc. Hence, in any case we come to a relation inside a quadrilateral on hyperbolic plane, which follows immediately from the relations shown in [FT]. See Fig 5.2.(b)–(d) for the corresponding planar quadrilaterals.

\psfrag

alpha \psfragbeta \psfraggamma \psfragdelta \psfragsigma \psfragtheta \psfragxi \psfragmu \psfragnu \psfragzeta \psfrageta \psfragphi \psfragpsi \psfraghi \psfraga(a) \psfragb(b) \psfragc(c) \psfragd(d)

Figure 5.2. To the proof of Lemma 5.2

For a horocycle centered at interior marked point (puncture) of denote by the hyperbolic length of . Following [FT] we define a conjugate horocycle around the same puncture by the condition . As in [FT], define a lambda length of a tagged arc using the distance to the conjugate horocycle: for ends of the arc tagged plain one takes the distance to the initial horocycle, for ends tagged notched one takes the distance to the conjugate horocycle. Reasoning as in Lemma 5.2 one can see that the similar Ptolemy relations hold for lambda lengths of tagged arcs.

Remark 5.3.

In Lemma 5.2 we discuss the basic Ptolemy relations only. More relations can be obtained in the same way by gluing some boundary edges in Fig. 5.1: some edges at the boundary can be attached to other ones, some edges can be self-identified producing pending arcs.

Now, let be a triangulation of with signed adjacency matrix . Choose an initial seed as follows

  • ;

  • ;

Consider the cluster algebra constructed by the initial seed . Lemma 5.2 combined with Theorem 4.19 lead to the following theorem.

Theorem 5.4.

Let be an s-decomposable skew-symmetrizable matrix. Let be a weighted orbifold and its triangulation constructed by an s-decomposition of . Suppose also that weights of all orbifold points of are equal to . Then the cluster algebra satisfies the following conditions:

  • the cluster variables of are lambda lengths of tagged arcs of triangulations of ;

  • the clusters consist of all lambda lengths of tagged arcs contained in the same triangulation of ;

  • the coefficients are lambda lengths of the boundary components of (the coefficient semifield is the tropical semifield generated by the lambda lengths of boundary components);

  • the exchange graph of coincides with the exchange graph of tagged triangulations of .

5.3. Orbifolds with orbifold points of weight

As in the previous section, let be a matrix with s-decomposable weighted diagram , let be a corresponding weighted orbifold. Suppose that all orbifold points in are of weight (this corresponds to matrices/weighted diagrams with all the outlets of weight ). In this section, we introduce a new object (a hyperbolic surface built from the orbifold ) and show that the lambda lengths of tagged arc on provide a realization of a cluster algebra with exchange matrix .

Definition 5.5 (Associated triangulated surface).

Let be a tagged triangulation of the orbifold . An associated triangulated surface is a surface with tagged triangulation built as follows: for each triangle containing an orbifold point we cut out and attach a triangulated disk with marked points as shown in Fig. 5.3, so that every pending arc in corresponds to a pair of conjugate arcs in .

The marked points arising in the procedure will be called special. The pairs of conjugate arcs arising in this procedure will be called associated (pairs of) arcs. Arcs in every associated pair should be flipped simultaneously.

Figure 5.3. Construction of the associated surface. Special marked points are encircled.
Remark 5.6.

The requirement that the associated arcs should be flipped simultaneously guarantees that the associated arcs always remain conjugate. Therefore, the procedure of building the associated surface from the triangulation of commutes with flips of . This implies that the exchange graph for tagged triangulations of coincides with the exchange graph for tagged triangulations of .

Recall that horocycles and are conjugate if , where is the hyperbolic length of . Given a hyperbolic structure on , a horocycle centered at interior marked point on is self-conjugate if it coincides with its conjugate , implying .

Definition 5.7 (Decorated Teichmüller space for associated surface).

A point in a decorated Teichmüller space of the associated surface is a hyperbolic structure on with a collection of horocycles, one around each marked point, satisfying the condition that the horocycles centered at special marked points are self-conjugate.

The lambda lengths of tagged arcs on associated surface are introduced in the usual way. It follows directly from the definition that the associated arcs have the same lambda lengths. Therefore, we may substitute a pair of associated tagged arcs by a single tagged arc . We call this single arc a pending arc, similar to the orbifold case and define .

Given an associated surface with special marked points, we can consider a similar surface whose marked points are not special. In the decorated Teichmüller space of consider a codimension subspace defined by any of the following two equivalent conditions:

  • for marked point corresponding to a special marked point of , the horocycle centered at is self-conjugate;

  • for each pair of conjugate arcs on corresponding to a pair of associated arcs of holds .

Clearly, the subspace coincides with the decorated Teichmüller space of the associated surface . The condition (2) also implies that the lambda lengths of tagged arcs of triangulations of (together with lambda lengths of boundary segments) parametrize .

\psfrag

alpha \psfragbeta \psfraggamma \psfragdelta \psfragsigma \psfragtheta \psfragxi \psfragmu \psfragnu \psfragzeta \psfrageta \psfragphi \psfragpsi \psfraghi \psfraga(a) \psfragb(b) \psfragc(c) \psfragd(d)

Figure 5.4. Notation for Ptolemy relations (Lemma 5.8). Pending arcs are drawn in bold.
Lemma 5.8.

In the notation of Fig. 5.4 the following Ptolemy relations hold: