Density of g -vector cones from triangulated surfaces
aa r X i v : . [ m a t h . R T ] J a n DENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES TOSHIYA YURIKUSA
Abstract.
We study g -vector cones associated with clusters of cluster algebras defined from amarked surface ( S, M ) of rank n . We determine the closure of the union of g -vector cones asso-ciated with all clusters. It is equal to R n except for a closed surface with exactly one puncture, inwhich case it is equal to the half space of a certain explicit hyperplane in R n . Our main ingredientsare laminations on ( S, M ), their shear coordinates and their asymptotic behavior under Dehn twists.As an application, if (
S, M ) is not a closed surface with exactly one puncture, the exchange graph ofcluster tilting objects in the corresponding cluster category is connected. If (
S, M ) is a closed surfacewith exactly one puncture, it has precisely two connected components. Introduction
Cluster algebras, introduced by Fomin and Zelevinsky in 2002 [FZ02], are commutative algebraswith generators called cluster variables. The certain tuples of cluster variables are called clusters.Their original motivation was to study total positivity of semisimple Lie groups and canonical bases ofquantum groups. In recent years, it has interacted with various subjects in mathematics, for example,representation theory of quivers, Poisson geometry, integrable systems, and so on.Let Q be a quiver without loops and 2-cycles, and let A ( Q ) be the associated cluster algebra withprincipal coefficients (see Subsection 3.1). We denote by cluster Q the set of clusters in A ( Q ). Eachcluster variable x in A ( Q ) has a numerical invariant g Q ( x ), called the g -vector of x [FZ07]. For each x ∈ cluster Q , one can define a cone C Q ( x ) := (cid:26)X x ∈ x a x g Q ( x ) | a x ∈ R ≥ (cid:27) in R n , called the g -vector cone of x . Note that these cones and their faces form a fan [Re14a, Theorem8.7]. We say that Q is finite type if cluster Q < ∞ . The following result is well-known. Theorem 1.1. [Re14a, Theorem 10.6]
If a quiver Q is finite type, then we have [ x ∈ cluster Q C Q ( x ) = R n . In this paper, we study an analogue of Theorem 1.1 for cluster algebras defined from marked surfacesthat were developed in [FG06, FG09, FST, FoT, GSV].Let (
S, M ) be a marked surface and T a tagged triangulation of ( S, M ) (see Subsection 2.1). Wedenote by | T | the number of tagged arcs of T . Fomin, Shapiro and Thurston [FST] constructed aquiver Q T associated with T . In A ( Q T ), cluster variables correspond to tagged arcs, and clusterscorrespond to tagged triangulations (Theorem 3.3). Our first aim is to give the following analogue ofTheorem 1.1. Theorem 1.2. If ( S, M ) is not a closed surface with exactly one puncture, then we have [ x ∈ cluster Q T C Q T ( x ) = R | T | , T. Yurikusa: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602 Japan
E-mail address : [email protected] . Key words and phrases. cluster algebra, marked surface, lamination, shear coordinate, cluster category, τ -tiltingtheory. G -VECTOR CONES FROM TRIANGULATED SURFACES where ( − ) is the closure with respect to the natural topology on R | T | . If ( S, M ) is a closed surface withexactly one puncture, then we have [ x ∈ cluster Q T C Q T ( x ) = [ x ∈ cluster Q op T C Q op T ( x ) = n ( a δ ) δ ∈ T ∈ R | T | | X δ ∈ T a δ ≥ o . The second aim of this paper is to apply Theorem 1.2 to representation theory. We considera non-degenerate potential W of Q T such that the associated Jacobian algebra J ( Q T , W ) is finitedimensional [DWZ]. Such a potential W exists (Proposition 4.5). The potential W and its Jacobianalgebra J = J ( Q T , W ) have been studied by a number of researchers (see e.g. [ABCP, CL, GLS, V]).We focus on the associated cluster category in this paper.Using the Ginzburg differential graded algebra Γ = Γ Q T ,W associated with ( Q T , W ) [G], Amiot [A]constructed a generalized cluster category C = C Q T ,W with cluster tilting object Γ. The g -vector ofeach rigid object (resp., τ -rigid pair) in C (resp., mod J ) is a certain element in the Grothendieck group K (add Γ) (resp., K (proj J )). The g -vectors of indecomposable direct summands of a cluster tiltingobject X (resp., a τ -tilting pair ( M, P )) form a cone C Γ ( X ) in K (add Γ) ⊗ Z R (resp., C J ( M, P ) in K (proj J ) ⊗ Z R ), called the g -vector cone of X (resp., ( M, P )). Note that these g -vector cones andtheir faces form a fan [DIJ]. Such a fan plays an important role in the study of scattering diagramsand their wall-chamber structures (see e.g. [B, BST, GHKK, GS, KS, Y18a]).We denote by c-tilt C (resp., s τ -tilt J ) the set of isomorphism classes of basic cluster tilting objects in C (resp., τ -tilting pairs in mod J ). We also denote by c-tilt + C (resp., c-tilt − C , s τ -tilt + J , s τ -tilt − J ) thesubset of c-tilt C (resp., c-tilt C , s τ -tilt J , s τ -tilt J ) consisting of mutation equivalence classes containingΓ (resp., Γ[1], ( J, , J )). We setc-tilt ± C := c-tilt + C ∪ c-tilt − C and s τ -tilt ± J := s τ -tilt + J ∪ s τ -tilt − J. The following analogues of Theorem 1.2 hold.
Theorem 1.3.
Let T be a tagged triangulation of a marked surface ( S, M ) . For a non-degeneratepotential W of Q T such that J = J ( Q T , W ) is finite dimensional, let C = C Q T ,W and Γ = Γ Q T ,W .Then we have the equalities [ U ∈ c-tilt ± C C Γ ( U ) = K (add Γ) ⊗ Z R and [ ( M,P ) ∈ s τ -tilt ± J C J ( M, P ) = K (proj J ) ⊗ Z R . This theorem means that g -vector cones are dense in the scattering diagram of J . It gives thefollowing application. Corollary 1.4.
Any basic cluster tilting object in C (resp., τ -tilting pair in mod J ) is contained in c-tilt ± C (resp., s τ -tilt ± J ). In particular, if ( S, M ) is not a closed surface with exactly one puncture,the exchange graph of c-tilt C (resp., s τ -tilt J ) is connected, thus c-tilt C = c-tilt + C = c-tilt − C (resp., s τ -tilt J = s τ -tilt + J = s τ -tilt − J ). Otherwise, it has precisely two connected components c-tilt + C and c-tilt − C (resp., s τ -tilt + J and s τ -tilt − J ). Notice that it was known by Plamondon [Pl13] and Ladkani [Lad13] that if (
S, M ) is a closed surfacewith exactly one puncture, then the exchange graph of c-tilt C is not connected. Also, it was knownby Qiu and Zhou [QZ] that if ( S, M ) has non-empty boundary, then the exchange graph of c-tilt C isconnected. Our proof is entirely different from theirs.To prove Theorem 1.2, our main ingredient is shear coordinates on ( S, M ). To study coefficientsin cluster algebras defined from T , Fomin and Thurston [FoT] used a certain class of curves in S ,called laminates, and finite multi-sets of pairwise non-intersecting laminates, called laminations (seealso [FG07, T]). To a laminate ℓ of ( S, M ), they associated an integer vector b T ( ℓ ) ∈ Z | T | whose entriesare shear coordinates of ℓ and defined b T ( L ) := P ℓ ∈ L b T ( ℓ ) ∈ Z | T | for a lamination L on ( S, M ). Theyshowed that the map L b T ( L ) induces a bijection between the set of laminations on ( S, M ) and Z | T | . Using this bijection, some properties of cluster algebras were given (see e.g. [MSW13, Re14b]).In this paper, we want to consider not only integer vectors but real vectors.For a multi-set L of laminates of ( S, M ), in the same way as g -vector cones, we can define a cone C T ( L ) in R | T | , called the shear coordinate cone of L with respect to T . Recall that there is a natural ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 3 injective map e from the set of tagged arcs of ( S, M ) to the set of laminates of (
S, M ) (see Subsection2.2). We denote by T the set of tagged triangulations of ( S, M ). The following result plays an importantrole to prove Theorem 1.2.
Theorem 1.5.
Let T be a tagged triangulation of a marked surface ( S, M ) . Then we have [ T ′ ∈ T C T ( e ( T ′ )) = R | T | . If ( S, M ) is a closed surface with exactly one puncture p , then we have [ T ′ ∈ T + C T ( e ( T ′ )) = [ T ′ ∈ T − ( − C T ( e ( T ′ ))) = n ( a δ ) δ ∈ T ∈ R | T | | X δ ∈ T a δ ≤ o , where T + (resp., T − ) is the set of tagged triangulations of ( S, M ) tagged at p in the same (resp.,different) way as T . It will be interesting to understand connections between our results and known results on Te-ichm¨uller spaces such as [FG11, Ro12, Ro13].This paper is organized as follows. In Section 2, we recall the notions of marked surfaces, laminations,and their shear coordinates. We study shear coordinates of laminates and their asymptotic behaviorunder Dehn twists, and prove Theorem 1.5. In Section 3, we recall cluster algebras defined fromtriangulated surfaces. We show that the shear coordinate of a laminate with respect to T correspondwith the g -vector of a cluster variable in A ( Q ) or A ( Q op ). Consequently, Theorem 1.2 follows fromTheorem 1.5. In Section 4, we recall τ -tilting theory, cluster tilting theory, and the relationshipsbetween them and cluster algebras. Finally, we prove Theorem 1.3 and Corollary 1.4. Acknowledgements . The author would like to thank his supervisor Osamu Iyama for his guidanceand helpful comments. He also thanks Daniel Labardini-Fragoso for valuable comments and the refereesfor fruitful suggestions. He is a Research Fellow of Society for the Promotion of Science (JSPS). Thiswork was supported by JSPS KAKENHI Grant Number JP17J04270.2.
Density of shear coordinate cones from triangulated surfaces
Marked surfaces and tagged triangulations.
We start with recalling the notions of [FST].Let S be a connected compact oriented Riemann surface with (possibly empty) boundary ∂S and M a non-empty finite set of marked points on S with at least one marked point on each boundarycomponent. We call the pair ( S, M ) a marked surface . Any marked point in the interior of S is calleda puncture . For technical reasons, we assume that ( S, M ) is neither a monogon with at most onepuncture, a digon without punctures, a triangle without punctures, nor a sphere with at most threepunctures.An arc γ of ( S, M ) is a curve in S with endpoints in M , considered up to isotopy, such that thefollowing conditions are satisfied: • γ does not intersect itself except at its endpoints; • γ is disjoint from M and ∂S except at its endpoints; • γ does not cut out an unpunctured monogon or an unpunctured digon.An arc with two identical endpoints is called a loop . Two arcs are called compatible if they don’tintersect in the interior of S . When we consider intersections of curves γ and δ , we assume that γ and δ intersect transversally in a minimum number of points. We denote by γ ∩ δ the set of theirintersection points. An ideal triangulation is a maximal collection of distinct pairwise compatible arcs.A triangle with only two distinct sides is called self-folded (see Figure 1). For an ideal triangulation T , a flip at an arc γ ∈ T replaces γ with another arc γ ′ / ∈ T such that ( T \ { γ } ) ∪ { γ ′ } is an idealtriangulation. Notice that an arc inside a self-folded triangle can not be flipped. To make flip alwayspossible, the notion of tagged arcs was introduced in [FST].A tagged arc δ of ( S, M ) is an arc whose each end is tagged in one of two ways, plain or notched ,such that the following conditions are satisfied: • δ does not cut out a monogon with exactly one puncture; • If an endpoint of δ lie on ∂S , then it is tagged plain; DENSITY OF G -VECTOR CONES FROM TRIANGULATED SURFACES p γ ′ oγ pι ( γ ′ ) o ⊲⊳ ι ( γ ) Figure 1.
A self-folded triangleand the corresponding tagged arcs δ ⊲⊳ ε δ ⊲⊳ ⊲⊳ ⊲⊳ ε Figure 2.
Pairs of conjugatearcs ( δ, ε ) • If δ is a loop, then the both ends are tagged in the same way.In the figures, we represent tagged arcs as follows:plain notched ⊲⊳ For an arc γ of ( S, M ), we define a tagged arc ι ( γ ) as follows: • If γ does not cut out a monogon with exactly one puncture, then ι ( γ ) is the tagged arc obtainedfrom γ by tagging both ends plain; • If γ is a loop at o ∈ M cutting out a monogon with exactly one puncture p , then there is aunique arc γ ′ that connects o and p and does not intersect γ . And then ι ( γ ) is the tagged arcobtained by tagging γ ′ plain at o and notched at p (see Figure 1).A pair of conjugate arcs is, for a self-folded triangle { γ, γ ′ } , ( ι ( γ ) , ι ( γ ′ )) or a pair obtained from( ι ( γ ) , ι ( γ ′ )) by simultaneous changing tags at each endpoint (see Figure 2).For a tagged arc δ , we denote by δ ◦ the arc obtained from δ by forgetting its tags. Two tagged arcs δ and ǫ are called compatible if the following conditions are satisfied: • The arcs δ ◦ and ǫ ◦ are compatible; • If δ ◦ = ǫ ◦ , then at least one end of ǫ is tagged in the same way as the corresponding end of δ ; • If δ ◦ = ǫ ◦ and they have a common endpoint o , then the ends of δ and ǫ at o are tagged in thesame way.A partial tagged triangulation is a collection of distinct pairwise compatible tagged arcs. If a partialtagged triangulation is maximal, then it is called a tagged triangulation . Recall that we denote by T the set of tagged triangulations of ( S, M ). We can define flips of tagged triangulations in the sameway as ones of ideal triangulations. In particular, any tagged arc can be flipped.
Theorem 2.1. [FST, Theorem 7.9, Proposition 7.10] If ( S, M ) is not a closed surface with exactlyone puncture, the exchange graph of T is connected, that is, any two tagged triangulations of ( S, M ) are connected by a sequence of flips. Otherwise, it has exactly two isomorphic components: one inwhich all ends of tagged arcs are plain and one in which they are notched. Laminations on marked surfaces.
We recall the notions of [FoT]. A laminate of (
S, M ) is anon-self-intersecting curve in S , considered up to isotopy relative to M , which is either • a closed curve, or • a curve whose ends are unmarked points on ∂S or spirals around punctures (either clockwiseor counterclockwise),and the following curves are not allowed (see Figure 3): • a curve cutting out a disk with at most one puncture; • a curve with two endpoints on ∂S such that it is isotopic to a piece of ∂S containing at mostone marked point; • a curve whose both ends are spirals around a common puncture in the same direction suchthat it does not enclose anything else. Definition 2.2.
We say that two laminates of (
S, M ) are compatible if they don’t intersect. A finitemulti-set of pairwise compatible laminates of (
S, M ) is called a lamination on (
S, M ) (see Figure 4).Let ℓ be a laminate of ( S, M ). For an ideal/tagged triangulation T of ( S, M ), we define the shearcoordinate b γ,T ( ℓ ) of ℓ with respect to γ ∈ T (see [FoT, Definition 12.2, 13.1]). ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 5 Figure 3.
Curves which are not laminates
Figure 4.
A lamination on anannulus with 2 puncturesFirst, we assume that T is an ideal triangulation. If γ ∈ T is not inside a self-folded triangle of T , then b γ,T ( ℓ ) is defined by a sum of contributions from all intersections of γ and ℓ as follows: Suchan intersection contributes +1 (resp., −
1) to b γ,T ( ℓ ) if a segment of ℓ cuts through the quadrilateralsurrounding γ as in the left (resp., right) diagram of Figure 5. Suppose that γ ∈ T is inside a self-+1 γ ℓ γℓ − Figure 5.
The contribution from a segment of the laminate ℓ on the left (resp., right)is +1 (resp., − { γ, γ ′ } of T , where γ ′ is a loop enclosing exactly one puncture p . Then we define b γ,T ( ℓ ) = b γ ′ ,T ( ℓ ( p ) ), where ℓ ( p ) is a laminate obtained from ℓ by changing the directions of its spiralsat p if they exist.Next, we assume that T is a tagged triangulation. If there is an ideal triangulation T satisfying T = ι ( T ), then we define b γ,T ( ℓ ) = b γ ,T ( ℓ ), where γ = ι ( γ ). For an arbitrary T , we can obtain a taggedtriangulation T ( p ··· p m ) from T by simultaneous changing all tags at punctures p , . . . , p m (possibly m = 0), in such a way that there is a unique ideal triangulation T satisfying T ( p ··· p m ) = ι ( T )(see [MSW11, Remark 3.11]). Then we define b γ,T ( ℓ ) = b γ ( p ··· pm ) ,T ( p ··· pm ) (cid:0) ( · · · (( ℓ ( p ) ) ( p ) ) · · · ) ( p m ) (cid:1) ,where γ ( p ··· p m ) corresponds to γ .For a multi-set L = L ′ ⊔ { ℓ } of laminates of ( S, M ), the shear coordinate b γ,T ( L ) of L with respectto γ ∈ T is inductively defined by b γ,T ( L ) = b γ,T ( L ′ ) + b γ,T ( ℓ ) . We denote by b T ( L ) a vector ( b γ,T ( L )) γ ∈ T ∈ Z | T | . Note that the shear coordinate cone C T ( L ) is acone spanned by b T ( ℓ ) for ℓ ∈ L . These vectors have the following property. Theorem 2.3. [FoT, Theorems 12.3, 13.6]
Let T be a tagged triangulation of ( S, M ) . The map sendinglaminations L to b T ( L ) induces a bijection between the set of laminations on ( S, M ) and Z | T | . Example 2.4.
For a digon (
S, M ) with exactly one puncture, all laminates are given as follows: ℓ ℓ ℓ ℓ ℓ ℓ . DENSITY OF G -VECTOR CONES FROM TRIANGULATED SURFACES We consider the following tagged triangulation T : T = 1 ⊲⊳ p , where T = 1 . The shear coordinate b ,T ( ℓ ) is given by b ,T ( ℓ ) = −
1. Since ℓ ( p )3 = ℓ , we have the equalities b ,T ( ℓ ) = b ,T ( ℓ ) = b ,T ( ℓ ( p )3 ) = b ,T ( ℓ ) = − . Similarly, for i ∈ { , } and j ∈ { , . . . , } , the shear coordinates b i,T ( ℓ j ) and b T ( ℓ j ) are given asfollows: i \ j 1 2 3 4 5 61 0 -1 -1 0 1 12 -1 -1 0 1 1 0 b T ( ℓ ) b T ( ℓ ) b T ( ℓ ) b T ( ℓ ) b T ( ℓ ) b T ( ℓ )In particular, we have [ j =1 C T ( { ℓ j , ℓ j +1 } ) = R , where ℓ = ℓ . On the other hand, all laminations on ( S, M ) are given by { mℓ j , nℓ j +1 } for j ∈{ , . . . , } and m, n ∈ Z ≥ . Since C T ( { ℓ j − , ℓ j } ) ∩ C T ( { ℓ j , ℓ j +1 } ) = C T ( { ℓ j } ) and b T induces abijection b T : {{ mℓ j , nℓ j +1 } | m, n ∈ Z ≥ } ←→ C T ( { ℓ j , ℓ j +1 } ) ∩ Z , there is a bijection between the set of laminations on ( S, M ) and Z .2.3. Elementary and exceptional laminates.
Non-closed laminates of (
S, M ) are divided into twotypes, elementary and exceptional. For a tagged arc δ of ( S, M ), we define an elementary laminate e ( δ ) as follows: • e ( δ ) is a laminate running along δ in a small neighborhood of it; • If δ has an endpoint o on a component C of ∂S , then the corresponding endpoint of e ( δ ) islocated near o on C in the clockwise direction as in the left diagram of Figure 6; • If δ has an endpoint at a puncture p , then the corresponding end of e ( δ ) is a spiral around p clockwise (resp., counterclockwise) if δ is tagged plain (resp., notched) at p as in the rightdiagram of Figure 6. δ e ( δ ) δ ⊲⊳ e ( δ ) Figure 6.
Elementary laminates of tagged arcsIt follows from the construction that the map e from the set of tagged arcs of ( S, M ) to the set oflaminates is injective. For an elementary laminate ℓ , we denote by e − ( ℓ ) a unique tagged arc δ suchthat e ( δ ) = ℓ . Note that, for a tagged arc δ , a lamination { e ( δ ) } is a reflection of the elementarylamination of δ defined in [FoT, Definition 17.2]. Our convention is more convenient for our aim.Elementary laminates have the following properties. ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 7 Proposition 2.5. (1) Let δ and δ ′ be tagged arcs such that δ ◦ = δ ′◦ . Then δ and δ ′ are compatible ifand only if e ( δ ) and e ( δ ′ ) are compatible.(2) The map e induces a bijection between the set of partial tagged triangulations of ( S, M ) withoutpairs of conjugate arcs and the set of laminations of ( S, M ) consisting only of distinct elementarylaminates.Proof. (1) Since e transforms δ and δ ′ just around the marked points of ( S, M ), it is enough to considerneighborhoods of their endpoints. In particular, if δ and δ ′ have no common endpoints, the assertionholds. Suppose that δ and δ ′ have at least one common endpoint. Since δ ◦ = δ ′◦ , ( δ, δ ′ ) is not a pairof conjugate arcs. Thus δ and δ ′ are compatible if and only if the ends of δ and δ ′ at each commonendpoint are tagged in the same way. By the definition of e , it is equivalent that e ( δ ) and e ( δ ′ ) arecompatible.(2) If two distinct tagged arcs δ and δ ′ satisfying δ ◦ = δ ′◦ are compatible, then ( δ, δ ′ ) is a pair ofconjugate arcs, in which case e ( δ ) and e ( δ ′ ) are not compatible. Therefore, the assertion follows from(1). (cid:3) Laminates which are neither closed nor elementary are called exceptional . They are characterizedas follows.
Proposition 2.6.
A laminate is exceptional if and only if it is one of the following curves (Figure 7): • a curve enclosing exactly one puncture whose both endpoints lie on a common boundary seg-ment; • a curve enclosing exactly one puncture whose both ends are spirals around a common puncturein the same direction.Proof. Applying the same transformation as e − to a non-closed laminate ℓ and forgetting its tags, weobtain a unique ideal arc. In general, an ideal arc γ is not obtained from a tagged arc by forgettingits tags if and only if γ is a loop cutting out a monogon with exactly one puncture. Therefore, ℓ is notelementary if and only if it is one of the desired cases. (cid:3) Figure 7.
Exceptional laminatesNote that exceptional laminates coincide with excluded curves for quasi-laminations in [Re14b]. Tointerpret shear coordinates of exceptional laminates as ones of elementary laminates, we introduce thefollowing notations. For an exceptional laminate ℓ of ( S, M ), elementary laminates ℓ p and ℓ q are givenby ℓ → ℓ p ℓ q , where = or or . In particular, ( e − ( ℓ p ) , e − ( ℓ q )) is a pair of conjugate arcs. For a lamination L on ( S, M ), we denoteby L pq the multi-set of elementary laminates obtained from L by replacing exceptional laminates ℓ ∈ L with ℓ p and ℓ q . Example 2.7.
In Example 2.4, ℓ and ℓ are exceptional, and ( ℓ ) p = ℓ , ( ℓ ) q = ℓ , ( ℓ ) p = ℓ and( ℓ ) q = ℓ . Thus we have the equalities b T ( ℓ ) = b T ( { ( ℓ ) p , ( ℓ ) q } ) and b T ( ℓ ) = b T ( { ( ℓ ) p , ( ℓ ) q } ) . In general, the same property as Example 2.7 holds for arbitrary exceptional laminates.
Lemma 2.8.
Let T be a tagged triangulation of ( S, M ) . For an exceptional laminate ℓ of ( S, M ) , wehave b T ( ℓ ) = b T ( { ℓ p , ℓ q } ) . DENSITY OF G -VECTOR CONES FROM TRIANGULATED SURFACES Proof.
By Proposition 2.6, there is a unique puncture p enclosed by ℓ . We only need to prove(2.1) b δ,T ( ℓ ) = b δ,T ( { ℓ p , ℓ q } )for any δ ∈ T . If δ ∈ T is not incident to p , then (2.1) is clear. We assume that δ is incident to p .Let δ , . . . , δ m be tagged arcs of T incident to p winding clockwisely around p such that the followingconditions are satisfied (see Figure 8): • ℓ crosses them at points p , . . . , p m in this order; • The segment of δ i from p to p i , that of δ i +1 from p to p i +1 , and that of ℓ from p i to p i +1 forma contractible triangle.Note that if these arcs contains a pair ( δ, ǫ ) of conjugate arcs, then we can choice the order of δ and ǫ . ℓp δ p m δ m p δ p Figure 8.
Local configuration around a puncture p Moreover, δ is different from δ m . Indeed, if δ = δ m , considering triangles with a side δ , there is atagged arc of T incident to p such that ℓ crosses it before p or after p m , a contradiction.The contributions to b δ i ,T ( ℓ ) at δ i ∩ ℓ except at p i coincide with the contributions to b δ i ,T ( ℓ p ) and b δ i ,T ( ℓ q ) at them. We denote by c i ( ℓ ) (resp, c i ( ℓ p ), c i ( ℓ q )) the contribution to b δ i ,T ( ℓ ) (resp., b δ i ,T ( ℓ p ), b δ i ,T ( ℓ q )) at p i for i ∈ { , . . . , m } . To prove (2.1), we only need to show(2.2) c i ( ℓ ) = c i ( ℓ p ) + c i ( ℓ q ) . First, we assume that neither ( δ , δ ) nor ( δ m − , δ m ) form a pair of conjugate arcs. Then it is easyto give the following values: c i ( ℓ ) = − i = 1 , i = m, i = 1 , m, c i ( ℓ p ) = (cid:26) − i = 1 , i = 1 , c i ( ℓ q ) = (cid:26) i = m, i = m. Therefore, (2.2) holds.Second, we assume that ( δ , δ ) is a pair of conjugate arcs tagged in the different ways at p , in whichcase m = 2. Then by exchanging δ and δ if necessary (see Figure 9), we have(2.3) c i ( ℓ p ) = (cid:26) c ( ℓ ) if i = 1 , i = 2 , c i ( ℓ q ) = (cid:26) i = 1 ,c ( ℓ ) if i = 2 . Therefore, (2.2) holds. ⊲⊳ δ δ ℓp δ ⊲⊳ δ ℓp Figure 9.
Pairs ( δ , δ ) of conjugate arcs tagged in the different ways at p such that(2.3) holdsFinally, we assume that ( δ , δ ) is a pair of conjugate arcs tagged in the same way at p . We define aset M as follows: If ( δ m − , δ m ) is a pair of conjugate arcs, then M = { m − , m } ; Otherwise, M = { m } .Then we have c i ( ℓ p ) = 0 = c j ( ℓ q ) for i / ∈ { , } and j / ∈ M , and c i ( ℓ ) = c i ( ℓ p ) if i ∈ { , } ,c i ( ℓ q ) if i ∈ M , . ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 9 Therefore, (2.2) holds. Moreover, it follows from the symmetry for the case that ( δ m − , δ m ) is a pairof conjugate arcs tagged in the same way at p . Consequently, (2.1) holds for any δ ∈ T . (cid:3) For a lamination L on ( S, M ), we have decompositions(2.4) L = L el ⊔ L ex ⊔ L cl = L nc ⊔ L cl , where L el (resp., L ex , L cl ) consists of all elementary (resp., exceptional, closed) laminates in L . Formulti-sets L and L ′ of laminates of ( S, M ), we define non-multi-sets e − ( L ) := { δ : a tagged arc | e ( δ ) ∈ L } and L \ L ′ := { ℓ ∈ L | ℓ / ∈ L ′ } . The following properties are used to prove Theorem 1.5 in Subsection 2.5.
Proposition 2.9.
Let L be a lamination on ( S, M ) with L cl = ∅ . Then the following properties hold: (1) C T ( L ) ⊆ C T ( L pq ) . (2) e − ( L pq ) is a partial tagged triangulation of ( S, M ) .Moreover, we take a set U of tagged arcs of ( S, M ) such that T ′ = e − ( L pq ) ⊔ U is a tagged triangulation.Then we have the equality (3) C T ( e T ′ ) = C T ( L pq ⊔ e U ) .Proof. (1) The assertion immediately follows from Lemma 2.8.(2) By Proposition 2.5(2), e − ( L el ) is a partial tagged triangulation of ( S, M ). Since L is a lamina-tion, any laminate in ( L ex ) pq is compatible with all laminates in L el \ ( L ex ) pq . Then, by Proposition2.5(1), any tagged arc of e − (( L ex ) pq ) is compatible with all tagged arcs of e − ( L el \ ( L ex ) pq ). More-over, e − (( L ex ) pq ) is a partial tagged triangulation since ( e − ( ℓ p ) , e − ( ℓ q )) is a pair of conjugate arcsfor ℓ ∈ L ex . Therefore, e − ( L pq ) = e − ( L el \ ( L ex ) pq ) ⊔ e − (( L ex ) pq )is a partial tagged triangulation of ( S, M ).(3) Since L pq coincides with the multiplicity of e e − ( L pq ), we have the equalities C T ( L pq ⊔ e U ) = C T ( e e − ( L pq ) ⊔ e U ) = C T ( e T ′ ) . (cid:3) .2.4. Shear coordinates and Dehn twists.
We consider the Dehn twist along a closed laminate andits effect on shear coordinates. In this subsection, we fix an ideal or tagged triangulation T , a closedlaminate ℓ c of ( S, M ) and its direction. We denote by T ℓ c the Dehn twist of ( S, M ) along ℓ c definedfrom the direction of ℓ c as follows: >> ℓ c T ℓc −−→ The aim of this subsection is to prove the following.
Theorem 2.10.
Let ℓ c be a closed laminate and ℓ a laminate of ( S, M ) intersecting with ℓ c , and let δ ∈ T . Then there is m ′ ∈ Z ≥ such that for any m ≥ m ′ , we have b δ,T ( T mℓ c ( ℓ )) = b δ,T ( T m ′ ℓ c ( ℓ )) + ( m − m ′ ) ℓ ∩ ℓ c ) b δ,T ( ℓ c ) . First, we assume that (
S, M ) is an annulus without punctures and T is its ideal triangulationconsisting of arcs τ , . . . , τ r crossing ℓ c in order of occurrence along ℓ c (we can have τ i = τ j even if i = j ), that is, T = τ ≫ τ ≫ τ τ r · · · ℓ c > > , G -VECTOR CONES FROM TRIANGULATED SURFACES where two vertical lines τ are identified. Any elementary laminate ℓ of ( S, M ) intersects with ℓ c atmost once since they intersect in a minimal number of points. We assume that ℓ intersects with ℓ c .We define the direction of ℓ as crossing ℓ c from left to right: T = ≫≫ ℓ c > > > ℓ Let s ∈ { , . . . , r } such that the starting point of ℓ is on the triangle of T with sides τ s − and τ s , where τ r + i = τ i . In particular, ℓ intersects at least one of τ s − and τ s . Thus ℓ intersects with the t ℓ ( ∈ Z ≥ )diagonals either τ s , τ s +1 , . . . , τ s + t ℓ − or τ s − , τ s − , . . . , τ s − t ℓ of T in order. In the former (resp., latter)case, we say that ℓ intersects with T in ascending (resp., descending ) order : τ s − τ s >ℓ ascendingorder τ s − τ s < ℓ descendingorder Proposition 2.11.
Let ( S, M ) be an annulus without punctures and ℓ an elementary laminate of ( S, M ) intersecting with ℓ c . (1) If ℓ intersects with T in ascending order, then so is T ℓ c ( ℓ ) and for δ ∈ T , we have b δ,T ( T ℓ c ( ℓ )) = b δ,T ( ℓ ) + b δ,T ( ℓ c ) . If ℓ and T ℓ c ( ℓ ) intersect with T in descending order, then for δ ∈ T , we have b δ,T ( T ℓ c ( ℓ )) = b δ,T ( ℓ ) − b δ,T ( ℓ c ) . (2) There is m ≫ such that T mℓ c ( ℓ ) intersects with T in ascending order.Proof. (1) We only prove the first assertion since the proof of the second assertion is similar. Supposethat ℓ intersects with T in ascending order. If t ℓ >
1, then the assertion holds since T ℓ c only transforms ℓ around an intersection point of ℓ and ℓ c as follows: > ℓℓ c >τ s τ s +1 τ s − τ s +2 T ℓc −−→ > T ℓ c ( ℓ ) > > τ s τ s +1 τ s − τ s +2 If t ℓ = 1, then ℓ and T ℓ c ( ℓ ) are given as follows: τ s − τ s τ s +1 > ℓ T ℓc −−→ τ s − τ s τ s +1 >ℓ> Then the assertion is directly given by enumerating their shear coordinates.(2) Suppose that ℓ intersects with T in descending order. If t ℓ < r , then T ℓ c ( ℓ ) intersects with T inascending order. If t ℓ ≥ r , then T ℓ c ( ℓ ) intersects with T in descending order and t T ℓc ( ℓ ) = t ℓ − r . Bythe induction, the assertion holds. (cid:3) Next, we consider an arbitrary marked surface (
S, M ) and its ideal triangulation T . For ℓ c inTheorem 2.10, we construct an annulus ( S ℓ c , M ℓ c ) and its triangulation T ℓ c as follows: Let τ , . . . , τ r be the arcs of T crossing ℓ c in order of occurrence along ℓ c ( τ i and τ j can be the same even if i = j ).Hence ℓ c crosses r triangles △ , . . . , △ r in this order. For i ∈ { , . . . , r } , let △ ′ i be a copy of thetriangle △ i , hence △ ′ i has the sides τ i and τ i +1 . Then an annulus ( S ℓ c , M ℓ c ) and its triangulation T ℓ c ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 11 are obtained by gluing △ ′ , . . . , △ ′ r along the edges τ i , that is,(2.5) T ℓ c = τ ≫ τ ≫ τ τ r · · ·△ ′ △ ′ r ℓ c > > in ( S ℓ c , M ℓ c ), where two vertical lines τ are identified. In particular, if τ i is inside a self-folded triangleof T , then the corresponding triangles are given by p τ i τ i − = τ i +1 ℓ c > in T −→ τ i − τ i +1 τ i ℓ c > in T ℓ c . For a laminate ℓ of ( S, M ) intersecting with ℓ c , let q ∈ ℓ ∩ ℓ c and ℓ q a laminate of ( S ℓ c , M ℓ c ) corre-sponding to the connected segment of ℓ in T ℓ c containing q as follows: ≫≫ ℓ c >ℓq q ′ in T −→ ≫≫ ℓ c >ℓ q ℓ q ′ q q ′ in T ℓ c Proposition 2.12.
Let T be an ideal triangulation of ( S, M ) , γ ∈ T , and ℓ a laminate of ( S, M ) intersecting with ℓ c . We assume that for all q ∈ ℓ ∩ ℓ c , ℓ q intersects with T ℓ c in ascending order. Thenwe have b γ,T ( T ℓ c ( ℓ )) = b γ,T ( ℓ ) + ℓ ∩ ℓ c ) b γ,T ( ℓ c ) . Proof.
The proof is divided into the following three cases (1)–(3).(1) If γ is not a side of triangles of T , then the assertion is clear.(2) We assume that γ is a side of some triangle of T and γ ∩ ℓ c = ∅ . If γ is not inside a self-foldedtriangle of T , then the construction of T ℓ c preserves the quadrilateral surrounding γ . Therefore, wehave b γ,T ( T ℓ c ( ℓ )) − b γ,T ( ℓ ) = X q ∈ ℓ ∩ ℓ c X ≤ i ≤ r,τ i = γ (cid:16) b τ i ,T ℓc ( T ℓ c ( ℓ q )) − b τ i ,T ℓc ( ℓ q ) (cid:17) . (1) = X q ∈ ℓ ∩ ℓ c X ≤ i ≤ r,τ i = γ b τ i ,T ℓc ( ℓ c ) = ℓ ∩ ℓ c ) b γ,T ( ℓ c ) . Suppose that γ is inside a self-folded triangle { γ, γ ′ } of T enclosing a puncture p . Recall that b γ,T ( ℓ ) = b γ ′ ,T ( ℓ ( p ) ), where ℓ ( p ) is a laminate obtained from ℓ by changing the directions of its spiralsat p if they exist (see Subsection 2.2). Thus the assertion follows from the previous case if for all q ∈ ℓ ( p ) ∩ ℓ c , ( ℓ ( p ) ) q intersects with T ℓ c in ascending order. This is checked as follows: If no ends of ℓ are spirals at p , then ℓ ( p ) = ℓ , hence it is clear. Otherwise, ℓ = ℓ ( p ) , and ℓ ∩ ℓ c and ℓ ( p ) ∩ ℓ c areidentified in the natural way. Then ℓ q and ( ℓ ( p ) ) q are only different in that their ends around p aregiven by γ ℓ ′ p and γ ℓ ′′ p in T ℓ c , where { ℓ ′ , ℓ ′′ } = { ℓ q , ( ℓ ( p ) ) q } .Therefore, ( ℓ ( p ) ) q also intersects with T ℓ c in ascending order.(3) We assume that γ is a side of some triangle △ i of T and γ ∩ ℓ c = ∅ . Then we prove b γ,T ( T ℓ c ( ℓ )) = b γ,T ( ℓ ). In this case, γ is not inside a self-folded triangle of T . Indeed, if γ is inside a self-folded triangle G -VECTOR CONES FROM TRIANGULATED SURFACES of T , then γ is either τ i or τ i +1 , hence it is a contradiction. Therefore, there is the quadrilateralsurrounding γ of T . Since for all q ∈ ℓ ∩ ℓ c , ℓ q intersects with T ℓ c in ascending order, the Dehn twist T ℓ c affects ℓ ∩ as follows: ℓ ∩ = τ i γ τ i +1 ℓ c > T ℓc −−→ T ℓ c ( ℓ ) ∩ = τ i γ τ i +1 ℓ c > Therefore, it gives b γ,T ( T ℓ c ( ℓ )) = b γ,T ( ℓ ). (cid:3) We are ready to prove Theorem 2.10.
Proof of Theorem 2.10.
First of all, we prove Theorem 2.10 for an ideal triangulation T . For q ∈ ℓ ∩ ℓ c ,by Proposition 2.11(2) there exists m q ∈ Z ≥ such that T m q ℓ c ( ℓ q ) intersects with T ℓ c in ascending order.Thus for m ≥ m ′ := max q ∈ ℓ ∩ ℓ c { m q } , T mℓ c ( ℓ q ) intersects with T ℓ c in ascending order for each q ∈ ℓ ∩ ℓ c . Therefore, by Theorem 2.12, we have b γ,T ( T m +1 ℓ c ( ℓ )) = b γ,T ( T mℓ c ( ℓ )) + ℓ ∩ ℓ c ) b γ,T ( ℓ c ) = · · · = b γ,T ( T m ′ ℓ c ( ℓ )) + ( m + 1 − m ′ ) ℓ ∩ ℓ c ) b γ,T ( ℓ c ) . For an arbitrary tagged triangulation T , we recall that there is a unique ideal triangulation T satisfying T ′ = ι ( T ), where T ′ is obtained from T by simultaneous changing all tags at some puncturesif necessary (see Subsection 2.2 for details). Then the shear coordinate b γ,T ( ℓ ) of a laminate ℓ is equalto b γ ,T ( ℓ ′ ), where ℓ ′ is a laminate obtained from ℓ by changing the directions of its spirals at thesepunctures if they exist. Since the change of directions of its spirals and the Dehn twist T ℓ c arecompatible, the proof of Theorem 2.10 comes down to the case of ideal triangulations. (cid:3) Proof of Theorem 1.5.
In this subsection, we fix a tagged triangulation T of ( S, M ). ByTheorem 2.3, to prove the first assertion of Theorem 1.5, we only need to show that for each lamination L on ( S, M ),(2.6) b T ( L ) ∈ [ T ′ ∈ T C T ( e ( T ′ )) . To prove (2.6), we need some preparation. We have decompositions (2.4) of L . By Proposition 2.9(2), e − (( L nc ) pq ) is a partial tagged triangulation of ( S, M ). Then we take a set U of tagged arcs of ( S, M )such that T L := e − (( L nc ) pq ) ⊔ U is a tagged triangulation. Lemma 2.13.
Let L be a lamination on ( S, M ) . (1) Any closed laminate ℓ in L cl does not intersect with tagged arcs of e − (( L nc ) pq ) , but it intersectswith at least one tagged arc of U . (2) C T ( e T ℓ ( T L )) = C T (( L nc ) pq ⊔ e T ℓ ( U )) .Proof. (1) Since L is a lamination, any closed laminate ℓ in L cl does not intersect with all laminatesin L nc . Thus ℓ does not intersect with all tagged arcs of e − (( L nc ) pq ) since e − and ( − ) pq transformlaminates just around the marked points of ( S, M ). Moreover, since T L is a tagged triangulation and ℓ is not contractible, ℓ intersects with at least one tagged arc of T L , hence it is of U .(2) By (1), we have T ℓ ( T L ) = e − (( L nc ) pq ) ⊔ T ℓ ( U ) and it is a tagged triangulation. The desiredequality is given by Proposition 2.9(3). (cid:3) Let ℓ , . . . , ℓ t be all distinct closed laminates in L cl and n i the multiplicity of ℓ i in L cl for i ∈{ , . . . , t } . By Lemma 2.13(1), N i := P ǫ ∈ U ℓ i ∩ ǫ ) is not zero. In particular, N i is equal to ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 13 P ǫ ∈ U ℓ i ∩ e ( ǫ )). We fix the direction of ℓ i for 1 ≤ i ≤ t and consider the Dehn twists T ℓ i . Since ℓ i are not intersect, T ℓ i are commutative. We set T := t Y i =1 T N ··· NtNi n i ℓ i . Proposition 2.14.
Let L be a lamination on ( S, M ) . Then we have b T ( L ) ∈ [ m ≥ C T ( e T m ( T L )) . Proof.
By Theorem 2.10, for m T ≫ m ≥ m T , we have b T ( e T m ( U )) = b T ( e T m T ( U )) + t X i =1 (cid:16) N · · · N t N i n i (cid:17) ( m − m T ) N i b T ( ℓ i )= b T ( e T m T ( U )) + ( m − m T ) N · · · N t t X i =1 n i b T ( ℓ i )= b T ( e T m T ( U )) + ( m − m T ) N · · · N t b T ( L cl ) . This equality gives lim m →∞ b T ( e T m ( U )) m − m T = N · · · N t b T ( L cl ) , thus b T ( L cl ) ∈ [ m ≥ C T ( e T m ( U )) . Since C T ( L nc ) ⊆ C T (( L nc ) pq ) by Proposition 2.9(1), we have b T ( L ) = b T ( L nc ) + b T ( L cl ) ∈ C T ( L nc ) + [ m ≥ C T ( e T m ( U )) ⊆ [ m ≥ C T (( L nc ) pq ⊔ e T m ( U )) = [ m ≥ C T ( e T m ( T L )) , where the last equality is given by Lemma 2.13(2). (cid:3) Proof of the first assertion of Theorem 1.5.
Since T m ( T L ) is a tagged triangulation of ( S, M ) for any m ∈ Z ≥ , Proposition 2.14 finishes the proof of (2.6). Hence the assertion holds. (cid:3) To prove the second assertion of Theorem 1.5, we give the following results in a more general setting.
Proposition 2.15.
Let T be an ideal triangulation of ( S, M ) without self-folded triangles. (1) For a laminate ℓ , we have X γ ∈ T b γ,T ( ℓ ) ∈ { , ± } . (2) For a tagged arc ǫ of ( S, M ) whose both endpoints are punctures, we have X γ ∈ T b γ,T ( e ( ǫ )) = − if both ends of ǫ are tagged plain , if both ends of ǫ are tagged notched , otherwise . Proof. (1) Fix a direction of ℓ . For a closed laminate ℓ , let T ℓ be in (2.5). For a non-closed laminate ℓ ,we define a polygon ( S ℓ , M ℓ ) and its ideal triangulation T ℓ consisting of triangles △ ′ , △ ′ , . . . , △ ′ r and τ i = △ ′ i − ∩ △ ′ i in the same way, where we need a slight modification at the spirals. If the starting(resp., ending) end of ℓ is a spiral around a puncture p , then the triangles of T incident to p are △ , . . . , △ s for 1 < s < r (resp., △ t , . . . , △ r for 1 < t < r ) as follows: · · · ℓ> > △ △ s − △ s △ r △ t +1 △ t in T −→ T ℓ = τ τ r · · ·△ ′ △ ′ r ℓ> > G -VECTOR CONES FROM TRIANGULATED SURFACES Let ℓ be an arbitrary laminate of ( S, M ) and we consider the ideal triangulation T ℓ consisting oftriangles △ ′ , . . . , △ ′ r ( △ ′ = △ ′ r if ℓ is closed). We call △ ′ i a left (resp., right ) triangle if a side of △ ′ i isa boundary segment of T ℓ on the left (resp., right) side of ℓ . Then we have b τ i ,T ℓ ( ℓ ) = △ ′ i − is a left triangle and △ ′ i is a right triangle , − △ ′ i − is a right triangle and △ ′ i is a left triangle , . Therefore, we have(2.7) r X k =1 b τ k ,T ℓ ( ℓ ) = △ ′ is a left triangle and △ ′ r is a right triangle , − △ ′ is a right triangle and △ ′ r is a left triangle , . On the other hand, since T has no self-folded triangles, we have X δ ∈ T b δ,T ( ℓ ) = r X k =1 b τ k ,T ℓ ( ℓ ) . Thus the assertion follows from (2.7).(2) We consider T e ( ǫ ) as above and define the direction of ǫ from ℓ by the obvious way. If the startingpoint of ǫ is tagged plain (resp., notched), then △ ′ is a right (resp., left) triangle. If the ending pointof ǫ is tagged plain (resp., notched), then △ ′ is a left (resp., right) triangle. Thus the assertion alsofollows from (2.7). (cid:3) Proof of the second assertion of Theorem 1.5.
Let (
S, M ) be a closed surface with exactly one punc-ture p and T its tagged triangulation. In this case, all ends of tagged arcs of T are tagged plain or theyare tagged notched. Thus we can assume that T is an ideal triangulation without self-folded triangles.Let ǫ and ǫ ′ be tagged arcs of ( S, M ) tagged plain and notched, respectively. We only need to showthat b T ( e ( ǫ )) ∈ n ( a γ ) γ ∈ T ∈ R | T | | X γ ∈ T a γ ≤ o and b T ( e ( ǫ ′ )) ∈ n ( a γ ) γ ∈ T ∈ R | T | | X γ ∈ T a γ ≥ o , that is, X γ ∈ T b γ,T ( e ( ǫ )) ≤ X γ ∈ T b γ,T ( e ( ǫ ′ )) ≥ . It immediately follows from Proposition 2.15(2). (cid:3)
Example of Proposition 2.14.
For an annulus (
S, M ) with exactly two marked points, alllaminates are given as follows: ℓ , · · · ℓ − ℓ − ℓ ℓ ℓ · · · , where ℓ is closed and ℓ m = T mℓ ( ℓ ) is elementary for m ∈ Z . Their shear coordinates with respect to T = 21are given by ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 15 b T ( ℓ ) = ( b ,T ( ℓ ) , b ,T ( ℓ )) = (1 , − ,b T ( ℓ m ) = (cid:26) ( m − , − m ) if m ≥ , ( − m − , m + 2) if m < . b T ( ℓ − ) b T ( ℓ ) b T ( ℓ − ) b T ( ℓ ) b T ( ℓ )... · · · For a tagged triangulation T ′ of ( S, M ), the shear coordinate cone C T ( e ( T ′ )) is given by { αb T ( ℓ j ) + βb T ( ℓ j +1 ) | α, β ∈ Z ≥ } for some j ∈ Z . Then the set of integer vectors which are not contained inthese shear coordinate cones is { b T ( { kℓ } ) | k ∈ Z > } . Taking L = { kℓ } and T L = T , Proposition 2.14means that b T ( kℓ ) = ( k, − k ) ∈ [ m ≥ C T ( e T m ( T )) = [ m ≥ C T ( { ℓ m , ℓ m +1 } ) . It is described in the above picture. 3.
Cluster algebras
Cluster algebras and triangulated surfaces.
We briefly recall cluster algebras with principalcoefficients [FZ07]. For that, we need to prepare some notations. Let n ∈ Z ≥ and F := Q ( t , . . . , t n )be the field of rational functions in 2 n variables over Q . Definition 3.1. (1) A seed with coefficients is a pair ( x , Q ) consisting of the following data:(a) x = ( x , . . . , x n , y , . . . , y n ) is a free generating set of F over Q .(b) Q is a quiver without loops and 2-cycles whose vertices are { , . . . , n } .Then we refer to x as the cluster , to each x i as a cluster variable and y i as a coefficient .(2) For a seed ( x , Q ) with coefficients, the mutation µ k ( x , Q ) = ( x ′ , Q ′ ) in direction k (1 ≤ k ≤ n )is defined as follows:(a) x ′ = ( x ′ , . . . , x ′ n , y , . . . , y n ) is defined by x k x ′ k = Y ( j → k ) in Q x j Y ( j → k ) in Q y j − n + Y ( j ← k ) in Q x j Y ( j ← k ) in Q y j − n and x ′ i = x i if i = k, where x n +1 = · · · = x n = 1 = y − n = · · · = y .(b) Q ′ is the quiver obtained from Q by the following steps:(i) For any path i → k → j , add an arrow i → j .(ii) Reverse all arrows incident to k .(iii) Remove a maximal set of disjoint 2-cycles.We remark that µ k is an involution, that is, we have µ k µ k ( x , Q ) = ( x , Q ). Moreover, it is elementarythat µ k ( x , Q ) is also a seed with coefficients.For a quiver Q without loops and 2-cycles whose vertices are { , . . . , n } . The framed quiver associ-ated with Q is the quiver ˆ Q obtained from Q by adding vertices { ′ , . . . , n ′ } and arrows { i → i ′ | ≤ i ≤ n } . We fix a seed ( x = ( x , . . . , x n , y , . . . , y n ) , ˆ Q ) with coefficients, called the initial seed . We alsocall each x i the initial cluster variable . Definition 3.2.
The cluster algebra A ( Q ) = A ( x , ˆ Q ) with principal coefficients for the initial seed( x , ˆ Q ) is a Z -subalgebra of F generated by the cluster variables and the coefficients obtained by allsequences of mutations from ( x , ˆ Q ).One of the remarkable properties of cluster algebras with principal coefficients is the strongly Laurentphenomenon [FZ07, Proposition 3.6], that is, A ( Q ) ⊆ Z [ x ± , . . . , x ± n , y , . . . , y n ]. We consider the Z n -grading in Z [ x ± , . . . , x ± n , y , . . . , y n ] given bydeg( x i ) = e i , deg( y j ) = ( { i → j in Q } − { i ← j in Q } ) ≤ i ≤ n , G -VECTOR CONES FROM TRIANGULATED SURFACES where e , . . . , e n are the standard basis vectors in Z n . Every cluster variable x of A ( Q ) is homogeneouswith respect to the Z n -grading, and its degree g Q ( x ) is called g -vector of x [FZ07, Proposition 6.1].We denote by cluster Q the set of clusters in A ( Q ) and by cl-var Q the set of cluster variables in A ( Q ).Let T be a tagged triangulation of ( S, M ). Fomin, Shapiro and Thurston [FST] constructed a quiver Q T without loops and 2-cycles as follows: Any tagged triangulation is obtained by gluing together anumber of puzzle pieces in Table 1 and by simultaneous changing all tags at some punctures (see [FST,Remark 4.2] for details). The vertices of Q T are arcs of T and its arrows are obtained as in Table 1 forpuzzle pieces of T , where we remove arrows incident to ∂S . Thus we have the cluster algebra A ( Q T )Puzzlepieces δ δ δ δ δ δ ⊲⊳ δ δ δ ⊲⊳ δ δ ⊲⊳ δ δ ⊲ ⊳ δ ′ δ ⊲⊳ δ ′ δ ⊲⊳ δ ′ Correspondingquivers δ δ δ δ δ δ δ δ δ δ δ δ δ δ ′ δ δ ′ δ δ ′ Table 1.
Puzzle pieces and the corresponding quiversassociated with T .We denote by T T the set of tagged triangulations of ( S, M ) obtained from T by sequences of flips,and by A T the set of tagged arcs of each tagged triangulation contained in T T . Cluster algebras definedfrom triangulated surfaces have the following properties. Theorem 3.3.
Let T be a tagged triangulation of ( S, M ) . (1) [FST, Theorem 7.11][FoT, Theorem 6.1] There is a bijection x ( − ) : A T ←→ cl-var Q T . Moreover, it induces a bijection x ( − ) : T T ←→ cluster Q T which sends T to the initial cluster in A ( Q T ) and commutes with flips and mutations. (2) [Lab10, Theorem 10.0.5][Lab09b, Theorem 7.1][Re14b, Proposition 5.2] For each δ ∈ A T , wehave − b T ( e ( δ )) = g Q T ( x δ ) . Note that, in another way, Theorem 3.3(2) can be directly given by the cluster expansion formulain [Y18b]. Moreover, it was proved in [FeT, Theorem 8.6] for orbifolds in the same way as [Re14b,Proposition 5.2].3.2.
Proof of Theorem 1.2.
We recall the following notion to prove Theorem 1.2.
Definition 3.4. [BQ] Let (
S, M ) be an arbitrary marked surface. The tagged rotation of a tagged arc δ of ( S, M ) is the tagged arc ρ ( δ ) defined as follows: • If δ has an endpoint o on a component C of ∂S , then ρ ( δ ) is obtained from δ by moving o tothe next marked point on C in the counterclockwise direction; • If δ has an endpoint at a puncture p , then ρ ( δ ) is obtained from δ by changing its tags at p .By Theorem 2.1, we have(3.1) T = (cid:26) T T ⊔ T ρT if ( S, M ) is a closed surface with exactly one puncture , T T otherwise . ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 17 Let S ∗ be the same surface as S oriented in the opposite direction and M ∗ = M . For a tagged arcor laminate γ of ( S, M ), we denote by γ ∗ the corresponding one of ( S ∗ , M ∗ ). In particular, the taggedtriangulation T ∗ of ( S ∗ , M ∗ ) is naturally induced by T ∈ T and we have Q T ∗ = Q op T . By Theorem3.3(1), the composition of maps ρ − ( − ), ( − ) ∗ and x ( − ) gives a bijection x ( ρ − ( − )) ∗ : A ρT ←→ cl-var Q op T . Moreover, it induces a bijection x ( ρ − ( − )) ∗ : T ρT ←→ cluster Q op T which sends ρT to the initial cluster in A ( Q op T ) and commutes with flips and mutations. Theorem 3.5.
Let T be a tagged triangulation of ( S, M ) . Then for each δ ∈ A ρT , we have b T ( e ( δ )) = g Q op T ( x ( ρ − ( δ )) ∗ ) . Proof.
For a tagged arc δ of ( S, M ), the equalities b T ( e ( ρ ( δ ))) = b T (( e ( δ ∗ )) ∗ ) = − b T ∗ ( e ( δ ∗ ))hold. Since Q T ∗ = Q op T , Theorem 3.3(2) gives − b T ∗ ( e ( δ ∗ )) = g Q T ∗ ( x δ ∗ ) = g Q op T ( x δ ∗ )for δ ∈ A T , hence b T ( e ( ρ ( δ ))) = g Q op T ( x δ ∗ ). (cid:3) Proof of Theorem 1.2.
By Theorems 3.3 and 3.5, we have [ T ′ ∈ T T C T ( e ( T ′ )) = [ x ∈ cluster Q T (cid:0) − C Q T ( x ) (cid:1) and [ T ′ ∈ T ρT C T ( e ( T ′ )) = [ x ∈ cluster Q op T C Q op T ( x ) . If (
S, M ) is a closed surface with exactly one puncture, then T T and T ρT coincide with T + and T − inTheorem 1.5, respectively. Therefore, the assertion follows from Theorem 1.5 and (3.1). (cid:3) Example for a cluster algebra.
For the tagged triangulation T in Subsection 2.6, the quiver Q T is the Kronecker quiver 1 ⇔
2. The set cluster Q T is described by( x , x ) (cid:18) x + y x , x (cid:19) (cid:18) x + y x , x + y y x + 2 y x + y x x (cid:19) · · · (cid:18) x , y x + 1 x (cid:19) (cid:18) x ′ , y x + 1 x (cid:19) (cid:18) x ′ , x ′′ (cid:19) · · · where x ′ = y y x + 2 y y x + x + y x x , x ′′ = y y x + 3 y y x + 2 y y x x + x + 3 y y x + 2 y x + y x x . The corresponding g -vectors are as follows:(1 , , (0 ,
1) ( − , , (0 ,
1) ( − , , ( − , · · · (1 , , (0 , −
1) ( − , , (0 , −
1) ( − , , ( − , · · · The g -vector cones of clusters are reflections of the corresponding shear coordinate cones in Subsection2.6 as follows: G -VECTOR CONES FROM TRIANGULATED SURFACES Representation theory τ -tilting theory and cluster tilting theory. In this subsection, we recall τ -tilting and clustertilting theory to prepare for the proofs of Theorem 1.3 and Corollary 1.4.First, we recall τ -tilting theory [AIR]. Let Λ be a finite dimensional algebra over a field. We denoteby mod Λ (resp., proj Λ) the category of finitely generated (resp., finitely generated projective) leftΛ-modules. We denote by τ the Auslander-Reiten translation of mod Λ and by | M | is the number ofnon-isomorphic indecomposable direct summands of M ∈ mod Λ. Let M ∈ mod Λ and P ∈ proj Λ.We say that a pair ( M, P ) is • τ -rigid if Hom Λ ( M, τ M ) = 0 = Hom Λ ( P, M ); • τ -tilting if ( M, P ) is τ -rigid and | Λ | = | M | + | P |• basic if M and P are basic; • a direct summand of ( M ′ , P ′ ) ∈ mod Λ × proj Λ if M is a direct summand of M ′ and P is adirect summand of P ′ ; • indecomposable if ( M, P ) is basic and | M | + | P | = 1.Recall that we denote by s τ -tilt Λ the set of isomorphism classes of basic τ -tilting pairs in mod Λ.For N ∈ s τ -tilt Λ and an indecomposable direct summand N ′ of N , there is a unique indecomposable τ -rigid pair N ′′ such that N/N ′ ⊕ N ′′ ∈ s τ -tilt Λ [AIR, Theorem 0.4]. Therefore, one can definemutations in s τ -tilt Λ.Let Λ = L ni =1 P i be a decomposition of Λ, where P i is an indecomposable projective Λ-module.Then [ P ] , . . . , [ P n ] form a basis for K (proj Λ), thus there is a natural bijection between K (proj Λ)and Z n . Let M ∈ mod Λ. There is a minimal projective presentation of MP → P → M → . We set g Λ ( M ) := [ P ] − [ P ] ∈ K (proj Λ) ≃ Z n , called the g -vector of M . We denote by i τ -rigid Λ the set of isomorphism classes of indecomposable τ -rigid pairs in mod Λ. The g -vector of ( M, P ) ∈ i τ -rigid Λ is g Λ ( M, P ) := g Λ ( M ) − g Λ ( P ).For our aim, we also need to consider the opposite algebra Λ op of Λ. For M ∈ mod Λ, the notationTr M denotes the transpose of M . We define ( − ) ∗ := Hom Λ ( − , Λ) : proj Λ ←→ proj Λ op . Then ( − ) ∗ gives K (proj Λ) ≃ K (proj Λ op ). Theorem 4.1. [AIR, Theorem 2.14][F, Subsection 3.4]
There is a bijection ϕ : i τ -rigid Λ ←→ i τ -rigid Λ op given by ( M, P ) (Tr M ⊕ P ∗ , M pr ) such that g Λ ( M, P ) = − g Λ op (Tr M ⊕ P ∗ , M pr ) , where M pr is a maximal projective direct summand of M . The map ϕ induces a bijection ϕ : s τ -tilt Λ ←→ s τ -tilt Λ op which sends (Λ , to (0 , Λ) and commutes with mutations. Next, we recall cluster tilting theory in 2-Calabi-Yau triangulated categories. Let C be a Hom-finiteKrull-Schmidt 2-Calabi-Yau triangulated category. We call X ∈ C rigid if Hom C ( X, X [1]) = 0. Wedenote by add U the category of all direct summands of finite direct sums of copies of U . We call U ∈ C cluster tilting if add U = { X ∈ C | Hom C ( U, X [1]) = 0 } . We denote by irigid C the set ofisomorphism classes of indecomposable rigid objects in C . Recall that we denote by c-tilt C the set ofisomorphism classes of basic cluster tilting objects in C . We assume that C has cluster tilting objects,that is, c-tilt C 6 = ∅ . In this case, any maximal rigid object in C is cluster tilting [ZZ, Theorem 2.6].Iyama and Yoshino [IY] gave mutations in c-tilt C (see also [BMRRT]).Let U = L ni =1 U i be a decomposition of U , where U i is indecomposable. Then [ U ] , . . . , [ U n ] forma basis for K (add U ), thus there is a natural bijection between K (add U ) and Z n . For U ∈ c-tilt C and X ∈ C , there is a triangle U → U → X → U [1] , ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 19 where U , U ∈ add U . We define g U ( X ) := [ U ] − [ U ] ∈ K (add U ) ≃ Z n , called the g -vector of X with respect to U .There is a close relationship between cluster tilting theory and τ -tilting theory as follows. Theorem 4.2. [AIR, Theorem 4.1]
Let U ∈ c-tilt C and Λ = End C ( U ) op . Then there is a bijection H := Hom C ( U, − ) : irigid C ←→ i τ -rigid Λ such that g U ( X ) = g Λ ( H ( X )) for X ∈ irigid C . Moreover, it induces a bijection H : c-tilt C ←→ s τ -tilt Λ which sends U to (Λ , and commutes with mutations. For • ∈ { + , −} , we denote by irigid • C (resp., i τ -rigid • Λ) the set of indecomposable direct summandsof an object in c-tilt • C (resp., s τ -tilt • Λ). Clearly, the map H in Theorem 4.2 gives bijectionsirigid • C ←→ i τ -rigid • Λ and c-tilt • C ←→ s τ -tilt • Λ . Representation theory and cluster algebras.
We consider a relationship between represen-tation theory and cluster algebras to prove Theorem 1.3 and Corollary 1.4. For a quiver with potential(
Q, W ), we have the associated Jacobian algebra J ( Q, W ), Ginzburg differential graded algebra Γ
Q,W ,and generalized cluster category C Q,W (see e.g. [A, DWZ, G, K08, K11] for details). The following isthe main result in the additive categorification of cluster algebras.
Theorem 4.3.
Let Q be a quiver without loops and -cycles and W a non-degenerate potential of Q such that J ( Q, W ) is finite dimensional. (1) [A, Theorem 2.1] The category C Q,W is a Hom-finite Krull-Schmidt -Calabi-Yau triangulatedcategory with a cluster tilting object Γ Q,W . (2) [FK, Theorem 6.3][CKLP, Corollary 3.5] There is a bijection X : irigid + C Q,W ←→ cl-var Q such that g Γ Q,W ( X ) = g Q ( X ( X )) for X ∈ irigid + C Q,W . Moreover, it induces a bijection X : c-tilt + C Q,W ←→ cluster Q which sends Γ Q,W to the initial cluster in A ( Q ) and commutes with mutations. Note that the map X in Theorem 4.3(2) is called the cluster character associated with ( Q, W ) (seee.g. [BY, CC, Pa, Pl11a, Pl11b]).We also study irigid − C Q,W and c-tilt − C Q,W . We haveEnd C Q,W (Γ Q,W ) op ≃ J ( Q, W ) and J ( Q, W ) op ≃ J ( Q op , W op ) , where Q op is the opposite quiver of Q and W op is a non-degenerate potential of Q op corresponding to W . Corollary 4.4.
Let Q be a quiver without loops and -cycles and W a non-degenerate potential of Q such that J ( Q, W ) is finite dimensional. Then there is a bijection X ′ : irigid − C Q,W ←→ cl-var Q op such that g Γ Q,W ( X ) = − g Q op ( X ′ ( X )) for X ∈ irigid − C Q,W . Moreover, it induces a bijection X ′ : c-tilt − C Q,W ←→ cluster Q op which sends Γ Q,W [1] to the initial cluster in A ( Q op ) and commutes with mutations. G -VECTOR CONES FROM TRIANGULATED SURFACES Proof.
Let X ′ be the following composition:irigid − C Q,W H −→ i τ -rigid − End C Q,W (Γ Q,W ) op ←→ i τ -rigid − J ( Q, W ) ϕ −→ i τ -rigid + J ( Q, W ) op ←→ i τ -rigid + J ( Q op , W op ) H − −→ irigid + C Q op ,W op X −→ cl-var Q op . By Theorems 4.1, 4.2 and 4.3, it induces a bijection between c-tilt − C Q,W and cluster Q op whichsends Γ Q,W [1] to the initial cluster in A ( Q op ) and commutes with mutations. Moreover, we have theequalities g Γ Q,W ( X ) = g J ( Q,W ) ( H ( X )) = − g J ( Q,W ) op ( ϕ H ( X )) = − g Γ Q op ,W op ( H − ϕ H ( X )) = − g Q op ( X ′ ( X ))for X ∈ irigid − C Q,W . (cid:3) For a tagged triangulation T of ( S, M ), we consider a non-degenerate potential W of Q T such that J ( Q T , W ) is finite dimensional. It is known that such a potential W exists. Proposition 4.5.
Let T be a tagged triangulation of ( S, M ) . Then there is a non-degenerate potential W of Q T such that J ( Q T , W ) is finite dimensional.Proof. For a sphere (
S, M ) with exactly four punctures, such a potential W was given in [GG] (seealso [GLS]). Suppose that ( S, M ) is not a sphere with exactly four punctures. Labardini-Fragoso[Lab09a, Lab16] defined a potential W of Q T for any tagged triangulation T of ( S, M ), and showedthat it is non-degenerate except for a sphere with exactly five punctures, which in this case wasproved in [GLS]. Finite dimensionally of J ( Q T , W ) was proved in [Lab09a] for ( S, M ) with non-emptyboundary and in [Lad12] for (
S, M ) with empty boundary, where it was proved independently in [TV]for spheres. (cid:3)
Proofs of Theorem 1.3 and Corollary 1.4.
We keep the notations in the previous subsection.Let Γ , U = L ni =1 U i ∈ c-tilt C and N = L ni =1 N i ∈ s τ -tilt Λ, where U i and N i are indecomposable. Wedefine g -vector cones C Γ ( U ) := (cid:26) n X i =1 a i g Γ ( U i ) | a i ∈ R ≥ (cid:27) and C Λ ( N ) := (cid:26) n X i =1 a i g Λ ( N i ) | a i ∈ R ≥ (cid:27) . Proof of Theorem 1.3.
Let T be a tagged triangulation of ( S, M ) and W a non-degenerate potentialof Q = Q T such that J ( Q, W ) is finite dimensional. By Theorem 4.2, we have C Γ Q,W ( U ) = C J ( Q,W ) ( H ( U ))for U ∈ c-tilt C Q,W . Therefore, we only need to prove the assertion for C Q,W . By Theorem 4.3(2) andCorollary 4.4, the equalities [ U ∈ c-tilt + C Q,W C Γ Q,W ( U ) = [ x ∈ cluster Q T C Q T ( x ) and [ U ∈ c-tilt − C Q,W C Γ Q,W ( U ) = [ x ∈ cluster Q op T (cid:0) − C Q op T ( x ) (cid:1) . hold. Thus the assertion follows from Theorem 1.2. (cid:3) Proof of Corollary 1.4. A g -vector cone C Γ Q,W ( U ) has dimension n for any U ∈ c-tilt C Q,W [DK,Theorem 2.4]. For U ≇ V ∈ c-tilt C Q,W , C Γ Q,W ( U ) and C Γ Q,W ( V ) have no intersections except for theirboundaries [DIJ, Corollary 6.7]. Thus there are no cluster tilting objects in c-tilt C Q,W \ c-tilt ± C Q,W by Theorem 1.3. The assertion follows from Theorem 2.1. (cid:3)
ENSITY OF g -VECTOR CONES FROM TRIANGULATED SURFACES 21 Example for representation theory.
For the tagged triangulation T in Subsection 2.6, thequiver Q T is the Kronecker quiver 1 ⇔
2. The set s τ -tilt J ( Q T ,
0) is as follows: (cid:16) ⊕
21 1 , (cid:17) (cid:16) ⊕
21 1 , (cid:17) (cid:16) ⊕ , (cid:17) · · · (cid:16) ,
21 1 (cid:17) (cid:16) , ⊕
21 1 (cid:17) (cid:0) , (cid:1) · · · The corresponding g -vectors(1 , , (0 ,
1) ( − , , (0 ,
1) ( − , , ( − , · · · (1 , , (0 , −
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