Gentle algebras arising from surfaces with orbifold points of order 3, Part I: scattering diagrams

To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero--Chapoton map defines a bijection between reachable $\tau$-rigid pairs and cluster monomials of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.


Introduction
The interplay between Fomin-Zelevinsky theory of cluster algebras [FZ02] and Teichmüller theory of surfaces with marked points originated in the works of Fock-Goncharov [FG07], Fomin-Shapiro-Thurston [FST08], Gekhtman-Shapiro-Vainshtein [GSV05] and Penner [Pen87].By allowing the cluster exchange relations to be dictated by polynomials more general than just binomials -thus defining generalized cluster algebras, Chekhov-Shapiro [CS14] have extended this interplay to surfaces with orbifold points.Generalized cluster algebras thus play the role of coordinate rings of decorated Teichmüller spaces of such orbifolds, with each triangulation providing a full coordinatization via lambda lengths, and with Chekhov-Shapiro's polynomials describing the change of coordinates corresponding to a flip of triangulations.
For generalized cluster algebras, however, so far there have been only a couple of works providing such representation-theoretic expressions for cluster variables, cf.[PS19,LFV19].
In [LFV19], Labardini-Velasco associate quivers with potential to the triangulations of a polygon with one orbifold point of order 3, and prove that the cluster monomials in the (coefficient-free) generalized cluster algebra of the latter are Caldero-Chapoton functions of τ -rigid pairs over the Jacobian algebras of those QPs.In the present note, we extend this to the framework of surfaces with marked points on the boundary and orbifold points of order 3, thus providing a vast generalization of Labardini-Velasco's work.
A bit more precisely, our construction possesses a few special features.Namely, the Jacobian algebras of the QPs we define are finite-dimensional and gentle, and the underlying quiver has loops arising naturally from the orbifold points.Furthermore, the representation theories of the QPs associated to triangulations related by a flip are closely related through adjoint pairs of reflection functors, which we construct for these QPs.These functors allow us to show that the Bridgeland stability scattering diagrams [Bri17] of the QPs satisfy what Gross-Hacking-Keel-Kontsevich [GHKK18] call mutation invariance.This fact leads to a cluster-like wall-crossing structure in the real vector space of stability conditions, whose walls we decorate with a representation-theoretic incarnation of Chekhov-Shapiro's generalized cluster transformations.As an upshot, we obtain our main result, namely, that for surfaces with marked points on the boundary and orbifold points of order 3, the generalized cluster monomials are precisely the Caldero-Chapoton functions of reachable τ -rigid pairs, see Theorem 1.1 below.
It would be interesting to find out how our work is related to Paquette-Schiffler's approach [PS19], which provides certain generalized cluster exchange relations in orbit cluster algebras arising from quivers with potential with admissible group actions.Our methods are direct, and different from those in [PS19], in the sense that we do not resort to covers of surfaces, quivers or algebras.
Let us give a more detailed summary of our results.
1.1.Generalized cluster algebras.In Section 2 we recall some of the basic facts surrounding Chekhov-Shapiro's generalized cluster algebras.One starts with certain cluster exchange data which includes a skew-symmetrizable matrix B, a tuple of positive divisors of the columns of B and a set of exchange polynomials.This data is used to define the notion of generalized seed mutation.The main difference with Fomin-Zelevinsky's seed mutation is that now the exchange polynomials are allowed to have more than just two terms.The generalized cluster algebra A(B) is defined to be the subalgebra generated inside the rational function field by all the clusters in the seeds obtained by applying all possible finite sequences of generalized seed mutations to a given initial seed.Afterwards, in Section 3 we describe how the triangulations κ of surfaces with marked points and orbifold points (Σ, M, O) give rise to generalized cluster algebras.Each triangulation κ has an associated skew-symmetrizable matrix B(κ) with rows and columns labeled by the arcs in κ.The essential observation is that triangulations κ and µ k (κ) related by a flip at k ∈ κ have matrices related by the corresponding matrix mutation of Fomin-Zelevinsky.The exchange polynomials are binomials or trinomials, depending on whether the corresponding arc is pending or not.The associated generalized cluster algebra A(Σ, M, O) = A(B(κ)) essentially depends only on the surface with marked points and orbifold points, not on the triangulation κ.As mentioned above, A(Σ, M, O) plays a relevant geometric role as coordinate ring of the decorated Teichmüller space of (Σ, M, O).
1.2.Gentle algebras associated to orbifolds.Let S = (Σ, M, O) be a surface Σ with non-empty boundary, a set of marked points M ⊂ ∂Σ and a set of interior points O ⊂ Σ \ ∂Σ which we call orbifold points of order 3.A triangulation of S is a maximal collection κ of compatible arcs with endpoints in M up to isotopy relative to M ∪ O.To such a triangulation κ, we associate a quiver (with loops) with potential (Q(κ), S(κ)) whose Jacobian algebra P(κ) is finite dimensional and gentle; see Definition 4.1.In the case without any orbifold points, this construction gives a quiver with potential whose flip/mutation dynamics was established in [LF09], and whose representation theory was studied in [ABCJP10].
1.3.Stability conditions and τ -tilting theory.We show that the category mod P(κ) of finite dimensional modules over P(κ) is informative for understanding A(κ) = A(B(κ)) in the sense of Theorem 1.1.The connection is made through scattering diagrams or wall-crossing structures originated in [GS11] and [KS06,KS14].We study Bridgeland's stability scattering diagram [Bri17] of the algebra P(κ), which in the case without orbifold points are shown in [Mou19] to have the same cluster structure as the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich [GHKK18].We study the following function Φ κ : R n → W(P(κ)), θ → mod θ-ss P(κ), which maps a King stability condition θ to the wide subcategory of θ-semistable modules over P(κ).
The structure of the level sets of this function reveals the cluster structure of A(κ).In particular, the so-called g-vector fan of A(κ) (whose rays are in bijection with cluster variables) can be seen from the support of Φ κ ; see Theorem 6.12.We develop BGP-type [BGP73] reflection functors F ± k : mod P(κ) → mod P(µ k (κ)) to relate Φ κ with Φ µ k (κ) where k does not have to be a sink or source.This key tool enables explicit investigations on Φ κ as well as any Φ σ for σ obtained by a sequence of flips from κ.In the case without orbifold points, these functors can be viewed as the degree zero cohomology of Keller-Yang's derived equivalences [KY11] which so far are only available for quivers without loops.Via the correspondence between τ -tilting theory and stability conditions of Brüstle-Smith-Treffinger [BST19], the τ -tilting theory of P(κ) can be related to the cluster structure of A(κ) through Φ κ .Our main result is the following.between the mutation graph of reachable support τ -tilting pairs of mod P(κ) and the exchange graph of unlabeled seeds of A(κ).
Remark 1.2.Sota Asai has informed us that Fu-Geng-Liu-Zhou [FGLZ21] recently proved that for all finite-dimensional gentle algebras, every indecomposable τ -rigid pair is reachable.Thus, the adjective "reachable" can be removed from the statement of Theorem 1.1 above.1.4.Motivic Hall algebras and Caldero-Chapoton functions.To prove Theorem 1.1, we make heavy use of Bridgeland's Hall algebra scattering diagram as a function φ Hall κ : R n → Ĥ(mod P(κ)).It can be seen as turning the category Φ κ (θ) = mod θ-ss P(κ) into an invertible element φ Hall κ (θ) = 1 ss (θ) in the motivic Hall algebra Ĥ(mod P(κ)); see Section 8. Another important ingredient is Joyce's integration map (Theorem 7.2) which turns 1 ss (θ) further into a formal exponential of a Poisson derivation.The integration map exists simply because our algebra P(κ) is the Jacobian algebra of a quiver with potential.After applying the integration map, we obtain a function φ κ : R n → Ĝ with values in Poisson automorphisms some of which precisely describe generalized cluster transformations.The function φ κ is what we call the stability scattering diagram of P(κ).
The above machinery makes it practical to express a non-initial cluster variable x i;t ∈ A(κ) in terms of representation theoretic information, in fact as the Caldero-Chapoton function of some indecomposable τ -rigid module M i;t .Our calculation is largely motivated by Nagao's work [Nag13] in the skew-symmetric case of ordinary cluster algebras.However the characterizations of the representation theoretic object corresponding to a cluster variable are very different as we rely on τ -tilting theory [AIR14] rather than Keller-Yang's derived equivalences [KY11].
1.5.Structure of the paper.In Section 2, we review the definitions of cluster algebras and their generalizations.In Section 3 and Section 4, we explain the construction of gentle algebras P(κ) associated to surfaces with orbifold points.In Section 5 and Section 6, we define reflection functors and use them to study stability conditions of P(κ), realizing cluster chamber structures.In Section 7, we review the motivic Hall algebras of quivers with relations.In Section 8, Section 9, and Section 10, we study the scattering diagrams associated to P(κ), relate them with the τ -tilting theory of mod P(κ) and prove our main theorem Theorem 1.1.In Section 11 we present in an explicit example in affine type C 2 , how the Caldero-Chapoton functions of support τ -tilting pairs related by an AIR-mutation satisfy a generalized exchange relation as stated in Theorem 1.1 (see Theorem 10.8 too).In the final Section 12, we summarize and outline the plan of the sequel [LFM].

Generalized cluster algebras
Here, we review Chekhov-Shapiro's definition of generalized cluster algebras [CS14].We only introduce the coefficient-free case which will suffice for our purposes.
Definition 2.1.Let F be the field of rational functions in n indeterminates with coefficients in C. A labeled seed in F is a pair (x, B) consisting of: • an ordered n-tuple x = (x 1 , . . ., x n ) of elements of F algebraically independent over C that generate F as a field; • a skew-symmetrizable matrix B ∈ Mat n×n (Z), i.e., a matrix for which there exists a diagonal matrix D = diag(d 1 , . . ., d n ) ∈ Mat n×n (Z) with positive diagonal entries such that DB + (DB) T = 0.
The tuple x is called the labeled cluster of the labeled seed (x, B), the elements x 1 , . . ., x n , are the cluster variables of the labeled seeds.
To define Chekhov-Shapiro's generalized mutation rule of labeled seeds, we first fix an n-tuple (r 1 , . . ., r n ) of positive integers such that for each j, the integer r j divides the j-th column of B, i.e., b ij /r j ∈ Z for any i and j.This property of (r 1 , . . ., r n ) is preserved under Fomin-Zelevinsky's matrix mutation.Denote the matrix B = (b ij ) := (b ij /r j ).For each i fix also a polynomial Definition 2.2.For each k ∈ {1, . . ., n}, the generalized mutation of (x, B) in direction k with respect to (r 1 , . . ., r n ) and (θ 1 , . . ., θ n ) is the labeled seed , where µ k (B) is the k th Fomin-Zelevinsky matrix mutation of B, and (2.1) Notice that the right hand side of (2.1) is the evaluation Definition 2.3.The (coefficient-free) generalized cluster algebra associated to the initial seed (x, B) with respect to the fixed data (r 1 , . . ., r n ) and (θ 1 , . . ., θ n ) is the subring A(x, B) of F generated by the set of all cluster variables that appear in the labeled seeds that can be obtained from (x, B) by applying arbitrary finite sequences of generalized cluster mutations.
We say that two labeled seeds (x ′ , B ′ ) and (x, B) are mutation equivalent, and write (x ′ , B ′ ) ∼ (x, B), if they can be obtained by a finite sequence of mutations from each other.
In [FZ07], Fomin and Zelevinsky introduce the notions of g-vector and F -polynomial in cluster algebras.Nakanishi [Nak15] has extended these notions to generalized cluster algebras.Let T n be the rooted n-regular tree.We denote its root as t 0 , and assume that each edge of T n has been labeled with a number from {1, . . ., n}, in such a way that whenever two distinct edges are incident at a vertex of T n , their labels are different.This allows us to assign to each vertex t ∈ T n a labeled seed ((x B;t0 1;t , . . ., x B;t0 n;t ), B(t)) in such a way that ((x B;t0 1;t0 , . . ., x B;t0 n;t0 ), B(t 0 )) := (x, B), and for each edge t k t ′ , the labeled seeds attached to t and t ′ are related by the k th generalized mutation.
Taking (r 1 , . . ., r n ) and (θ 1 , . . ., θ n ) as fully fixed data, and B as our initial matrix (attached to the root t 0 of T n ), to each vertex t of T n we associate n-tuples (c B;t0 1;t , . . ., c B;t0 n;t ), (g B;t0 1;t , . . ., g B;t0 n;t ), of vectors in Z n , and an n-tuple (F B;t0 1;t , . . ., F B;t0 n;t ) of elements of the rational function field Q(y 1 , . . ., y n ).Whenever it is clear which matrix B is being attached to t 0 , we will take the liberty of possibly omitting the superscript B; t 0 from the notation.These tuples are defined recursively as follows.For each i = 1, . . ., n, c B;t0 i;t0 := e i g B;t0 i;t0 := e i and F B;t0 i;t0 := 1 (e i is the i th standard basis vector).Furthermore, whenever we have an edge t k t ′ in T n , we have where for t ∈ T n , b ki (t) is the (k, i)-th entry of the matrix B(t), b j (t) denotes the j-th column of B(t), and c B;t0 k,j;t is the j th entry of c Remark 2.5.
(2) In [FZ07] and [Nak15], the definition of g-vectors and F -polynomials is given rather in terms of their properties in a (generalized) cluster algebra with principal coefficients.Under this approach, the fact that they can be obtained through recursions becomes a theorem instead of a definition.
For a vertex t ∈ T n and a permutation σ of the index set {1, . . ., n}, set , where the (i, j) th entry of the matrix σ • (B(t)) ∈ Z n×n is defined to be b σ −1 (i)σ −1 (j) (t).If t and s are vertices of T n such that (x B;t0 s , B(s)) = σ • (x B;t0 t , B(t)), then for every k ∈ {1, . . ., n} we have the equality of labeled seeds which means that the automorphism of T n that sends t to s and the edge t k ⋆ to the edge s σ(k) * is compatible with mutations of labeled seeds.Let G(B) be the group of graph automorphisms of T n that arise this way.
Definition 2.6.The exchange graph of unlabeled seeds of the generalized cluster algebra A = A(x, B) is the simple graph E(A) obtained as the quotient T n /G(B).

Surfaces with orbifold points of order 3
Chekhov-Shapiro [CS14] have shown that surfaces with marked points and orbifold points give rise to generalize cluster algebras.In this paper we will restrict our attention to the situation where all marked points are contained in the boundary of the surface and all orbifold points have order 3. Definition 3.1.An unpunctured surface with marked points and orbifold points of order 3 is a triple (Σ, M, O) consisting of: • a compact connected oriented two-dimensional real differentiable manifold Σ with non-empty boundary ∂Σ; • a finite subset M ⊂ ∂Σ containing at least one element from each connected component of ∂Σ; The elements of M will be called marked points, whereas the elements of O will be called orbifold points (of order 3).
• the image i([0, 1]) must and only intersect the boundary ∂Σ at its endpoints i(0) and i(1); • i has no self intersections except possibly at endpoints; • if i cuts a monogon, then such a monogon contains exactly one orbifold point.In this case, we call i a pending arc.
We consider arcs up to isotopy relative to M ∪ O. Two arcs are called compatible if there are representatives in their isotopy classes whose images as curves do not intersect in Σ \ M. Whenever two arcs are compatible, we will directly take the corresponding representatives.Definition 3.3.A triangulation of (Σ, M, O) is a maximal collection of pairwise compatible arcs.If κ is a triangulation of (Σ, M, O), we define the flip at any i ∈ κ to be the unique triangulation It is easy to see from the definition that flips are involutive.
Definition 3.4.The flip graph E(Σ, M, O) has the triangulations of (Σ, M, O) as vertices; two triangulations are connected by an edge precisely when they are related by a flip.
For the next definition, consider the subset of Σ obtained as the union of the arcs in a triangulation κ.The complement of this union in Σ is a disjoint union of connected components.Definition 3.5.Suppose κ is a triangulation of (Σ, M, O).
(1) The topological closure in Σ of any of the connected components mentioned in the preceding paragraph will be called a triangle of κ; (2) a triangle not containing orbifold points will be called non-singular triangle; (3) a triangle containing an orbifold point will be called singular triangle.
Thus, a non-singular triangle always has three distinct sides, and a singular triangle always has exactly one side.See Figure 1.Each side of a non-singular triangle is either an arc (even possibly a pending one) or a segment of the boundary of Σ.Furthermore, at most two sides of a non-singular triangle can be pending arcs.The unique side of a singular triangle is pending.
Recall that a quiver is a quadruple Recall also that we draw any a ∈ Q 1 as t(a) a − → h(a) or a : t(a) → h(a).
Definition 3.6.Following [ABCJP10] and [GLF20, Definition 3.2], we define Q(κ) to be the quiver whose vertices are the arcs in κ, that is, Q 0 (κ) := κ, and whose arrows of are induced by the nonsingular triangles of κ through the orientation of Σ.More precisely, for each non-singular triangle △ of κ and every pair i, j ∈ κ of arcs contained in △ such that i immediately follows j in △ with respect to the clockwise sense in △ induced by the orientation of Σ, we draw a single arrow from j to i.
Notice that we are not including the boundary segments in the quiver Q(κ).Take a triangulation κ, and fix a labeling of the arcs in κ by the numbers 1, . . ., n := |κ|.We say then, that κ is a labeled triangulation.We associate to κ a skew-symmetrizable matrix B(κ) as follows.The quiver Q(κ) has an associated skew-symmetric matrix B = (b ij ) (i,j)∈κ×κ whose entries are given by the rule The following lemma is easy to check.Lemma 3.7.If κ and σ are two triangulations, and σ is the flip µ i (κ) of κ at i ∈ κ, then we have We associate a generalized cluster algebra to B(κ) by taking (3.1) r j = d j and θ j = rj ℓ=0 u ℓ v rj −ℓ for every j = 1, . . ., n.
By construction, for each j = 1, . . ., n, r j divides all entries of the j th column of B(κ).
There is a bijection under which the arcs in κ correspond to the initial cluster variables.Furthermore, this bijection induces a graph isomorphism between the flip graph of (Σ, M, O) and the exchange graph of unlabeled seeds of A(κ), making the triangulation κ correspond to the seed (x, B(κ)).Thus, denoting by x j the cluster variable corresponding to each arc j on (Σ, M, O), for every triangulation σ and every arc k ∈ σ, if k ′ denotes the unique arc on x j j:b jk (σ)<0 x j + j:b jk (σ)>0 x 2 j if k is pending.
Remark 3.9.The role that the orbifold points play in this paper is overwhelmingly combinatorial.They play a geometric role, although very subtly, as follows.Chekhov-Shapiro [CS14] have shown that within the ring of real-valued functions on the decorated Teichmüller space of a surface with marked points and orbifold points of arbitrary orders, the λ-lengths of arcs satisfy an exchange rule (2.1), hence the subring generated by them is (isomorphic to) a generalized cluster algebra (with so-called boundary coefficients).When the orders of the orbifold points are all taken to be equal to three, the exchange relation (2.1) obeyed by the λ-lengths takes the specific form (3.2).One of our main results in this paper, is that it is this specific form that the Caldero-Chapoton functions of indecomposable τ -rigid pairs satisfy whenever one mutates a basic support τ -tilting pair.

The gentle algebra associated to a triangulation
Let Q be a quiver.Denote by k Q the path algebra of Q over a field k.Denote by e i the primitive So it can be viewed as a linear sum of oriented cyclic paths (called cycles) in Q and each cycle is considered up to cyclic permutations.
Assume that k is algebraically closed of characteristic zero.We define the Jacobian ideal ∂S to be the (two-sided) ideal in k Q generated by all cyclic derivatives ∂S : Definition 4.1.Let S = (Σ, M, O) and κ be an ideal triangulation of S. The quiver Q(κ) is defined by adding an arrow ε j : j → j (also called a loop) for each pending arc j ∈ κ.We define a potential where the first sum runs over all internal non-singular triangles of κ and the second sum runs over all pending arcs of κ.
Proof.The cyclic derivatives are either of the form 3ε 2 j for j pending or a • b for a and b two arrows of Q(κ) in the same internal triangle of κ.It is then clear that there exists some N ∈ N such that every path of length greater than N vanishes modulo the relations generated by the cyclic derivatives.Thus the algebra P(κ) is of finite dimension over k.
Recall that an algebra k Q /I is called gentle if (1) each vertex of Q is incident with at most two incoming and at most two outgoing arrows; (2) the ideal I is generated by paths of length two; (3) for each b ∈ Q 1 there is at most one a ∈ Q 1 and at most one c ∈ Q 1 such that ab ∈ I and bc ∈ I; (4) for each b ∈ Q 1 there is at most one a ∈ Q 1 and at most one c ∈ Q 1 such that ab / ∈ I and bc / ∈ I.
The conditions (1) and (2) hold for Q(κ) and I = ∂S(κ) from the constructions of the quiver and the potential.Conditions (3) and (4) are easily checked by studying all possible local configurations of a triangulation.
For such a Jacobian algebra P(κ), to each vertex k ∈ κ, we associate the subalgebra (1) In the absence of orbifold points, the gentle algebras P(κ) were introduced in [ABCJP10] and [LF09].
(2) Our definition of the algebras P(κ) in terms of triangulations of (Σ, M, O) can be seen as a special case of the much more general geometric construction of gentle algebras in terms of dissections of surfaces.See, for instance, [OPS18].
(3) The canonical inclusion of k Q(κ) in the complete path algebra k Q(κ) induces an algebra isomorphism between P(κ) and the quotient of k Q(κ) by the m-adic topological closure of the two-sided ideal generated by ∂S(κ).

Reflection functors
In this section, we define BGP-type reflection functors [BGP73] between module categories of the gentle algebras associated to triangulations related by a flip.
Denote by mod A the category of finitely generated left modules of a k-algebra A. Let κ and σ be two labeled triangulations related by a flip at k ∈ κ.We will define two functors 2), the category mod P(κ) is well-known to be equivalent to rep(Q(κ), ∂S(κ)), the category of finite dimensional k-linear representations of Q(κ) satisfying relations in ∂S(κ).Therefore in the following we treat any P(κ)-module equivalently as an object in rep(Q(κ), ∂S(κ)).For a representation M of a quiver Q, denote by M i the vector space associated to i ∈ Q 0 and by M a : M t(a) → M h(a) the linear map associated to a ∈ Q 1 .All tensor products ⊗ in this section will be of vector spaces over k.
Flipping κ at k will result an arrow a * : we identify a * * with a for any a incident to k in Q 1 (κ).See the diagrams in Case 1 and Case 2 in Section 5.1.

Sink reflection F +
k .Let M be in mod P(κ).Consider the k-linear map α : where in each component y ⊗ x is mapped to y • M a (x) for y ∈ H k and x ∈ M i .Here y acts on M k by the H k -module structure of M k .By construction, the map α is an H k -module homomorphism.Define M ′ k := ker α as an H k -module.For each a : i → k, there is an induced map

the natural inclusion of the kernel composed by the projection to the component
as a k-linear space and identify it with , we first need to specify M ′ i for every i.We define We still need to determine the actions of the arrows in Q 1 (σ) for F + k (M ).For the arrows shared by Q 1 (σ) and Q 1 (κ) (except ε k ), it is easy to see that their domains and targets do not change.We keep the actions of these arrows the same as before.
For each For each b : k → j in Q 1 (κ), there are two cases.If b does not belong to an internal triangle, then we define M ′ b * = 0. Otherwise b in an internal triangle has the local configuration Note that a and c satisfy To show it is actually a module over P(σ) we need to check that the relations ∂S(σ) are satisfied.
In fact, as any relation in ∂S(σ) are local to each internal triangle or pending arc, we only need to check at the triangles which contain the arc k where the flip µ k takes place.There are essentially two cases of local configurations.
Case 1: k is not pending.
Note that some vertices of {1, 2, 3, 4} may be identical or pending, in which case we are not drawing the loops in the diagram.We check the following relations.We use the label of an arrow to represent its action for simplicity.
. This is clear from the following diagram and that g • b = 0.
Case 2: k is pending.
Note that (exactly) one of the vertices 1 and 2 may also be pending.Denote ε = ε k and simplify the notation k • ε ⊗ M 1 = ε ⊗ M 1 .We check the following relations.
(2) a * • b * = 0.This follows from the diagram As before, let M be in mod P(κ).Consider the k-linear map Then f b further restricts to a linear map on k k by a unique map from coker β to M i which we denote by M ′ a * : M ′ k → M i .The rest of the actions of arrows are defined in the same way as for the case F + k .As in the case of F + k , we then need to check that the actions satisfy the relations given by ∂S(σ) so that F − k (M ) becomes a module over P(σ).We leave the details in this case to the reader.
Remark 5.1.If k is a sink (resp.source) of Q 1 (κ) and not pending, then F + k (resp.F − k ) is (defined in the same way as) the classical BGP reflection functor [BGP73].In the case where Q(κ) is acyclic and k is a sink or source (possibly pending), F ± k are the same as the reflection functors of [GLS17] in the corresponding situations.When there is no orbifold point, the functors F ± k fit into a more general setting of quivers with non-degenerate potentials as in [Mou19].Although definitely our F ± k are motivated by the mutation of decorated representations of Derksen-Weyman-Zelevinsky [DWZ08], the F ± k here are partial versions of the corresponding DWZ-mutations in the existence of loops which we shall develop in [LFM].

Adjunction between F +
k and F − k .Let k ∈ κ be a vertex.Consider the two reflection functors Proposition 5.2.The functors F + k and F − k are adjoint to each other, i.e., there is a bifunctorially isomorphism ) for any N ∈ mod P(σ) and M ∈ mod P(κ).
Proof.There are two cases on whether k is pending.
Case 1: k is not pending.
For any two representations M and N of a quiver Q and a path a from i to j (i.e. a composition of arrows from i to j), denote by S a (M, N ) the equation The space of morphisms in mod P(κ) has the following description Note that for On the other hand, we can express Hom P(σ) (N, M ′ ) (writing M ′ = F + k (M )) as the set of tuples of linear maps (z i : By the constructions of N ′ and M ′ , there are following equivalences of equations and S y (N, M ′ ) ⇔ S y (N, M ′ ) for any other y shared by Q 1 (κ) and Q 1 (σ).Then the two spaces Hom P(κ) (N ′ , M ) and Hom P(σ) (N, M ′ ) are canonically identified and this identification is bifunctorial.
Case 2: k is pending.
We keep the same notations such as N ′ and M ′ as in Case 1. Similarly to Case 1, the space Hom P(κ) (N ′ , M ) consists of tuples of linear maps (x i : and hence x k : N ′ k → M k can be well-defined by the commutative diagram Analogously, the space Hom P(σ) (N, M ′ ) is the set of tuples for any other y = ε k shared by Q 1 (κ) and Q 1 (σ), these two homomorphism spaces are canonically identified bifunctorially.
Combining Case 1 and Case 2, we have proven the adjunction between F + k and F − k .
For a labeled triangulation κ, denote by S k the simple module in mod P(κ) corresponding to k ∈ κ.We define the following two full subcategories Proposition 5.3.The functor F + k restricts to an equivalence from ⊥ S k ⊂ mod P(κ) to S ⊥ k ⊂ mod P(σ) with F − k being its inverse equivalence.
Proof.By the adjunction Proposition 5.2, there are two natural transformations the counit for M ∈ mod P(κ) and the unit k ), the counit ε M (resp.the unit η N ) is an isomorphism.

Stability conditions under reflections
In this section we show that the reflection functors F ± k relate stability conditions for the abelian categories mod P(σ) and mod P(κ) in a certain way when σ and κ are related by a flip.6.1.Stability conditions.Definition 6.1 (King [Kin94]).Let A be a finite dimensional k-algebra and θ : K 0 (mod A) → R an additive function on the Grothendieck group of mod A. An object M ∈ mod A is called θ-semistable (resp.θ-stable) if θ(M ) = 0 and for any non-zero proper submodule The full subcategory of all θ-semistable modules is denoted by mod θ-ss A. It is standard that mod θ-ss A is closed under kernels, cokernels and extensions.We will call a full subcategory of mod A satisfying those properties a wide subcategory.
We will from now on call such θ a (King's) stability condition of mod A. Let k be algebraically closed.Then it is well-known that mod A is equivalent to the module category of a basic algebra.In other words, we could assume that A comes from a quiver Q with admissible relations, i.e.A = k Q /I where I ⊂ k Q is an ideal spanned by linear combinations of paths of length at least two.Now the category mod A is equivalent to rep(Q, I) the category of finite dimensional k-linear representations of Q satisfying relations in I.
Identifying Q 0 with {1, . . ., n} where n = |Q 0 |, the Grothendieck group K 0 (mod A) is then isomorphic to Z n with the basis of (isoclasses) of simple modules {S i | i ∈ Q 0 } identified with the standard basis {e i | i = 1, . . ., n} of Z n .A stability condition θ can be expressed as a tuple Recall that we have associated the matrix B(κ) = (b ij ) to a labeled triangulation κ.Let θ = (θ i ) i be a stability condition of mod P(κ).Fix k ∈ κ.Consider the flipped triangulation σ = µ k (κ) and the associated gentle algebra P(σ).We define two stability conditions θ ′ and θ ′′ on mod P(σ) in terms of θ as follows.Define θ ′ = (θ ′ i ) i by (6.1) Theorem 6.2.Let κ, k, σ be as above.Let θ = (θ i ) i be a stability condition of mod P(κ) such that Proof.Let M ∈ mod P(κ) be θ-semistable with θ k > 0. The θ-semistability implies that M does not have any quotient isomorphic to S k .Thus M belongs to ⊥ S k .
Denote by d = (d i ) i the class of M in the Grothendieck group in the basis of S i and The module M is θ-semistable if and only if any such L 1 ֒→ M satisfies θ(L 1 ) ≥ 0. Thus it is equivalent to that any such ≤ 0, which is the same condition of M ′ being θ ′semistable.In fact, here such a quotient L ′ 2 of some M ′ ∈ S ⊥ k has the characterization that for a quotient M ′ ։ N , there is a quotient module L ′ 2 of N cogenerated by the quotient vector space N/N k , which guarantees that L ′ 2 belongs to S ⊥ k .Then M ′ is θ ′ -semistable if and only if any such quotient Then proving the if part for F + k amounts to show the only if part for F − k , which is similar to what we have done for the only if part for F + k .
The following is a direct corollary of Theorem 6.2.
Let A = k Q /I as in the beginning of the section.Let N = K 0 (mod A) and M = Hom(N, Z).
Denote by W(A) the collection of all wide subcategories of mod A. Consider the function We denote the wide subcategory of only zero objects simply by 0. For example, for θ ∈ M R generic, we have Φ A (θ) = 0 as θ = (θ i ) i does not satisfy any linear equation i θ For M ∈ mod A, denote by Ab(M ) the smallest wide subcategory of mod A containing M .
Proof.By definition any object in Ab(S k ) is θ-semistable.If there exists M / ∈ Ab(S k ) ⊂ mod P(κ) that is θ-semistable.Let N be the submodule of M generated by i =k M i , which by definition is again θ-semistable and belongs to ⊥ S k .Then it can be verified as in the proof of Theorem 6.2 that Denote by C + the first (closed) quadrant in M R with the basis (e * i ) i dual to (e i ) i , i.e., The subset C + is a closed polyhedral cone in M R .Denote by C • the interior of any closed cone C in M R .
(2) Any θ ∈ (C + ∩ e ⊥ i ) • satisfies θ j > 0 for j = i and θ i = 0. Thus θ can only have zero evaluation on M ∈ Ab(S i ).Any M ∈ Ab(S i ) is obviously θ-semistable.(3) If θ avoids the countable union of codimension one subsets of the form e ⊥ i ∩ d ⊥ for d ∈ N not proportional to e i , any module M satisfying θ(M ) = 0 must be in Ab(S i ), which is θ-semistable.
Let κ and σ be two triangulations related by a flip at k. Denote by the linear map sending θ to θ ′ (resp.θ ′′ ) as in (6.1) and (6.2).Define two (open) half spaces in M R We define the piecewise linear map Remark 6.6.We note that the piecewise linear transformation T k defined above should not be confused with the one in [GHKK18, Definition 1.22] which we denote by Tk below.In the current situation, their Tk : M R → M R reads as follows.Let θ ′ = (θ ′ i ) i = Tk (θ).Then Thus the maps T k and Tk are related by The following proposition rephrases Corollary 6.3 in terms of the function Φ A .
6.2.Cluster chamber structure.Applying Proposition 6.4, Lemma 6.5 and Proposition 6.7 iteratively for sequences of mutations, one can build the so-called cluster chamber structure on a subset of M R from the function Φ κ as follows.
Let σ = κ(t) be the triangulation associated to t ∈ T n (with κ associated to t 0 ).Suppose the unique path from t 0 to t takes the sequence (k 1 , k 2 , . . ., k l ) of flips.Denote κ j := µ kj • • • µ k2 µ k1 (κ) with κ 0 = κ and κ l = σ.Proposition 6.8.There is a unique choice of signs (ǫ 1 , ǫ 2 , . . ., ǫ l ) ∈ {+, −} l with ǫ l = + such that for any j = 2, . . ., l, the cone Proof.We prove the proposition by induction backwards on j from j = l.In fact, the cone T + k l ;κ l (C + ) is contained either in H + k or H − k for any k.This is because by Proposition 6.7, for any θ ∈ (C + ) • , we have Therefore the hyperplane e ⊥ k does not intersect with T + k l ;κ l (C + ) in its interior since a generic point on e ⊥ k supports Ab(S k ) for the function Φ κ l−1 (by (3) of Lemma 6.5).So we determine ǫ l−1 such that k l−1 .Now suppose for induction that we have determined the sign ǫ j such that that C j is contained in H ǫj kj and that Φ κj (θ) = 0 for any θ ∈ C • j .Consider the reflection functor F ǫj kj : mod P(κ j ) → mod P(κ j−1 ).By Proposition 6.7, for any θ ∈ C Proof.This is a direct consequence of (the proof of) Proposition 6.8.In fact, it is clear from the induction that for any θ ∈ C • j for j = 1, 2, . . ., l − 1, we have Φ κj (θ) = 0.It is extended to j = 0 for one more step of the induction.
Note that each map T ǫj kj ;κj is an automorphism of M ∼ = Z n .So each C j is again a maximal simplicial cone whose n rays has the enumeration by {1, . . ., n} inherited from that of C + .We denote the integral generators of these rays by r j i ∈ M for i ∈ {1, . . ., n}, which form an integral basis of M .The n (relatively open) facets {d j i | i = 1, . . ., n} of C j also has the enumeration such that r j i / ∈ dj i .The normal vectors of these facets dual to r j i are denoted by n j i ∈ N .As already did in the proof of Corollary 6.9, we continue to denote C 0 = C t and d 0 i = d i;t .We next consider the function value of Φ κ in the interior of a facet d i;t .Construction 6.10.We construct a subcategory T t (Ab(S i )) = T κ;t0 t (Ab(S i )) of mod P(κ) iteratively as follows.First consider Ab(S i ) ⊂ mod P(σ), which is the smallest wide subcategory containing S i .We have Assume for some j that for any i, the function Φ κj is constant on d j i .For In either case, we see that The above construction gives an inductive way to describe the subcategory Φ κj (d j i ) for j = 0, . . ., l.We define T t (Ab(S i )) ⊂ mod P(κ) to be Φ κ (d i;t ) (corresponding to j = 0).It is clear from the iterative construction of T t (Ab(S i )) that it is equivalent to Ab(S i ) ⊂ mod P(σ) which is equivalent to (1) mod Summarizing Corollary 6.9 and Construction 6.10, we have Proposition 6.11 (Chamber structure).The function Φ κ : M R → W(P(κ)) (1) takes value 0 in the interior of C t for any t ∈ T n , and (2) is constant in the interior of any facet d i;t of C t with value T t (Ab(S i )).Now we relate the cones C t with the g-vectors and c-vectors associated to the generalized cluster algebra A(κ).Namely we will show that the chamber structure in Proposition 6.11 coincide with the cluster combinatorics of A(κ), thus bearing the name cluster chamber structure.We denote r i;t ∈ M for the integral generators of the rays of C t and n i;t ∈ N the dual normal vectors of the facets d i;t .Theorem 6.12.We have for any t ∈ T n and and i ∈ {1, . . ., n} the equalities of vectors Proof.This theorem is proved inductively.Initially, r i;t0 = g i;t0 = e * i ∈ M and n i;t0 = c i;t0 = e i for all i.We show that the recursive formula for r i;t is the same as that of g i;t and the one of n i;t the same as that of c i;t .Note that by definition (n i;t ) i is the dual basis of (r i;t ) i .
A remarkable feature is that for any n i;t , its coordinates (in the basis e i ) always have the same sign (called sign-coherent), i.e., they are either all non-negative or non-positive.This is because by Proposition 6.11 the corresponding facet d i;t constantly supports the wide subcategory T t (Ab(S i )) via the function Φ κ so it must be orthogonal to the dimension vector d ∈ N + of the generator of T t (Ab(S i )).So the normal vector n i;t must be sign-coherent.
With the duals n i;t known to be sign-coherent, the vectors r i;t satisfy the following recursion (for t ′ k t in T n ): This result can be extracted from [GHKK18, Corollary 5.5 and 5.9] which only relies on the definitions of r i;t as T κ;t0 t (e i ).A more direct proof can be found in [Mou21, Proposition 7.7].Accordingly the dual vectors n i;t have the recursion Using the sign-coherence of the vectors n i;t , one sees their recursion is exactly the same as that of the c-vectors in Section 2 by induction on the distance from t to t 0 .This leads to the equality n i;t = c i;t for any t and i.Now with the c-vectors known to be sign-coherent, it is clear that the g-vectors defined in Section 2 have the same recursion as the vectors r i;t .
The following corollary is an easy consequence of the sign-coherence of c-vectors showed in the proof of Theorem 6.12.Corollary 6.13.The matrix B(t) = (b ij (t)) can be recovered from the c-vectors by the formula Definition 6.14.We define ∆ + κ to be the set of the cones {C t = C κ;t0 t | t ∈ T n }.We note that two cones C t and C s are regarded equal if they are the same subset of M R .Remark 6.15.By Proposition 6.11, the interior of C t has the characterization of being a connected component of M R \ supp(Φ κ ) (thus shall be called a chamber ).Two such cones C t and C s are hence either identical or their interiors do not intersect.When t k t ′ in T n , we have from their constructions.In this case, we say that they are related by mutation at their k-th facet.There is then an n-regular graph E(∆ + κ ) whose vertex set is ∆ + κ such that two cones are joint by an edge if and only if they intersect in codimension one.
Let σ = κ(t) be the labeled triangulation associated to t ∈ T n (with κ associated to t 0 ∈ T n ).Applying the previous constructions, but with t as a new root of T n , and with σ sitting on this new root, we obtain ∆ + σ = {C σ;t s | s ∈ T n } and the n-regular graph E(∆ + σ ).We now establish the concrete relation between this data with t as root, and the data ∆ + κ and E(∆ + κ ) with t 0 as root.Recall the notations at the beginning of Section 6.2.Consider the piecewise linear map Proposition 6.16.The map T κ;t0 t : R n → R n sends the cones in ∆ + σ to ∆ + κ bijectively such that for any s ∈ T n , we have Moreover, it induces an isomorphism between the graphs Proof.By definition, C σ;t s = T σ;t s (C + ) and C κ;t0 s = T κ;t0 s (C + ).Since T κ;t0 t • T σ;t s = T κ;t0 s , the equation (6.3) follows.For the rest of the statement, it suffices to show the case where σ and κ are related by a single flip.By Proposition 6.7, the map T κ;t0 t sends chambers to chambers with the inverse being T σ;t t0 .Here ∆ + κ is in fact the subset of chambers that are connected to C + via mutations at facets; see Remark 6.15.Hence the bijection follows.The graph isomorphism follows from that T κ;t0 t is a bijection on vertices and is also a local isomorphism between graphs.

The motivic Hall algebra of (Q, I)
In this section, we briefly review the motivic Hall algebra (as introduced by Joyce [Joy07]) of a quiver with relations following [Bri12] and [Bri17, Section 5].We note that quivers may have oriented cycles such as loops and two-cycles.There is another algebraic stack M (2) parametrizing short exact sequences 0 → E 1 → E 2 → E 3 → 0 in mod A up to certain equivalence ([Bri17, Section 4.4]).There are morphisms of stacks π i : M (2) → M sending a short exact sequence to its constituent E i for i = 1, 2, 3.In the following diagram, the morphism (π 3 , π 1 ) is of finite type and π 2 is representable and proper ([Bri17, Lemma 4.5]).There is a convolution product * on K(St/M) defined as where h fits into the Cartesian square . The algebra We complete H with respect to N + ⊂ N ⊕ obtaining the completed motivic Hall algebra Ĥ = d∈N ⊕ H d .Consider the N + -graded Lie subalgebra (under the commutator bracket) and its completion ĝHall = d∈N + H d .There is a corresponding pro-unipotent group ĜHall whose elements are formal symbols exp(x), which are in one-to-one correspondence with x ∈ ĝHall .Note that there is an embedding ĜHall ֒→ Ĥ by sending exp(x) ∈ ĜHall to the exponential series 1 This embedding respects multiplications because the multiplication in ĜHall is defined through the Baker-Campbell-Hausdorff formula.
7.2.Integration map.Let K(Var/C) be the Grothendieck ring of varieties over C. Consider the regular subring where L denote the class of the affine line A 1 and P n the projective space of dimension n in K(St/C).
We define H reg to be the submodule of H generated over K reg (St/C) by elements [X → M] with X a variety.
It follows from the above theorem that H reg can be equipped with the Poisson bracket Thus g reg := (H reg ) >0 /(L − 1) ⊂ H is a Lie algebra with the commutator bracket.Denote by ĝreg its completion and Ĝreg the corresponding pro-unipotent group.
extending the one on H reg .Then the N ⊕ -graded Poisson homomorphism (the integration map in Theorem 7.2) I : H reg → C[N ⊕ ] extends to I : C → B (as well as to the completions I : Ĉ → B).
For an element exp(a) ∈ Ĝreg , the following lemma describes how I(exp(a)) ∈ Ĝ acts on B as an automorphism.Note that via the embedding ĜHall ⊂ Ĥ, exp(a) can be viewed as an invertible element in Ĥ.

Scattering diagrams of P(κ)
Bridgeland [Bri17] introduced the Hall algebra scattering diagram for any quiver with relations (Q, I), and in the case where I is a Jacobian ideal the stability scattering diagram by applying the integration map (Theorem 7.2).The aim of this section is to apply this construction to the algebra P(κ), where we stress that in the presence of loops in the quiver, the construction remains valid.
Let A = C Q /I be as in the last section and θ ∈ M R a stability condition.
There is an open substack M ss (θ) ⊂ M parametrizing θ-semistable modules; see [Bri17, Lemma 6.3].Denote Note that the degree zero term of 1 ss (θ) is the unit 1 since the zero module is θ-semistable.So the element 1 ss (θ) belongs to the group ĜHall = 1 + Ĥ(Q, I) >0 where Ĥ(Q, I) >0 = d∈N + H(Q, I) d .Definition 8.1.Following [Bri17], we define the Hall algebra scattering diagram of A to be the function Remark 8.2.The function φ Hall A can be viewed as an algebraic avatar of The following absence of poles theorem is due to Joyce [Joy07].Here we present the form taken in [Bri17, Theorem 5.3] and [Bri11, Theorem 6.3].
Suppose that I is a Jacobian ideal.Hence by Theorem 7.2 the integration maps I : ĝreg → g and I : Ĝreg → Ĝ are valid.Furthermore, according to Joyce's Theorem 8.3, the range of the Hall algebra scattering diagram Definition 8.4.We define stability scattering diagram for A to be the function To describe the group element φ A (θ) = I(1 ss (θ)) for some θ, we express it by its Poisson action on the completion for some power series f (y) in a single variable y.In fact, it is easy to see that if To compute the power series f (y) for φ A (θ), we use the framed stability introduced below.Two m-framed A-modules M and N are said to be equivalent if there exists an isomorphism f : M → N intertwining their framings.
There is a fine moduli scheme F (d, m, θ) whose C-points parametrize m-framed θ-stable A-modules of dimension d up to equivalence; see [Bri17, Section 8.2] for a detailed description.In the case where I = 0, these moduli schemes were studied by Engel and Reineke earlier in [ER09].Now we present Bridgeland's formula on the wall-crossing action using framed representations.A closely related formula but in a slightly different setting is obtained by Reineke for acyclic quivers in [Rei10].
Theorem 8.6 ([Bri17, Theorem 10.2 and (11.1)]).Let m ∈ M ⊕ and d ∈ N + be a primitive dimension vector.Let θ be a general point in d ⊥ ⊂ M R (such that any n ∈ N orthogonal to θ is parallel to d).
Then the action of φ A (θ) on x m ∈ B is given by Here χ(X) takes the Euler characteristic of X(C) (in the analytic topology) for a C-scheme X.
Remark 8.7.Our setting for the above theorem is slightly more generalized than that of [Bri17] as we allow quivers to have loops.However, as long as the ideal I is the Jacobian ideal from a potential, the integration map I is valid as noted in the proof of Theorem 7.2.The rest then carries over to the present setting.
Next we apply the above theory to the case when A = P(κ) := C Q(κ) / ∂S(κ) coming from a triangulation κ of some (Σ, M, O).We write φ κ for the function φ P(κ) for simplicity.The motivic Hall algebra is written as H(mod P(κ)).
For part (2), suppose that the facet d is d i;t of C t .Then by Proposition 6.11, there is an embedding ι : mod H i → mod P(κ) such that the image is exactly Φ κ (θ).Here H i is the algebra C[ε]/ε 2 if i is pending or C if i is not pending.The simple module S i ∈ mod H i embeds as ι(S i ) the generator of Φ κ (θ).The embedding induces an injective algebra homomorphism ι : Ĥ(mod H i ) → Ĥ(mod P(κ)) .Take m = e * 1 and θ = 0 is trivial.The only dimension vectors that can appear are the multiples of e 1 .In the case where i is not pending, it follows from the exact same computation in [Bri17,Lemma 11.4] that the action is given by x → x(1 + y).So in this case f (y d ) = 1 + y d .Suppose that i is pending.Then we have (1) For d = 0, the only e * 1 -framed 0-stable module is the 0 module (with the framing fr being the 0 map), thus χ(F (0, e * 1 , 0)) = χ(pt) = 1.(2) For d = e 1 , up to equivalence there is only the simple module C with the framing being the quotient fr : (3) For d = 2e 2 , up to isomorphism there are two H i -modules H i and C ⊕ C where the later can never be e * 1 -framed 0-stable because the image of a framing is always proper (thus breaking the stability condition).The former H i with the framing fr : H i → H i being an isomorphism is e * 1 -framed 0-stable but every such is equivalent to each other.So we have χ(F (2e 1 , e * 1 , 0) = χ(pt) = 1.(4) For all other ke 1 for k ≥ 3, the moduli scheme F (ke 1 , e * 1 , θ) is empty.This is because the image of any framing must be a proper submodule, which however violates the stability.Now we conclude that in the case where i is pending, the power series f (y d ) is 1 + y d + y 2d .
We associate an element p Hall t ∈ ĜHall to each t ∈ T n and will see that its integration p t := I(p Hall t ) ∈ Ĝ is closely related to the mutations of cluster variables from x to x t .Recall the notations in Construction 6.10.Define where ǫ i := sgn(n i ki ) ∈ {±} (as these must be sign-coherent).Proof.This lemma completely follows from Bridgeland's theorem [Bri17, Theorem 6.5] that Φ Hall κ is consistent in the following sense.For any θ i ∈ M R in the interior of some C i ∈ ∆ + κ for i = 1, 2, there is an associated element φ Hall κ (θ 1 , θ 2 ) := 1 T (θ1) * 1 −1 T (θ2) ∈ ĜHall .In the meantime, one could consider a path γ going from C 1 to C 2 by crossing facets of cones in ∆ + κ .Since E(∆ + κ ) is connected, such paths always exist.Consider the product Here from left to right of the product, the sequence d i record the facets that γ crosses in order and we determine the signs ǫ i = +1 if the cross is in the same direction as the positive normal vector of d i and otherwise ǫ i = −1.A direct implication of the consistency of Φ κ is that p γ = φ Hall κ (θ 1 , θ 2 ) and in particular this does not depend on the path γ (see [Bri17, (6.7)]).The current lemma corresponds to the particular situation where θ 1 = θ and θ 2 is some interior point in C + .Proposition 8.10.We have the identity for any t ∈ T n and any i where g i;t is the g-vector associated to x i;t and F i;t (y) is the corresponding F -polynomial.
Proof.Note that the action of p t on x m is taken in the algebra B. The result follows from the chamber structure and the description of wall-crossings of the stability scattering diagram φ κ from Theorem 8.8.We prove it by induction on the distance from t to t 0 .Here the Poisson bracket on C[N ⊕ ] is given by the skew-symmetric matrix B = B(κ).
Suppose the identity is true for t and let t ′ k t be an edge in T n .Notice that since all wall-crossings act on x Bn y n for any n ∈ N ⊕ trivially, the identity implies for any i, We show in the following the identity is true for t ′ and k in the case where k is pending and the vector c k;t is positive; other cases are similar.In this case, we have the recursion The action of φ(d k;t ) on x gi;t for i = k is trivial because c k;t is orthogonal to any of those g i;t .Thus we have (by Theorem 8.8 and induction) The last step uses the recursive definition of F -polynomials in Section 2. 9. τ -tilting theory of P(κ) In this section we briefly review the τ -tilting theory of Adachi-Iyama-Reiten [AIR14] and study the case of P(κ).Definition 9.1.Let A be a finite dimensional algebra over an algebraically closed field k.A finitely generated A-module is said to be τ -rigid if Hom A (M, τ M ) = 0 where τ : mod A → mod A is the Auslander-Reiten translation.
In what follows we assume that A is of the form k Q /I with I an admissible ideal.Let n = |Q 0 |.Definition 9.2.
(1) A τ -rigid pair (M, P ) is a τ -rigid module M and a projective A-module such that We say that an A-module M is basic if its indecomposable summands are pairwise non-isomorphic.We say a pair (M, P ) basic if M and P are both basic.Denote by sτ -tilt A the set of all isomorphism classes of basic support τ -tilting pairs of A. It is clear that both (A, 0) and (0, A) are support τ -tilting pairs.
For two τ -rigid pairs (M 1 , P 1 ) and (M 2 , P 2 ), simply write (M 1 , P 1 )⊕(M 2 , P 2 ) for (M 1 ⊕M 2 , P 1 ⊕P 2 ).We say (M, P ) indecomposable if it is of the form (M, 0) or (0, P ) with M or P being indecomposable.The two support τ -tilting pairs in Theorem 9.3 are said to be AIR-mutations of each other.We use the notation (N, Q) = µ AIR k (M, P ) to indicate that the support τ -tilting pairs (M, P ) and (N, Q) are related by AIR-mutation of their k th indecomposable summand.One could regard AIR-mutation as an operation on sτ -tilt A. For example, write (M, P ) ∈ sτ -tilt A as (M 1 ⊕ M 2 , P ) such that M 2 is indecomposable.Then by Theorem 9.3, there exists either a unique M 3 ∈ mod A non-isomorphic to M 2 or a unique P 1 ∈ proj A such that (M 1 ⊕ M 3 , P ) or (M 1 , P ⊕ P 1 ) is support τ -tilting.
From now on we will often use a single letter (e.g.M instead of (M, P )) for a τ -rigid pair for simplicity.We shall call (0, A) the initial support τ -tilting pair.Suppose that the vertex set Q 0 has been identified with I = {1, . . ., n}.Then the indecomposable summands of (0, A) are (0, P i ) for i ∈ I, thus indexed by I.For any M ∈ sτ -tilt A whose indecomposable summands are indexed by I, denote by µ k (M) its mutation at the corresponding summand.Immediately one sees there is an association of M t ∈ sτ -tilt A to a vertex t ∈ T n such that M t0 = (0, A) and for t k t ′ , M t = µ k (M t ′ ).Denote by M i;t the indecomposable summand of M t of index i.We call M ∈ sτ -tilt A reachable if it is isomorphic to some M t .The set of isomorphism classes of reachable τ -rigid pairs is denoted by rsτ -tilt A. A τ -rigid pair is reachable if all its indecomposable summands are that of some reachable M t .
Let M := K 0 (proj A) be the Grothendieck group of the category of finitely generated projective modules over A. It is a lattice with a basis consisting of the classes of the indecomposable projective modules P 1 , P 2 , . . ., P n .
The lattice M is naturally dual to N := K 0 (mod A) by the pairing The basis e i := S i for N is then dual to e * i := P i for M .For T ∈ mod A, let Q 1 → Q 0 → T → 0 be its minimal projective presentation.We define the g-vector of T to be identified with a vector in Z n when expressed in the basis {e * i | i ∈ I}.To each τ -rigid pair M = (T, P ), we associate a g-vector g(M) := g(T ) − g(P ).
Note that our convention is opposite to that in [AIR14] where they define g(M) := g(P ) − g(T ).In our convention, the g-vector of (0, P i ) is e * i .
(1) Any τ -rigid pair is always a direct summand of some M ∈ sτ -tilt A.
(2) The g-vectors of indecomposable direct summands of any M ∈ sτ -tilt A form a Z-basis of M .
(3) The map M → g(M) is injective from the set of (the isoclasses of ) τ -rigid pairs to M .
For M any τ -rigid pair, define C M to be the closed rational polyhedral cone in M R generated by the g-vectors of its indecomposable summands.By Theorem 9.4, the cone C M is simplicial of maximal dimension if M is support τ -tilting, and the collection of all these cones {C M | M ∈ sτ -tilt A} together with all their faces form a simplicial fan in M R considered in [DIJ19].Two maximal cones C M and C M ′ intersect at one of their common facet if they are related by a mutation.We denote the dual graph of the reachable component of this simplicial fan by E(rsτ -tilt A).In other words, the vertices of E(rsτ -tilt A) are reachable objects in rsτ -tilt A up to isomorphism and two are joint by an edge if and only if they are obtained by a mutation from each other.Thus we call E(rsτ -tilt A) the exchange graph of rsτ -tilt A.
The following important theorem of Brüstle, Smith and Treffinger relates the cones C M with stability conditions of mod A. Suppose that M i;t is of the form (M i;t , 0).We use the following identity provided by Lemma 10.5 in the motivic Hall algebra: We show the following identity in the motivic Hall algebra to complete the proof of Theorem 10.4.
Notice that here N = N 1 are the same quotient module of M i;t (if N = N 1 , the set is empty).Consider another stack Hom(M i;t , T ) (and with a natural map to M T ) parametrizing morphisms from M i;t to T (see for example the construction in [Nag13, Section 6.1(D)]).Then there is a morphism of stacks from Z to Hom(M i;t , T ) over M T sending a short exact sequence above to the composition M i;t → N → E which induces an equivalence from the groupoid Z(C) to Hom(M i;t , T )(C).which finishes the proof.
In Table 1 we can see the quivers and skew-symmetrizable matrices of the triangulations, as well as the images r j 1 , r j 2 and r j 3 of the standard basis vectors under the transformations T + k1;κ1 Thus, with the notation of Proposition 6.8, Proposition 6.16 tells us that the cone C j = {λ 1 r j 1 + λ 2 r j 2 + λ 3 r j 3 | λ 1 , λ 2 , λ 3 ≥ 0} should be regarded as sitting inside the τ -tilting fan of the gentle algebra P(κ j ).

⋆Figure 1 .
Figure 1.The two types of triangles of a triangulation.Left: a non-singular triangle.Right: a singular triangle.

7. 1 .
Motivic Hall algebras.Let A be a finite dimensional algebra of the form C Q /I with I admissible.Recall the notations below Corollary 6.3.There is an algebraic C-stack M (for example, as defined in [Bri17, Section 4.2]) that parametrizes all objects in mod A. The stack M can be decomposed into a disjoint union M = d∈N ⊕ M d where each M d is an open and closed substack parametrizing objects of dimension vector d ∈ N ⊕ ; see [Bri17, Lemma 4.1].
π1) Let K(St/M) be the relative Grothendieck group of finite type C-stacks (with affine stabilizers) over M as in [Bri17, Definition 5.1].It is naturally a module over the Grothendieck ring K(St/C).The elements in K(St/M) are of the form [X f − → M] where X is in St/C and f is a morphism of C-stacks.
linear and makes K(St/M) into an associative K(St/C)-algebra; see [Bri12, Section 4].The algebra (K(St/M), * ) is the motivic Hall algebra of (Q, I) which we denote by H = H(Q, I).

.
Let m ∈ M such that m(d) ≥ 0 for any d ∈ N ⊕ .Let θ ∈ M R be a King's stability condition.Denote by P m the projective A-module P m := Definition 8.5.An m-framed θ-stable A-module is M ∈ mod A with a morphism fr : P m → M (the framing) such that(1) M is θ-semistable;(2) Any proper submodule M ′ ⊂ M containing the image of fr satisfies θ(M ′ ) > 0.

Theorem 8. 8 .
The stability scattering diagram φ κ : M R → Ĝ has the following properties.(1)Let C be any chamber in ∆ + κ .Then the function φ κ has constant value 1 in C • the interior of C;(2) Let d be any (relatively open) facet of C. Then the power series f (y d ) corresponding to φ κ (θ) = I(1 ss (θ)) for any θ ∈ d (as in (8.1)) is actually a polynomial(a) 1 + y d if Φ κ (θ) ∼ = mod C; (b) 1 + y d + y 2d if Φ κ (θ) ∼ = mod C[ε]/ε 2where d ∈ N + is the dimension vector of the generator of Φ κ (θ).Proof.Part (1) is true because Φ κ (θ) = 0 for any θ ∈ C • by Proposition 6.11.Thus we have φ κ Meanwhile the algebra C[[y]] embeds in C[[N ⊕ ]] by sending y to y d where d ∈ N + is the dimension vector of ι(S i ).We further have I(log[M id − → M])| y=y d = I(log(1 ss (θ))) ∈ ĝ.Therefore to compute the power series f (y d ) in y d for I(1 ss (θ)) ∈ Ĝ, we only need to compute the one of I([M id − → M]) for Ĥ(mod H i ) as a power series in a single variable y, and replace y with y d .Thus it amounts to the calculation for the action of I([M id − → M]) on the Poisson algebra C[[y]] ⊗ C C[x] with {y, x} = yx.By Theorem 8.6, the action of I(1 ss (θ)) on a monomial x m is given by x m → x m • θ(d)=0 χ(F (d, m, θ)) • y d .Now we are in mod C[ε]/ε 2 or mod C, and [M id − → M] = 1 ss (0)
Theorem 9.3 ([AIR14, Theorem 0.4]).Any basic almost complete support τ -tilting pair for A is a direct summand of exactly two basic support τ -tilting pairs.
Theorem 9.5 ([BST19, Proposition 3.13 and Theorem 3.14]).The function Φ A : M R → W(A) is constant in the interior of C M for any τ -rigid pair M = (T, P ) and has the valueJ M := T ⊥ ∩ ⊥ τ T ∩ P ⊥ .Moreover, the subcategory J M contains exactly n − |T | − |P | non-isomorphic stable modules.In particular, Φ A (θ) = 0 for any θ in the interior of C M for M ∈ sτ -tilt A.
d∈N ⊕ L −m(d) 1 T (θ),d = 1 T (θ) * d∈N ⊕ [Gr(M i;t , d) → M d ] where m = g(M i;t ).Proof.For M ∈ mod A, denote the disjoint union of quiver Grassmannians of all dimensions by Gr(M ) := d∈N ⊕ Gr(M, d).Denote T := T (θ).So the right-hand side of the identity can be rewritten as[M T → M] * [Gr(M i;t ) → M] = [Z π2•h − −− → M] where Z fits into the following Cartesian square Z M (2) M M T × Gr(M i;t ) M × M h π2 (π3,π1)A C-point of the stack Z by definition represents a short exact sequence in mod A0 → N → E → T → 0where N is a quotient module of M i;t and T ∈ T (which implies E ∈ T ).The set of morphisms between two such objects are pairs of isomorphisms (a, b) such that the following diagram commutes Thus we have [Z → M] = [Hom(M i;t , T ) → M] in the motivic Hall algebra (see [Bri17, Section 5.2]).By [AIR14, Proposition 2.4], for any T ∈ T (θ) of dimension vector d, we have the formula−g(M i;t )(d) = hom (M i;t , T ) − hom (T, τ (M i;t )).However since T ∈ T = Fac(M t ) (by Proposition 9.6) and M i;t is τ -rigid, we have hom (T, τ (M i;t )) = 0.Thus the fiber at any C-point in M T ,d of the map Hom(M i;t , T (θ)) → M T is isomorphic to C −m(d) .Therefore we have d∈N ⊕ L −m(d) 1 T ,d = [Hom(M i;t , T ) → M]

Figure 2 .
Figure 2. Five triangulations of the digon with two orbifold points of order 3. Adjacent triangulations are related by a flip.