Locally free Caldero-Chapoton functions via reflections

We study the reflections of locally free Caldero-Chapoton functions associated to representations of Geiss-Leclerc-Schr\"oer's quivers with relations for symmetrizable Cartan matrices. We prove that for rank 2 cluster algebras, non-initial cluster variables are expressed as locally free Caldero-Chapoton functions of locally free indecomposable rigid representations. Our method gives rise to a new proof of the locally free Caldero-Chapoton formulas obtained by Geiss-Leclerc-Schr\"oer in Dynkin cases. For general acyclic skew-symmetrizable cluster algebras, we prove the formula for any non-initial cluster variable obtained by almost sink and source mutations.


Introduction
Cluster algebras are invented by Fomin and Zelevinsky [FZ02] in connection with dual canonical bases and total positivity.A cluster algebra A(B) associated to a skew-symmetrizable matrix B is a subalgebra of Q(x 1 , . . ., x n ) generated by a distinguished set of generators called cluster variables obtained by certain iterations called mutations.A first remarkable feature is that they turn out to be Laurent polynomials with integer coefficients.Much effort has been taken to give formulas or interpretations of these Laurent polynomials since the invention of cluster algebras.
The classification of finite type cluster algebras is identical to the Cartan-Killing classification of finite root systems [FZ03].In particular, non-initial cluster variables are naturally in bijection with positive roots of the corresponding root system.Meanwhile, Gabriel's theorem [Gab72] states that the indecomposable representations of a Dynkin quiver are in bijection with positive roots, thus further in bijection with non-initial cluster variables.Caldero and Chapoton [CC06] showed that any non-initial cluster variable can be obtained directly from its corresponding quiver representation as the generating function of Euler characteristics of quiver Grassmannians of subrepresentations, which we now call the Caldero-Chapoton function.
Caldero and Keller [CK06] have extended the above correspondence to cluster algebras associated to acyclic quivers, that is, non-initial cluster variables of A(Q) are in bijection with real Schur roots 2020 Mathematics Subject Classification.13F60, 16G20. in the root system associated to Q, and are again equal to the Caldero-Chapoton functions of the corresponding indecomposable rigid representations.
Geiß, Leclerc and Schröer [GLS17] have defined a class of Iwanaga-Gorenstein algebras H associated to acyclic skew-symmetrizable matrices, generalizing the path algebras of acyclic quivers.These algebras are defined over arbitrary fields so certain geometric constructions valid for quivers carry over to them.The authors introduced locally free Caldero-Chapoton functions for locally free H-modules and showed that in Dynkin cases those of locally free indecomposable rigid modules are exactly noninitial cluster variables [GLS18].Their proof however does not explicitly interpret mutations of cluster variables in terms of representations but actually relies on [GLS16] a realization of the positive part of the enveloping algebra of a simple Lie algebra using locally free H-modules and a known connection between cluster algebras of Dynkin types and (dual) enveloping algebras [YZ08].
In this paper, we study the recursion of locally free Caldero-Chapoton functions of modules under reflection functors.These functors, introduced in [GLS17] for H-modules, generalize the classical Bernstein-Gelfand-Ponomarev reflection functors [BGP73] for representations of Dynkin quivers.We show that this recursion coincides with cluster mutations that happen at a sink or source, leading to our main results: (1) Non-initial cluster variables of a rank 2 cluster algebra are exactly locally free Caldero-Chapoton functions of locally free indecomposable rigid H-modules.
(2) In Dynkin cases, we obtain a new proof of the aforementioned correspondence in [GLS18] which does not rely on results in [GLS16] and [YZ08].
(3) In general, any non-initial cluster variable obtained from almost sink and source mutations is expressed as the locally free Caldero-Chapoton function of a unique locally free indecomposable rigid H-module.
We next provide a more detailed summary of this paper.
1.1.Rank 2 cluster algebras.Let b and c be two non-negative integers.The cluster algebra A(b, c) is defined to be the subalgebra of Q(x 1 , x 2 ) generated by cluster variables {x n | n ∈ Z} satisfying relations n n is even.
Every cluster variable x n is viewed as a rational function of x 1 and x 2 .The cluster algebras A(b, c) are said to be of rank 2 because the cardinality of each cluster {x n , x n+1 } is 2. Let c 1 and c 2 be two positive integers such that c 1 b = c 2 c.Let g := gcd(b, c).Let Q be the quiver .
Following [GLS17], we define H = H(b, c, c 1 , c 2 ) to be the path algebra CQ modulo the ideal Denote by rep H the category of finitely generated left H-modules.For any M ∈ rep H and i ∈ {1, 2}, the subspace M i := e i M is a finitely generated module over the algebra H i := e i He i ∼ = C[ε]/(ε ci ).We say that M ∈ rep H is locally free (l.f. for short) if M i is a free H i -module for i = 1, 2. For such M , we define its rank vector where m i denotes the rank of M i as a finitely generated free H i -module.Let E 1 (resp.E 2 ) be the locally free module with rank vector (1, 0) (resp.(0, 1)).
To any locally free M ∈ rep H with rank M = (m 1 , m 2 ), we associated a Laurent polynomial (1.1) where Gr l.f.(r, M ) is the locally free quiver Grassmannian (see Definition 4.1) which is a quasi-projective complex variety parametrizing locally free submodules of M with rank vector r, and χ(•) denotes the Euler characteristic in complex analytic topology.The Laurent polynomial X M is the locally free Caldero-Chapoton function associated to M .Our first main result is Theorem 1.1 (Theorem 5.7).For bc ≥ 4, there is a class of locally free indecomposable rigid In fact, this equality gives a bijection between all locally free indecomposable rigid H-modules (up to isomorphism) and non-initial cluster variables of A(b, c).
Remark 1.2.When bc < 4, the cluster variables x n are periodic, that is, there are only finitely many distinguished x n .These cases actually fall into another class of cluster algebras of finite types (or Dynkin types), which will be discussed in Section 1.3.
Let T n be the infinite simple n-regular tree emanating from a given root t 0 such that the n edges incident to any vertex are numbered by {1, . . ., n}.We associate Σ to t 0 , and inductively if Σ t = (B t = (b t ij ), (x 1;t , . . ., x n;t )) is associated to some vertex t ∈ T n , then is associated to t ′ for t k t ′ in T n , where µ k (B t ) is Fomin-Zelevinsky's matrix mutation of B t in direction k and In this way, each t ∈ T n is associated with a well-defined seed (B t , (x 1;t , . . ., x n;t )) where B t is an n × n integral skew-symmetrizable matrix and each rational function x i;t ∈ Q(x 1 , . . ., x n ) is called a cluster variable.The cluster algebra A(B) is then defined to be the subalgebra of Q(x 1 , . . ., x n ) generated by all cluster variables.The exchange between Σ t and Σ t ′ for t k t ′ is usually called a cluster mutation.
1.3.Locally free Caldero-Chapoton formulas.Let (C, D, Ω) be an n × n symmetrizable Cartan matrix C, a symmetrizer D, and an acyclic orientation Ω (see Section 2 for precise definitions).Geiß, Leclerc and Schröer [GLS17] have associated a finite dimensional K-algebra H = H K (C, D, Ω) to the triple (where K is a field), generalizing the path algebra of an acyclic quiver.Similar to the rank 2 case, there are locally free H-modules, forming the subcategory rep l.f.H ⊂ rep H. Analogously, each M ∈ rep l.f.H has its rank vector rank M ∈ N n .Let E i be the locally free module with rank vector We define the bilinear form −, − H : Z n ⊗ Z n → Z such that on the standard basis The skew-symmetrization of −, − H (on the basis (α i ) i ) defines a skew-symmetric matrix DB (thus defining a skew-symmetrizable matrix B = B(C, Ω) = (b ij ) actually having integer entries), i.e. explicitly, we have Definition 1.4 ([GLS18, Definition 1.1]).For a locally free H C (C, D, Ω)-module M , the associated locally free Caldero-Chapoton function is where Suppose that k ∈ {1, . . ., n} is a sink of Ω and let s k (H) := H(C, D, s k (Ω)) be the reflection of H at k.There is the (sink) reflection functor (see Section 3) which generalizes the classical BGP reflection functor on quiver representations.
The following proposition gives an algebraic identity on Caldero-Chapoton functions under reflections.It is the key recursion that makes connection with cluster mutations.Proposition 1.5 (Proposition 4.7 and Corollary 4.8).Let M be a locally free H-module such that the map M k,in is surjective.Then the reflection M ′ := F + k (M ) ∈ rep s k (H) is also locally free, and where For B = B(C, Ω), an easy calculation shows that when k is a sink or source, µ k (B) = B(C, s k (Ω)).This hints that the recursion in Proposition 1.5 is closely related to cluster mutations as in (1.2) at sink or source, which actually leads to our next main result Theorem 1.6 (Theorem 7.4).For any non-initial cluster variable x in A(B) obtained by almost sink and source mutations, there is a unique locally free indecomposable rigid H-module M such that x = X M .
If C is of Dynkin type, it is known that any non-initial cluster variable can be obtained by almost sink and source mutations.Therefore our method provides a new proof of the following theorem of Geiß-Leclerc-Schröer [GLS18, Theorem 1.2 (c) and (d)].
Theorem 1.7 (Theorem 6.3).If C is of Dynkin type, then the map M → X M induces a bijection between isomorphism classes of locally free indecomposable rigid H-modules and the non-initial cluster variables of A(B).
1.4.Other related work.Caldero and Zelevinsky [CZ06] studied how the Caldero-Chapoton functions of representations of generalized Kronecker quivers behave under reflection functors and used them to express cluster variables of skew-symmetric rank 2 cluster algebras.Our result in rank 2 can thus be seen as a generalization to the skew-symmetrizable case.
We remark that the recursion in Proposition 1.5 has already been achieved in the skew-symmetric case for any reflection, not necessarily at sink or source, of any quiver by Derksen-Weyman-Zelevinsky [DWZ08,DWZ10].Extending their theory, especially obtaining Caldero-Chapoton type formulas, to the skew-symmetrizable case in full generality remains an open problem; see for example [Dem, LFZ16, GLF17, GLF20, LA19, BLA21].
There are several earlier work generalizing Caldero-Chapoton type formulas (or in the name of cluster characters) to the skew-symmetrizable case.Demonet [Dem11] has obtained cluster characters for acyclic skew-symmetrizable cluster algebras by extending [GLS11] to an equivariant version.Rupel [Rup11,Rup15] has used representations of valued quivers over finite fields to obtain a quantum analogue of Caldero-Chapoton formula for quantum acyclic symmetrizable cluster algebras.The representation theories used in those work are however different from the one initiated in [GLS17] which we follow in this paper.
Fu, Geng and Liu [FGL20] have obtained locally free Caldero-Chapoton formulas for type C n cluster algebras with respect to not necessarily acyclic clusters.In the upcoming work [LFMb] with Labardini-Fragoso, we prove locally free Caldero-Chapoton formulas with respect to any cluster for cluster algebras associated to surfaces with boundary marks and orbifold points.1.5.Organization.The paper is organized as follows.In Section 2, we recall the algebras H(C, D, Ω) defined by Geiß-Leclerc-Schröer and review some necessary notions including locally free H-modules.In Section 3, we review the definition of reflection functors for H-modules and their properties.In Section 4 we study the reflections of F -polynomials of locally free modules, leading to a cluster type recursion of locally free Caldero-Chapoton functions.In Section 5, Section 6 and Section 7, we apply the results obtained in Section 4 to rank 2, Dynkin, and general cases respectively to obtain locally free Caldero-Chapoton formulas of cluster variables for skew-symmetrizable cluster algebras.

Acknowledgement
The author would like to thank Daniel Labardini-Fragoso for helpful discussions.LM is supported by the Royal Society through the Newton International Fellowship NIF\R1\201849.

The algebras H(C, D, Ω)
In this section, we review the algebras H(C, D, Ω) defined in [GLS17] and some relevant notions.
Following [GLS17], we define (over some ground field K) the algebra H := H K (C, D, Ω) to be the path algebra KQ modulo the ideal generated by the relations The opposite orientation of Ω is which clearly is an orientation of C. We denote H * := H(C, D, Ω * ).

2.2.
From now on, we will always assume that Ω is acyclic.For i ∈ I, let As a right H j -module, i H j is free of rank −c ji with the basis given by For more details, we refer to [GLS17, Section 5.1].

2.3.
Let rep H denote the category of finitely generated left H-modules.We will often treat rep H as the equivalent category of quiver representations of Q satisfying relations in I.For M ∈ rep H and i ∈ I, the subspace M i := e i M is a finitely generated module over H i .
Definition 2.1.We say that M ∈ rep H is locally free if for each i ∈ I, the H i -module M i is free, i.e. is isomorphic to H ⊕ri i for some r i ∈ N.
Denote the full subcategory of locally free H-modules by rep l.f.H.For M ∈ rep l.f.H, define its rank vector where r i stands for the rank of e i M as a free H i -module.Let E i be the locally free H-module such that rank We remark that H i itself is the only (non-zero) indecomposable projective (also injective) 2.4.Any M ∈ rep H is determined by the H i -modules M i for i ∈ I and the H i -module homomorphisms for any (i, j) ∈ Ω.We will later describe an H-module M by specifying the data When there is no ambiguity, the subscript H j under the tensor product will be omitted, hence the simplified notation i H j ⊗ M j .

Reflection functors
The Bernstein-Gelfand-Ponomarev reflection functors [BGP73] are firstly defined to relate representations of an acyclic quiver Q with that of the reflection of Q at a sink or source vertex.These functors have been generalized to act on representations of H(C, D, Ω) in [GLS17].In this section, we recall their definitions and review some useful properties.
For an orientation Ω of C and i ∈ I, the reflection of Ω at i is the following orientation of We denote s i (H) := H(C, D, s i (Ω)).Denote The only cases we will need are reflections at a sink or source.
3.1.Sink reflection.Let k be a sink of Ω.In this subsection we define the sink reflection functor The isomorphism ρ : k H j → Hom Hj ( j H k , H j ) (2.1) induces the isomorphism Then further by the tensor-hom adjunction, we have under which β jk corresponds explicitly to the map (3.1) where is defined by setting f ′ i = f i for i = k and f ′ k to be naturally induced between kernels.Thus F + k is functorial.
3.2.Source reflection.For k a source of Ω, we define the source reflection functor where each component M jk : M k → k H j ⊗M j for (k, j) ∈ Ω is defined through the structure morphism M jk as follows.In fact, by the tensor-hom adjunction, we have the canonical isomorphism where the later is further identified with Hence there is some We define where Analogously to F + k , it is clear that F − k is also functorial.
3.3.Some properties of reflection functors.For i ∈ I, let S i be the simple H-module supported at the vertex i.Note that S i is at the same time the socle and the top (or head) of E i .The following lemma is straightforward.
Lemma 3.1.For any M ∈ rep H, we have Proposition 3.2 ([GLS17, Proposition 9.1 and Corollary 9.2]).The pair of reflection functors are (left and right) adjoint (additive) functors.They define inverse equivalences on subcategories We now focus on the actions of reflection functors on locally free modules.
Lemma 3.3 ([Gei18, Lemma 3.6]).Suppose that k is a sink (resp.source) of Ω and M a locally free rigid H-module, with no direct summand isomorphic to E k .Then we have Hom H (M, E k ) = 0 (resp.Hom H (E k , M ) = 0).In particular, the map M k,in (resp.M k,out ) is surjective (resp.injective).
Proof.The case where k is a sink is [Gei18, Lemma 3.6].The other case is simply a dual version.
Let L = Z n .We think of rank vectors of locally free H-modules as living in L via N n ⊂ Z n .For i ∈ I, define the reflection Proposition 3.4 ([GLS17, Proposition 9.6] and [Gei18, Lemma 3.5]).If k is a sink (resp.source) of Ω and M is a locally free rigid H-module, then Remark 3.5.We remark that the functor Remark 4.2.It is clear that the set Gr l.f.(r, M ) can be realized as a locally closed subvariety of the product of ordinary Grassmannians i∈I Gr(c i r i , M i ).We take its analytic topology and denote by χ(•) the Euler characteristic.
Definition 4.3.For M ∈ rep l.f.H, we define its locally free F -polynomial as Recall that we have defined in Section 1.3 the bilinear form −, − H : Z n × Z n → Z and the skewsymmetrizable matrix B = (b ij ) associated to (C, Ω).Definition 4.4.For M ∈ rep l.f.H with rank M = (m i ) i∈I , the associated locally free Caldero-Chapoton function is the Laurent polynomial where v i := x 1/ci i .
Remark 4.5.Using the F -polynomial F M , the Caldero-Chapoton function X M can be rewritten as We note that every term in the summation is an actual monomial since [−b ij ] + m j + b ij r j ≥ 0 because we need r j ≤ m j for the quiver Grassmannian to be non-empty.Moreover, for r = 0 and r = rank M , χ(Gr l.f.(r, M )) = 1 and the two corresponding monomials are coprime.Therefore n i=1 x mi i can be characterized as the minimal denominator when expressing X M = f /g as a quotient of a polynomial f and a monomial g.We thus call rank M the d-vector of the Laurent polynomial X M .
Example 4.6.For k ∈ I and E k ∈ rep l.f.H, the only non-empty locally free quiver Grassmannians are Gr l.f.(0, E k ) = {0} and Gr l.f.(α k , E k ) = {E k }.Thus we have The key recursion.The following is the key proposition on the recursion of F -polynomials under reflections.
Proposition 4.7.Let M ∈ rep l.f.H be of rank (m i ) i∈I and k be a sink of H. Suppose that the map Their F -polynomials satisfy the equation where Proof.The first half of the statement is simply a recast of Proposition 3.4 without the hypothesis and conclusion on the rigidity.Explicitly, we observe that M ′ k and M k naturally fit into the following exact sequence (of free H k -modules) Since M j is free over H j of rank m j , we have that for any j ∈ Ω(k, −), the bimodule k H j ⊗ M j is isomorphic to k H ⊕mj j , thus a free left H k -module of rank b kj m j .Then the calculation on m ′ k follows from the exact sequence.
Next we prove the recursion on F -polynomials.
Step I. Let e = (e i ) i ∈ N I be a rank vector.Decompose Gr l.f.(e, M ) into constructible subsets Ze;r (M ) as follows.Let N ⊂ M be a locally free submodule.Denote where E(•) denotes the injective hull (of an H k -module).Then Gr l.f.(e, M ) is a disjoint union of (finitely many) Ze;r (M ) when r runs over N and thus χ(Gr l.f.(e, M )) = r∈N χ( Ze;r (M )).
Step II.Meanwhile for M ′ ∈ rep l.f.s k (H), a rank vector e and s ∈ N, let Xe;s (M ′ ) be the constructible subset of Gr l.f.(e, M ′ ) consisting of locally free submodules N ⊂ M ′ such that where we denote and F (•) stands for the (isomorphism class of) maximal free submodule of an H k -module.Decomposing Gr l.f.(e, M ′ ) into subsets Xe;s (M ′ ) where s runs over N, we have χ(Gr l.f.(e, M ′ )) = s∈N χ( Xe;s (M ′ )).
Step III.Let e ′ denote the rank vector (with n − 1 entries) obtained from e by forgetting the k-th component.Define for r ∈ N the subset (1) (N i ) i =k is closed under the actions of arrows in (the quiver of) H that are not incident to k; (2) The injective hull of M k,in (N (k,−) ) is of rank r.
There is the natural forgetful map The fiber over a point The space V is computed in Proposition 4.9.According to its description in the proof, V is determined (up to isomorphism) by the isomorphism class of M k,in (N (k,−) ) as H k -module.This means we can if necessary decompose Z e ′ ;r (M ) further into (finitely many) locally closed subsets j∈J Z j so that each π −1 (Z j ) π − → Z j is a fiber bundle.It is shown in Proposition 4.9 that the Euler characteristic of a fiber V is m k −r e k −r .Therefore we have For M ′ and s ∈ N, we define the subset (1) (N i ) i =k is closed under the actions of arrows in (the quiver of) s k (H) not incident to k; (2) The rank of a maximal free H k -submodule of (M ′ k,out ) −1 (N (−,k) ) is s.Then we have the forgetful map having Euler characteristic, according to Corollary 4.11, χ(W) = s e k .Analogous to (4.2), we have Step IV.Now recall that M and M ′ are reflections of each other, i.e.M ′ = F + k (M ), with rank vectors (m i ) i and (m ′ i ) i respectively such that We claim that for any e ′ = (e i ) i =k (4.4) b kj e j .
In fact, let N i be a free submodule (of rank e i ) of M i for any i = k and then we have from (4.1) the following short exact sequence of H k -modules where by our abuse of notation, the H k -module N (−,k) (which is for the orientation s k (Ω)) is actually the same as where B/E(A) is free.The number of indecomposable summands of E(A)/A is easily seen to be rank E(A) − rank F (A).The number of indecomposable summands of C, which equals rank E(C), is just rank B − rank F (A).We now have the equality b kj e j .
Then one sees from their definitions that X e ′ ,s (M ′ ) and Z e ′ ;r (M ) for any r and s such that r + s = rank N (k,−) define the exact same tuples (N i ) i =k in i =k Gr l.f.(e i , M i ), hence (4.4).
Now we rewrite the F -polynomials as Let Corollary 4.8.In the setting of the above Proposition 4.7, we have where Proof.We first derive from Proposition 4.7 that where ŷj : In fact, this equality directly follows from the algebraic equations Then we spell out the two sides of the desired equation in the form of Remark 4.5.Now it amounts to show which is straightforward to check.
We finish this section by proving the following proposition (and Corollary 4.11) which has been used in the proof of Proposition 4.7.Proposition 4.9.Let M be a (finitely generated) C[ε]/(ε n )-module whose (any) maximal free submodule F (M ) is of rank m.Let L ⊂ M be a submodule such that E(L) the injective hull of L is of rank ℓ.Assume further that L is contained in a free submodule of M .Then for any integer e between ℓ and m, the variety Proof.For any It is clear that M (k) is contained in M (k) .These vector spaces fit into short exact sequences and filtrations is a C-vector subspace of dimension ie satisfying the following conditions: (1) A point in F n clearly determines a submodule N (n) of M containing L, which is free of rank e simply for dimension reasons.A free basis can be obtained by choosing a (vector space) basis of N (1) and taking a lift in N (n) through ε n−1 .Sending N ∈ V to the filtration given by (N (i) ) n i=1 induces an isomorphism from V to F n .There are maps π k+1,k : F k+1 → F k forgetting the largest subspace N (k+1) in a flag.We next show that (1) F 1 is isomorphic to the Grassmannian Gr(e − ℓ, m − ℓ); (2) each π k+1,k is a fiber bundle with fiber being an affine space.
The assumption that E(L) is of rank ℓ implies L (1) ∼ = C ℓ , and that Let (N (i) ) k i=1 be a point in F k .We collect several auxiliary vector subspaces of M (k+1) ) to be used later.
• Let P (k) be the preimage of N (k) in M (k+1) of the (surjective) map ε : M (k+1) → M (k) .It fits in the short exact sequence • The two subspaces L (k+1) and N (k) intersect to give exactly L (k) .Their span W (k+1) := L (k+1) + N (k) thus has dimension ke + ℓ k+1 .Notice W (k+1) ∩ K (k+1) = N (1) .In fact, for any a ∈ N (k) and b ∈ L (k+1) such that ε(a+b) = 0, we have ε We claim that X k+1 is isomorphic to the affine space A (e−ℓ k+1 )•(m k+1 −e) , which follows from Proof.Quotient by E the common subspace for all, the space X is identified with It is then standard that the space X ′′ (of vector subspaces of a given dimension transversal to a fixed vector subspace of the complimentary dimension) is isomorphic to Now in the context of the above lemma, let , and We see that X k+1 ∼ = A (e−ℓ k+1 )•(m k+1 −e) .Now that the fiber of each π k+1,k is an affine space, the variety V ∼ = F n is then homotopic to the base Proof.In the setting of Proposition 4.9, letting L = 0, the result follows.

The rank 2 case
The purpose of this section is to prove Theorem 1.1 (Theorem 5.7).In fact, the construction of the algebra H in Section 1.1 can be seen as within the general framework introduced in Section 2, which we explain in below.
Let D = diag(c 1 , c 2 ) be a symmetrizer of C. One easily sees that H := H(C, D, Ω) is the same as H(b, c, c 1 , c 2 ) defined in Section 1.1, where we denote the arrow α k there by α Denote H * := H(C, D, Ω * ).Then there are reflection functors We omit the subscripts in the reflection functors since the sign ± already specifies which vertex the reflection is performed at.Next we define a class of modules obtained from iterative reflections.Definition 5.1.We define for n ≥ 0 the following H-modules Remark 5.2.Let us clarify the above construction of M (n + 3).For any n ≥ 0 and 0 ≤ k ≤ n, let Now we have a sequence of functors F (k) := F + : rep H (k+1) → rep H (k) for 0 ≤ k ≤ n − 1.Then M (n + 3) is obtained by iteratively applying F (k) , i.e.
if n is even, The modules M (−n) are defined using F − in a similar way.
Proof.It follows from Proposition 3.4 that any M (n + 3) or M (−n) is locally free and rigid because so is E 1 or E 2 .Now assume that for any 0 ≤ k ≤ n the modules M (k + 3) and M (−k) are all indecomposable and that the map M 1,out is surjective for M (k + 3) and M 2,in is injective for any M (−k).Denote the rank vectors by α(n) := rank M (n).Now by the construction of M (n + 4) and M (−(n + 1)) and Proposition 3.4, we have that these two modules are locally free and rigid, and where n ∈ {1, 2} is congruent to n modulo 2. It is then known that (in the case bc ≥ 4) both α(n + 4) and α(−(n + 1)) are real positive roots of C (other than the simple roots α 1 and α 2 ) and in particular are strictly positive linear combinations of α 1 and α 2 ; see for example [SZ04, Section 3.1].By Remark 3.5, M (n + 4) and M (−(n + 1)) are also indecomposable.So they cannot have any summand isomorphic to E 1 or E 2 .Now by Lemma 3.3, the induction is completed.
Remark 5.4.In fact, by [GLS20], locally free indecomposable rigid H-modules are parametrized by their rank vectors as real Schur roots of C (depending on Ω).Since the rank vectors α(n) = rank M (n) for n ≤ 0 and n ≥ 3 are exactly the real Schur roots (see for example [SZ04]), we know that {M (n) | n ≤ 0 or n ≥ 3} fully lists locally free indecomposable rigid H-modules.
We next calculate M = M (5 which is a free H 1 -module of rank 5 having the basis Thus M 1 ∼ = C 15 as a vector space with the basis {ε k 1 e j | 1 ≤ j ≤ 5, 0 ≤ k ≤ 2}.The action of α 21 on this basis is calculated in table below (only non-zero terms shown).
For example, we have by (3.1) that Recall the definition of the cluster variables x n of rank 2 cluster algebras given in Section 1.1.We rewrite Corollary 4.8 in the current rank 2 situation.
The following is the main result of this section.
Theorem 5.7.The map Proof.We prove X M(n+3) = x n+3 and X M(−n) = x −n for n ≥ 0 by induction on n using the recursion Corollary 5.6.For n = 0, we have M (0) = E 2 and M (3) = E 1 , thus Assume the statement is true for some n ≥ 0. By the obvious symmetry between H and H * by switching the orientation, we have that where the notation M (n + 3, H * ) stresses that the module M (n + 3, H * ) is constructed for H * instead of H.We would like to apply Corollary 5.6 to M := M (n + 3, H * ) and the sink reflection functor The condition that M 1,in is surjective is guaranteed by Lemma 5.3 (applied to the algebra H * ).Then by Corollary 5.6, we have 2 ).Immediately we obtain The proof for M (−n) for n ≥ 0 uses a similar induction.Now that x n = X M(n) is a Laurent polynomial in x 1 and x 2 , the unique minimal common denominator (up to a scalar) is easily seen to be x α(n) .Since the positive roots α(n) are distinct, so are the cluster variables x n .
Remark 5.8.To a pair (b, c) ∈ Z 2 >0 , one can also associate an algebra H(b, c) defined as the path algebra CQ of the quiver Q = 1 2 ε1 α ε2 modulo the relations ε c 1 = 0 and ε b 2 = 0.When b and c are coprime, the algebra H(b, c) is the same as H(C, D, Ω) for D = [ c 0 0 b ].However, when b and c are not coprime, the algebra H is not included in the construction of [GLS17].We note that Theorem 1.1 can be easily adapted to using the algebras H(b, c).In the case b = c = 2, the algebra H(b, c) coincides with a construction in [LFMa] where the ordinary Caldero-Chapoton functions are shown to give cluster variables of a generalized cluster algebra.

Dynkin cases
The purpose of this section is to give a new proof of Theorem 1.7 (Theorem 6.3).Let C be of Dynkin type and B = B(C, Ω) the associated skew-symmetrizable matrix.Denote by ∆ + C the set of positive roots associated to C.
The following lemma is well-known; see for example [Kir16, Chapter 3].
Lemma 6.2.Let β be a positive root for C and Ω be an orientation.Then there always exists a sequence i = (i 1 , . . ., i k+1 ) adapted to Ω for β such that It is clear that for such a sequence i in the above lemma, i k+1 must not be equal to i k .To any sequence i = (i 1 , . . ., i k+1 ), consider the following path in T n Recall the cluster mutations as introduced in Section 1.3 which generate cluster variables.Recursively performing cluster mutations from t 0 to t k+1 we obtain an n-tuple (x 1;t k+1 , . . ., x n;t k+1 ) of cluster variables associated to t k+1 .we denote x i := x i k+1 ;t k+1 .Suppose that i is adapted to Ω for a positive root β as in Lemma 6.2.Note that i ℓ is a source for the orientation We have source reflection functors Theorem 6.3 ([GLS18, Theorem 1.2]).The map M → X M induces a bijection between isomorphism classes of locally free indecomposable rigid H(C, D, Ω)-modules and the non-initial cluster variables of the cluster algebra A(B).
Proof.For i adapted to Ω for a positive root β, we show by induction on the length of i that (⋆) M i is locally free, indecomposable and rigid with rank vector β, and that X M i = x i .
If i = (i) is of length one, then M i = E i .Notice that i is not necessarily a sink or source.As in Example 4.6, we have x [bji ]+ j Assume that (⋆) is true for any i of length no greater than k ∈ N. Let i = (i ℓ ) k+1 ℓ=1 and i ′ be the sequence (i 2 , i 3 , . . ., i k+1 ) ∈ I k , which is adapted to the orientation s i1 (Ω).By assumption, the module H) is locally free, rigid and indecomposable with rank vector β ′ := s i2 . . .s i k (α i k+1 ) and that X M i ′ = x i ′ ∈ A(µ i1 (B)).Since s i1 (β ′ ) = β and β ′ ∈ ∆ + (C), the positive root β ′ cannot be a positive multiple of α i1 .Thus the indecomposable module M i ′ does not have any direct summand isomorphic to E i1 .By Lemma 3.3, the map (M i ′ ) i1,out is injective.By Proposition 3.4, we have that M i = F − i1 (M i ′ ) is locally free, rigid and indecomposable with rank vector β = s i1 (β ′ ).
Hence X M i = x i ∈ Z[x ± 1 , . . ., x ± n ], which completes the induction and proves (⋆).By [GLS17, Theorem 1.3], the module M i constructed from i only depends on the positive root β and the (thus well-defined) map β → M i induces a bijection from ∆ + C to locally free indecomposable rigid H-modules (up to isomorphism).Thus the formula x i = X M i implies that the cluster variable x(β) := x i also only depends on β.In view of Remark 4.5, each x(β) has d-vector β.By [FZ03], these x(β) are exactly the non-initial cluster variables of A(B), hence the desired bijection.

Beyond Dynkin and rank 2 cases
For (C, D, Ω) which is neither of Dynkin type nor in rank 2, in general we will not be able to reach all locally free indecomposable rigid modules by reflections.In this section, we prove locally free Caldero-Chapoton formulas for cluster variables that can be obtained by almost sink and source mutations.In particular, any cluster variable on the bipartite belt [FZ07] can be obtained this way.
Definition 7.1 (cf.[Rup11]).A sequence i = (i 1 , . . ., i k+1 ) ∈ I k+1 is called admissible to an orientation Ω of C if i 1 is a sink or source for Ω, i 2 is a sink or source for s i1 (Ω), . . .i k is a sink or source for Let B = B(C, Ω) and A(B) be the (coefficient-free) cluster algebra associated to B. As defined in Section 6, for an arbitrary sequence i, there is the cluster variable x i ∈ A(B) by successive cluster mutations.
Definition 7.2.We say that the cluster variable x i ∈ A(B) corresponding to a sequence i = (i ℓ ) ℓ is obtained by almost sink or source mutations if i is admissible to Ω.
Remark 7.3.We note that by definition the last index i k+1 can be arbitrary in I.It is the only step in the mutation sequence (µ i1 , . . ., µ i k+1 ) that may not be at a sink or source, thus the term almost.
The following is our main result in this section.
Theorem 7.4.For any admissible sequence i, either the cluster variable x i is an initial one or there is a locally free indecomposable rigid H(C, D, Ω)-module M i such that X M i (x 1 , . . ., x n ) = x i .Moreover, the module M i is uniquely determined (up to isomorphism) by x i .
Proof.We slightly modify the functors F ± i to define the operations x j if M = x j for j = i.
For an admissible sequence i, let

2. 1 .
Let (C, D, Ω) be a symmetrizable Cartan matrix C, a symmetrizer D of C, and an acyclic orientation Ω of C. Let I = {1, . . ., n}.Precisely, the matrix C = (c ij ) ∈ Z I×I satisfies that • c ii = 2 for any i ∈ I and c ij ≤ 0 for i = j, and • there is some symmetrizer D out is injective) by Proposition 3.2 and Lemma 3.1.4. Locally free Caldero-Chapoton functions 4.1.Locally free Caldero-Chapoton functions.Let H = H C (C, D, Ω).Definition 4.1.For M ∈ rep l.f.H and a rank vector r = (r i ) i∈I , the locally free quiver Grassmannian is Gr l.f.(r, M ) := {N | N is a locally free submodule of M and rank N = r}.

Lemma 4. 10 .
Let 0 ≤ e ≤ a ≤ b ≤ c and e ≤ d be non-negative integers such that b + d − e = c.Let E ⊂ A ⊂ C be vector spaces of dimensions e, a, and c respectively.Let D ⊂ C be a subspace of dimension d such that D ∩ A = E. Then the space where D/E denotes the image of D/E under π.Now we have dim (C/E)/(A/E) = c− a, dim (D/E) = d − e, and thus dim (C/E)/(A/E) = dim (D/E) + dim B ′′ .

Corollary 4. 11 .
Let M be a C[ε]/ε n -module whose (any) maximal free submodule is of rank m.Then for any 0 ≤ e ≤ m, the variety W(e, M ) := {N | N ⊂ M, N is free of rank e} has Euler characteristic m e .