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Triangular bases in quantum cluster algebras and monoidal categorification conjectures

We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parametrized by tropical points as expected in the Fock-Goncharov conjecture. As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez-Leclerc and Fomin-Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.

1. Introduction 1.1. Background. Cluster algebras were invented by Sergey Fomin and Andrei Zelevinsky around the year 2000 in their seminal work [FZ02]. These are commutative algebras with generators defined recursively called cluster variables. The quantization was later introduced in [BZ05]. Fomin and Zelevinsky aimed to develop a combinatorial approach to the canonical bases in quantum groups (discovered by Lusztig [Lus90] and Kashiwara [Kas90] independently) and the theory of total positivity in algebraic groups (by Lusztig [Lus93] [Lus94]). They conjectured that the cluster structure should serve as an algebra framework for the study of the "dual canonical bases" in various coordinate rings and their q-deformations. In particular, they proposed the following conjecture.
Conjecture. All monomials in the variables of any given cluster (the cluster monomials) belong to the dual canonical basis.
This claim would imply the positivity conjecture of the Laurent phenomenon of skew-symmetric cluster algebras, which was recently proved by Lee and Schiffler [LS13] by elementary algebraic computation.
It soon turns out that (quantum) cluster algebras are related to many other areas. For example, by using cluster algebras, Bernhard Keller has proved the periodicity conjecture of Y -systems in physics, cf. [Kel13]. We refer the reader to B. Keller's introductory survey [Kel12] for a review of applications of cluster algebras. However, despite the success in other areas, the original motivation to study cluster algebras remains largely open. In previous literature, the following bases have been constructed and shown to contain the (quantum) cluster monomials.
(i) By using preprojective algebras, cf. [GLS11], Geiss, Leclerc, and Schröer have shown that, if G is a semi-simple complex algebraic group and N ⊂ G a maximal nilpotent subgroup, then the coordinate algebra C[N] admits a canonical cluster algebra structure. They further established the generic basis of this algebra, which contains the cluster monomials. As an important consequence, they identified the generic basis with Lusztig's dual semicanonical basis, which is a variant of the dual canonical basis, cf.
Conjecture. All cluster monomials belong to the basis of the simple modules.
Partial results in type A and D were due to [HL10] [HL13b]. Inspired by the work of Hernandez and Leclerc, Hiraku Nakajima used graded quiver varieties to study cluster algebras [Nak11] and verified the monoidal categorification conjecture for cluster algebras arising from bipartite quivers. His work was later generalized to all acyclic quivers by Yoshiyuki Kimura and the author [KQ12] (cf. [Lee13] for a different and quick proof).
By the work of Khovanov, Lauda [KL09], and Rouquier [Rou08], a quantum group U q (n) admits a categorification by the modules of quiver Hecke algebras, in which the dual canonical basis vectors correspond to the finite dimensional simple modules. Therefore, Fomin and Zelevinsky's conjecture can also be viewed a monoidal categorification conjecture of the corresponding (quantum) cluster algebras.
1.2. Construction, results and comments. We shall consider quantum cluster algebras A that are injective-reachable (Assumption 2), by which we mean, for any principal quiver Q(t) (with a generic potential) of a seed t, we can obtain the "tilting object" formed by the injective right-modules of its Jacobian algebra via a mutation sequence Σ, cf. Section 5.1. Starting from the new seed Σt, also denoted by t[1], we can again consider the corresponding injective modules and obtain a new seed t [2]. Repeating this process, we obtain the notion of an injectivereachable chain of seeds {t[d]} d∈Z .
Fix any seed t, we define the "pointed set" I t consisting of normalized products of quantum cluster variables in t and t[1]. The triangular basis L t for the seed t, if it exists, is defined to be the unique barinvariant basis such that every element of I t has a (≺ t , m)-unitriangular decomposition into L t , where ≺ t is certain order and m = q − 1 2 Z[q − 1 2 ], cf. Section 3.1 5.3.
Our existence theorem 6.4.1 states that if there exists a common triangular basis in the seeds {t[d]}, d ∈ Z, such that it has positive structure constants, satisfies certain local conditions, and its parametrization by the tropical points on these seeds are compatible with the tropical transformations, then this basis lifts to the unique common triangular basis L with the same properties for all seeds.
By construction, the common triangular basis L contains all the quantum cluster monomials. Therefore, if we can show that the basis of the simple modules (or the dual canonical basis) produces the common triangular basis, the monoidal categorification conjecture is verified.
Notice that the seeds t[d] for all d have isomorphic principal quivers. Consequently, we have the reduced existence theorem 6.4.7 as one possible reduction of the existence theorem 6.4.1, whose crucial part is the finite criterion that the cluster variables along the mutation sequence from t to t[1] and t[−1] are contained in the triangular basis L t of t.
As applications, we prove the existence of the common triangular basis in the following cases, cf. Theorem 9.4.1 9.4.2 9.4.3.
(i) Assume that A arises from the quantum unipotent subgroup A q (n(w)) associated with a reduced word w (called type (i)). If the Cartan matrix is of Dynkin type ADE, or if the word w is inferior to an adaptable word under the left or right weak order, then the dual canonical basis gives the common triangular basis after normalization and localization at the frozen variables. (ii) Assume that A arises from representations of quantum affine algebras (called type (ii)). Then, after localization at the frozen variables, the basis of the simple modules gives the common triangular basis.
We also find an easy proof of Conjecture 8.3.3 about cluster structures on all integral quantum unipotent subgroups, cf. Proposition 9.1.3.
Remark 1.2.1 (Key points in the proofs). The existence theorem 6.4.1 is established by elementary algebraic computation and the combinatorics of cluster algebras. We control the triangular bases by the positivity of their structure constants as well as their expected compatibility with the tropical transformations. On the one hand, the expected compatibility implies that the new quantum cluster variables obtained from one-step mutations are contained in the triangular basis of the original seed, cf. Example 6.2.4 Proposition 6.2.3. On the other hand, if these new quantum cluster variables are contained in the original triangular basis, we construct the triangular bases in the new seeds which are compatible with the original one, cf. Proposition 6.3.5. When we consider applications to a quantum cluster algebra A of type (i) or (ii), most initial conditions are obtained by induction based on the properties of the standard basis (or dual PBW basis) and the T -systems. By the reduced existence theorem 6.4.7, it remains to verify that finite many quantum cluster variables are contained in the initial triangular basis L t 0 . Some of them are known to be Kirillov-Reshetikhin modules, cf. [Nak03][GLS13] [HL13b]. We compare the rest with the characters of some variants of Kirillov-Reshetikhin modules in the language of graded quiver varieties, cf. Section 7.4. We simplify our treatment by freely changing coefficient pattern with the help of the correction technique introduced in [Qin13], cf. Section 4.
Remark 1.2.2. The idea of constructing the common triangular basis of a quantum cluster algebra first appeared in the work of Berenstein and Zelevinsky [BZ12], whose construction was over acyclic quivers. For general acyclic quivers, it is not clear if their construction agrees with the triangular basis in this paper or not.
Remark 1.2.3. The existence theorem is stated for quantum cluster algebras associated with skew-symmetrizable matrix under several assumptions, which essentially demand good properties of the combinatorics of cluster algebras, cf. Assumption 1 3 4.
Such properties are well known for the quiver case (the skew-symmetric case) thanks to the categorification by cluster categories. The recent paper [GHKK14] found lots of results about combinatorics of the skewsymmetrizable case, which might be useful in considering the corresponding triangular bases.
The recent paper by Gross, Hacking, Keel and Kontsevich [GHKK14] provided another positive basis of commutative cluster algebras parametrized by tropical points and a more precise form of the Fock-Goncharov dual basis conjecture was verified. Their work also implied the positivity of the Laurent phenomenon of commutative cluster algebras.
When the author was preparing the present article, Seok-Jin Kang, Masaki Kashiwara, Myungho Kim and Se-Jin Oh established an approach to cluster algebras on quantum unipotent subgroups by studying simple modules of quiver Hecke algebras, which reduced the verification of the corresponding monoidal categorification conjecture to one-step mutations in one seed [KKKO14]. This result allows them to verify the conjecture for quantum cluster algebras arising from quantum unipotent subgroups in a forthcoming paper.
1.3. Contents. In the first part of Section 2, we recall basic definitions and properties of quantum cluster algebras A and its categorifications (including the Calabi-Yau reduction). This part serves as a background for the construction of the common triangular basis.
In Section 3, we consider the lattice D(t) of leading degrees for every seed t of a cluster algebra together with its dominance order ≺ t . We then define and study pointed elements in the quantum torus T (t), whose name is borrowed from [LLZ14]. We recall the tropical transformation expected by [FG09]. We end the section by a natural discussion of dominant degrees.
In Section 4, we recall and reformulate the correction technique introduced in [Qin13], which is easy but useful to keep track of equations involving pointed elements when the coefficients and quantization change.
In Section 5, we construct and study the injective pointed set I t consisting of normalized products of the cluster variables and the injectives in any given seed t.
In section 6, we introduce the triangular bases L t of a quantum cluster algebra such that they are unitriangular to the injective pointed sets I t respectively. We present local conditions on t, such as positivity and unitriangularity of the basis L t (Section 6.1). As a crucial step, we study one step mutations (Section 6.2). Then we discuss when the triangular basis in one seed becomes the triangular basis in a new seed (Section 6.3). We conclude the section by proving the existence theorem 6.4.1 for the common triangular basis.
We devote the rest of the paper to the existence and applications of the common triangular bases.
In Section 7, we review the theory of graded quiver varieties for acyclic quivers and the associated Kirillov-Reshetikhin modules. As the main result, we compute qt-characters of a special class of simple modules. These slightly generalize Nakajima's notations and results [Nak04] [Nak03]. A reader unfamiliar with this topic might skip this section and refer to Nakajima's works.
In Section 8, we review the notion of monoidal categorifications, the quantum cluster algebras A(i) associated with a word i, and the monoidal categorification conjecture in type (i). We then introduce the quantum cluster structure A(i) on a subring R(i, a) of the graded Grothendieck ring over graded quiver varieties, where the word i and grading a are adaptable. We present the corresponding monoidal categorification conjecture.
In section 9, we show that the basis of the simple modules produces the initial triangular bases of the corresponding quantum cluster algebra A(i) after localization. We verify local conditions in the initial seed. In order to show that it lifts to the common triangular basis, it remains to check that certain cluster variables are contained in this basis. We mainly work in the case when i is the power c N +1 of an acyclic Coxeter word c. Consequences are summarized in the end.

Acknowledgements
Special thanks are due to Bernhard Keller and Christof Geiss for their encouragements. The author is grateful to Hiraku Nakajima and Yoshiyuki Kimura for important discussion and comments. He thanks David Hernandez, Pierre-Guy Plamondon, and Pavel Tumarkin for interesting discussion on quantum affine algebras, cluster algebras and Coxeter groups. He also thanks Andrei Zelevinsky and Dylan Rupel for suggesting the name "pointed element". He thanks Yoshiyuki Kimura, David Hernandez and Bernhard Keller for remarks on a previous version of this paper.
The author thanks the Mathematical Sciences Research Institute at Berkeley for the financial support, where part of this work was done during his stay in 2012.

Preliminaries on additive categorifications
2.1. Quantum cluster algebras. We review the definition of quantum cluster algebras introduced by [BZ05]. Our notations will be similar to those in [Qin12] [KQ12].
Let m ≥ n be two non-negative integers. Let B be an m × n integer matrix. Its entry in position (i, j) is denoted by b ij . We define the principal part and the coefficient pattern of the matrix B to be its upper n × n submatrix B and lower (m − n) × n submatrix respectively. We shall always assume the matrix B to be of full rank n and its principal part matrix skew-symmetrizable.
A quantization matrix Λ is an m×m skew-symmetric integer matrix.
for some n × n diagonal matrix D = Diag(d 1 , d 2 , . . . , d n ) whose diagonal entries are strictly positive integers.
By [BZ05], the product DB is skew symmetric. The quantization matrix Λ gives the bilinear form Λ( , ) on the lattice Z m such that Λ(g, h) = g T Λh for g, h ∈ Z m . Here we use ( ) T to denote the matrix transposition.
Let q be an indeterminate. Define q 1 2 to be its square root such that (q 1 2 ) 2 = q. The quantization matrix Λ makes the usual Laurent polynomial ring into a quantum torus.
Definition 2.1.2. The quantum torus T is the Laurent polynomial ring Z[q ± 1 2 ][X ± 1 , . . . , X ± m ] endowed with the twisted product * such that X g * X h = q 1 2 Λ(g,h) X g+h , for any g, h ∈ Z m .
We denote the usual product of T by ·. The notation X g denotes the monomial 1≤i≤m X g i i of the X-variables X i with respect to the usual product. Recall that the Y -variables Y k , 1 ≤ k ≤ n, are defined as X Be k , where e k is the k-th unit vector in the lattice Z m . The monomials Y v , v ∈ N n , are similarly defined as k Y v k k . By [BZ05], the quantum torus T is contained in its skew-field of fractions F .
The bar involution ( ) on T is the anti-automorphism of T sending q 1 2 to q − 1 2 and X g to X g . A sign ǫ is an element in {−1, +1}. For any 1 ≤ k ≤ n and sign ǫ, we associate to B the m × m matrix E ǫ whose entry at position (i, j) is , and the n × n matrix F ǫ whose entry at position (i, j) is We only consider triples (Λ, B, X) such that (Λ, B) is a compatible pair and X = {X 1 , . . . , X m } the X-variables in the corresponding quantum torus T .
. Fix a sign ǫ and an integer 1 ≤ k ≤ n. The mutation µ k on a triple (Λ, B, X) is the operation that generates the new triple (Λ ′ , B ′ , X ′ ): The last equation above is called the exchange relation at the vertex k. By this definition, the quantum torus T and the new quantum torus T ′ associated with the new triple share the same skew-field of fractions.
Let (Λ, B, X) be a given triple. Let T n denote an n-regular tree with root t 0 whose edges will be labeled by {1, . . . , n},. There exists a unique way of associate a triple (Λ(t), B(t), X(t)) with each vertex t of T n such that (i) (Λ(t 0 ), B(t 0 ), X(t 0 )) = (Λ, B, X).
(ii) If two vertices t and t ′ are related by an edge labeled k, then the triple (Λ(t ′ ), B(t ′ ), X(t ′ ) is obtained from (Λ(t), B(t), X(t)) via the mutation µ k . We call the triples (Λ(t), B(t), X(t)) the quantum seeds. The X-variables X i (t), 1 ≤ i ≤ m, are called quantum cluster variables. The quantum cluster monomials are the monomials of quantum cluster variables from the same seed. Notice that, when j > n, the cluster variables X j (t) remain the same for all seeds t. Consequently, we call them the frozen variables or coefficients and simply denote them by X j . The vertices 1, 2, . . . , n are said to be exchangeable and form the set ex. Let Z[q 1 2 ± ]P denote the coefficient ring Z[q ± 1 2 ][X ± j ] n<j≤m . The frozen variables commute with any quantum cluster variable up to to a power of q 1 2 , which situation is called q-commute or quasi-commute.
Given any quantum seed parametrized by a vertex t, we call it the seed t for simplicity. We also use B t and Λ t to denote B(t) and Λ(t) respectively.
Definition 2.1.4. The quantum cluster algebra A is the Z[q ± 1 2 ]-subalgebra of F generated by the quantum cluster variables and the inverses of the frozen variables, where we use the twisted product.
The specialization A| q 1 2 →1 gives us the commutative cluster algebra, which is denoted by A Z . We also define the quantum cluster algebra A † as the subalgebra of A generated by the quantum cluster variables The quantum cluster algebra is contained in the quantum torus T (t) for any seed t, cf. the quantum Laurent phenomenon [FZ02] [BZ05] Fix the initial seed t 0 . For any given quantum cluster variable, we can define its extended g-vector g i (t) in Z m , cf. [Tra11] [FZ07]. Its restriction to the first n components is called the corresponding gvector, which is the grading of the cluster variable X i (t) when the B-matrix is of principal coefficients type, namely, when We refer the reader to the survey [Kel12] for more details on the combinatorics of cluster algebras.
2.2. Quivers and cluster categories. We review the necessary notions on the additive categorification of a cluster algebra by cluster categories, more details could be found in [Kel08] [Pla11b].
Let B be the m × n matrix introduced before. Assume that its principal part B is skew-symmetric.
Then we can construct the quiver Q = Q( B). This is an oriented graph with the vertices 1, 2, . . . , m and [b ij ] + arrows from i to j, where 1 ≤ i, j ≤ m. We call Q an ice quiver, its vertices 1, . . . , n the exchangeable vertices, and its vertices n + 1, . . . , m the frozen vertices. Its principal part is defined to be the full subquiver Q = Q(B) on the vertices 1, 2, . . . , n. For any given arrow h in a quiver, we let s(h) and t(h) to denote its start and terminal respectively.
The quiver Q is called acyclic if it contains no oriented cycles, and bipartite if any of its vertices is either a source point or a sink point.
Example 2.2.1. Figure 1 provides an example of an ice quiver, where we use the diamond nodes to denote the frozen vertices.
We fix the base field to be the complex field C. Choose a generic potential W for the ice quiver Q in the sense of [DWZ08]. Associate the Ginzburg algebra Γ( Q, W ) to the quiver with potential ( Q, W ). The restriction of ( Q, W ) to the vertices [1, n] is called the principal part and denoted by (Q, W ). Claire Amiot introduced the generalized cluster category C in [Ami09] as the quotient category where per Γ is the perfect derived category of dg-modules Γ and D f d Γ its full subcategory whose objects have finite dimensional homology.
The natural functor π : per Γ → C gives us By Pierre-Guy Plamondon's work [Pla11b], the presentable cluster category D is defined as the full subcategory of C which consists of objects M such that there exist triangles with M 1 , M 0 , M 0 , M 1 in add T , and S is the shift functor in C. Following [Pal08], we define the index and coindex of M as where [ : T ] denote the multiplicities of an object in add T with respect to the indecomposables Let F denote 1 the functor Ext 1 C (T, ) from D to the category of right modules of End C T . Then, similar to [Pal08, Lemma 2.1 2.3], [Pla11b, Lemma 3.6, Notation 3.7], we have An object M in C is said to be coefficient-free if (i) it contains no direct summand T j for j > n, and (ii) the module F M is supported at the principal quiver Q.
By [DWZ10] [Pla11b], we define the cluster characters for coefficientfree objects M in D as where Gr e F M denotes the variety of the e-dimensional submodules of F M and χ the Euler-Poincaré characteristic.
By [Pla11b], there exists a unique way of associating an object if two vertices t and t ′ are related by an edge labeled k, then the object T (t ′ ) is obtained from T (t) by replacing one summand in a unique way. Nag13]). The commutative cluster variables X i (t)| q →1 agree with the cluster characters x T i (t) .
2.3. Calabi-Yau reduction. In this subsection, we assume that that the Jacobian algebra of the quiver with potential ( Q, W ) is finite dimensional. Consider the full subcategory of U consisting of the coefficientfree objects. Take the ideal (Γ F ) consisting of the morphisms factoring through some T j , j > n. Then, by [IY08][Pla13], U = U/(Γ F ) is equivalent to the generalized cluster category of (Q, W ). For any coefficient-free object X, denote its image in U by X. Let Σ denote the shift functor in U/(T F ). Any given ΣT k , 1 ≤ k ≤ n, has the lift ΣT k by [IY08]. Define ΣT = (⊕ n k=1 ΣT k ) ⊕ (⊕ n<j≤m T j ). For any given 1 ≤ k ≤ n, by taking S −1 M = Σ −1 T k in (2), we get Proof. T k fits into a triangle of the following type ([Pla13, Lemma 3.15]): where T F is some direct sum of T j , j > n. Rewrite it as the triangle The claim follows by definition.
The following equality follows as a consequence Notice that Ext 1 C (T, Σ −1 T k ) is the k-th projective right-module of the Jacobian algebra principal of (Q, W ). We will later discuss this equality in Example 5.1.6 with more details.

Dominance order and pointed sets
3.1. Constructions. Let t be any vertex of the n-regular tree T n .
Define the degree lattice D(t) to be Z m .
Definition 3.1.1 (Dominance order). For any given g ′ , g ∈ D(t), we say g ′ ≺ t g (or g dominates g ′ ) if we have g ′ = g + B(t)v for some 0 = v ∈ N n . We denote g ′ t g if g ′ ≺ t g or g ′ = g. Lemma 3.1.2. For any g 1 ≺ t g 0 , there exists finitely many g 2 such that g 1 ≺ t g 2 ≺ t g 0 .
Proof. Notice that B(t) is of full rank. The claim follows from definition.
Recall that the quantum torus T (t) is the Laurent polynomial ring generated by X i (t), 1 ≤ i ≤ m.
Definition 3.1.3 (Leading term). For any Laurent polynomial Z ∈ T (t), the leading terms are those Laurent monomials of Z such that, among all the Laurent monomials of Z with non-zero coefficients, they have maximal multidegrees with respect to the order ≺ t .
Definition 3.1.4 (Pointed element). If a Laurent polynomial Z ∈ T (t) has a unique leading term whose degree is η, we say its leading (or maximal) degree is deg t Z = η. If its leading degree has coefficient q α for some α ∈ 1 2 Z, we define the normalization of Z in the quantum torus T (t) to be [Z] t = q −α Z.
We say Z is pointed at η (or η-pointed) if Z has a unique leading term, and this term has degree η with coefficient 1.
For simplicity, we often denote deg t Z and [Z] t by deg Z and [Z] respectively when the context is clear.
Definition 3.1.5 (Pointed set). Let W be a subset of D(t). A subset L of T (t) is said to be pointed at W (or W -pointed) if it consists of pointed elements, and, if the map sending Z ∈ L to deg t Z ∈ D(t) is a bijection from L to W .
We also use L t to denote the D(t)-pointed set L. For the element of L with leading degree g ∈ W , denote it by L( g; t), L t ( g), or simply L( g).
It immediately follows from the definition that any pointed set is Z[q 1 2 ]-linearly independent.
Example 3.1.6. By [Pla13], for any commutative cluster algebra studied in [GLS11], the dual semicanonical basis, after localization at the frozen variables, is a D(t)-pointed set for any seed t.
Assume that we are given a collection of elements Z i in the quantum torus T (t) such that, at any Laurent monomial m, there are finitely many Z i with non-zero coefficient z i,m . Then we let the sum i Z i denote the element m ( i z i,m )m in the completion of the quantum torus T (t).
Definition 3.1.7 (Unitriangularity). Let L be a given W -pointed set, W ⊂ D(t). Any pointed element Z ∈ T (t) is said to be ≺ t -unitriangular to L, if it has the following (possibly infinite) expansion into L: A pointed subset of T (t) is said to be ≺ t -unitriangular (resp. (≺ t , m)-unitriangular) to L if all its elements have this property. We might write the notation ≺ t as ≺ or simply omit it, when this order is clear from the context. Remark 3.1.8. It follows from Lemma 3.1.2 that the coefficients b g ′ in (4) are uniquely determined by Z. In fact, denote the coefficient of the Laurent expansion of Z at any given degree g ′′ ≺ t deg t Z by z g ′′ and that of L( g ′ ) by a g ′ , g ′′ . We must have the following equation: It follows that the coefficient b g ′′ is determined by those b g ′ appearing. By Lemma 3.1.2, this is a finite algorithm for any given g ′′ .
In general, there could exist many infinite expansion of Z into L if we don't require them to take the unitriangular form (4).
Lemma 3.1.9 (Expansion properties). Let L be any D(t)-pointed subset of T (t). Let Z be any pointed element in T (t). Then the following statements are true.
(i) Z is ≺ t -unitriangular to L.
(ii) If Z and L are bar-invariant and Z is (≺ t , m)-unitriangular to L, then Z equals L(deg Z; t).
(iii) If Z a Z[q ± 1 2 ]-linear combination of finitely many elements of L, then such a combination is unique and takes the unitriangular form (4).
Proof. (i)(ii) The statements are obvious.
(iii) Suppose that some L t (η), η ∈ D(t), appears in the expansion of Z into L with non-zero coefficient b(η), such that η ⊀ t deg t Z. Because the expansion is finite, we can always find a degree η such that it is maximal among those having this property. Then the coefficient of X(t) η of Z equals b(η) = 0. But the coefficient of this Laurent monomial in the pointed element Z must vanish. This contradiction shows that such η cannot exist.
Lemma 3.1.10 (Inverse transition). Let M, L be any two D(t)-pointed subsets of T (t). If M is (≺ t , m)-unitriangular to L, then L is (≺ t , m)unitriangular to M as well.
Proof. For any g ∈ D(t), let us denote M( g; t) = L( g; t) + g 1 ≺t g c g, g 1 L( g 1 ; t), c g, g 1 ∈ m, L( g; t) = M( g; t) + Expand RHS of the second equation by using the first equation. Then the coefficient of any factor L( g 2 ; t), g 2 ≺ t g, appearing in RHS is the following three terms' sum c g, g 2 + b g, g 2 + g 2 ≺t g 1 ≺t g b g, g 1 c g 1 , g 2 , which must vanish.
We verify the claim by induction on the order of the degrees g 2 such that g 2 ≺ t g. If g 2 is maximal among such degrees, the last term vanishes and we have b g, g 2 ∈ m. Assume b g,( g 2 ) ′ ∈ m for any ( g 2 ) ′ such that g 2 ≺ t ( g 2 ) ′ . Then the first and last term belong to m. Consequently, b g, g 2 ∈ m belong to m as well.

Tropical transformation.
Definition 3.2.1 (Tropical transformation, [FG09][FZ07, (7.18)]). For any 1 ≤ k ≤ n, we consider the piecewise linear map φ µ k t,t : D(t) → D(µ k t), such that any given degree g = ( g j ) 1≤j≤m ∈ D(t) has the image g ′ = ( g ′ j ) 1≤j≤m = φ µ k t,t ( g) with the following components On the cluster algebra side, Fomin and Zelevinsky wrote this transformation in [FZ07,(7.18)] and conjectured that it describes the transformation of the g-vectors of cluster variables. When the cluster algebra admits a categorification by a cluster category, this formula (5) can be interpreted as the transformation rule of the indices of object in the cluster category, cf. [DK07][KY11] [Pla13]. We refer the reader to [FZ07,Remark 7.15] [Pla13, Section 3.5] [FG09]for more details.
The following observation is obvious.
Lemma 3.2.3 (Sign coherent transformation). Assume that two vectors η 1 , η 2 in D(t) are sign coherent at the k-th component, namely, we have simultaneously η 1 k , η 2 k ≥ 0 or η 1 k , η 2 k ≤ 0. The the tropical transformation (5) is additive on them: We shall keep the following assumption throughout this paper.
Assumption 1 (Categorification). The combinatorics of the quantum cluster algebra A can be categorified. More precisely, there is a category C, such that (i) the cluster variables X i (t), 1 ≤ i ≤ m, t ∈ T n , correspond to objects M i (t) in C. (ii) Given any two seeds t 1 , t 2 and any mutation sequence from t 1 to t 2 , we define φ t 2 ,t 1 to be the composition of the transformations (5) along this sequence. Then φ t 2 ,t 1 is independent of the choice of the sequence. Moreover, there exists an index map Ind t from M i (t) to Z m for every seed t, such that the indices of the objects M i (t), t ∈ T n , are related by the map φ t 2 ,t 1 : (iii) In any quantum torus T (t 0 ), the cluster variables X i (t) has the following Laurent expansion where the coefficients c v (q 1 2 ) ∈ Z[q ± 1 2 ]. In particular, (iv) The quantization matrices satisfy, for any t, t 0 ∈ T n , . Remark 3.2.4. When B(t) is skew-symmetric, Assumption 1 is known to hold where the categorification is provided by the (generalized) cluster category in Section 2.2. The verification of most conditions could be found in Plamondon's works [Pla11b][Pla11a] [Pla13]. By translating (5) into multiplication by E ǫ in Definition 2.1.3, we see that the last condition can be deduced from the sign coherence of g-vectors. This condition implies that the quantization matrices Λ(t) lift to a bilinear form λ on categories, cf. [Qin12].
Let pr n : Z m → Z n denote the projection onto the first n-components. For any g in D(t), we denote the projection pr n g by g. Define pr f : Z m → Z m to be the projection onto the last m − n components.
As the coefficient-free version of the piecewise linear map φ t ′ ,t , we have the following restriction map.
Lemma 3.2.5. The restriction pr n φ t ′ ,t | Z n : Z n → Z n is a bijection of sets.
Proof. It suffices to verify the statement for the case t ′ = µ k t, where we have (pr n φ t ′ ,t )(pr n φ t,t ′ ) = 1.
We are interested in the pointed sets whose leading degree parametrization is compatible with the tropical transformation rules.
Definition 3.2.6 (Compatible pointed sets). Let t 1 , t 2 be two seeds. Let L t 1 be a D(t 1 )-pointed set and L t 2 a D(t 2 )-pointed set. We say L t 1 is compatible with L t 2 if, for any η ∈ D(t 1 ), we have L(φ t 2 ,t 1 η; t 2 ) = L(η; t 1 ).
By Assumption 1(ii), this compatibility relation is reflexive and transitive.

Dominant degrees.
Definition 3.3.1 (Dominant degree cone). For any given seed t, define the dominant degree cone D † (t) ⊂ D(t) to be the cone generated by the leading degrees of all cluster variables deg t (X i (t ′ )), for any 1 ≤ i ≤ m and t ′ ∈ T n . Its elements are called the dominant degrees of the lattice D(t).
By (6) in Assumption 1, we have D † (t) = φ t,t 0 D † (t 0 ), ∀ t, t 0 ∈ T n . The following observation is obvious. Lemma 3.3.2. Let Z be any element in A † . Then, for any seed t, the ≺ t -maximal degrees of the Laurent expansion of Z in T (t) are contained in D † (t).
Definition 3.3.3 (Bounded from below). We say that (≺ t , D † (t)) is bounded from below if for any g ∈ D † (t), there exists finitely many g ′ ∈ D † (t) such that g ′ ≺ t g.
Remark 3.3.4. When A is of type (i) or (ii) in the sense of Section 8, take the canonical initial seed t 0 . In Section 9.1, we shall see that The following question arises naturally.
Question 3.3.5. If (≺ t 0 , D † (t 0 )) is bounded from below, is it true that (≺ t , D † (t)) is bounded from below for all t ∈ T n ?

Correction technique
The correction technique was introduced in [Qin13] in order to keep track of bases when the quantization and the coefficient pattern of a quantum cluster algebra change. In this section, we recall and reformulate this correction technique. It will help us simplify many arguments by coefficient changes. 4.1. Similarity and variation. Let var * be any isomorphism of finite n-elements sets var * : ex (2) ∼ → ex (1) . We associate to it an isomorphism var of the lattices var : jk ) be an I (i) × ex (i) matrix, i = 1, 2, where I (i) has m (i) elements and ex (i) ⊂ I (i) has n elements. The matrices B (1) and B (2) are called similar, if there exists an isomorphism var * : ex (2) ∼ → ex (1) such that the matrix entries satisfy b (2) var * (j) var * (k) , ∀j, k ∈ ex (2) . In this case, we denote the principal part B (2) as var B (1) . We denote by pr ex (i) : Z I (i) → Z ex (i) the natural projection onto the coordinates at ex (i) .
For any given full rank matrices B (i) , i = 1, 2, consider the Laurent polynomial rings . Assume that there is an embedding from I (1) to I (2) such that its restriction induces an isomorphism var * : ex (2) ∼ → ex (1) . Then we can naturally embed Z I (1) into Z I (2) by adding 0 on the extra coordinate. Denote this embedding by var.
By abuse of notation, define the variation map var to be the map from the set of pointed elements in T (1) to that of T (2) such that, for any pointed element M (1) , its image is the unique pointed element M (2) which is similar to M (1) and satisfies var η (1) = η (2) . Now, let us take into account of quantization.
We say these two compatible pairs are similar, if there exists an isomorphism var * : ex (2) → ex (1) , such that the principal B-matrices B (2) = var B (1) , and, in addition, there exists a positive integer δ such that D (2) = δ var D (1) .
Definition 4.1.3 (Similar pointed elements). We say the pointed elements Definition 4.1.4 (variation map). Let the embedding I (1) → I (2) be given as before. The variation map var is the injective application from the set of the pointed elements of T (1) to that of T (2) sending any pointed element M (1) to the unique similar pointed element M (2) subject to var(η (1) ) = η (2) .
Example 4.1.5. Let us take two quantum seeds t (1) 0 with the following two compatible pairs respectively These compatible pairs are similar via the identification of vertices var * (i) = (i), i = 1, 2. The following quantum cluster variables obtained by the same mutation sequence are similar: The variation map var would send X 2 (µ 2 µ 1 t Correction. We consider algebraic equations involving the pointed elements. Fix two compatible pairs ( B (i) , Λ (i) ), i = 1, 2. As in the previous discussion, assume that there is an embedding from I (1) to I (2) whose restriction induces an isomorphism var * : Let [ ] (i) denote the normalization in the quantum torus T (i) and deg (i) ( ) denote the leading degree of a pointed element in T (i) . Then [Qin13, Thm 9.2] can be reformulated as follows.
. Fix an integer s ≥ 1. Assume that we have equations for pointed elements P , Q, M i and (possibly infinitely many) N j in the quantum torus Then, for any P (2) , Q (2) , M (2) i , N (2) j in T (2) similar to P, Q, M i , N j , we have the following equations in T (2) : where f M , f j are the Laurent monomials in the frozen variables X i , n < i ≤ m (2) , given by are given by image of the the variation map in Definition 4.1.4, we have Proof. We consider Laurent expansions of (7) (8) and compare their coefficients term by term. Details are given in [Qin13, Theorem 9.2].
The following observations are immediate consequences of the correction technique 4.2.1 and the existence of (quantum) F -polynomials [Tra11] [FZ07].
(iii) Assume that B(t) is skew-symmetric. Let Z be any given pointed element in the quantum torus T (t). Assume that it has a (possibly infinite) unitriangular expansion where N j are normalized twisted products of quantum cluster variables of where N ′ j are normalized products of the quantum cluster variables of A(t ′ ) associated with the same mutation sequences, the coefficients f j are given by Theorem 4.2.1.

Injective pointed sets
, for some sequence of mutations Σ and a permutation σ of {1, . . . , n}, if we have the following equalities Let us denote X σ(i) (Σt) by I i (t) and call it an injective quantum cluster variable for t (or simply an injective for t). The meaning of "injective" will be explained in Remark 5.1.5.
We trivially extend the action of σ to a permutation of {1, . . . , m}. Let σ act naturally on Z m by pull back such that σe i = e σ −1 i .
We make the following assumptions for the rest of this paper.
Assumption 2. The cluster algebra contains an injective-reachable seed.
We always read mutation sequences µ is · · · µ i 2 µ i 1 from right to left in this paper, which we also denote by (i s , . . . , i 2 , i 1 ) for simplicity.
Definition 5.1.2 (Injective-reachable cluster algebra). A cluster algebra is called injective-reachable if all its seeds are injective-reachable.
Proposition 5.1.3. Assumption 3 is equivalent to the claim that, for any 1 ≤ k ≤ n, we have Proof. First assume that (9) holds. Omitting the symbol (t), we can write And Assumption 3 follows from the following straightforward calculation: Conversely, assume that Assumption 3 is true. Because the map pr n φ t ′ ,t is invertible on Z n , the above calculation implies we must have On the other hand, consider the exchange rule in the seed Σt:

FAN QIN
View this equation in T (t). Because X σk (µ σk Σt) is a pointed element whose unique leading term has coefficient 1, its leading degree must be chosen from either Their images in Z n under the projection pr n are different except the trivial case when σk is an isolated point in Q(Σt). (10) implies that we should choose the former one. This choice gives us (9). Let us impose the principal coefficients on t, namely cf. [FZ07]. We can calculate the leading degrees of the cluster variables appearing in the assumption, following the mutation rule of (extended) g-vectors [FZ07, (6.12)][Tra11, (3.32)]. As a consequence, Assumption 3 is equivalent to the claim that the matrix B(Σt) satisfies We denote X i (t) by P σi (Σt) and call it a projective quantum cluster variable for Σt (or simply a projective for Σt). We make the following assumption on the Laurent expansions of I i (t) and P σi (Σt) for the rest of this paper. Its meaning will be clear in Remark 5.1.5.
Assumption 4 (Cluster expansions). Let t be any given seed. When q 1 2 specializes to 1, the Laurent expansion of I i (t)| q 1 2 →1 in T (t) takes the following form: and the Laurent expansion of P i (t)| q 1 2 →1 in T (t) takes the following form: Moreover, the degrees satisfy Remark 5.1.5 (Injectives and projectives). Assume that the cluster algebra admits the categorification in Section 2. We rewrite (3) as Recall that T i correspond to the quantum cluster variable X i (t). Thanks to the Calabi-Yau reduction, ΣT i corresponds to the i-th injective right module of the Jacobian algebra associated with the principal quiver Q(t) and its generic potential. So ΣT i corresponds to the quantum cluster variable I i (t) = X σi (t[1]) on the vertex σi in Q(t[1]). On the other hand, thanks to Calabi-Yau reduction, T i correspond to the indecomposable projective of the Jacobian algebra associated with Q(t[1]) and its generic potential at this vertex.
The above equation holds when we replace ΣT and T by their direct summands ΣT j and T j respectively, 1 ≤ j ≤ m. Notice that the multiplicity of ΣT j contributes the σj-th component in the degree lattice where p(σi, t[1]) is the dimension of the σi-th projective right module of the Jacobian algebra associated with the quiver Q(t[1]) and its generic potential. Equation (13) follows.
The seed t[−1] is obtained from t by applying (σΣ i ) −1 = (4, 1, 2, 3, 4). Its ice quiver is given in Figure 3 In the quantum torus T (t[−1]), we have It is straightforward to check that the projective right modules of the Jacobian algebra of (Q(t), W (t)) have dimensions The Y -variables of the seed t in the quantum torus T (t) are given by The projectives for t, obtained by applying the sequence (σΣ i ) −1 , take the following form in the quantum torus T (t): This example meets the expectation of Assumption 4. 5.2. Injective-reachable chain. Let t be any seed which is injectivereachable via an associated pair (Σ t , σ t ). We denote the seed t by t[0].
For any integer d ≥ 0, by Lemma 5.1.1, we recursively obtain that Let ord σ t denote the order of the permutation σ t . Then t[ord σ t − 1] is injective-reachable via Moreover, the principal part of the quiver of the seed t[ord σ t ] is the same as that of t. Since injective-reachable is independent of the coefficient part, the seed t 5.3. Injective pointed set. For any g ∈ Z n , denote g + = (max{(g) i , 0}) 1≤i≤n and g − = (max{(−g) i , 0}) 1≤i≤n . We will also view g, g ± ∈ Z n as elements of Z m by putting zero at the last m − n components.
Define the injective pointed set I t to be {I( g; t), g ∈ D(t)} such that for any g ∈ D(t), We also view g f as an element in Z m−n by removing its first n components.
By definition, I t is a D(t)-pointed Z[q ± 1 2 ]-linearly independent subset of T (t). The following lemma suggests that its elements behave well under one-step mutations.
Lemma 5.3.2 (Neighboring injective pointed set). For any given t ∈ T n and 1 ≤ k ≤ n, define t ′ to be µ k t. Then the set I t ′ is D(t)-pointed in T (t) with respect to the order ≺ t . Furthermore, for any g ∈ D(t ′ ), we have Proof. The degrees in D(t ′ ) change by the piecewise linear formula (5) which depends on the sign of the k-th components. In our situation, the degrees of the factors of I( They are sign coherent at the k-th component. Therefore, we can use Lemma 3.2.3 and deduce that It remains to check that the coefficient of the leading term remains to be 1. We have to verify the following equality: Decompose the degrees appearing above into sum of unit vectors by Lemma 3.2.3. Then it suffices to verify the following equations for any pair of unit vectors (e i , e j ), 1 ≤ i, j ≤ m, sign-coherent at the k-th component: These equations follow from the mutation rule of quantization matrices, ∀i, j = k: Remark 5.3.3. The notion of leading terms depends on the choice of seed and so does the normalization factor. Different pointed elements Z i ∈ T (t 1 ) might have identical leading degrees in the quantum torus T (t 2 ) of the new seed t 2 , and a pointed element Z in T (t 1 ) might not be pointed in the T (t 2 ). The normalization factor of the twisted product Z 1 * Z 2 might also change if they are not quantum cluster variables in the same cluster.
Example 5.3.4. We consider the initial acyclic quiver Q = Q(t 0 ) The set I t 0 consists of the normalized ordered products of X i (t 0 ) and I j (t 0 ) such that X i (t 0 ) and I i (t 0 ) do not appear simultaneously. It is apparently D(t 0 ) = Z 2 pointed.

FAN QIN
Consider the neighboring seed µ 1 t 0 , its quantization matrix is Λ(µ 1 t 0 ) = −B. In the quantum torus T (µ 1 t 0 ), we have the following Laurent expansions It is straightforward to check that I t 0 is a D(µ 1 t 0 )-pointed set.
If we proceed to the seed It is straightforward to check that the set of leading degrees of The normalization factors of X 1 * I 2 are also different in T (t 0 ) and T (t 0 [1]). 1 2 Figure 4. A quiver Q of type A 2 .

Triangular bases
In this section, we consider the triangular basis L t with respect to the injective pointed set I t . In the existence theorem 6.4.1, we shall prove that a common triangular basis for all seeds exists, provided some initial local conditions are satisfied.
6.1. Local conditions. For any 1 ≤ k ≤ n, let us denote X k (µ k t) and X σ(k) (µ σ(k) Σt) by X ′ k (t) and I ′ k (t) respectively. Definition 6.1.1 (Local conditions). For any given vertex t ∈ T n and 1 ≤ k ≤ n, we consider the following local conditions on t: There exists a bar-invariant D(t)-pointed basis L t in A, which satisfies L(0; t) = 1 and factors through the frozen variables: We also consider the following auxiliary conditions: . For simplicity, we often omit the the notation t and, if a condition L?(k) is satisfied by k from a set of vertices E, we denote this property by L?(E); Remark 6.1.2. Notice that we allow infinite expansions in L3 ′ , L3 ′′ , L4 ′ , L5 ′ . Also, Condition L0 implies that the Laurent monomials of the frozen variables are contained in L t . We propose Condition L2 as an analog of the transition property between dual PBW bases and dual canonical bases and use Condition L3 to strengthen L2. If the seed t has acyclic principal part with appropriate coefficient pattern, then I t is the dual PBW basis, and we can take L t to be the dual canonical basis. The conditions L0(t) L1(t) L2(t) are well known results and, moreover, L t contains all quantum cluster monomials by [KQ12].
The following important lemma shows that left multiplying X k or right multiplying I k will preserve a (≺ t , m)-unitriangular expansion.
Proof. The first equation follows from the definition of compatible pairs, and it implies the remaining equations. Proof. The expansion coefficients can be interpreted as q Lemma 6.1.6. Assume that a vertex t ∈ T n satisfies L0 L2. ( (iii) If t further satisfies L1, then L4(k) and L5(k) are equivalent to the claim that X ′ k (t), I ′ k (t) are contained in L t respectively. Proof. The statements (i)(ii) follow from the bar-invariance of the triangular basis. The statement (iii) follows from the positive Laurent expansions of L t in the seeds t and Σt respectively. We give a proof that X ′ k (t) ∈ L t implies L4(k) here. The proof for L5(k) is same. For simplicity, we specialize q 1 2 to 1 and omit the symbol (t). Consider the following product of basis elements in L t By Lemma 6.1.5, the positive Laurent expansion of each basis element L( g; t) appearing must take the form where F g (Y k ) is a polynomial in Y k whose coefficients are no larger than those of (1 + Y k ) d k at all degrees.
Then its coefficients are also no larger than those of (1 + Y k ) d k .
Consider the Laurent expansion of L( g; t) in T (µ k t). We obtain This is an Laurent polynomial in T (µ k t) if and only if . Lemma 6.1.6(iii) is not essential to this paper. It simplifies Proposition 6.2.3 such that the statement for the case d k = 1 is sufficient.
Lemma 6.1.7. Let us consider any t ∈ T n . Assume that it verifies L0, L2, then the following claims are true.
Proof. (ii)(iii) are obvious by L2 and bar-invariance. We give a proof for (i).
Assume that we have L3(k; t). By L2, it implies that [X k * I t ] t is m-unitriangular to L t and, consequently, m-unitriangular to I t . Therefore, I t is m-unitriangular to [X k * I t ] t .
Assume that we have L3 ′ (k; t). Let S be any element in L t . Then S is m-unitriangular to I t : Using L3 ′ (k), we see that X k * S is m-unitriangular to I t and, consequently, to L t .
If we have L3 ′′ . For any given g ∈ D(t) and 1 ≤ k ≤ n, take (f, d X , d I ) such that the normalized product I(f, d X , d I ; t) equals [X k * I( g, t)] t , then it is (≺ t , m)-unitriangular to I t . For the given k, L3 ′ (k) follows by taking the inverse transition and it implies L3(k).
(i) Let t and t ′ be two similar seeds such that ( B(t), Λ(t)) is similar to ( B(t ′ ), Λ(t ′ )) via the variation map var. Assume we have a basis L t of A(t) so that t satisfies L0(t) and L2(t). Let L t ′ denote the set of the pointed elements in T (t ′ ) similar to the elements of L t . Then it is the basis of A(t ′ ), such that t ′ satisfies L0(t ′ ) and L2(t ′ ).
(ii) If t also satisfies L1 or L3, the same holds for t ′ .
(iii) Furthermore, for some given mutation sequence ← − µ = µ var * is · · · µ var * i 1 , r ∈ N, assume that the the quantum cluster variable Proof. (i) L t ′ is D(t ′ )-pointed and therefore linearly independent. It factors through the frozen variables by construction. Any given basis element b ∈ L t is a polynomial of quantum cluster variables of A(t) with coefficients in Z[q ± 1 2 ]P . Then any element b ′ ∈ L t ′ similar to b is a polynomial of quantum cluster variables of A(t ′ ) with coefficients in Z[q ± 1 2 ]P ′ by Lemma 3.1.9(iii) and 4.2.2(iii). Therefore, b ′ is contained in A(t ′ ). We see that L t ′ spans A(t ′ ) by Theorem 4.2.1 and the condition L2(t ′ ) by Lemma 4.2.2(iii).
(ii) The conditions follow from Theorem 4.2.1 and Lemma 4.2.2(iii) respectively.
(iii) Recall that X is (var ← − µ t ′ ) is similar to X var * is ( ← − µ t) by Lemma 4.2.2. The claim follows from the construction of L t ′ .
6.2. One step mutations. We shall show in Proposition 6.2.3 that the expected transformation rule (5) for the leading degrees of triangular basis elements implies the important conditions L4, L5: quantum cluster variables obtained by one step mutations are contained in the triangular bases.
We will control the Laurent expansions of the related triangular basis elements with the help of the positivity of the structure constants of the basis L t and Assumption 4. To warm up, we start with a straightforward calculation.
Proposition 6.2.1. For any given t in T n and 1 ≤ k ≤ n, assume that we have L0, L1, L2, L3([1, n]), then the Laurent expansion of L t (deg t X ′ k (t)) in T (t) contains that of X ′ k (t). Proof. Omit the notation t for simplicity.
Denote e + = j [b kj ] + e j , X + = X e + . Then we have deg t X ′ k = e + −e k and X ′ k = X −1 k X + (1 + Y k ). We consider the normalized twisted product [X − deg t I k −e k * X + * I k ], where the first factor is a Laurent monomial of the frozen variables. By Lemma 6.1.6(ii), I k and the cluster monomials in t belong to the basis L t . By Lemma 6.1.5, the Laurent expansion of I k has non-negative coefficients. Further applying Assumption 4, we see that the Laurent expansion of I k must take the form Therefore, the Laurent expansion of [X − deg t I k −e k * X + * I k ] takes the form On the other hand, by L3, we have where the coefficients c η ∈ N[q ± 1 2 ] ∩ m. Comparing (19) with the Laurent expansion of (20). They all have non-negative coefficients in N[q ± 1 2 ]. By analyzing these coefficients, we easily deduce that the monomial X + Y k can only be the contribution of L t (e + − e k ).
Therefore, the Laurent expansion of L t (e + − e k ) takes the form Remark 6.2.2. In the proof of Proposition 6.2.1, the calculation of the desired basis element L t (deg t X ′ k (t)) is based on the following two steps: (i) put the desired basis element L t (deg t X ′ k (t)) in the product of cluster variables contained in L t (20), and (ii) analyze its Laurent expansion by using various positivity properties, as well as partially known cluster expansions in Assumption 4. However, Proposition 6.2.1 can not precisely control L t (deg t X ′ k (t)), because the desired basis element appears as the leading term in (20). To fix this, in Proposition 6.2.3, we put it as the last term in a similar but less intuitive equation.
Proof. The quantum cluster variable X ′ k (t) is given by the exchange relation Recall that σe i = e σ −1 i . In T (t[−1]), the above equation becomes By (9), we have Let us denote Correspondingly, we have the following expansion of a product of elements in L t[−1] : where the non-negative coefficients p η belong to m by L3([1, n]; t[−1]), and the degree α given by Notice that, by using Assumption 1(iv), α can also be written as where we use (13) in the last equality. We now view equation (22) in the quantum torus T (t), which becomes By (13), we also have By (12) and L1(t), the Laurent expansion of P k (t) takes the form Consequently, the Laurent expansion of LHS of (23) becomes where α d = 1 2 Λ t (deg t Y (t) −d , i) ≤ 0 for any d > e k . Notice that the last two terms sum to X ′ k (t). Similar to the discussion in Proposition 6.2.1, we compare this Laurent expansion to the Laurent expansion of the RHS of (23), which has non-negative coefficients by Lemma 6.1.5.
First, L t (deg X ′ k (t)) contains X deg t X ′ k (t) because it needs to have the leading degree deg t X ′ k (t). Notice that the Laurent expansion of L t (deg X ′ k (t)) is contained in (24), and the only term with degree dominated by is contained in some other basis elements L t[−1] (η) in (23), this will contradicts to the properties that p η ∈ m and barinvariance of L t[−1] (η). Therefore, the desired basis element . Example 6.2.4 (Control Laurent expansion). We continue Example 5.1.6. Take k = 4. We want to control the Laurent expansion of L t (deg t X ′ k (t); t) by locating it as the last few terms of a Laurent polynomial in T (t) and deduce that it is the triangular basis element L t (deg t X ′ k (t))). We have the following quantum cluster variables in T (t): Then the desired basis element L t (deg t X ′ 4 (t)) appears in the (≺ t[−1] , m)-unitriangular expansion of the following normalized product in Compute the quantization degree )(e 1 − e 5 − e 8 , −e 4 + e 5 + e 6 + e 7 + e 8 ) Now, view the twisted product (25) in T (t), we obtain Recall that the expansion of the normalized product into L t[−1] takes the form with non-negative coefficients p η ∈ m. It follows that X ′ 4 (t) must agree with L t (deg t X ′ 4 (t)). 6.3. Triangular bases for new seeds. We shall show in Proposition 6.3.2 that, if the quantum cluster variables X ′ k (t), I ′ k (t) are contained in the triangular basis L t , then L t implies the existence of the triangular basis L t ′ for the new seed t ′ = µ k t and they are compatible. We further verify the local conditions for the new triangular basis in Proposition 6.3.5. Lemma 6.3.1. Let t, t ′ be two given seeds. Assume that we have L0(t) and L0(t ′ ) with L t ′ = L t given by L( g; t ′ ) = L(φ t,t ′ ( g); t), ∀ g ∈ D(t ′ ). Let Z ∈ A be any element which has a unique leading degree in each of T (t) and T (t ′ ), such that we have deg t Z = η Z , deg t ′ Z = g Z = φ t ′ ,t (η Z ), for some η ∈ D(t).
If [Z] t is (≺ t , m)-unitriangular to L t , then it is also pointed in T (t ′ ) and (≺ t ′ , m)-unitriangular to L t ′ as well.
Proof. By the assumption, in T (t), we have Because Z ∈ A, RHS is a finite sum. View this equation in T (t ′ ) and rewrite RHS in terms of L t ′ , we obtain Notice that φ t ′ ,t η ′ = φ t ′ ,t η Z = g Z . By this equation, [Z] t has coefficient 1 at its leading degree g Z . In particular, [Z] t is pointed at g Z in T (t ′ ). Moreover, this expansion is a finite sum. By applying Lemma 3.1.9(i)(iii), we see it must take the form Proposition 6.3.2 (triangular basis in new seed). Let t, t ′ be seeds related by a mutation at some vertex k. Assume that we have L0(t) L1(t) L2(t) L3([1, n]; t) as well as L4(k; t) and L5(k; t). Furthermore, assume that, in the quantum torus T (t), I t ′ is (≺ t , m)-unitriangular to I t . Then the following claims are true.
(i) The quantum cluster monomials in t ′ belong to L t .
In other words, we have the bar-invariant D(t ′ )-pointed basis L t ′ defined by L( g; t ′ ) = L(φ tt ′ g; t) for any g ∈ D(t ′ ).
The statement follows by taking those d X with the k-th coordinate vanishing.
(ii) It follows from Lemma 5.3.2 that I( g; t ′ ) is pointed at degree φ t,t ′ g in T (t). By the hypothesis in the Proposition, it is (≺ t , m)unitriangular to I t and, consequently, to L t . So we have View (27) in T (t ′ ) from now on. Since I( g; t ′ ) belong to A, this linear combination is a finite sum. Furthermore, because X k (t ′ ) and I k (t ′ ) belong to L t by L4(k; t) L5(k; t), the coefficients c ηη ′ are structure constants of the basis L t and thus non-negative by L1(t). In addition, by the same proof of Lemma 6.1.5, L4(k; t) and L5(k; t) imply that the Laurent expansion of each term of RHS in the quantum torus T (t ′ ) must have non-negative coefficients. Finally, notice that LHS has a unique leading term of degree g with coefficient 1. Therefore, there exists a unique leading term of degree g in RHS with coefficient 1, which must be the contribution from L(η; t). Consequently, L(η; t) is pointed at g in T (t ′ ).
(iii) We obtain the claim by using (27) for each element I t ( g; t ′ ).
(iv) I t is (≺ t ′ , m)-unitriangular to L t ′ by Lemma 5.3.2 and Lemma 6.3.1. L t ′ is (≺ t ′ , m)-unitriangular to I t ′ by (ii) and Lemma 3.1.10. The claim follows.
Remark 6.3.3. The positivity condition L1(t) in Proposition 6.3.2 is only used for proving the claim (ii).
We enhance Proposition 6.3.2 by verifying the local conditions for the new seed t ′ equipped with the new triangular basis L t ′ .
Lemma 6.3.4. Let t be any given seed. Assume that we have L0(t) L2(t) L3([1, n]; t). Let Z be a element in T (t) with a unique maximal degree.
If the normalization Proof. Notice that we have L3 ′′ (t) by Lemma 6.1.7. Substitute the factor [Z] t by its expansion in I t . By using Lemma 6.1.4 and L3 ′′ (t), we obtain the desired result. Proposition 6.3.5 (local conditions for new seeds). Let t, t ′ be seeds related by a mutation at some given vertex k.
Proof. For simplicity, we omit the notation (t) inside X(t) d , X ′ k (t),

Proof of claim (i):
By Definition 5.3.1, for any given g ∈ D(t ′ ), we have, in T (t), where g k ∈ Z n is obtained from g by imposing g k = 0. By Lemma 5.3.2, it is normalized in T (t) as well: We substitute the factors (X ′ k ) [g k ] + or (I ′ k ) [−g k ] + appearing by using the equations in L4 ′ (k; t), L5 ′ (k; t). Because at most one of them could appear, by using Lemma 6.3.4, we obtain that, for any g ∈ D(t ′ ), where η = φ t,t ′ ( g). Namely, I t ′ is (≺ t , m)-unitriangular to I t . By Proposition 6.3.2, t ′ satisfies L0 L1 L2, such that L( g; t ′ ) = L(φ t,t ′ g; t).
Proof of claim (ii): Take any S ∈ L t ′ . First, take any i = k. We have In T (t ′ ), the twisted product [X i (t) * S] t has the unique maximal degree deg t ′ (X i (t)) + deg t ′ S. This maximal degree turns out to be , thanks to the sign coherence at the k-th components. Therefore, we can apply Lemma 6.3.1 and deduce that [X i (t ′ ) * S] t is pointed and (≺ t ′ , m)-unitriangular to L t ′ in T (t ′ ).
It remains to prove that the normalized twisted product [X Denote deg t ′ S = g. By Proposition 6.3.2(ii)(iii)(iv), S is (≺ t ′ , m)unitriangular to I t . Therefore, by applying Lemma 6.1.4 in T (t ′ ), we have Let us write where η k is obtained from pr n η by setting the k-th component to 0, η f is the degree of coefficients such that the leading terms of both side agree. This element is also normalized in T (t ′ ) by Lemma 5.3.2: (Case ii-a) If η k ≤ 0, then I(η; t) does not contain any factor X k . Consider the following normalized product in T (t) The leading degrees deg t X ′ k and deg t I(η; t) are sign coherent at the k-th components. It follows that deg t ′ (X ′ k * I(η; t)) = φ t ′ ,t deg t (X ′ k * I(η; t)).
In addition, X ′ k ∈ L t ′ = L t is (≺ t , m)-unitriangular to I t and, by Lemma 6.3.4, X ′ k * I(η; t) is (≺ t , m)-unitriangular to I t after normalization in T (t). Therefore, we can change the seed from t to t ′ by applying Lemma 6.3.1 and obtain that [ By the exchange relation at vertex k, we have, in the quantum torus Notice that the leading term of the pointed element [ where, by the sign coherence at the k-th components and the mutation rule of quantization matrices, we have Now, in the quantum torus T (t), the leading degrees of the factors in both terms are sign coherent at the k-th components. It follows that By L3 ′′ (t), these terms, when normalized in T (t), are (≺ t , m)-unitriangular to L t . Applying Lemma 6.3.1, their normalization in T (t ′ ) remain the same, and, when normalized in T (t ′ ), they become (≺ t ′ , m)-unitriangular to L t ′ . 6.4. Existence theorem. We present the existence theorem about the common triangular basis. The criterion can be reduced to a finite problem, as we shall see in the reduced existence theorem 6.4.7.
Observe that, by Proposition 6.2.3 and Lemma 6.
By Proposition 6.3.5, we obtain that for t ′ = µ k t, the chain (t ′ [d]) d∈Z verifies all the local conditions in the theorem. Moreover, the common triangular basis on it is compatible with the common triangular basis on (t[d]).
We verify the theorem by repeatedly applying the above arguments for all seeds along any mutation sequence.
Definition 6.4.2. Under the assumption of the Theorem 6.4.1, the common injective-triangular basis L t , t ∈ T n , is called the common triangular basis of A, which we can denote by L without confusion.
Remark 6.4.3. By Theorem 6.4.1, the elements of the common triangular basis L have distinct unique leading terms in every quantum torus, and their degrees change under the piecewise linear formula (5). Therefore, L possesses the property of the basis expected by Fock-Goncharov conjecture [FG06].
For the rest of this subsection, we require that the cluster algebra is associated with skew-symmetric matrices, so that the quantum Fpolynomials are known to exist.
A change of the coefficient pattern or quantization will not affect the existence of the common triangular basis.
Corollary 6.4.4. Assume B(t 0 ) is skew-symmetric. Let t 0 , t ′ 0 be two initial seeds with similar compatible pairs in the sense of Section 4 and A(t 0 ), A(t ′ 0 ) the corresponding quantum cluster algebras respectively. Assume that A(t 0 ) verifies all the conditions in Theorem 6.4.1.
Let L t ′ 0 denote the initial triangular basis of A(t ′ 0 ) constructed as similar elements of the initial basis L t 0 ⊂ A(t 0 ), cf. 6.1.8. Then A(t ′ 0 ) verifies all the conditions in Theorem 6.4.1 such that L t ′ 0 lifts to the common triangular basis of A(t ′ 0 ). Proof. We can prove it based on Theorem 6.4.1 and the correction technique in Section 4.
Alternatively, we can deduce it from Theorem 6.4.7 and the existence of quantum F -polynomials.
Next, we use the correction technique in Section 4 to reduce the existence theorem 6.4.1. Proof. We know these conditions are satisfied in t[0] = t. Any given seed t[d], d ∈ Z, is similar to t[0]. So we obtain its triangular basis as similar elements to the elements of L t[0] by Lemma 6.1.8. We deduce that the local conditions are satisfied the correction technique in Section 4. Lemma 6.4.6. Let µ be any mutation sequence and (t[d]) d∈Z a chosen injective-reachable chain of seeds on which L0 and L2 hold. If for some 1 ≤ k ≤ n, the cluster variable X k (µt) belongs to the basis L t , then for any d ∈ Z, the cluster variable Proof. The compatible pairs of the seeds t[d], d ∈ Z are similar. Because X k (µt) is unitriangular to I t , we have a similar unitriangular expansion of X σ d k The following theorem is a possible reduction of the existence theorem 6.4.1, which reduces the criterion to a finite problem.
Theorem 6.4.7 (Reduced existence theorem). Assume that t is a seed injective-reachable via (Σ, σ) and B(t) a skew-symmetric matrix. Assume that t verifies the local conditions L0(t), L1(t), L2(t), L3([1, n]; t), and, furthermore, its triangular basis L t contains the quantum cluster variables obtained along the mutation sequences Σ and σΣ −1 starting from t.
Proof. By Proposition 6.4.5 and the correction technique 4.2.1, the existence of L t implies the existence of the triangular basis L t[d] , d ∈ Z, which satisfies L0, L1, L2, L3 as well. It remains to show that these bases are compatible.
Repeatedly using Proposition 6.3.5, we can construct the triangular bases L t ′ for the seeds t ′ obtained along the mutation sequence σΣ −1 from t to t[−1]. These bases are compatible with L t . In particular, L t[−1] and L t are compatible. By using Lemma 6.4.6, we deduce the similar result for L t[d] , d ∈ Z. Now, we have found all conditions demanded by Theorem 6.4.1. The claim follows.

Graded quiver varieties and minuscule modules
Fix a symmetric generalized Cartan matrix C. As a preparation for the statement of monoidal categorification in Section 8.4 and the proof of theorem 9.4.1, we review graded quiver quivers, working in a framework slightly generalized than that of [Nak03]. Following the arguments of [Nak03], we obtain characters of some simple modules as our main result Proposition 7.4.3(i)(ii).
A reader unfamiliar with graded quiver varieties, quantum affine algebras, or only interested in type ADE might admit the construction in [Nak03][Nak04] and skip this section, whose main result will only be used in the proof of Proposition 9.2.4. 7.1. A review of graded quiver varieties. We start with a review of the graded quiver varieties used in [KQ12][Qin13], whose construction follows from that of [Nak01][Nak04]. We choose its grading as in [Qin13] [KQ12].
Graded quiver varieties. Given any generalized symmetric Cartan matrix, we associate the diagram Γ to the Cartan matrix such that its set of vertices is I and has −C ij edges between any two different vertices i and j. We then choose an orientation Ω of its edges.
Fix an acyclic quiver (Γ, Ω) associated with C. For any arrow h in Ω, we denote its opposite arrow by h. Let Ω denote the set of arrows opposite to the arrows in Ω. Define the H to be the union of Ω and Ω.
We follow the torus grading in [Qin13, Section 4.3] [KQ12]. Choose a map ξ from I to {1, 2, . . . , r}, whose images are denoted by ξ i for i ∈ I, such that ξ i > ξ j whenever there exists a nontrivial path from i to j in the acyclic quiver (Γ, Ω). Define ǫ ij ∈ {±1} to be the sign of ξ i − ξ j and Example 7.1.1. Consider the acyclic quiver (Γ, Ω) in Figure 5. We have ǫ 21 = ǫ 23 = ǫ 34 = ǫ 45 = 1, ǫ 12 = ǫ 32 = ǫ 43 = ǫ 54 = −1. The q-analog of Cartan matrix C q is defined to be the linear map from Z I×Z to Z I×Z , such that any η = (η i,b ) has the image given by For any finitely supported v ∈ N V , w ∈ N W , we say the pair ( For any finite supported w, define C −1 q w to be the unique vector such that (C −1 q w) i,b = 0 when b ≪ 0 and C q (C −1 q w) = w. We refer the reader to [HL11, Example 2.2] for an example, where a comparison of notations is presented in Remark 7.2.1.

FAN QIN
Example 7.1.2. Consider the acyclic quiver (Γ, Ω) in Figure 6. Then we can only choose ξ 1 = 3, ξ 2 = 2, ξ 3 = 1. We have ǫ 12 = ǫ 23 = ǫ 13 = 1, ǫ 21 = ǫ 32 = ǫ 31 = −1, and the following equation The component η 2,a+1 contributes to the components of degree (2, a), (2, a + 2), (1, a + 1 + ǫ 21 ), (3, a + 1 + ǫ 23 ) via the map C q . For any d ∈ Z, define the shift operator [d] to be the linear map from Z I×Z to Z I×Z such that, for any η ∈ Z I×Z , we have Take any finitely supported dimension vector v, v ′ , in N V and w in N W . Denote the associated graded C-vector by V , V ′ and W . We define the following vector spaces , w), whose coordinates will be denoted by Notice that Rep(Ω; v, w) can be naturally viewed as the vector space of (v, w)-dimensional representations of a quiver with vertices V ⊔ W. We denote this quiver by Γ Ω can call it the framed repetition quiver associated with the acyclic orientation Ω.
Different acyclic orientations produce isomorphic framed repetition quivers. Choose and fix such a quiver Γ Ω from now on.
The analog of the moment map µ is the linear map from Rep(Ω; v, w) to L(v, v[−1]) given by Example 7.1.3. Let the acyclic quiver (Γ, Ω) be given by Figure 5. Part of the the framed repetition quiver Γ Ω is shown in Figure 7 1,a+6  The variety µ −1 (0) has the natural action by the reductive algebraic group G v = (i,b)∈V GL(V i,b ) such that, for any given g = (g i,b ), the action is given by Define M • (v, w) to be the geometric invariant quotient of the G vvariety µ −1 (0) associated with χ v in the sense of Mumford, cf. [MFK94]. For any v ′ < v, there is a natural embedding of M 0 Grothendieck ring and characters. We have intersection cohomology sheaves . Define the operation of Z[t ± ] on the Grothendieck ring of the derived category of constructible sheaves over C-vectors spaces on M 0 • (w) such that t acts as the shift functor. Let K w denote the free abelian group generated by the classes of the sheaves IC(v, w) over Define the quantum torus Y to be the Laurent polynomial ring for any w 1 , w 2 ∈ N W ,cf. [KQ12][Qin13, (40)]. We let [ ] to denote the normalization by t-factors in Y.
There exists a natural multiplication on R t , which is derived from the geometric restriction functor and the Euler twist E( , ), cf. [Nak11]. Let R t denote the Grothendieck ring equipped with this multiplication. Furthermore, the structure constants of its natural basis {S(w)} are given by such that a v (t) ∈ N[t ± ], a 0 = 1. Let R t=1 denote the specialization of R t by taking t = 1.
Let χ t,q denote the t-analog of the q-characters (qt-character for short) χ t,q is an injective ring homomorphism with respect to the twisted product * , cf. [Nak04][VV03] [Her04]. We refer the reader to [Qin13] for its precise definition in our convention. In particular, define the Laurent monomial Then the character of any S(w) is a formal series pair (v, w) is. We say the leading term of χ t,q S(w) is Y w . The simple module S(w) is called minuscule if Y w is the only one l-dominant monomial appearing in its character.
Truncated characters. We consider truncated qt-characters which were first introduced in [HL10].
For any function c ∈ (2Z) I . Denote its value at i ∈ I by c i . When the function c takes a constant value c ∈ 2Z, we simply denote it by c. We say c is Ω ′ -adaptable (or adaptable for short), if the full subquiver of the fixed framed repetition quiver Γ Ω on the vertices (i, c i −1), i ∈ I, is the acyclic quiver (Γ, Ω ′ ). We shall always assume c to be adaptable for some Ω ′ from now on.
Let χ t,q ≤c denote the truncation of χ t,q at the subring Remark 7.2.1. When taking the quiver (Γ, Ω) to be bipartite and choosing any j ∈ I, we can identify our graded framed quiver with the corresponding quiver in [Nak03] by identifying our vertex (i, s − ǫ ij ) with the vertex (i, s) in [Nak03]. We can then translate the results of [Nak03] to Section 7.2.
For any Laurent monomial m = 1 in Y and j ∈ I, define r j (m) = max s ∈ Q|m contains the factor Y i,s+ξ ji for some i ∈ I .
Here we use ξ ij to distinguish the grading of the vertices in (Γ, Ω). It follows that r j (m) + ξ ji = r i (m), for any i, j, and that the non-positive criterion in the following definition is independent of the choice of j.
Definition 7.2.2 (right negative). The monomial m is said to be right negative if, for some j ∈ I, the factors Y i,r j (m)+ξ ji has non-positive powers for any i ∈ I.
Example 7.2.3. Let us take the quiver (Γ, Ω) in Figure 6. Then the Notice that the l-dominant monomials are not right negative. The product of two right negative monomials is still right negative.
In [Nak03], Nakajima computed truncated qt-characters of Kirillov-Reshetikhin modules by studying right negative monomials and using combinatorial properties of qt-characters defined over graded quiver varieties. His arguments remain effective for our graded quiver varieties and imply the following results.
Theorem 7.2.4 ([Nak03]). (i) All monomials in χ t,q W (i) k,a are right negative except its leading term Y w (i) k,a . (ii) Let m be a right negative monomial appearing in the truncated qt-character χ t,q ≤2k W (i) k,a , then we have where 0 ≤ s ≤ k − 1.
(iii) We have the following equality in the quantum torus Y; Recall that [ ] denote the normalization in the quantum torus such that the leading term has coefficient 1.
Notice that m becomes the leading term Y w (i) k,a of χ t,q W (i) k,a if we take s = k. The following Lemma was used in the proof of Theorem 7.2.4.
Lemma 7.2.5 ( [Nak03]). Take monomials m and m ′ from χ t,q W (i) k,a and χ t,q W (i) 1,a+2k respectively. If the product mm ′ is not right negative, then the following claims are true.
(i) The monomials take the form is 1 if s = k and t −1 otherwise.
As a consequence of the properties of Kirillov-Reshetikhin modules in Theorem 7.2.4, we deduce the following T -systems.
Theorem 7.2.6 ([Nak03]). The following equation holds in the quantum torus Y: k+1,a agrees with the simple module S(w In order to compare Grothendieck rings with quantum groups, we will need a different t-deformation (introduced in [Her04] as an algebraic approach to qt-characters) and work with the corresponding extended Grothendieck ring R = R t ⊗Z[t ± 1 2 ] as in [HL11]. To be more precise, as in [KQ12, Section 6.1], we define the quantum torus Y H t 1 2 as the Laurent polynomial ring Z[t ± 1 2 ][Y ± i,a ] (i,a)∈W equipped with the twisted product * defined by for any w 1 , w 2 ∈ N W ,cf. [KQ12, Section 6]. Replacing the Euler form −E( , ) in the construction of the twisted multiplication in R by 1  2 N ( , ), we obtain the extended Grothendieck ring R H t 1 2 , which will be simply denoted by R. Denote the corresponding qt-character from R to Y H Assume that the quiver (Γ, Ω) is of bipartite orientation. Choose and fix some j ∈ I. Then our vertices (i, b − ǫ ij ) are labeled by (i, b) in [HL11], and C −1 q is translated as follows:

Consequently, (39) is translated into
Then it identifies with [HL11, (6)] because the matrix C is symmetric.
Lemma 7.3.2. For any c ∈ 2N, d, d ′ ∈ N, and i ∈ I, we have Proof. With the help of [Qin13, Lemma 4.3], for any (i, a) ∈ W and (j, b) ∈ V, we make the following computation.
The desired claim follows.
7.4. Computation of characters. We compute qt-characters of some simple modules which will be useful in the proof of Theorem 9.4.1.
Recall that we have calculated the Euler form E in Lemma 7.3.2. Inspired by Lemma 7.2.5, we consider the following product. Results of this type was found in [FH14].
Lemma 7.4.1. Fix integers h, k ∈ N such that k ≥ 2, k > h ≥ 1. Take monomials m and m ′ from χ t,q W (i) k,a and χ t,q W (i) h,a+2(k−h+1) respectively. If the product mm ′ is not right negative, then the following claims are true.
We compute the truncated qt-characters of the Kirillov-Reshetikhin modules For simplicity, specialize t to 1. It is easy to compute the following difference We observe that the difference has only one l-dominant monomial. In fact, we will show it is the truncated character of the minuscule module W For 0 ≤ h ≤ k, we define: Then W (iii) We have the following equality in the quantum torus Y; where [ ] denote the normalization by t-factors.
(iv) In the Grothendieck ring R t , W k+1,a agrees with S(w Definitions. We refer the reader to [HL10] and [Nak01] for more details. Let A † Z be any given commutative cluster algebra. We give the following definition following [HL10]. Definition 8.1.1. We say that A † Z admits a monoidal categorification if there exists a tensor category (U, ⊗), such that (i) There exists a ring isomorphism κ from A † Z to the Grothendieck ring R t=1 of U.
(ii) κ sends the cluster variables (resp. the cluster monomials) to classes of prime real simple objects (resp. classes of real simple objects).
Assume that the set of simple objects (simples for short) of U are parametrized as {S(w)} where w are some parameters. By abuse of of notation, we denote an object and its class in the Grothendieck ring R t=1 by the same symbol. Denote the multiplication in the Grothendieck ring R t=1 by ⊗.
Next, let a quantum cluster algebra A † be given. Let (U, ⊗) be a monoidal category and R t its graded Grothendieck ring equipped with a twisted product * and a bar-involution ( ) such that t = t −1 . Assume that R t has a Z[t ± ]-basis {S(w)}, called the simple basis, where S(w) are classes of simple objects and bar-invariant. Here the t-grading in R t comes from the natural grading of objects in U or from choosing a t-deformation of the previous Grothendieck ring R t=1 . We assume that the simple basis in the graded Grothendieck ring has positive structure constants with respect to the twisted product * : By the convention of this article, we will take d = 2. Definition 8.1.2. We say that U provides a monoidal categorification of A † if (i) There exists a ring isomorphism κ from A † to the graded Grothendieck ring R of U such that κ(q 1 2 ) = t 1 d and it commutes with the barinvolutions.
(ii) κ sends the quantum cluster variables (resp. the quantum cluster monomials) to classes of prime real simple objects (resp. classes of real simple objects).
Standard basis and properties. In this article, we shall work with the case that the set of parameters w is ⊕ l k=1 Nβ k , whose basis vector β k is called the k-th root vectors. The object S(β k ) is called the k-th fundamental object.
Endow N l with the natural lexicographical order ≺ w such that for any w = (w k ), w ′ = (w ′ k ), w ≺ w w ′ if and only if there exists some 1 ≤ p ≤ l, such that w k = w ′ k for any k < l and w p < w ′ p , cf. [Kim12,4.3.5].
Let M(w) denote the class in R defined as where α w is some special integer. Denote m = t − 1 2 Z[t − 1 2 ]. We shall further assume that the set of classes {M(w)} decomposes (≺ w , m)unitriangularly into the basis {S(w)} in the sense of Definition 3.1.7. Then it is also a Z[t ± 1 2 ]-basis of R, which is called the standard basis. Assume κ is given and we fix the initial seed t 0 . If the elements κ −1 S(w) have distinct leading degrees deg S(w) ∈ D(t 0 ) = Z m in the quantum torus T (t 0 ) (Definition 3.1.3), we define the map θ −1 sending w to deg κ −1 S(w). Denote the elements κ −1 S(w) and κ −1 M(w) in A † by S(w) and M(w) respectively.
The following observation states that a monoidal categorification of A Z implies that of A.
Proposition 8.1.3. Let U be a monoidal category which satisfies (47) with its graded Grothendieck ring R isomorphic to a quantum cluster algebra A † by identifying some simples with the initial quantum cluster variables. Assume that U has the commutative Grothendieck ring R t=1 as the specialization of R at t 1 2 = 1 and, moreover, U provides a monoidal categorification of the commutative cluster algebra A † Z . Then U provides a monoidal categorification of A † as well.
Proof. Let t be any seed and {S i (t)} the collection of simples in R t=1 identified with the commutative cluster variables {x i (t)}. Then, for any index k, there exists some simple module S k (t) * such that we have the following exchange relation in R t=1 : where S + and S − are monomials in {S i (t)}. In particular, S + and S − are simples. Now, assume that after the quantization, {S i (t)} is identified with {X i (t)} the set of the quantum cluster variables in t. The twisted product takes the form 2 ] by (47). It follows that f + and f − must be powers of t ± 1 2 so that they specialize to 1. Next, left multiplying S k (t) −1 , by the bar-invariance of S k (t) * , we must get where [ ] t denote normalization in the quantum torus Z[t ± 1 2 ][S i (t) ± ], cf. Definition 3.1.4. Therefore, S k (t) * is identified with the quantum cluster variable X k (µ k t).
Starting from the initial seed t 0 and repeating the above argument, we obtain the desired claim.
8.2. Quantum cluster algebras associated with words. We shall recall various quantum cluster algebras associated with words, which are expected to admit monoidal categorifications. More details could be found in [GLS11].
Ice quiver of type (i)(ii). Fix an non-empty vertex set I = {1, . . . , r} and a generalized Cartan matrix C = (C i,j ) i,j∈I . For any i ∈ I, let s i denote the simple reflection at the i-th simple root. By choosing an acyclic orientation Ω on the diagram Γ associated with C. We obtain an acyclic quiver (Γ, Ω).
Let i = (i l , i l−1 , . . . , i 2 , i 1 ) be a non-empty word of length l with elements in I.
We say the word i is reduced, if the Weyl group element w(i) = s i l · · · s i 2 s i 1 has length l.
We say the word i is Ω-adaptable (or adaptable for short), if there exists an acyclic quiver (Γ, Ω) associated with the Cartan matrix C, such that (i l , . . . , i 2 , i 1 ) is a sink sequence of the quiver when read from right to left, cf. [GLS07, Section 3.7] for more details.
An adaptable word (i r , . . . , i 2 , i 1 ) which contains every i ∈ I exactly once is called an acyclic Coxeter word.
We consider the following types of the pair (C, i): (i) The generalized Cartan matrix C is symmetric, and i is reduced.
(ii) The generalized Cartan matrix C is of type ADE, and the word i is Ω-adaptable for some acyclic quiver (Γ, Ω) associated C. Without loss of generality, we can assume that the diagram Γ associated with C is connected.
For any 1 ≤ k ≤ l, i ∈ I, we define the following notations (cf. [GLS11, Section 9.8] [BFZ05]) For any interval [a, b] = {k ∈ Z|a ≤ k ≤ b} in [1, l], define the multiplicities is called the level of the vertex k.
If we define the following subset of [a, b]: Then the cardinality of [1, l](i) is just m(i k ). The subset [1, l](i) is naturally a sequence (read from right to left) whose elements are ordered as the following: Following 3 [GLS11, Section 2.4] [BFZ05], we associate the quiver Γ i to i as the following: • The vertices of Γ i are 1, 2, . . . , l.
• For 1 ≤ s, t ≤ l, there are −C isit arrows from t to s if t + ≥ s + > t > s. These are called the ordinary arrows of Γ i . • For any 1 ≤ s ≤ l, there is an arrow from s to s + if s + ≤ l.
These are called the horizontal arrows of Γ i . By default, we consider words satisfying the following assumption.
Assumption 5. We assume that every element of I appears in the word i at least once.
Define the mutation sequence (read from right to left) We also define a permutation σ i of [1, l] such that, for any k ∈ [1, l], We always view the quiver Γ i as an ice quiver by freezing the vertices max i, i ∈ I. The number of exchangeable vertices is n = l − r. We associate the initial l × n matrix B(t 0 ) to the quiver Γ i such that, at position (i, j), its entry b ij equals the difference between the number of arrows from i to j and the number of arrows from j to i. Let A Z (i) = A Z (t 0 ) denote the commutative cluster algebra arising from this initial matrix B(t 0 ).
We use L(k min , k) to denote the initial cluster variables X k (t 0 ), 1 ≤ k ≤ l. When ← − µ k is nontrivial, applying it to the seed ← − µ k−1 · · · ← − µ 2 ← − µ 1 t 0 , we obtain the new cluster variables which we will denote by L(k + , k[d] + ). The mutation sequence Σ i is the composition of the following mutation sequences (read from right to left): By applying the mutation sequence ← − µ 1 to t 0 , we obtain the new cluster variables (from right to left): L(5, 10), L(5, 8), L(5, 5).
8.3. Monoidal categorification conjecture: type (i). We refer the reader to [Kim12] and [GLS13] for precise definitions in this subsection.
Quantum cluster algebra structure. When the pair (C, i) is of type (i), A † Z (i) is isomorphic to the coordinate ring of the unipotent subgroup N(w i ), cf. [BFZ05] [GLS11]. Moreover, the canonical initial seed t 0 has a natural quantization matrix Λ(t 0 ) by [GLS13]. We define the rational quantum cluster algebra to be A † . It has the dual PBW basis and the dual canonical basis as the restrictions from that of U Z[v ± ] (n). We refer the reader to [Kim12] for more details. Denote A denote the dual canonical basis of the quantum unipotent subgroup A Q(v) (n(w i )). The map κ transports the bar-involution on T (t 0 ) to a bar-involution on A Q(v) (n(w i )). By multiplying v 1 2 -factors, we obtain a bar-invariant basis S(w)|w ∈ N l . Then, the rational quantum cluster algebra has the bar-invariant basis {S(w)} = {κ −1 S(w)}. 4 In [GLS13], the square roots q 1 2 and v 1 2 did not appear because the quantum cluster algebra is rescaled and its bar-involution is defined differently.
By the work of Khovanov, Lauda [KL09] and Rouquier [Rou08], the quantum group U(n) can be viewed as the Grothendieck ring of modules of the corresponding quiver Hecke algebra, such that v-shifts of the dual canonical basis corresponds to the classes of finite dimensional simple modules [VV11] [Rou12]. It follows that the integral form A Z[v ± ] (n(w i )) is isomorphic to the graded Grothendieck ring R(w(i)) of certain monoidal subcategory U(w i ), and each S(w) represent the class of a simple module.
The category U(w i ) satisfies the assumptions in Section 8.1. In particular, R has the basis {M(w)} with expected properties, which corresponds to the normalization of the dual PBW basis. Here, we use Kashiwara's bilinear form following the convention of the [GLS13]. Proof. Notice the algebra A † is generated by the quantum cluster variables over Z[q ± 1 2 ]. Therefore, κA † is generated by elements in {S(w)} and, consequently, contained in the integral form A Z[v ± 1 2 ] (n(w i )). But the integral form is generated by the fundamental modules M(β k ) over Z[v ± ], which are images of quantum cluster variables. Therefore, κA † is the integral form.
Moreover, the Lusztig's braid group symmetry T i 1 at vertex i 1 maps where h i 1 , ξ k linearly depends on the homogeneous grading ξ k of D s i 2 ···s i k ̟ i k ,s i 2 ···s i k ̟ i k . And it will send the dual canonical basis of A 2 ) (n(w i )) modulo these factors depending on homogeneous gradings, cf. [Kim12]. Consequently, after rescaling, we have the embedding from R(w i−i 1 ) into R(w i ) sending S(β k ) to S(β k ) and the simple basis into the simple basis. Notice that, if we work with Lusztig's bilinear form instead, then the above factor does not appear, cf. [Lus93, 38.2.1].
Therefore, there is an natural embedding from R(w i−i 1 −i l ) to R(w i ). Denote the images of the above embeddings by R(w i ) l , R(w i ) 1 , R(w i ) 1, l respectively. 8.4. Monoidal categorification conjecture: adaptable word. When the pair (C, i) is of type (ii), we refer the reader to [HL10] for the original monoidal categorification conjecture of A Z (i) in terms of finite dimensional representation of quantum affine algebras. In this section, we shall present a quantized and slightly generalized version of this conjecture in terms of graded quiver varieties, where i is adaptable and C any generalized symmetric Cartan matrix.
Let i be any given Ω ′ -adaptable word for an acyclic quiver (Γ, Ω ′ ). Fix a multidegree a = (a k ) 1≤k≤l ∈ (2Z) l such that a k[d+1] = a k[d] − 2. Define the corresponding map ι a which sends the vertices 1 ≤ k ≤ l of Γ i to (i k , a k ). It follows that We further assume the multidegree a to be Ω ′ -adaptable (or adaptable for simplicity) such that the full subquiver of Γ Ω on the vertices ( min i, amin i − 1), where i ∈ I, is isomorphic to (Γ, Ω ′ ). Because the word i is adaptable, an adaptable embedding multidegree a always exists. The associated map ι a is called an adaptable embedding.
Example 8.4.1. We take the framed repetition quiver Γ Ω associated with (Γ, Ω) in Figure 5. Part of this quiver is drawn in 7.
Fix an adaptable embedding ι a . Define R(i, a) to be the subring of the extended graded Grothendieck ring R generated by the simples W (i k ) 1,a k , i ∈ I, 1 ≤ k ≤ l. It has the simple basis and standard basis as expected in Section 8.1. Quantum cluster algebra structure. We denote by θ a : T (t 0 ) → Y H t 1 2 the embedding of the Laurent polynomial rings such that It follows that we have then induces the twisted multiplication on T (t 0 ) via the morphism θ a such that s +1,as ). By Lemma 7.3.2, it is compatible with the matrix B such that We denote the quantum cluster algebra obtained via this quantization by A(i). By comparing T -systems with specific exchange relations, we obtain the following result, whose proof goes the same as that of [GLS13, Proposition 12.1].
Theorem 8.4.2. Assume that the word i and the multidegree a are adaptable. Then there exists an injective algebra homomorphism κ −1 from the Grothendieck ring R(i, a) to the quantum cluster algebra A † (i), such that κ −1 (t 1 2 ) = q 1 2 and, for any If the Cartan matrix C is of Dynkin type ADE, R t (i, a) and R(i, a) = R H t 1 2 (i, a) are t-deformations of the Grothendieck ring of the monoidal subcategory U(i, a) of finite dimensional representations of the the quantum affine algebras U ε ( g), where each S(w) represents a simple module as constructed by [Nak01].
The corresponding monoidal categorification conjecture takes the following form. Embeddings of Grothendieck rings. Let a 1 to be the multidegree obtained from a by deleting the component a 1 . Similarly, construct a l and a 1, l by deleting a l and a 1 , a l from a respectively. By construction, we have natural embeddings of R(i−i 1 , a 1 ), R(i−i l , a l ), R(i−i 1 −i l , a 1, l ) into R(i, a), which send S(β k ) to S(β k ) and the simple bases into the simple basis. As before, we denote the images by R(i, a) 1 , R(i, a) l , R(i, a) 1, l respectively.

Monoidal categorification conjectures
We refer the reader to Section 2 8 for the necessary notations and definitions used in this section.
Fix a pair (C, i) of type (i) or (ii). We want to know if the quantum cluster algebra A = A(i) has the common triangular basis or not. We assume that every i ∈ I appears at least once in i (Assumption 5).
Recall that we have the corresponding extended Grothendieck ring R(i) which can be taken as an extension of a quantum unipotent subgroup if i is reduced or a subring R(i, a) for some adaptable multidegree a if i is adaptable. 9.1. Local conditions in the initial seed. We work in the canonical initial seed t 0 associated with (C, i) in this section.
Define the injective linear map θ −1 from N l = ⊕ 1≤k≤l Nβ k to D † (t 0 ) such that, for any i ∈ I, we have Lemma 9.1.1. θ −1 is an isomorphism.
Proof. Recall that there exists an isomorphism κ −1 from A † Z (i) ⊗ Q to R t=1 (i) ⊗ Q. It sends the standard basis element M(w) to a pointed element κ −1 M(w) with leading degree θ −1 w. Notice that the simple basis element κ −1 S(w) is also pointed at θ −1 w.
Any element Z in A † Z (i) ⊗ Q has the expansion into the simple basis elements κ −1 S(w). Consequently, the ≺ t 0 -maximal degrees of Z take the form θ −1 w, w ∈ N l . It follows by definition that the dominant degree lattice D † (t 0 ) equals θ −1 N l .
Notice that, for any g 1 , g 2 ∈ D † (t 0 ), the condition g 1 ≺ t 0 g 2 implies that g 1 ≺ w g 2 with respect to the lexicographical order ≺ w on ⊕Nβ k . We have the following useful consequence.
With the help of Lemma 9.1.1 9.1.2, we obtain the following proposition, which verifies Conjecture 8.3.3 for type (i) quantum cluster algebras. It was already known for type (ii) quantum cluster algebras by [HL13a, Theorem 5.1].
Proposition 9.1.3. . The injective homomorphism κ −1 from R(i) to A † (i) is an isomorphism. In particular, S(w) is a basis of A † (i).
Proof. Work in the initial quantum torus T (t 0 ). Observe that any element of A † (i) has maximal degrees in D † (t 0 ). For any given element Z of A † (i), combining this observation and the expansion algorithm in Remark 3.1.8, we obtain a unique ≺ t 0 -unitriangular expansion of Z into the D † (t 0 )-pointed set {S(w)}: Then this expansion is finite because (≺ t 0 , D † (t 0 )) is bounded from below. Therefore, {S(w)} is a basis of A † (i). The claim follows.
For any 1 ≤ a ≤ b ≤ l with i a = i b , use S(a, b) to denote the simple module S(β a + β a[1] + . . . + β b ). Then we have the quantum cluster variables X k (t 0 ) = S(θe k ) = S(k min , k) and, for any k = k max , I k (t 0 ) = S(k[1], k max ).
Remark 9.1.4. When the word i is adaptable, we can construct the graded Grothendieck ring R(i) = R(i, a) as in Section 7 for some choice of adaptable multidegree a = (a k ) ∈ Z l , such that the root vectors β k corresponds to the unit vectors e i k ,a k . In this situation, the map θ identify the X-variables X k and Y -variables i k ,a k −1 respectively in the convention of Section 7, cf. Section 8.4.
For any w, the basis element S(w) = κ −1 S(w) has the leading degree g = θ −1 w and we denote it by L( g). Denote the basis {S(w)} by L †,t 0 .
It is (≺ w , m)-unitriangular to the basis L †,t 0 . By Lemma 3.1.9(iii), it is (≺ t 0 , m)-unitriangular to L †,t 0 as well. From now on, by default, we shall only consider the dominance orders ≺ t associated with a seed t (or denoted by ≺ when there is no confusion). Notice that N l = ⊕ k Nβ k inherits the order ≺ t 0 via the isomorphism θ. In type (i), this property was proved in [Kim12]. With the help of the standard basis and T -systems, we will give a proof by induction in the end of this subsection .
Corollary 9.1.6. By taking into account of the inverses of S(θe s ), where s max = s, the simple basis {S(w)} generates a basis of the quantum cluster algebra A(i), which we denote by L t 0 . Its structure constants are contained in N[q ± 1 2 ].
We use t 0 l − to denote the seed from t 0 by freezing l − . Because the only exchangeable vertex connected to l is l − , the frozen vertex l does not connect to any exchangeable vertex in the corresponding ice quiver Q(t 0 l − ). Consequently, the quantum cluster algebra A † (t 0 l − ) is generated by A † (i − i l ) and the frozen variable X l . It follows that A † (t 0 l − ) has the following basis L † (t 0 l − ) = X d l L(η)|L(η) ∈ L † (t 0 (i − i l )), d ∈ N .
As a consequence, we obtain the following Lemma.
where (w 2 ) l = 0, e(w, w 1 , w 2 ) is contained in m if w 1 l + w 2 = w + θe k and equals to 1 if w 1 l β l + w 2 = w + θe k . Recall that the standard basis of A †,t 0 is unitriangular to its simple basis. The claim follows.
Corollary 9.1.9. (triangular basis) The basis L t 0 verifies Condition L2(t 0 ). Namely, it is the unique triangular basis for the seed t 0 .
We finish this subsection by a proof of Proposition 9.1.5.
Lemma 9.1.10. Take any Z in R. If Z is (≺, m)-unitriangular to the standard basis, then for any w 1 , w l ∈ N, the normalized product [Z * S(w l β l )] and [S(w 1 β 1 ) * Z] are (≺, m)-unitriangular to the standard basis.
Proof. Notice that we can replace Z by a unitriangular decomposition into the standard basis. The claim follows from Lemma 6.1.4.
Lemma 9.1.11. Take any Z = a i S(w i ) in R such that a i ∈ N[t ± 1 2 ]. If there exists some simple module S(w) such that Z * S(w) or S(w) * Z is a simple module up to t 1 2 -shift, then Z is a simple module.
Proof. The claim follows by the positivity of the structure constants of the simple basis.
Proof of Proposition 9.1.5. We prove the claim by induction.
If the adaptable word has length l = |I|, the statement is trivial. Assume that the Proposition has been verified for any adaptable word of length less than l. For any given word i of length l, w = n k=1 w k β k with w k ∈ N, and any 1 ≤ s ≤ l such that s = s max , we want to show the normalized product [S(θe s ) * S(w)] is a simple module. By its bar-invariance, it suffices to show that this product is (≺, m)-unitriangular to the simple basis, or, equivalently, to the standard basis.
Let w 1 , w l , w 1, l denote the dimension vector obtained from w by subtracting w 1 β 1 , w l β l , and w 1 β 1 + w l β l respectively.
Notice that the module S(w) is (≺, m)-unitriangular to the standard basis: where we have used induction hypothesis R 1 ∼ → R(i − i 1 ) in the last equality. Here, the coefficients are contained in m except the leading term with w 1 = w, w 2 = w 1 1 equals to 1. By Lemma 9.1.10, the above expression is (≺, m)-unitriangular to the standard basis.
(ii-2) Assume that i l = i 1 . Consider the exchange relation at vertex l[−1] in the seed µ l[−2] · · · µ 1[1] µ 1 t 0 of the quantum cluster algebra A(i) (cf. [GLS11] for its ice quiver). Via the isomorphism κ −1 , it corresponds to the following T -system in R: (53) Denote the last simple module 1≤t<l,t=t max S(θe t ) −C i l i t by S(q) Let us take any w ′ such that w ′ 1 = w ′ l = 0. Notice that, in the quantum torus T (t 0 ), the leading monomials of [S(θe l[−1] ) * S(1[1], l)] and S(q) differ by Y l[−1] . . . Y l Y 1 , which commutes with the monomial X θ −1 w ′ by the definition of compatible pair. If we deal with the type (ii), this commutativity also follows from (40)(41). Therefore, by using the above T -system, we obtain . Because [S(θe l ) * S(w ′ )] is a simple module, the above expansion of S(θe l ) * S(w) is (≺, m)-unitriangular to the standard basis by Lemma 9.1.10. The desired claim follows by its bar-invariance. 9.2. Compatibility. By Theorem 6.4.7, in order to apply the existence theorem to the quantum cluster algebra A(i), it remains to check the following property.
Conjecture 9.2.1. The quantum cluster variables obtained along the sequence (σΣ i ) −1 from t 0 to t 0 [−1] are contained in L t 0 .
In Section 9.3, we will give an example as a different quick proof of this property for type A 4 based on the level-rank duality.
In this section, we verify that the triangular bases in t 0 and t 0 [−1] are compatible when i = c N +1 for some acyclic Coxeter word c and N ∈ N.
Let i − i l denote the word obtained from i by removing the last element i l and Σ i−i l the corresponding sequence. Then we have Σ i = (σ ← − µ l ) −1 Σ i−i l = ← − µ l Σ i−i l . This observation inductively implies the following result.
Lemma 9.2.3. For any d ∈ Z, the seed t[d] is injective-reachable via either (σ d Σ i , σ) or (σ d Σ i , σ). Moreover, the sets of the quantum cluster variables obtained along both sequences starting from t[d] are the same. definition of mutations, this property is preserved by the quivers appearing along AB or BA. Therefore, the mutation sequences AB and BA give identical set of vertices. The claim follows by recursively swapping A and B for all d.
Moreover, any such quiver appearing is a union of small blocks in Figure 14 for i = j ∈ I, 0 ≤ d ′ ≤ N − 2, with at most one diagonal edge of multiplicity −C ij .  Proof. We have shown that the triangular basis L t 0 in t 0 satisfies the conditions L0, L1, L2, L3. Observe that the all quantum cluster variables along the mutation sequence Σ i from t 0 to t 0 [1] are known to be Kirillov-Reshetikhin modules, and, in particular, contained in the 4 7 9 10 1 2 3 5 6 8 Figure 15. Ice quiver Q(t ′ 0 ) obtained by rotating the quiver in Figure 9 and changing coefficient pattern.
We have seen that the simple basis {S(w)} provides the initial triangular basis in the seed t 0 after localization. Combining Proposition 9.2.7, we have verified all the conditions in Theorem 6.4.7 for the word c N +1 . Notice that, to any adaptable word i, we can associate an initial subword i ′ of some c N +1 , such that w i = w i ′ and, consequently, A(i) agrees with A(i ′ ). Notice that i ′ is adaptable as well. By working with type (ii) quantum cluster algebras, we have the embedding from R(i ′ , a ′ ) to R(c N +1 , a) and the following result.
Theorem 9.4.1. Assume that C is symmetric and i an adaptable word. Then the simple basis {S(w)}, after localization at the frozen variables, gives the common triangular basis for all seeds of the quantum cluster algebra A(i) in the sense that it verifies all the conditions in Theorem 6.4.1. In particular, it is parametrized by tropical points as expected by Fock-Goncharov, and it contains all quantum cluster monomials.
The following result is a direct consequence.
Theorem 9.4.2. Conjecture 8.4.4 holds true. In particular, the quantum cluster monomials in A of type (ii) are simple modules of quantum affine algebras.
For quantum cluster algebras arising from quantum unipotent subgroups, we have verified Conjecture 8.3.3 in Proposition 9.1.3. Moreover, Theorem 9.4.1 implies that the quantum cluster monomials of A of type (i) are contained in the dual canonical basis for the following cases.
Theorem 9.4.3. Conjecture 8.3.2 holds true for any reduced word w which is inferior than some adaptable word with respect to the left or right weak order. In particular, it holds true for any reduced word w when the Cartan type is ADE.
Proof. The first claim follows from Theorem 9.4.1 and compositions of the embeddings of Grothendieck ring in the end of Section 8.3.
For type ADE, the claim holds for the longest word because it is adaptable. Any reduced word w is inferior than the longest word with respect to the left and right weak order, cf. [BB05]. The second claim follows.
Remark 9.4.4. When the quantum cluster algebra is of type (i). The claim that all seeds t satisfy condition L3(t) might be compared with the following weaker form of Leclerc's conjecture [Lec03, Section 3.1].
Let b 1 , b 2 be elements of the dual canonical basis B * . Suppose that b 1 corresponds to a cluster variable. Then the expansion of b 1 , b 2 on B * is of the following form: where s ∈ Z, γ c b 1 b 2 ∈ q s−1 Z[q −1 ]. Leclerc's conjecture was recently proved in [KKKO14].