An analog of Leclerc's conjecture for bases of quantum cluster algebras

Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We introduce an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the $t$-analogs of $q$-characters of simple modules of quantum affine algebras.


FAN QIN
Dual canonical bases and cluster theory. Let g denote a Kac-Moody algebra with a symmetrizable Cartan datum, and U q = U q (g) the corresponding quantized enveloping algebra, where q is not a root of unity. The negative (or positive) part U q − of U q possesses the famous canonical bases [Lus90] [Lus91] [Kas90]. The corresponding dual basis B up also has fascinating properties and is related to the theory of total positivity [Lus94].
Fomin and Zelevinsky invented cluster algebras as a combinatorial framework to understand the total positivity [Lus94] and the dual canonical bases B up . We refer the reader to the survey [Kel08] for further details of cluster algebras.
Let there be given any Weyl group element w ∈ W . Then the dual canonical basis B up of U q − restricts to a basis B up (w) = B up ∩A q [N − (w)] for the quantum unipotent subgroup A q [N − (w)], see [Kim12]. Notice that, if g is a finite dimensional semi-simple Lie algebra, then A q [N(w 0 )] agrees with U q − where w 0 denotes the longest element in W . Thanks to previous works (such as [BFZ05] [BZ05] [GLS11] [GLS13] [GY16] [GY20]), it is known that the quantum unipotent subgroup A q [N − (w)] is a (partially compactified) quantum cluster algebra A q (t 0 ), where the initial seed t 0 = t 0 ( − → w ) is constructed using a reduced word − → w of w. By Fomin and Zelevinsky [FZ02], the dual canonical basis B up (w) is expected to contain all quantum cluster monomials, which was formulated as the quantization conjecture for Kac-Moody cases in [Kim12]. This conjecture has been verified for acyclic cases by [HL10] [Nak11] [KQ14], for symmetric semisimple cases and partially for symmetric Kac-Moody cases by [Qin17], for all symmetric Kac-Moody cases by [KKKO18], and recently, for all symmetrizable Kac-Moody cases by [Qin20].
Leclerc's conjecture. A basis element b ∈ B up ⊂ U q − is said to be real if b 2 ∈ q Z B up . Leclerc proposed the following conjecture regarding the multiplication by a real element of B up , which is analogous to Kashiwara crystal graph operator.
Conjecture 1.1.1 (Leclerc's Conjecture [Lec03, Conjecture 1]). Assume that b 1 is a real element of B up . Then, for any b 2 ∈ B up such that b 1 b 2 / ∈ q Z B up , the expansion of their product on B up takes the form 1.2. Main results. By [Qin17] [KK19] [Qin20], after localization and rescaling, the dual canonical basis B up (w) agrees with the common triangular basis of the corresponding quantum cluster algebra in the sense of [Qin17]. Correspondingly, we formulate the following analog of Leclerc's conjecture.
Conjecture 1.2.1 (Conjecture 5.1.3). Conjecture 1.1.1 is true if we replace the dual canonical basis by the common triangular basis.
Recall that the quantum cluster monomials provide a subset of the real elements in the dual canonical basis B up (w) (we conjecture that all real elements take this form, see Conjecture 5.2.2). Our first main result is the following weaker form of the analogous conjecture.
Theorem 1.2.2 (Theorem 5.1.2). Conjecture 1.2.1 is true for the real basis elements corresponding to quantum cluster monomials.
Theorem 1.2.2 implies a triangularity property for the t-analogs of q-characters of simple modules of quantum affine algebras (Theorem 5.1.4) and a possible categorical interpretation (Remark 5.1.5).
Our second main result follows as a consequence of Theorem 1.2.2.
Theorem 1.2.3 (Theorem 5.2.1). If we consider the dual canonical basis B up (w) of the quantum unipotent subgroup A q [N − (w)], then Conjecture 1.1.1 holds true for the real elements corresponding to quantum cluster monomials.
In order to study the analog of Leclerc's conjecture and prove Theorem 1.2.2, we will consider not only triangularity with respect to degrees but also triangularity with respect to codegrees. Correspondingly, we introduce the notion of double triangular bases (Definition 4.1.5). We show that the common triangular basis is necessarily the double triangular basis with respect to every seed (Theorem 4.3.2).
It is worth remarking that, if the cluster algebra is categorified by a rigid monoidal category, then degrees and codegrees are related to the two different ways of taking the dual objects in the category, see [KK19].
1.3. Contents. In Section 2,we briefly review basic notions in cluster theory needed by this paper.
In Sections 3.1, 3.2, we review notions and techniques introduced and studied by [Qin17] [Qin19] such as dominance orders, (co)degrees, (co)pointed functions. In Section 3.3, we define tropical transformation for codegrees in analogous to that for degrees. In Section 3.4, we review the notion of injective-reachability, and define the set of distinguished functions I t , P t for seeds t, and we present some related statements.
In Section 4, we define various bases whose degrees or codegrees satisfy certain properties. In particular, we introduce the notion of double triangular bases. We discuss the relation between double triangular bases and (common) triangular bases. We prove that common triangular bases have good properties on their codegrees (Theorem 4.3.2).
In Section 5, we propose an analog of Leclerc's conjecture for common triangular bases (Conjecture 5.1.3) and show a weaker form holds true (Theorem 5.1.2). We discuss its consequence for modules of quantum affine algebras (Theorem 5.1.4, Remark 5.1.5). We deduce that the weaker form is satisfied by the dual canonical bases of U q (w) (Theorem 5.2.1).

Basics of cluster algebras
We briefly review notions in cluster theory necessary for this paper following [Qin17] [Qin19] [Qin20]. A reader unfamiliar with cluster theory is referred to [Kel08] [BZ05] for background materials. Denote . Notice that we have a natural bar involution ( ) on k which sends v to v −1 . Let ( ) T denote the matrix transposition and [ ] + denote the function max(0, ).
2.1. Seeds. Fix a finite set of vertices I and its partition I = I uf ⊔ I f into the unfrozen and frozen vertices.
Let there be given a quantum seed t = ( B(t), Λ(t), (X i (t)) i∈I ) where X i (t) are indeterminate, the integer matrices B(t) = (b ij (t)) i∈I,j∈I uf and Λ(t) = (Λ ij (t)) i,j∈I form a compatible pair, i.e. there exists some diagonal matrix D = diag(d k ) k∈I uf with strictly positive integer diagonals, such that B(t) T Λ(t) = D 0 . X i (t) are called the i-th X-variables or quantum cluster variables associated to t, B(t) the B-matrix, Λ(t) the Λ-matrix, and B(t) := (b ij (t)) i,j∈I uf the principal part of B(t) or the B-matrix.
(2) The matrix B(t) is of full rank |I uf |.
Define the following lattices (of column vectors): where f i (t), e k (t) denote the i-th and k-th unit vectors respectively.
i∈I with the usual addition and multiplication (+, ·), where we denote The quantum Laurent polynomial ring (also called the quantum torus) LP(t) associated to t is defined as the commutative algebra k[M • (t)] further endowed with the twisted product * : By the algebraic structure on LP(t), we mean (+, * ) unless otherwise specified.
The monomials X(t) m , m ∈ N I , are called the quantum cluster monomials associated to t. The Laurent monomials X(t) m , m ∈ N I uf ⊕ Z I f , are called the localized quantum cluster monomials associated to t. Define We also define F (t) to be the skew field of fractions of LP(t).
For simplicity, we often omit the symbol t when there is no confusion.

2.2.
Mutations. For any k ∈ I uf , we have an operation called mutation µ k which gives us a new seed [BZ05] for precise definitions of B(t ′ ), Λ(t ′ ) . Recall that we have µ 2 k t = t. Given any initial seed t 0 , we let ∆ + t 0 denote the set of seeds obtained from t 0 by iterated mutations. Then we have ∆ + t 0 = ∆ + t if t ∈ ∆ + t 0 . Throughout this paper, we will always work with seeds from the same set ∆ + = ∆ + t 0 where the initial seed t 0 is often omitted for simplicity. For simplicity, denote t = ( B, Λ, (X i )) and Recall that there is an algebra isomorphism µ * k : F (t ′ ) ≃ F (t) called the mutation birational map, such that Notice that we can also write µ * where ← − µ t ′ ,t = ← − µ = µ kr · · · µ k 2 µ k 1 is a sequence of mutations (read from right to left). We define the muta- for all t ′ ∈ ∆ + . Correspondingly, we call X i (t ′ ), i ∈ I f , t ′ ∈ ∆ + , the frozen variables, and denote them by X i for simplicity. Define the set of frozen factors to be P = {X m |m ∈ Z I f }.
2.3. Cluster algebras. Let there be given a quantum seed t ∈ ∆ + . Definition 2.3.1. The (partially compactified) quantum cluster algebra A q (t) is defined to be the k-subalgebra of LP(t) generated by the quantum cluster variables ← − µ * t ′ ,t X i (t ′ ), i ∈ I, t ′ ∈ ∆ + . The (localized) quantum cluster algebra A q (t) is defined to be the localization of A q (t) at P.
The upper quantum cluster algebra . It is sometimes convenient to forget the symbols t, t ′ by viewing ← − µ *

Dominance orders and pointedness
In this section, we recall the notions and some basic results concerning dominance orders and pointed functions from [Qin17] [Qin19]. We also describe properties of codegrees and copointed functions in analogous to those of degrees and pointed functions.
Definition 3.1.1 (Dominance order). We denote g ′ t g if there exists some n ∈ N ≥0 uf (t) such that g ′ = g + p * n. In this case, we say g ′ is dominated by g, or g ′ is inferior to g.
The meanings of symbols ≺ t , ≻ t , t are given in the obvious way.
Notice that LP(t) has a subring k[N ≥0 Elements in LP(t) will be called functions or formal Laurent series.
Similarly, we consider the subring k[Y −1 k (t)] k∈I uf of LP(t) and its completion k[Y −1 k (t)] k∈I uf with respect to the maximal ideal generated by Y −1 k (t), k ∈ I uf . We define the following completion of LP(t): By a formal sum, we mean a possibly infinite sum. Let Z denote a formal sum Z = m∈M • (t) b m X(t) m . Notice that it belongs to LP(t) (resp. LP(t)) if and only if its Laurent degree support supp M • (t) Z = {m|b m = 0} has finitely many ≺ t -maximal elements (resp. finitely many ≺ t -minimal elements).
Definition 3.2.1 ((Co)degrees and (co)pointedness). The formal sum Z is said to have degree g if supp M • (t) Z has a unique ≺ t -maximal element g, and we denote deg t Z = g. It is said to be pointed at g or g-pointed if we further have b g = 1.
The formal sum Z is said to have codegree η if supp M • (t) Z has a unique ≺ t -minimal element η, and we denote codeg t Z = η. It is said to be copointed at η or η-copointed if we further have b η = 1.
Let there be given a set S. It is said to be M Similarly, if Z has codegree η, we define its codegree normalization in LP(t) ⊗ k F (k) to be: Let there be given a (possibly infinite) collection of formal sums Z j . Notice that their formal sum j Z j is well-defined if, at each Laurent degrees, only finitely many of them have non-vanishing coefficients.
Notice that, Lemma 3.1.2 implies that a degree ≺ t -unitriangular sum is a well-defined sum in LP(t) and, similarly, a codegree ≻ tunitriangular sum is a well-defined sum in LP(t).
Lemma 3.2.5. (1) [Qin17] Let there be given a M • (t)-pointed set S, then any pointed function Z ∈ LP(t) can be written uniquely as a (degree) ≺ t -unitriangular sum in terms of S.
(2) Let there be given a M • (t)-copointed set S, then any copointed element Z ∈ LP(t) can be written uniquely as a codegree ≻ t -unitriangular sum in terms of S. (2) can be proved similarly, or we can deduce it from (1) by using the map ι defined in (3.2).
In the cases of Lemma 3.2.5, we say Z is (degree) ≺ t -unitriangular to S or codegree ≻ t -unitriangular to S respectively. It is further said to be (degree) (≺ t , m)-unitriangular to S or codegree (≻ t , m)-unitriangular to S respectively, if its decomposition in S has such properties.
3.3. Tropical transformations and compatibility. As before, let there be given seeds Denote the i-th cluster variables associated to t and t ′ by X i and X ′ i respectively. Let f i , f ′ i denote the i-th unit vectors associated to t and t ′ respectively.
In general, we define the (degree) tropical transformation φ t ′ ,t : as the composition of the tropical transformations for adjacent seeds along the mutation sequence ← − µ from t to t ′ . By [GHK15], φ t ′ ,t is the tropicalization of certain birational maps between the split algebraic tori associate to t, t ′ and, consequently, independent of the choice of ← − µ .
. It has the following property.
Consider the following set of Laurent polynomials The following very useful result shows that certain mutation sequences swap pointedness and copointedness. (2) Let there be given Let S denote a set consisting of g-pointed functions S g ∈ LP(t) for distinct g ∈ M • (t). If S g are compatibly pointed at t, t ′ for all g, we say S is compatibly pointed at t, t ′ , or the pointed sets S and ← − µ * t,t ′ S are (degree) compatible.
Definition 3.3.6 (Codegree tropical transformation). For any seeds t ′ = µ k t, k ∈ I uf , we define the codegree tropical transformation φ op t ′ ,t : In general, we define the codegree tropical transformation φ op t ′ ,t : as the composition of the codegree tropical transformations for adjacent seeds along the mutation sequence ← − µ from t to t ′ .
Let us justify our definition of the codegree tropical transformation.
To any given seed t = ( B, Λ, (X i ) i∈I ), we associate the opposite seed For any given k ∈ I uf , we have µ k (t op ) = (µ k t) op . It is straightforward to check the commutativity of the following diagram: We have the following result.
Lemma 3.3.7. Let there be given seeds Proof. By the commutativity between ι and mutations, it suffices to check the claim for adjacent seeds t ′ = µ k t, which follows from definition.
Notice that we have LP(t; t ′ ) = ιLP(t op ; (t ′ ) op ) and U q (t) = ι(U q (t op )) by the commutativity between ι and mutations.
If Z belongs to U q (t) ⊂ LP(t), then Z is said to be compatibly copointed at ∆ + if it is compatibly copointed at t, t ′ for any t ′ ∈ ∆ + .
Let S denote a set consisting of η-copointed elements S η ∈ LP(t) for distinct η ∈ M • (t). If S η are compatibly copointed at t, t ′ for all η, we say S is compatibly copointed at t, t ′ , or the copointed sets S and ← − µ * t,t ′ S are (codegree) compatible.
Remark 3.3.9. We refer the reader to [KK19, Section 3.5] for a categorical view of the degrees and the codegrees together with their tropical transformations, which are obtained by taking dual objects in the module category of quiver Hecke algebras.
3.4. Injective-reachability and distinguished functions. Let σ denote a permutation of I uf . For any mutation sequence ← − µ = µ kr · · · µ k 1 , we define σ ← − µ = µ σkr · · · µ σk 1 . Let pr I uf and pr I f denote the natural projection from Z I to Z I uf and Z I f respectively.
In this case, we denote t ′ = t[1] and say it is shifted from t (by [1]) with the permutation σ. Similarly, we denote t = t ′ [−1] and say it is the shifted from t ′ (by [−1]) with the permutation σ −1 .
Let there be given an injective-reachable seed t. Recursively, we construct a chain of seeds {t[d]|d ∈ Z} called an injective-reachable chain, such that t Since a quantum cluster monomial is pointed, it is also copointed by [FZ07] (we can also see this using the map ι). It follows that Notice that if t is injective-reachable, then so is any seed t ′ ∈ ∆ + . Such properties is equivalent to the existence of a green to red sequence. See [Qin17] [Qin19] for more details.
For any g = (g i ) i∈I ∈ Z I ≃ M • (t), denote [g] + = ([g i ] + ) i∈I . We have the following g-pointed element in LP(t): for some frozen factor p g ∈ P. Define the following set of distinguished pointed functions Notice that (3.6) appears in [Qin17,(18)] as an assumption. Replacing t by t[−1] in the above argument, we obtain for any k ∈ I uf and some u ′ k ∈ Z I uf . Correspondingly, for any η ∈ Z I ≃ M • (t), we have the following η-copointed element in LP(t): for some frozen factor p η ∈ P. Define the following set of distinguished copointed functions   Proof.
(1) Recall that ιP k (t) is a quantum cluster variable contained in LP(t op ). By (3.7), ιP k (t) is pointed at −f k + u for some u ∈ Z I f . The claim follows.
(2) By the commutativity between mutations and ι, Using the commutativity between ι : LP(t op ) ≃ LP(t) and mutations, its cluster variables have the following Laurent expansion in LP(t op ): Proof.
We have the following relation between degree and codegree tropical transformations, which will be useful for studying properties of double triangular bases (Proposition 4.3.1).
Proof. It suffices to check the claim for the case t ′ = µ k t, k ∈ I uf . Notice that, in this case, we have t ′ Notice that the maps in the diagram are isomorphisms for u ∈ Z I f . In view of the piecewise linearity of φ t ′ ,t and φ op t ′ [1],t[1] , it remains to check the claim that, for i ∈ I uf , and also It follows that these two vectors in M • (t ′ ) agree.
(ii) For the non-trivial case i = k, we have Notice that I k (t) and I k (t ′ ) are related by an exchange relation for the seeds (t[1], t ′ [1]). It follows that we have see [Qin17,(14)].
(iii) By (3.5) and the linearity of ψ t,t[1] , for i = k in I uf , we have (3.5) implies that the two vectors in M • (t ′ ) agree.
(iv) For the non-trivial case i = k, we have ) and the claim follows.
Consequently, we obtain a relation between the degree compatibility and the codegree compatibility.
Proposition 3.4.5. Let there be given seeds t, t ′ ∈ ∆ + and Z ∈ . Then Z is compatibly pointed at t[1], t ′ [1] ∈ ∆ + if and only if it is compatibly copointed at t, t ′ .
, and similar statements hold in LP(t ′ [1]) and LP(t ′ ). The claim follows from Proposition 3.4.4.

Bidegrees and bases
Let there be given an injective-reachable quantum seed t and a subalgebra A(t) ⊂ U q (t). Assume that A(t) possesses a k-basis L. Then A(t) naturally gives rise to a subalgebra A(t ′ ) : And L naturally gives rise to a basis ← − µ * t,t ′ L of A(t ′ ). We sometimes omit the symbols t, t ′ , identifying A(t) and A(t ′ ), L and ← − µ * t,t ′ L.

Bases with different properties.
Definition 4.1.1 (Degree-triangular basis). A k-basis L of A(t) is said to be a degree-triangular basis with respect to t if the following conditions hold: (1) X i (t) ∈ L for i ∈ I.
(4) (Degree triangularity) For any basis element L g , i ∈ I, the decomposition of the pointed function [X i (t) * L g ] t in terms of L is degree (≺ t , m)-unitriangular: The basis is said to be a cluster degree-triangular basis with respect to t, or a triangular basis for short, if it further contains the quantum cluster monomials in t and t[1].
It is not clear if a degree-triangular basis is unique or not. Nevertheless, a triangular basis must be unique if it exists, see [Qin17, Lemma 6.3.2]. By definition, I t is (≺ t , m)-unitriangular to the triangular basis.
We now propose the dual version below.
Definition 4.1.2 (Codgree-triangular basis). A k-basis L of A(t) is said to be a codegree-triangular basis with respect to t if the following conditions hold: (1) X i (t) ∈ L for i ∈ I.
(3) (Codegree parametrization) L is M • (t)-copointed, i.e., it takes the form L = {L η |η ∈ M • (t)} such that L η is η-copointed. (4) (Codegree triangularity) For any basis element L η , i ∈ I, the decomposition of the copointed function {L η * X i (t)} t in terms of L is codegree (≻ t , m)-unitriangular: The basis is said to be a cluster codegree-triangular basis with respect to t if it further contains the quantum cluster monomials in t and t[−1].
By definition, P t is codegree (≻ t , m)-unitriangular to the cluster codegree-triangular basis. Similar to [Qin17, Lemma 6.3.2], we can show that the cluster codegree-triangular basis is unique.
(2) Let there be given a codegree-triangular basis L.
Definition 4.1.4 (Bidegree-triangular basis). If L is both degree-triangular and codegree-triangular with respect to t, we call it a bidegree-triangular basis with respect to t.
Definition 4.1.5 (Double triangular basis). If L is bidegree-triangular with respect to t and further contains the quantum cluster monomials in t, t[−1], t[1], we call it a cluster bidegree-triangular basis of A(t) or a double triangular basis with respect to t.
Definition 4.1.6 (Common triangular basis). Assume that L is the triangular basis of A(t) with respect to t.
with respect to t ′ and is compatible with L for any t ′ ∈ ∆ + , we call L the common triangular basis.

From triangular bases to double triangular bases.
Proposition 4.2.1. Let there be given the triangular basis L t of A(t) with respect to the seed t. If L t[−1] := ← − µ * t,t[−1] L t is the triangular basis with respect to t[−1], then L t is the double triangular basis with respect to t.
Proof. By assumption, L t contains the quantum cluster monomials in t, t[1], t[−1]. It remains to check that L t satisfies the defining conditions of a codegree triangular basis for t.
(ii-a) Take any i ∈ I f . Then for any V ∈ L t which is bipointed by (ii-b) Take any k ∈ I uf and any η-copointed element V ∈ L t . Then Applying the mutation ← − µ * t[−1],t , we obtain Proposition 3.3.4 implies that Z ′ is copointed and, for any j > 0, Then this is a codegree (≻ t , m)-unitriangular decomposition in terms of the copointed set L t .
We prove the following inverse result, although it will not be used in this paper. ( (ii-a) Take any i ∈ I f . Then for any (ii-b) Take any k ∈ I uf and g-pointed m)-unitriangular to P t and, consequently, is codegree (≻ t , m)-unitriangular to L t . We obtain a finite codegree (≻ t , m)unitriangular decomposition Applying the mutation ← − µ * t,t[−1] , we obtain Proposition 3.3.4 implies that Z ′ is pointed and, for any j < r, we have  Proof. Notice that ι sends (quantum) cluster monomials ← − because it commutes with mutations. In particular, it gives a bijection between the sets of cluster monomials.
Because the common triangular basis L gives rise to the double triangular bases for all seeds by Proposition 4.2.1, it gives rise to a codegree triangular bases L t ′ ⊂ LP(t ′ ) for any seed t ′ ∈ ∆ + . Then ιL t ′ ⊂ LP((t ′ ) op ) is a degree triangular bases containing all cluster monomials. Therefore, ιL t ′ is the triangular basis with respect to (t ′ ) op .
Moreover, for any t, t ′ ∈ ∆ + , because the elements of L are compatibly pointed at t[1], t ′ [1], the elements of L are compatibly copointed at t, t ′ by Proposition 3.4.5. It follows that the elements of ιL are compatibly pointed at t op , (t ′ ) op . Therefore, ιL is the common triangular basis by definition.
Recall that a common triangular basis is necessarily compatibly pointed at ∆ + . We have the following results.
Theorem 4.3.2. Let there be a k-subalgebra A(t) of the upper quantum cluster algebra U q (t). Assume that A(t) possesses the common triangular basis L. Then the following statements are true.

5.
1. An analog of Leclerc's conjecture. Let there be given an injective-reachable seed t and a k-subalgebra A(t) of the upper quantum cluster algebra U q (t).
Proposition 5.1.1. Assume that A(t) possesses a bidegree-triangular basis L. Take any i ∈ I and g ∈ M • (t). Denote the codegree of the g-pointed basis element L g by η. Then we have either X i (t) * L g ∈ v Z L or Moreover, we have s = λ(f i , g), h = λ(f i , η).
Proof. Omit the symbol t for simplicity.
Denote the codegree of L g by η = g + Bn, where n ∈ N ≥0 uf (t) ≃ N I uf . Then X i * L g has degree Because L is a degree-triangular basis, we have a degree (≺ t , m)unitriangular decomposition with finitely many S (0) , · · · , S (r) ∈ L : (i) Assume n i = 0, then v −s X i * L g is pointed and bar-invariant. Because every basis elements S (j) appearing in (5.1) are bar-invariant and b (j) ∈ m, it follows that v −s X i * L g = S (0) ∈ L.
(ii) Assume n i = 0. Then h < s. In addition, v −s X i * L g is pointed but not bar-invariant, because it has the Laurent monomial v h−s X η+f i at the codegree.
Notice that v −h X i * L g is copointed. Multiplying the decomposition (5.1) by v s−h and applying the bar involution, we get a decomposition of copointed elements Because L is a codegree-triangular basis and v h L g * X i is copointed, the above decomposition must be codegree (≻ t , m)-unitriangular. But v h−s S (0) is not copointed since S (0) ∈ L is copointed but h < s. Relabeling S (j) , j > 0, if necessary, we assume codeg S (j) ≻ t codeg S (r) for j < r. Then the codegree term X η+f i is contributed from S (r) and S (r) is copointed at codeg(L g * X i ) = η + f i with decomposition coefficient 1 = v h−s b (r) . In addition, the remaining terms S (j) , 0 < j < r must have coefficients v h−s · b (j) in m. It follows that b j := b (j) v s belongs to v h+1 Z[v] for 0 < j < r. The claim follows by taking S = S (0) , H = S (r) , L (j) = S (j) for 0 < j < r.
Theorem 5.1.2. Let there be given a k-subalgebra A(t) of the upper quantum cluster algebra U q (t). Assume that it has the the common triangular basis L. Then, for any i ∈ I, V ∈ L, and any localized quantum cluster monomial R, we have either R * V ∈ v Z L or , and S, L (j) , H are finitely many distinct elements of L.
Proof. Since L is the common triangular basis, Proposition 4.2.1 implies that ← − µ * t,t ′ L is the double triangular basis (and thus bidegreetriangular) of A(t ′ ) = ← − µ * t,t ′ A(t) for any seed t ′ ∈ ∆ + . We apply Proposition 5.1.1 for localized quantum cluster monomials associated to t ′ .
Theorem 5.1.2 is a weaker form of the following analog of Leclerc's conjecture.
Conjecture 5.1.3. Assume that L is the common triangular basis. Assume that R is a real basis element in L (i.e. R 2 ∈ L). Then, for any V ∈ L, we have either R * V ∈ v Z L or , and S, L (j) , H are finitely many distinct elements of L.
Choose any l ∈ N. Let C l denote a level-l subcategory of the monoidal category of the finite dimensional modules of a quantum affine algebra U q ( g) in the sense of [HL10], where g is a Lie algebra of type ADE. Let K t (C l ) denote its t-deformed Grothendieck ring, t a quantum parameter. By [Qin17], K t (C l ) is a (partially compactified) quantum cluster algebra A q . Notice that K t (C l ) has a bar-invariant basis {[S]} where S are simple modules. By [Qin17], {[S]} becomes the common triangular basis of the corresponding quantum cluster algebra A q after localization at the frozen factors.
A simple module R in C l is called real if R ⊗ R remains simple. Theorem 5.1.2 implies the following result.
Theorem 5.1.4. Let R be any real simple module in C l corresponding to a cluster monomial. Then, for any simple modules V ∈ C l , either R ⊗ V is simple, or there exists finitely many distinct simple modules S, L (j) , H in C l such that the following equation holds in the deformed Grothendieck ring K t (C l ): . Notice that we can replace [S] by the t-analog of q-character of S and embed K t (C l ) into the completion of a quantum torus, see [Nak04] [VV03] [Her04]. Correspondingly, Theorem 5.1.4 gives an algebraic relation for such characters.
Remark 5.1.5. Assume that the quantum cluster algebra arises from a quantum unipotent subgroup of symmetric Kac-Moody type, which possesses the dual canonical basis correspond to the set of self-dual simple modules of the corresponding quiver Hecke algebra. In this case, up to v-power rescaling, S and H correspond to the simple socle and simple head of the convolution product R • V respectively. See [KKKO18, Section 4] for more details.
From this view, Theorem 5.1.4 suggests that an analog of Leclerc's conjecture might hold for the deformed Grothendieck ring K t (C l ) of quantum affine algebra and, in addition, it might have a categorical interpretation in analogous to that in [KKKO18, Section 4].

5.2.
Properties of dual canonical bases. Let us consider the quantum unipotent subgroup A q [N − (w)] of symmetrizable Kac-Moody types in the sense of [Kim12] [Qin20]. It is isomorphic to a (partially compactified) quantum cluster algebra after rescaling, see [GY16] [GY20] or [Qin20]. Theorem 5.1.2 implies the following weaker version of Conjecture 1.1.1. If b 1 ∈ B up (w) corresponds to a quantum cluster monomial after rescaling, then for any b 2 ∈ B up (w), either b 1 b 2 ∈ q Z B up (w) or (1.1) holds true.
Proof. By [Qin20], after rescaling and localization at the frozen factors, the dual canonical basis B up (w) of A q [N − (w)] becomes the common triangular basis of the corresponding quantum cluster algebra. Therefore, elements of B up (w) satisfy the algebraic relation (5.2) after rescaling. Notice that the rescaling factors depends on the natural root-lattice grading of U q , which is homogeneous for ← − µ * t ′ ,t X i (t ′ ) * V, S, L (j) , H in (5.2), because the Y -variables have 0-grading [Qin20, Section 9.1]. The claim follows from Theorem 5.1.2.
Theorem 5.2.1 would implies Conjecture 1.1.1 if the following multiplicative reachability conjecture can be proved.
is real (i.e. b 2 ∈ q Z B up (w)), then it corresponds to a quantum cluster monomial after rescaling.
Conjecture 5.2.2 can be generalized as the following, which implies Conjecture 5.1.3 by Theorem 5.1.2.
Conjecture 5.2.3 (Multiplicative reachability conjecture). Let L denote a common triangular basis. If b ∈ L is real (i.e. b 2 ∈ L), then it corresponds to a localized quantum cluster monomial.
Remark 5.2.4 (Reachability conjectures). When the cluster algebra admits an additive categorification by triangulated categories (cluster categories), we often expect that the rigid objects (objects with vanishing self-extensions) correspond to the (quantum) cluster monomials. If so, such objects can be constructed from the initial cluster tilting objects via (categorical) mutations. Let us call such an expectation the additive reachability conjecture. This conjecture is not true for a general cluster algebra because the cluster algebra seems too small for the cluster category.
When the cluster algebra admits a monoidal categorification by monoidal categories, we similarly expect that the real simple objects correspond to the (quantum) cluster monomials (see [HL10]). If so, such objects can be constructed from the an initial collection of real simple objects via (categorical) mutations. Let us call such an expectation the multiplicative reachability conjecture. Conjecture 5.2.2 is related to the special case for A q [N − (w)].
We also conjecture an equivalence between the additive reachability conjecture and the multiplicative reachability conjecture, which can be viewed as an analog of the open orbit conjecture [GLS11, Conjecture 18.1]. See [Nak11, Section 1] for a comparison between additive categorification and monoidal categorification.
All these conjectures are largely open.