An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras
aa r X i v : . [ m a t h . QA ] M a y AN ANALOG OF LECLERC’S CONJECTURE FORBASES OF QUANTUM CLUSTER ALGEBRAS
FAN QIN
Abstract.
Dual canonical bases are expected to satisfy a certain(double) triangularity property by Leclerc’s conjecture. We pro-pose an analogous conjecture for common triangular bases of quan-tum cluster algebras. We show that a weaker form of the analogousconjecture is true. Our result applies to the dual canonical basesof quantum unipotent subgroups. It also applies to the t -analogsof q -characters of simple modules of quantum affine algebras. Contents
1. Introduction 11.1. Background 11.2. Main results 31.3. Contents 32. Basics of cluster algebras 42.1. Seeds 42.2. Mutations 52.3. Cluster algebras 63. Dominance orders and pointedness 63.1. Dominance orders 63.2. Pointedness 63.3. Tropical transformations and compatibility 83.4. Injective-reachability and distinguished functions 114. Bidegrees and bases 154.1. Bases with different properties 164.2. From triangular bases to double triangular bases 174.3. Properties of common triangular bases 195. Main results 205.1. An analog of Leclerc’s conjecture 205.2. Properties of dual canonical bases 22References 241.
Introduction
Background.
Dual canonical bases and cluster theory.
Let g denote a Kac-Moodyalgebra with a symmetrizable Cartan datum, and U q = U q ( g ) the cor-responding quantized enveloping algebra, where q is not a root of unity.The negative (or positive) part U q − of U q possesses the famous canon-ical bases [Lus90] [Lus91] [Kas90]. The corresponding dual basis B up also has fascinating properties and is related to the theory of totalpositivity [Lus94].Fomin and Zelevinsky invented cluster algebras as a combinatorialframework to understand the total positivity [Lus94] and the dualcanonical bases B up . We refer the reader to the survey [Kel08] forfurther details of cluster algebras.Let there be given any Weyl group element w ∈ W . Then the dualcanonical 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]. Noticethat, if g is a finite dimensional semi-simple Lie algebra, then A q [ N ( w )]agrees with U q − where w 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 ),where the initial seed t = t ( −→ 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 formu-lated 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 symmetricKac-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 realif b ∈ q Z B up . Leclerc proposed the following conjecture regarding themultiplication by a real element of B up , which is analogous to Kashiwaracrystal graph operator. Conjecture 1.1.1 (Leclerc’s Conjecture [Lec03, Conjecture 1]) . As-sume that b is a real element of B up . Then, for any b ∈ B up such that b b / ∈ q Z B up , the expansion of their product on B up takes the form b b = q h b ′ + q s b ′′ + X c = b ′ ,b ′′ γ cb ,b c (1.1) where b ′ = b ′′ , h < s ∈ Z , γ cb ,b ∈ q h +1 Z [ q ] ∩ q s − Z [ q − ] . This conjecture was proved by [KKKO18] for symmetric Kac-Moodycases using quiver Hecke algebras [KL09] [KL10] [Rou08].
Main results.
By [Qin17] [KK19] [Qin20], after localization andrescaling, the dual canonical basis B up ( w ) agrees with the commontriangular basis of the corresponding quantum cluster algebra in thesense of [Qin17]. Correspondingly, we formulate the following analogof Leclerc’s conjecture. Conjecture 1.2.1 (Conjecture 5.1.3) . Conjecture 1.1.1 is true if wereplace the dual canonical basis by the common triangular basis.
Recall that the quantum cluster monomials provide a subset of thereal elements in the dual canonical basis B up ( w ) (we conjecture thatall real elements take this form, see Conjecture 5.2.2). Our first mainresult 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 realbasis 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 (Theorem5.1.4) 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 canonicalbasis B up ( w ) of the quantum unipotent subgroup A q [ N − ( w )] , then Con-jecture 1.1.1 holds true for the real elements corresponding to quantumcluster monomials. In order to study the analog of Leclerc’s conjecture and prove The-orem 1.2.2, we will consider not only triangularity with respect to de-grees 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 doubletriangular basis with respect to every seed (Theorem 4.3.2).It is worth remarking that, if the cluster algebra is categorified bya rigid monoidal category, then degrees and codegrees are related tothe 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 clustertheory needed by this paper.In Sections 3.1, 3.2, we review notions and techniques introducedand studied by [Qin17] [Qin19] such as dominance orders, (co)degrees,(co)pointed functions. In Section 3.3, we define tropical transformationfor codegrees in analogous to that for degrees. In Section 3.4, we reviewthe notion of injective-reachability, and define the set of distinguishedfunctions I t , P t for seeds t , and we present some related statements.In Section 4, we define various bases whose degrees or codegrees sat-isfy certain properties. In particular, we introduce the notion of doubletriangular bases. We discuss the relation between double triangular FAN QIN bases and (common) triangular bases. We prove that common trian-gular bases have good properties on their codegrees (Theorem 4.3.2).In Section 5, we propose an analog of Leclerc’s conjecture for commontriangular bases (Conjecture 5.1.3) and show a weaker form holds true(Theorem 5.1.2). We discuss its consequence for modules of quantumaffine algebras (Theorem 5.1.4, Remark 5.1.5). We deduce that theweaker form is satisfied by the dual canonical bases of U q ( w ) (Theorem5.2.1). 2. Basics of cluster algebras
We briefly review notions in cluster theory necessary for this paperfollowing [Qin17] [Qin19] [Qin20]. A reader unfamiliar with clustertheory is referred to [Kel08] [BZ05] for background materials.Denote k = Z [ q ± ] = Z [ v ± ], where v = q is a formal parameter.Define m = v − Z [ v − ]. Notice that we have a natural bar involution( ) on k which sends v to v − . Let ( ) T denote the matrix transpositionand [ ] + 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 = ( e B ( t ) , Λ( t ) , ( X i ( t )) i ∈ I ) where X i ( t ) are indeterminate, the integer matrices e B ( t ) = ( b ij ( t )) i ∈ I,j ∈ I uf andΛ( t ) = (Λ ij ( t )) i,j ∈ I form a compatible pair, i.e. there exists some diag-onal matrix D = diag( d k ) k ∈ I uf with strictly positive integer diagonals,such that e B ( t ) T Λ( t ) = (cid:0) D (cid:1) . X i ( t ) are called the i -th X -variablesor quantum cluster variables associated to t , e B ( t ) the e B -matrix, Λ( t )the Λ-matrix, and B ( t ) := ( b ij ( t )) i,j ∈ I uf the principal part of e B ( t ) orthe B -matrix. Lemma 2.1.1 ([BZ05]) . (1) We have d i b ik ( t ) = − d k b ki ( t ) for i, k ∈ I uf .(2) The matrix e B ( t ) is of full rank | I uf | . Define the following lattices (of column vectors): M ◦ ( t ) = ⊕ i ∈ I Z f i ( t ) ≃ Z I N uf ( t ) = ⊕ k ∈ I uf Z e k ( t ) ≃ Z I uf , where f i ( t ), e k ( t ) denote the i -th and k -th unit vectors respectively.Denote N ≥ uf ( t ) = ⊕ k ∈ I uf N e k ( t ) ≃ N I uf .Define the linear map p ∗ : N uf ( t ) → M ◦ ( t ) such that p ∗ n = e B ( t ) n .Let λ denote the bilinear form on M ◦ ( t ) such that λ ( g, g ′ ) = g T Λ( t ) g ′ . Lemma 2.1.2.
For any i ∈ I , k ∈ I uf , we have λ ( f i ( t ) , p ∗ e k ( t )) = − δ ik d k . The group algebra of M ◦ ( t ) is the Laurent polynomial ring k [ M ◦ ( t )] := k [ X ( t ) m ] m ∈ M ◦ ( t ) = k [ X i ( t ) ± ] i ∈ I with the usual addition and multipli-cation (+ , · ), where we denote X ( t ) f i ( t ) = X i ( t ).The quantum Laurent polynomial ring (also called the quantumtorus) LP ( t ) associated to t is defined as the commutative algebra k [ M ◦ ( t )] further endowed with the twisted product ∗ : X ( t ) m ∗ X ( t ) m ′ = v λ ( m,m ′ ) X ( t ) m + m ′ . By the algebraic structure on LP ( t ), we mean (+ , ∗ ) unless otherwisespecified.The monomials X ( t ) m , m ∈ N I , are called the quantum clustermonomials 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 the Y -variables to be Y k ( t ) := X ( t ) p ∗ e k ( t ) , k ∈ I uf . Denote Y ( t ) n = X ( t ) p ∗ n for n ∈ N uf ( t ).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 muta-tion µ k which gives us a new seed t ′ = µ k t = ( e B ( t ′ ) , Λ( t ′ ) , ( X i ( t ′ )) i ∈ I )where X ′ i := X i ( t ′ ) are indeterminate. See [BZ05] for precise definitionsof e B ( t ′ ), Λ( t ′ ) . Recall that we have µ k t = t .Given any initial seed t , we let ∆ + t denote the set of seeds obtainedfrom t by iterated mutations. Then we have ∆ + t = ∆ + t if t ∈ ∆ + t .Throughout this paper, we will always work with seeds from the sameset ∆ + = ∆ + t where the initial seed t is often omitted for simplicity.For simplicity, denote t = ( e B, Λ , ( X i )) and t ′ = ( e B ′ , Λ ′ , ( X ′ i )).Denote v k = v d k . Recall that there is an algebra isomorphism µ ∗ k : F ( t ′ ) ≃ F ( t ) called the mutation birational map, such that µ ∗ k ( X ′ i ) = ( X i i = kv λ ( f k , P j ∈ I [ − b jk ] f + j ) X − k ∗ ( X P j ∈ I [ − b jk ] + f j + v − k X P i ∈ I [ b ik ] + f i ) i = k . Notice that we can also write µ ∗ k ( X ′ i ) = X − f k + P j ∈ I [ − b jk ] + f j · (1 + Y k ).Recall that ( µ ∗ k ) is an identity.Let there be given any seed t ′ = ←− µ t ′ ,t t , where ←− µ t ′ ,t = ←− µ = µ k r · · · µ k µ k is a sequence of mutations (read from right to left). We define the muta-tion birational map ←− µ ∗ t ′ ,t : F ( t ′ ) ≃ F ( t ) as the composition µ ∗ k · · · µ ∗ k r .It is known that ←− µ ∗ t ′ ,t is independent of the choice for the mutationsequence ←− µ t ′ ,t from t to t ′ . Define ←− µ t,t ′ = ←− µ − t ′ ,t . Then it is clear that ←− µ ∗ t,t ′ = ( ←− µ ∗ t ′ ,t ) − .Notice that, if i ∈ I f , we have ←− µ ∗ t ′ ,t X i ( t ′ ) = X i ( t ) for all t ′ ∈ ∆ + .Correspondingly, we call X i ( t ′ ), i ∈ I f , t ′ ∈ ∆ + , the frozen variables, FAN QIN and denote them by X i for simplicity. Define the set of frozen factorsto 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 alge-bra A q ( t ) is defined to be the k -subalgebra of LP ( t ) generated by thequantum cluster variables ←− µ ∗ t ′ ,t X i ( t ′ ) , i ∈ I , t ′ ∈ ∆ + .The (localized) quantum cluster algebra A q ( t ) is defined to be thelocalization of A q ( t ) at P .The upper quantum cluster algebra U q ( t ) is defined to be ∩ t ′ ∈ ∆ + ←− µ ∗ t ′ ,t LP ( t ′ ) . Recall that we have A q ( t ) ⊂ A q ( t ) ⊂ U q ( t ). Moreover, for t, t ′ ∈ ∆ + ,we have ←− µ ∗ t ′ ,t U q ( t ′ ) = U q ( t ), ←− µ ∗ t ′ ,t A q ( t ′ ) = A q ( t ), ←− µ ∗ t ′ ,t A q ( t ′ ) = A q ( t ).It is sometimes convenient to forget the symbols t, t ′ by viewing ←− µ ∗ t ′ ,t as an identification.3. Dominance orders and pointedness
In this section, we recall the notions and some basic results con-cerning dominance orders and pointed functions from [Qin17] [Qin19].We also describe properties of codegrees and copointed functions inanalogous to those of degrees and pointed functions.3.1.
Dominance orders.Definition 3.1.1 (Dominance order) . We denote g ′ (cid:22) t g if there existssome n ∈ N ≥ uf ( t ) such that g ′ = g + p ∗ n . In this case, we say g ′ isdominated by g , or g ′ is inferior to g . The meanings of symbols ≺ t , ≻ t , (cid:23) t are given in the obvious way. Lemma 3.1.2 ([Qin17]) . For any g, g ′ ∈ M ◦ ( t ) , there exist finitelymany g ′′ such that g ′′ (cid:22) t g and g ′′ (cid:23) t g ′ . Pointedness.
Let there be given a quantum seed t .Notice that LP ( t ) has a subring k [ N ≥ uf ( t )] := k [ Y k ( t )] k ∈ I uf . Let \k [ Y k ( t )] k ∈ I uf denote the completion of k [ Y k ( t )] k ∈ I uf with respect to themaximal ideal generated by Y k ( t ), k ∈ I uf . The formal completion of LP ( t ) is defined to be d LP ( t ) = LP ( t ) ⊗ k [ Y k ( t )] k ∈ I uf \k [ Y k ( t )] k ∈ I uf . Elements in d LP ( t ) will be called functions or formal Laurent series.Similarly, we consider the subring k [ Y − k ( t )] k ∈ I uf of LP ( t ) and itscompletion \k [ Y − k ( t )] k ∈ I uf with respect to the maximal ideal generatedby Y − k ( t ), k ∈ I uf . We define the following completion of LP ( t ): g LP ( t ) : = LP ( t ) ⊗ k [ Y − k ( t )] k ∈ I uf \k [ Y − k ( t )] k ∈ I uf . By a formal sum, we mean a possibly infinite sum. Let Z denote aformal sum Z = P m ∈ M ◦ ( t ) b m X ( t ) m . Notice that it belongs to d LP ( t )(resp. g 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. finitelymany ≺ 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 -maximalelement 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 aunique ≺ t -minimal element η , and we denote codeg t Z = η . It is saidto 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 ◦ ( t ) -pointed if it takesthe form S = { S g | g ∈ M ◦ ( t ) } where S g are g -pointed functions in d LP ( t ) . Similarly, it is said to be M ◦ ( t ) -copointed, if it takes the form S = { S η | η ∈ M ◦ ( t ) } where S η are η -copointed functions in g LP ( t ) . Definition 3.2.2 (Normalization) . Let F ( k ) denote the fraction fieldof k . If Z has degree g , we define its (degree) normalization in d LP ( t ) ⊗ k F ( k ) to be [ Z ] t : = b − g Z. Similarly, if Z has codegree η , we define its codegree normalization in g LP ( t ) ⊗ k F ( k ) to be: { Z } t : = b − η Z. Let there be given a (possibly infinite) collection of formal sums Z j .Notice that their formal sum P j Z j is well-defined if, at each Laurentdegrees, only finitely many of them have non-vanishing coefficients. Definition 3.2.3 (Degree triangularity) . A formal sum P j b j Z j ofpointed elements Z j ∈ d LP ( t ) , b j ∈ k , is said to be degree ≺ t -unitriangular,or ≺ t -unitriangular for short, if { deg t Z j | b j = 0 } has a unique ≺ t -maximal element deg t Z j and b j = 1 . It is further said to be degree ( ≺ t , m ) -unitriangular, or ( ≺ t , m ) -unitriangular for short, if we furtherhave b j ∈ m for j = j . Definition 3.2.4 (Codegree triangularity) . A formal sum P j b j Z j ofcopointed elements Z j ∈ g LP ( t ) , b j ∈ k , is said to be codegree ≻ t -unitriangular if { codeg t Z j | b j = 0 } has a unique ≺ t -minimal element codeg t Z j and b j = 1 . It is further said to be codegree ( ≻ t , m ) -unitriangular, if we further have b j ∈ m for j = j . FAN QIN
Notice that, Lemma 3.1.2 implies that a degree ≺ t -unitriangularsum is a well-defined sum in d LP ( t ) and, similarly, a codegree ≻ t -unitriangular sum is a well-defined sum in g LP ( t ). Lemma 3.2.5. (1) [Qin17]
Let there be given a M ◦ ( t ) -pointed set S ,then any pointed function Z ∈ d 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 el-ement Z ∈ g LP ( t ) can be written uniquely as a codegree ≻ t -unitriangularsum in terms of S .Proof. (1) is proved as in [Qin17, Lemma 3.1.10(i)], see also [Qin19,Definition-Lemma 4.1.1].(2) can be proved similarly, or we can deduce it from (1) by usingthe map ι defined in (3.2). (cid:3) 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 )-unitriangularto S respectively, if its decomposition in S has such properties.3.3. Tropical transformations and compatibility.
As before, letthere be given seeds t ′ = ←− µ t , where ←− µ = ←− µ t ′ ,t is a sequence of muta-tions. Denote ←− µ t,t ′ = ←− µ − t ′ ,t . Denote the i -th cluster variables associatedto t and t ′ by X i and X ′ i respectively. Let f i , f ′ i denote the i -th unitvectors associated to t and t ′ respectively. Definition 3.3.1 (Tropical transformation) . If t ′ = µ k t , k ∈ I uf , wedefine the (degree) tropical transformation φ t ′ ,t : M ◦ ( t ) ≃ M ◦ ( t ′ ) suchthat, for any g = ( g i ) i ∈ I ∈ M ◦ ( t ) ≃ Z I , its image φ t ′ ,t g = ( g ′ i ) i ∈ I ∈ M ◦ ( t ′ ) ≃ Z i is given by g ′ i = − g k i = kg i + b ik [ g k ] + i = k, b ik ≥ g i + b ik [ − g k ] + i = k, b ik < . In general, we define the (degree) tropical transformation φ t ′ ,t : M ◦ ( t ) ≃ M ◦ ( t ′ ) as the composition of the tropical transformationsfor adjacent seeds along the mutation sequence ←− µ from t to t ′ . By[GHK15], φ t ′ ,t is the tropicalization of certain birational maps betweenthe split algebraic tori associate to t, t ′ and, consequently, independentof the choice of ←− µ .Recall that ←− µ ∗ t,t ′ X i is a pointed Laurent polynomial in LP ( t ′ ) by[DWZ10] [Tra11] [GHKK18]. Definition 3.3.2 (Degree linear transformation [Qin19, Definition 3.3.1]) . Define ψ t ′ ,t : M ◦ ( t ) ≃ M ◦ ( t ′ ) to be the linear map such that ψ t ′ ,t ( f i ) = deg t ′ ←− µ ∗ t,t ′ X i . By [Qin19, Lemma 3.3.4], the mutation map ←− µ ∗ t,t ′ : F ( t ) ≃ F ( t ′ )induces an injective algebra homomorphism b µ : LP ( t ) ֒ → d LP ( t ′ ). Ithas the following property. Lemma 3.3.3 ([Qin19]) . For any m ∈ Z I , b µX m is a well-definedfunction in d LP ( t ′ ) pointed at degree ψ t ′ ,t m . Moreover, for Z ∈ LP ( t ) ∩ ←− µ ∗ t ′ ,t LP ( t ′ ), we have b µ ( Z ) = ←− µ ∗ t,t ′ Z , see[Qin19, Lemma 3.3.4]. Correspondingly, denote b µ by ←− µ ∗ t,t ′ for simplic-ity.Consider the following set of Laurent polynomials LP ( t ; t ′ ) : = LP ( t ) ∩ ←− µ ∗ t ′ ,t LP ( t ′ ) . Then LP ( t ; t ′ ) is a k -algebra, such that ←− µ ∗ t,t ′ LP ( t ; t ′ ) = LP ( t ′ ; t ).The following very useful result shows that certain mutation se-quences swap pointedness and copointedness. Proposition 3.3.4 (Swap [Qin19, Propositions 3.3.9, 3.3.10]) . (1) Forany g, η ∈ M ◦ ( t ) , we have η (cid:22) t g if and only if ψ t [ − ,t η (cid:23) t ψ t [ − ,t g .(2) Let there be given Z ∈ LP ( t ; t [ − ⊂ LP ( t ) . Then Z is η -copointed if and only if ←− µ ∗ t,t [ − Z is ψ t [ − ,t η -pointed. Definition 3.3.5 (Compatibility) . If Z belongs to LP ( t ; t ′ ) ⊂ LP ( t ) ,then Z is said to be compatibly pointed at t, t ′ if it is g -pointed for some g ∈ M ◦ ( t ) , and ←− µ ∗ t,t ′ Z is φ t ′ ,t g -pointed.If Z belongs to U q ( t ) ⊂ LP ( t ) , then Z is said to be compatibly pointedat ∆ + if it is compatibly pointed at t, t ′ for any t ′ ∈ ∆ + .Let S denote a set consisting of g -pointed functions S g ∈ d LP ( t ) fordistinct g ∈ M ◦ ( t ) . If S g are compatibly pointed at t, t ′ for all g , wesay 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 : M ◦ ( t ) ≃ M ◦ ( t ′ ) as such that, for any g = ( g i ) i ∈ I ∈ M ◦ ( t ) ≃ Z I , itsimage φ op t ′ ,t g = ( g ′ i ) i ∈ I ∈ M ◦ ( t ′ ) ≃ Z i is given by g ′ i = − g k i = kg i − b ik [ g k ] + i = k, b ik ≤ g i − b ik [ − g k ] + i = k, b ik > . In general, we define the codegree tropical transformation φ op t ′ ,t : M ◦ ( t ) ≃ M ◦ ( t ′ ) as the composition of the codegree tropical trans-formations 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 = ( e B, Λ , ( X i ) i ∈ I ), we associate the opposite seed ι ( t ) := t op = ( − e B, − Λ , ( X i ∈ I )). Then [Qin19, Lemma 2.2.5] impliesthat, for any mutation sequence ←− µ , we have ( ←− µ t ) op = ←− µ ( t op ).Let us define ι : M ◦ ( t ) ≃ M ◦ ( t op ) as an isomorphism on Z I suchthat ι ( f i ( t )) = ι ( f i ( t op )). Correspondingly, by defining ι ( X m ) = X m ,we obtain natural k -algebra anti-isomorphisms ι : LP ( t ) ≃ LP ( t op )(3.1) ι : d LP ( t ) ≃ g LP ( t op )(3.2) ι : g LP ( t ) ≃ d LP ( t op ) . (3.3)Notice that ι : LP ( t ) ≃ LP ( t op ) induces an anti-isomorphism ι : F ( t ) ≃ F ( t op ).For any given k ∈ I uf , we have µ k ( t op ) = ( µ k t ) op . It is straightforwardto check the commutativity of the following diagram: F ( t ) ι −→ F ( t op ) ↑ µ ∗ k ↑ µ ∗ k F ( µ k t ) ι −→ F ( µ k ( t op )) . (3.4)In particular, ι ( µ ∗ k X i ( µ k t )) = µ ∗ k ( ιX i ( µ k t )) is given by X i ( t op ) if i = k ,or X ( t op ) − f k ( t op )+ P j [ − b jk ] + f j ( t op ) + X ( t op ) − f k ( t op )+ P i [ b ik ] + f i ( t op ) if i = k .Notice that Y ( t ) n = X e Bn while Y ( t op ) n = X − e Bn . It follows that Z ∈ d LP ( t ) is g -pointed if and only if ιZ ∈ g LP ( t op ) is g -copointed. Wehave the following result. Lemma 3.3.7.
Let there be given seeds t ′ = ←− µ t ′ ,t t . Then the codegreetropical transformation φ op t ′ ,t : M ◦ ( t ) ≃ M ◦ ( t ′ ) equals the composition M ◦ ( t ) ι −→ M ◦ ( t op ) φ ( t ′ )op ,t op −−−−−→ M ◦ (( t ′ ) op ) ι −→ M ◦ ( t ′ ) . In particular, it isindependent of the choice of ←− µ t ′ ,t .Proof. By the commutativity between ι and mutations, it suffices tocheck the claim for adjacent seeds t ′ = µ k t , which follows from defini-tion. (cid:3) Notice that we have LP ( t ; t ′ ) = ι LP ( t op ; ( t ′ ) op ) and U q ( t ) = ι ( U q ( t op ))by the commutativity between ι and mutations. Definition 3.3.8 (Codegree compatibility) . If Z belongs to LP ( t ; t ′ ) ⊂LP ( t ) , then Z is said to be compatibly copointed at t, t ′ if it is η -copointed for some η ∈ M ◦ ( t ) , and ←− µ ∗ t,t ′ Z is φ op t ′ ,t η -copointed. If Z belongs to U q ( t ) ⊂ LP ( t ) , then Z is said to be compatibly co-pointed at ∆ + if it is compatibly copointed at t, t ′ for any t ′ ∈ ∆ + .Let S denote a set consisting of η -copointed elements S η ∈ g 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 cate-gorical view of the degrees and the codegrees together with their tropicaltransformations, which are obtained by taking dual objects in the mod-ule category of quiver Hecke algebras.
Injective-reachability and distinguished functions.
Let σ denote a permutation of I uf . For any mutation sequence ←− µ = µ k r · · · µ k ,we define σ ←− µ = µ σk r · · · µ σk .Let pr I uf and pr I f denote the natural projection from Z I to Z I uf and Z I f respectively. Definition 3.4.1 ([Qin17, Definition 5.1.1]) . A seed t is said to beinjective-reachable if there exists a mutation sequence ←− µ = ←− µ t ′ ,t and apermutation σ of I uf , such that the seed t ′ = ←− µ t ′ ,t t satisfies b σi,σj ( t ′ ) = b ij ( t ) for i, j ∈ I uf and, for any k ∈ I uf , deg t ←− µ ∗ t ′ ,t X σk ( t ′ ) = − f k + u k (3.5) for some u k ∈ Z I f .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 ′ [ − and say it isthe shifted from t ′ (by [ − ) with the permutation σ − . Let there be given an injective-reachable seed t . Recursively, weconstruct a chain of seeds { t [ d ] | d ∈ Z } called an injective-reachablechain, such that t [ d ] = ( σ d − ←− µ ) t [ d − I k ( t ) = ←− µ ∗ t [1] ,t X σk ( t [1]) and P k ( t ) = ←− µ ∗ t [ − ,t X σ − ( k ) ( t [ − d ∈ N I uf , define the cluster monomial I ( t ) d := [ Q k I k ( t ) d k ] t and P ( t ) d := [ Q k P k ( t ) d k ] 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 I ( t ) d = { Q k I k ( t ) d k } t and P ( t ) d = { Q k P k ( t ) d k } t .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 havethe following g -pointed element in LP ( t ): I tg = [ p g ∗ X ( t ) [ g ] + ∗ I ( t ) [ − pr I uf g ] + ] t for some frozen factor p g ∈ P . Define the following set of distinguishedpointed functions I t := { I tg | g ∈ M ◦ ( t ) } . Denote t ′ = t [1]. By (3.5), the linear map ψ t,t ′ : M ◦ ( t ′ ) ≃ M ◦ ( t ) isdetermined by ψ t,t ′ ( f ′ σk ) = − f k + u k , k ∈ I uf ψ t,t ′ ( f ′ i ) = f i, i ∈ I f . Using Proposition 3.3.4, we deduce that codeg t [1] P σk ( t [1]) = codeg t [1] ←− µ ∗ t,t [1] X k ( t ) = ψ − t,t [1] f k = − f σk ( t [1]) + u k . (3.6)Notice that (3.6) appears in [Qin17, (18)] as an assumption. Replacing t by t [ −
1] in the above argument, we obtain codeg t P σk ( t ) = − f σk + u ′ k (3.7)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 ): P t,η = [ p η ∗ X ( t ) [ η ] + ∗ P ( t ) [ − pr I uf η ] + ] t for some frozen factor p η ∈ P . Define the following set of distinguishedcopointed functions P t := { P t,η | η ∈ M ◦ ( t ) } . The two kinds of distinguished functions are related by the followingresult. At a categorical level, it can be viewed as the duality betweeninjective representations and projective representations for a pair ofopposite quivers, see [Qin17, Section 5.3] for more discussion.
Lemma 3.4.2.
Denote ←− µ = ←− µ t [1] ,t . The following claims are true.(1) For any k ∈ I uf , we have ιP k ( t ) = I k ( t op ) .(2) We have t [ − op = ( t op )[1] = ←− µ − t op , which is shifted from t op with the permutation σ − .(3) We have t [1] op = ( t op )[ −
1] = ←− µ t op , which is shifted from t op withthe permutation σ .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 . Theclaim follows.(2) By the commutativity between mutations and ι , we have t [ − op =( ←− µ − t ) op = ←− µ − t op .The seed t [ − op has the principal B -matrix given by b ij ( t [ − op ) = − b ij ( t [ − − b σ − i,σ − j , i, j ∈ I uf . Using the commutativity between ι : LP ( t op ) ≃ LP ( t ) and mutations, its cluster variables have thefollowing Laurent expansion in LP ( t op ):( ←− µ − ) ∗ ( X σ − k ( ←− µ − t op )) = ι (( ←− µ − ) ∗ X σ − k ( ←− µ − t ))= ι ( P k ( t )) ,k ∈ I uf , which are pointed at − f k + u , u ∈ Z I f . It follows that t [ − op is a shifted seed t op [1] with the permutation σ − .(3) Notice that t [1] op = ( ←− µ t ) op = ←− µ ( t op ) by the commutativitybetween mutations and ι . Since ←− µ − t op = t op [1] with the permutation σ − by (2), we have ←− µ t op = t op [ −
1] with the permutation σ . (cid:3) Lemma 3.4.3 (Substitution) . (1) [Qin17, Lemma 6.2.4] Assume that [ X ( t ) d ∗ I ( t ) d ′ ] t is ( ≺ t , m ) -unitriangular to I t for any d ∈ N I uf ⊕ Z I f and d ′ ∈ N I uf . If Z is ( ≺ t , m ) -unitriangular to I t , then the normalizedproducts [ X ( t ) d ∗ Z ∗ I ( t ) d ′ ] t are ( ≺ t , m ) -unitriangular to I t too.(2) Assume that { P ( t ) d ′ ∗ X ( t ) d } t is codegree ( ≻ t , m ) -unitriangularto P t for any d ∈ N I uf ⊕ Z I f and d ′ ∈ N I uf . If Z is codegree ( ≻ t , m ) -unitriangular to P t , then the codegree normalized products { P ( t ) d ′ ∗ Z ∗ X ( t ) d } t are codegree ( ≻ t , m ) -unitriangular to P t too.Proof. (1) has been proved in [Qin17]. We can prove (2) using similararguments as those for (1), or deduce (2) from (1) by using the map ι . (cid:3) We have the following relation between degree and codegree tropicaltransformations, which will be useful for studying properties of doubletriangular bases (Proposition 4.3.1).
Proposition 3.4.4.
For any t, t ′ ∈ ∆ + , the following diagram com-mutes: (3.8) M ◦ ( t [1]) ψ t,t [1] −−−→ M ◦ ( t ) ↓ φ op t ′ [1] ,t [1] ↓ φ t ′ ,t M ◦ ( t ′ [1]) ψ t ′ ,t ′ [1] −−−−→ M ◦ ( t ′ ) . Proof.
It suffices to check the claim for the case t ′ = µ k t , k ∈ I uf . Noticethat, in this case, we have t ′ [1] = µ σk ( t [1]) and ←− µ ∗ t,t ′ I i ( t ) = I i ( t ′ ) for i = k .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 checkthe claim that, for i ∈ I uf , φ t ′ ,t ψ t,t [1] ( ± f σi ( t [1])) = ψ t ′ ,t ′ [1] φ op t ′ [1] ,t [1] ( ± f σi ( t [1])) . (i) By definition, for i = k in I uf , we have φ t ′ ,t ψ t,t [1] ( f σi ( t [1])) = deg t ′ ←− µ ∗ t,t ′ I i ( t ) = deg t ′ I i ( t ′ ) and also ψ t ′ ,t ′ [1] φ op t ′ [1] ,t [1] ( f σi ( t [1])) = ψ t ′ ,t ′ [1] ( f σi ( t ′ [1]))= deg t ′ I i ( t ′ )It follows that these two vectors in M ◦ ( t ′ ) agree.(ii) For the non-trivial case i = k , we have φ t ′ ,t ψ t,t [1] ( f σk ( t [1])) = deg t ′ ←− µ ∗ t,t ′ I k ( t ) ψ t ′ ,t ′ [1] φ op t ′ [1] ,t [1] ( f σk ( t [1]))= ψ t ′ ,t ′ [1] ( − f σk ( t ′ [1]) + X i ∈ I uf [ − b σi,σk ( t [1])] + f σi ( t ′ [1]) + X s ∈ I f [ − b s,σk ( t [1])] + f s )= − deg t ′ I k ( t ′ ) + X i ∈ I uf [ − b σi,σk ( t [1])] + deg t ′ I i ( t ′ ) + X s ∈ I f [ − b s,σk ( t [1])] + f s = − deg t ′ I k ( t ′ ) + X i ∈ I uf [ b ik ( t ′ )] + deg t ′ I i ( t ′ ) + X s ∈ I f [ b s,σk ( t ′ [1])] + f s Notice that I k ( t ) and I k ( t ′ ) are related by an exchange relation for theseeds ( t [1] , t ′ [1]). It follows that we have deg t ′ ←− µ ∗ t,t ′ I k ( t ) = − deg t ′ I k ( t ′ ) + X i ∈ I uf [ b ik ( t ′ )] + deg t ′ I i ( t ′ ) + X s ∈ I f [ b s,σk ( t ′ [1])] + f s , see [Qin17, (14)].(iii) By (3.5) and the linearity of ψ t,t [1] , for i = k in I uf , we have φ t ′ ,t ψ t,t [1] ( − f σi ( t [1])) = φ t ′ ,t ( f i ( t ) − u i )= f i ( t ′ ) − u i and also ψ t ′ ,t ′ [1] φ op t ′ [1] ,t [1] ( − f σi ( t [1])) = ψ t ′ ,t ′ [1] ( − f σi ( t ′ [1]))= − deg t ′ I i ( t ′ )(3.5) implies that the two vectors in M ◦ ( t ′ ) agree.(iv) For the non-trivial case i = k , we have ψ t,t [1] ( − f σk ( t [1])) = − deg t I k ( t ) ψ t ′ ,t ′ [1] φ op t ′ [1] ,t [1] ( − f σk ( t [1]))= ψ t ′ ,t ′ [1] ( f σk ( t ′ [1]) − X i ∈ I uf [ b σi,σk ( t [1])] + f σi ( t ′ [1]) − X s ∈ I f [ b s,σk ( t [1])] + f s )= deg t ′ I k ( t ′ ) − X i ∈ I uf [ b σi,σk ( t [1])] + deg t ′ I i ( t ′ ) − X s ∈ I f [ b s,σk ( t [1])] + f s = deg t ′ I k ( t ′ ) − X i ∈ I uf [ b ik ( t )] + deg t ′ I i ( t ′ ) − X s ∈ I f [ b s,σk ( t [1])] + f s It follows that φ t,t ′ ψ t ′ ,t ′ [1] φ op t ′ [1] ,t [1] ( − f σk ( t [1]))= deg t ←− µ ∗ t ′ ,t I k ( t ′ ) − X i ∈ I uf [ b ik ( t )] + deg t I i ( t ) − X s ∈ I f [ b s,σk ( t [1])] + f s Notice that I k ( t ) and I k ( t ′ ) are related by an exchange relation for theseeds ( t [1] , t ′ [1]). It follows that we have deg t ←− µ ∗ t ′ ,t I k ( t ′ ) = − deg t I k ( t ) + X i ∈ I uf [ b ik ( t )] + deg t I i ( t ) + X s ∈ I f [ b s,σk ( t [1])] + f s , see [Qin17, (14)]. Consequently, we get ψ t,t [1] ( − f σk ( t [1])) = φ t,t ′ ψ t ′ ,t ′ [1] φ op t ′ [1] ,t [1] ( − f σk ( t [1]))and the claim follows. (cid:3) Consequently, we obtain a relation between the degree compatibilityand the codegree compatibility.
Proposition 3.4.5.
Let there be given seeds t, t ′ ∈ ∆ + and Z ∈LP ( t ) ∩ ←− µ ∗ t ′ ,t LP ( t ′ ) ∩ ←− µ ∗ t [1] ,t LP ( t [1]) ∩ ←− µ ∗ t ′ [1] ,t LP ( t ′ [1]) . Then Z is com-patibly pointed at t [1] , t ′ [1] ∈ ∆ + if and only if it is compatibly copointedat t, t ′ .Proof. By Proposition 3.3.4, ←− µ ∗ t,t [1] Z is η -copointed in LP ( t [1]) if andonly if Z is ψ t,t [1] η -pointed in LP ( t ), and similar statements hold in LP ( t ′ [1]) and LP ( t ′ ). The claim follows from Proposition 3.4.4. (cid:3) Bidegrees and bases
Let there be given an injective-reachable quantum seed t and a sub-algebra 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 ′ ) := ←− µ ∗ t,t ′ A ( t ) ⊂ U q ( t ′ ) = ←− µ ∗ t,t ′ U q ( t ). And L naturally gives rise to a basis ←− µ ∗ t,t ′ L of A ( t ′ ). Wesometimes 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 ) issaid to be a degree-triangular basis with respect to t if the followingconditions hold:(1) X i ( t ) ∈ L for i ∈ I .(2) (Bar-invariance) L is invariant under the bar involution.(3) (Degree parametrization) L is M ◦ ( t ) -pointed, i.e., it takes theform L = { L g | g ∈ M ◦ ( t ) } such that L g is g -pointed.(4) (Degree triangularity) For any basis element L g , i ∈ I , thedecomposition of the pointed function [ X i ( t ) ∗ L g ] t in terms of L is degree ( ≺ t , m ) -unitriangular: [ X i ( t ) ∗ L g ] t = X g ′ (cid:22) t g + f i b g ′ L g ′ where b g + f i = 1 , b g ′ ∈ m for g ′ ≺ t g + f i .The basis is said to be a cluster degree-triangular basis with respectto t , or a triangular basis for short, if it further contains the quantumcluster monomials in t and t [1] . It is not clear if a degree-triangular basis is unique or not. Neverthe-less, a triangular basis must be unique if it exists, see [Qin17, Lemma6.3.2]. 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 ) issaid to be a codegree-triangular basis with respect to t if the followingconditions hold:(1) X i ( t ) ∈ L for i ∈ I .(2) (Bar-invariance) L is invariant under the bar involution.(3) (Codegree parametrization) L is M ◦ ( t ) -copointed, i.e., it takesthe 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 , thedecomposition of the copointed function { L η ∗ X i ( t ) } t in termsof L is codegree ( ≻ t , m ) -unitriangular: { L η ∗ X i ( t ) } t = X η ′ (cid:23) t η + f i c η ′ L η ′ where c η + f i = 1 , c η ′ ∈ m for η ′ ≻ t η + f i .The basis is said to be a cluster codegree-triangular basis with respectto t if it further contains the quantum cluster monomials in t and t [ − . By definition, P t is codegree ( ≻ t , m )-unitriangular to the clustercodegree-triangular basis. Similar to [Qin17, Lemma 6.3.2], we canshow that the cluster codegree-triangular basis is unique. Lemma 4.1.3 (Factorization) . (1) [Qin17, Lemma 6.2.1] Let there begiven a degree-triangular basis L . Then [ X i ( t ) ∗ S ] t = [ S ∗ X i ( t )] t ∈ L for any i ∈ I f .(2) Let there be given a codegree-triangular basis L . Then { X i ( t ) ∗ S } t = { S ∗ X i ( t ) } t ∈ L for any i ∈ I f . Definition 4.1.4 (Bidegree-triangular basis) . If L is both degree-triangularand codegree-triangular with respect to t , we call it a bidegree-triangularbasis with respect to t . Definition 4.1.5 (Double triangular basis) . If L is bidegree-triangularwith respect to t and further contains the quantum cluster monomialsin t, t [ − , t [1] , we call it a cluster bidegree-triangular basis of A ( t ) ora double triangular basis with respect to t . Definition 4.1.6 (Common triangular basis) . Assume that L is thetriangular basis of A ( t ) with respect to t . If ←− µ ∗ t,t ′ L is the triangularbasis of A ( t ′ ) = ←− µ ∗ t,t ′ A ( t ) with respect to t ′ and is compatible with L forany 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 [ − := ←− µ ∗ t,t [ − L t is the triangular basiswith respect to t [ − , then L t is the double triangular basis with respectto t .Proof. By assumption, L t contains the quantum cluster monomials in t, t [1] , t [ − L t satisfies the defining conditionsof a codegree triangular basis for t .(i) Since L t [ − is M ◦ ( t [ − L t = ←− µ ∗ t [ − ,t L t [ − is M ◦ ( t )-copointed by Proposition 3.3.4.(ii-a) Take any i ∈ I f . Then for any V ∈ L t which is bipointed by(i), we have { V ∗ X i ( t ) } t = [ V ∗ X i ( t )] t = [ X i ( t ) ∗ V ] t ∈ L t by Lemma4.1.3.(ii-b) Take any k ∈ I uf and any η -copointed element V ∈ L t . Then ←− µ ∗ t,t [ − X k ( t ) = I σ − k ( t [ − ←− µ ∗ t,t [ − V is pointed at g = ψ t [ − ,t η .Since ←− µ ∗ t,t [ − V belongs to the triangular basis ←− µ ∗ t,t [ − L t = L t [ − ,the normalized product [ ←− µ ∗ t,t [ − V ∗ I σ − k ( t [ − t [ − = v α ←− µ ∗ t,t [ − V ∗ I σ − k ( t [ − α ∈ Z , is ( ≺ t [ − , m )-unitriangular to I t [ − by Lemma3.4.3. Therefore, it is ( ≺ t [ − , m )-unitriangular to L t [ − . Then it hasthe following finite ( ≺ t [ − , m )-unitriangular decomposition in L t [ − : Z := v α ←− µ ∗ t,t [ − V ∗ I σ − k ( t [ − S (0) + r X j =1 b ( j ) S ( j ) with b ( j ) ∈ m , r ∈ N , deg t [ − S ( j ) ≺ t [ − deg t [ − S (0) = deg t [ − Z for j > Applying the mutation ←− µ ∗ t [ − ,t , we obtain Z ′ : = ←− µ ∗ t [ − ,t Z = v α V ∗ X k ( t ) = ←− µ ∗ t [ − ,t S (0) + r X j =1 b ( j ) ←− µ ∗ t [ − ,t S ( j ) . Proposition 3.3.4 implies that Z ′ is copointed and, for any j > codeg t ←− µ ∗ t [ − ,t S ( j ) ≻ t codeg t ←− µ ∗ t [ − ,t S (0) = codeg t Z ′ . Then this is acodegree ( ≻ t , m )-unitriangular decomposition in terms of the copointedset L t . (cid:3) We prove the following inverse result, although it will not be used inthis paper.
Proposition 4.2.2.
Assume that L t is the double triangular basis of A ( t ) with respect to the seed t . Then L t [ − := ←− µ ∗ t,t [ − L t is the triangu-lar basis with respect to t [ − .Proof. By assumption, L t [ − contains the quantum cluster monomi-als in t [ − , t . It remains to check that L t [ − satisfies the definitioncondition of a degree triangular basis for t [ − L t is M ◦ ( t )-copointed, L t [ − = ←− µ ∗ t,t [ − L t is M ◦ ( t [ − i ∈ I f . Then for any ( g − f i )-pointed element V ∈ L t [ − , we have X i ∗ V = v α X i · V = v α V ∗ X i for some α ∈ Z . Since X i · V is g -pointed, it agrees with [ X i ∗ V ] t [ − . Moreover, ←− µ ∗ t [ − ,t ( X i · V ) is η -copointed by Proposition 3.3.4, where η = ψ − t [ − ,t g .Therefore, ←− µ ∗ t [ − ,t ( v − α X i ∗ V ) = v − α X i ∗ ←− µ ∗ t [ − ,t V agrees with thecopointed function { X i ∗ ←− µ ∗ t [ − ,t V } t . Using Lemma 4.1.3, we deducethat ←− µ ∗ t [ − ,t [ X i ∗ V ] t [ − = { X i ∗ ←− µ ∗ t [ − ,t V } t is contained in codegreetriangular basis L t . Consequently, [ X i ∗ V ] t [ − belongs to L t [ − .(ii-b) Take any k ∈ I uf and g -pointed element V ∈ L t [ − . Then ←− µ ∗ t [ − ,t X k ( t [ − P σk ( t ) ∈ L t , and ←− µ ∗ t [ − ,t V is copointed at η = ψ − t [ − ,t g . The function ←− µ ∗ t [ − ,t [ X k ( t [ − ∗ V ] t [ − is copointed by Propo-sition 3.3.4, i.e., ←− µ ∗ t [ − ,t [ X k ( t [ − ∗ V ] t [ − = { P σk ( t ) ∗ ←− µ ∗ t [ − ,t V } t .Since L t is a double triangular basis, ←− µ ∗ t [ − ,t V is codegree ( ≻ t , m )-unitriangular to P t . Lemma 3.4.3 implies that { P σk ( t ) ∗ ←− µ ∗ t [ − ,t V } t is codegree ( ≻ t , m )-unitriangular to P t and, consequently, is code-gree ( ≻ t , m )-unitriangular to L t . We obtain a finite codegree ( ≻ t , m )-unitriangular decomposition Z := { P σk ( t ) ∗ ←− µ ∗ t [ − ,t V } t = r − X j =0 b ( j ) S ( j ) + S ( r ) with r ∈ N , b ( j ) ∈ m , codeg S ( j ) ≻ t codeg t S ( r ) = codeg t Z for j < r .Applying the mutation ←− µ ∗ t,t [ − , we obtain Z ′ : = ←− µ ∗ t,t [ − Z = [ X k ( t [ − ∗ V ] t [ − = r X j =1 b ( j ) ←− µ ∗ t,t [ − S ( j ) + ←− µ ∗ t,t [ − S ( r ) . Proposition 3.3.4 implies that Z ′ is pointed and, for any j < r , we have deg t [ − ←− µ ∗ t,t [ − S ( j ) ≺ t [ − deg t [ − ←− µ ∗ t,t [ − S ( r ) = deg t [ − Z ′ . Therefore,this decomposition is a degree ( ≺ t [ − , m )-unitriangular decompositionin L t . (cid:3) Properties of common triangular bases.
Define the subalge-bra A ( t op ) = ι A ( t ) ⊂ U q ( t op ). Proposition 4.3.1. If A ( t ) possesses the common triangular basis L ⊂ LP ( t ) , then A ( t op ) possesses the common triangular basis ι L ⊂LP ( t op ) .Proof. Notice that ι sends (quantum) cluster monomials ←− µ ∗ t ′ ,t X ( t ′ ) m to(quantum) cluster monomials ←− µ ∗ ( t ′ ) op ,t op X (( t ′ ) op ) m , m ∈ N I uf , becauseit commutes with mutations. In particular, it gives a bijection betweenthe sets of cluster monomials.Because the common triangular basis L gives rise to the doubletriangular bases for all seeds by Proposition 4.2.1, it gives rise to acodegree triangular bases L t ′ ⊂ LP ( t ′ ) for any seed t ′ ∈ ∆ + . Then ι L t ′ ⊂ LP (( t ′ ) op ) is a degree triangular bases containing all clustermonomials. Therefore, ι L t ′ is the triangular basis with respect to ( t ′ ) op .Moreover, for any t, t ′ ∈ ∆ + , because the elements of L are compat-ibly pointed at t [1] , t ′ [1], the elements of L are compatibly copointedat t, t ′ by Proposition 3.4.5. It follows that the elements of ι L arecompatibly pointed at t op , ( t ′ ) op .Therefore, ι L is the common triangular basis by definition. (cid:3) Recall that a common triangular basis is necessarily compatiblypointed at ∆ + . We have the following results. Theorem 4.3.2.
Let there be a k -subalgebra A ( t ) of the upper quantumcluster algebra U q ( t ) . Assume that A ( t ) possesses the common triangu-lar basis L . Then the following statements are true.(1) ←− µ ∗ t,t ′ L is the double triangular basis of A ( t ′ ) = ←− µ ∗ t,t ′ A ( t ) for anyseed t ′ ∈ ∆ + .(2) L is compatibly copointed at ∆ + .Proof. (1) The claim follows from Proposition 4.2.1. (2) By Proposition 4.3.1, ι L is the common triangular basis of A ( t op ),which is necessarily compatibly pointed at (∆ + ) op . Applying ι again,we deduce that L = ι ( ι L ) is compatibly copointed at ∆ + . (cid:3) Main results
An analog of Leclerc’s conjecture.
Let there be given aninjective-reachable seed t and a k -subalgebra A ( t ) of the upper quan-tum cluster algebra U q ( t ). Proposition 5.1.1.
Assume that A ( t ) possesses a bidegree-triangularbasis 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 X i ( t ) ∗ L g = v s S + X j b j L ( j ) + v h H such that s > h ∈ Z , b j ∈ v h +1 Z [ v ] ∩ v s − Z [ v − ] , and S, L ( j ) , H arefinitely many distinct elements of L with deg t H, deg t L ( j ) ≺ t deg t S = f i + g, codeg t S, codeg t L ( j ) ≻ t codeg t H = f i + η. 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 + e Bn , where n ∈ N ≥ uf ( t ) ≃ N I uf .Then X i ∗ L g has degree f i + g with coefficient v s := v λ ( f i ,g ) , codegree f i + η with coefficient v h := v λ ( f i ,η ) . It follows that h = s + λ ( f i , e Bn ) ≤ s where h = s if and only if n i = 0.Because L is a degree-triangular basis, we have a degree ( ≺ t , m )-unitriangular decomposition with finitely many S (0) , · · · , S ( r ) ∈ L :[ X i ∗ L g ] t = v − s X i ∗ L g = S (0) + X j> b ( j ) S ( j ) (5.1)such that b ( j ) ∈ m , deg S ( j ) ≺ deg S (0) = f i + g for j > 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-invariantand 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 pointedbut 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 decompositionof copointed elements v h L g ∗ X i = v h − s S (0) + X j> v h − s · b ( j ) S ( j ) . 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 . Rela-beling 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 coefficient1 = v h − s b ( r ) . In addition, the remaining terms S ( j ) , 0 < j < r musthave 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 . (cid:3) Theorem 5.1.2.
Let there be given a k -subalgebra A ( t ) of the upperquantum cluster algebra U q ( t ) . Assume that it has the the commontriangular basis L . Then, for any i ∈ I , V ∈ L , and any localizedquantum cluster monomial R , we have either R ∗ V ∈ v Z L or R ∗ V = v s S + X j b j L ( j ) + v h H (5.2) such that s > h ∈ Z , b j ∈ v h +1 Z [ v ] ∩ v s − Z [ v − ] , and S, L ( j ) , H arefinitely many distinct elements of L .Proof. Since L is the common triangular basis, Proposition 4.2.1 im-plies that ←− µ ∗ t,t ′ L is the double triangular basis (and thus bidegree-triangular) of A ( t ′ ) = ←− µ ∗ t,t ′ A ( t ) for any seed t ′ ∈ ∆ + . We apply Propo-sition 5.1.1 for localized quantum cluster monomials associated to t ′ . (cid:3) Theorem 5.1.2 is a weaker form of the following analog of Leclerc’sconjecture.
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 ∈ L ). Then, forany V ∈ L , we have either R ∗ V ∈ v Z L or R ∗ V = v s S + X j b j L ( j ) + v h H such that s > h ∈ Z , b j ∈ v h +1 Z [ v ] ∩ v s − Z [ v − ] , and S, L ( j ) , H arefinitely many distinct elements of L . Choose any l ∈ N . Let C l denote a level- l subcategory of the monoidalcategory of the finite dimensional modules of a quantum affine algebra U q ( b 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 param-eter. By [Qin17], K t ( C l ) is a (partially compactified) quantum clusteralgebra 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 tri-angular basis of the corresponding quantum cluster algebra A q afterlocalization 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 correspondingto 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 deformedGrothendieck ring K t ( C l ) : [ R ] ∗ [ V ] = t s [ S ] + X j b j [ L ( j ) ] + t h [ H ] where s > h ∈ Z , b j ∈ t h + Z [ t ] ∩ t s − Z [ t − ] . 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 algebraicrelation for such characters. Remark 5.1.5.
Assume that the quantum cluster algebra arises from aquantum unipotent subgroup of symmetric Kac-Moody type, which pos-sesses the dual canonical basis correspond to the set of self-dual simplemodules of the corresponding quiver Hecke algebra. In this case, up to v -power rescaling, S and H correspond to the simple socle and sim-ple 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’sconjecture might hold for the deformed Grothendieck ring K t ( C l ) ofquantum affine algebra and, in addition, it might have a categoricalinterpretation in analogous to that in [KKKO18, Section 4] . Properties of dual canonical bases.
Let us consider the quan-tum unipotent subgroup A q [ N − ( w )] of symmetrizable Kac-Moody typesin the sense of [Kim12][Qin20]. It is isomorphic to a (partially com-pactified) quantum cluster algebra after rescaling, see [GY16] [GY20]or [Qin20]. Theorem 5.1.2 implies the following weaker version of Con-jecture 1.1.1. Theorem 5.2.1.
Consider the dual canonical basis B up ( w ) of A q [ N − ( w )] .If b ∈ B up ( w ) corresponds to a quantum cluster monomial after rescal-ing, then for any b ∈ B up ( w ) , either b b ∈ q Z B up ( w ) or (1.1) holdstrue. 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 tri-angular 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-latticegrading 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]. Theclaim follows from Theorem 5.1.2. (cid:3) Theorem 5.2.1 would implies Conjecture 1.1.1 if the following multi-plicative reachability conjecture can be proved.
Conjecture 5.2.2. If b ∈ B up ( w ) ⊂ A q [ N − ( w )] is real (i.e. b ∈ q Z B up ( w ) ), then it corresponds to a quantum cluster monomial afterrescaling. Conjecture 5.2.2 can be generalized as the following, which impliesConjecture 5.1.3 by Theorem 5.1.2.
Conjecture 5.2.3 (Multiplicative reachability conjecture) . Let L de-note a common triangular basis. If b ∈ L is real (i.e. b ∈ L ), then itcorresponds to a localized quantum cluster monomial. Remark 5.2.4 (Reachability conjectures) . When the cluster algebraadmits an additive categorification by triangulated categories (clustercategories), we often expect that the rigid objects (objects with vanish-ing self-extensions) correspond to the (quantum) cluster monomials. Ifso, such objects can be constructed from the initial cluster tilting objectsvia (categorical) mutations. Let us call such an expectation the additivereachability conjecture. This conjecture is not true for a general clus-ter algebra because the cluster algebra seems too small for the clustercategory.When the cluster algebra admits a monoidal categorification by monoidalcategories, we similarly expect that the real simple objects correspond tothe (quantum) cluster monomials (see [HL10] ). If so, such objects canbe constructed from the an initial collection of real simple objects via(categorical) mutations. Let us call such an expectation the multiplica-tive reachability conjecture. Conjecture 5.2.2 is related to the specialcase for A q [ N − ( w )] .We also conjecture an equivalence between the additive reachabilityconjecture and the multiplicative reachability conjecture, which can beviewed as an analog of the open orbit conjecture [GLS11, Conjecture18.1] . See [Nak11, Section 1] for a comparison between additive cate-gorification and monoidal categorification.All these conjectures are largely open. References [BFZ05] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky,
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