A quantum cluster algebra approach to representations of simply-laced quantum affine algebras

We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q,t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q,t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.

1 Introduction of the Kazhdan-Lusztig type, to compute the pq, tq-characters, and so the q-characters of all simple finite-dimensional U q pĝq-modules.
This algorithm is theoretically computable, but as noted in [Nak10], trying to compute it in reality can easily exceed the size of computer memory available. The first step of the algorithm is to compute the pq, tq-characters of the fundamental representations, and for example, for g of type E 8 , the 5th fundamental representation requires 120Go of memory to compute.
In [HL10] Hernandez and Leclerc introduced a new point of view on representations of quantum affine algebras, using the theory of cluster algebras that was developed by Fomin and Zelevinsky in the early 2000's [FZ02], [FZ03], [BFZ05], [FZ07]. In [HL16a] they established a new algorithm to compute q-characters of a particular class of irreducible modules, called Kirillov-Reshetikhin modules, which include the fundamental modules, using the cluster algebra structure of the Grothendieck ring of a subcategory of the category of finite-dimensional U q pĝq-modules. The picture is completed when put into the broader context of the category O`of representations of U q pĝq, introduced by Hernandez-Jimbo in [HJ12]. In [HL16b], Hernandez and Leclerc showed that the Grothendieck ring of this category, which contains the finite-dimensional representations, is isomorphic to a cluster algebra built on an infinite quiver, while explicitly giving the identification.
In a previous work [Bit19b], the author defined the quantum Grothendieck ring for this category O`of representations as a quantum cluster algebra, as defined by Berenstein and Zelevinsky [BZ05]. However, the question of whether this quantum Grothendieck ring contained the quantum Grothendieck of the category of finite-dimensional representation, as used by Nakajima, was only proven in type A, and remained conjectural for other types.
In this Chapter we propose to show that, when g is of simply-laced type, the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional U q pĝq-modules has a quantum cluster algebra structure (Proposition 7.3.3). The proof relies heavily on a family of relations satisfied by the pq, tq-characters of the Kirillov-Reshetikhin modules called quantum T -systems proved in [Nak03a]. These relations are t-deformations of the T -systems relations, first stated in [KNS94]. These relations have not been generalized to non-simply-laced types, except for type B n in [HO19]. This is the main reason why the results of this chapter are limited to ADE types. This quantum cluster algebra approach gives a new algorithm to compute the pq, tq-characters of the Kirillov-Reshetikhin modules, and in particular of the fundamental modules (see Proposition 6.3.1). This algorithm seems more efficient, at least in terms of number of steps, than the Frenkel-Mukhin algorithm (see Remark 7.3.6).
For certain subcategories of the category of finite-dimensional U q pĝq-modules generated by a finite number of fundamental modules, Qin obtained in [Qin17] in a different context results similar to some results whose direct proofs are given here (see Remark 5.2.5). In this present work, we give explicit sequences of mutations to obtain pq, tq-characters of fundamental modules.
Next, we use this new result to prove a conjecture that was stated by the author in the aforementionned work [Bit19b]. This previous work dealt with a category O`of representations of the Borel subalgebra of the quantum affine algebra, which contains the finitedimensional U q pĝq-modules. The quantum Grothendieck ring of this category was defined as a quantum cluster algebra, and it was conjectured that this ring contained the quantum Grothendieck ring of the category of finite-dimensional representations. Here, we show that the quantum cluster algebra considered in [Bit19b] can be seen as a twisted version (in the sense of [GL14]) of the quantum cluster algebra occurring in the finite-dimensional case (see Proposition 7.2.3). As an application, the pq, tq-characters of the fundamental modules are obtained as quantum cluster variables in the quantum Grothendieck ring of the category O`(Proposition 7.3.3), and the inclusion of quantum Grothendieck rings conjectured in [Bit19b] follows naturally (Theorem 7.3.1 and Corollary 7.3.5).
Note that these results extend the algorithm to compute pq, tq-characters of some simple modules in the category O`. However, for this category of representations, the question of defining analogs of standard modules remains open. The author tackled this question in a previous work [Bit19a], and gave a complete answer when the underlying simple Lie algebra is g " sl 2 . This work is also a partial answer to the first point of Nakajima's "to do" list from [Nak11].
The author would also like to note that in type A, parallel results to the ones presented here were proven in [Tur18], via a different approach. In this work, pq, tq-characters of Kirillov-Reshetikhin modules are also obtained as quantum cluster variables in some quantum cluster algebra, the method uses a generalization of the tableaux-sum notations introduced by Nakajima in [Nak03b].
Finally, we use this algorithm to explicitly compute, when g is of type D 4 , the pq, tqcharacter of the fundamental representation at the trivalent node. This paper is organized as follows. In Sections 2 and 3 we recall notations and results regarding finite-dimensional representations of quantum affine algebras. In Section 4, we recall results regarding the t-deformation of Grothendieck rings, such as pq, tq-characters and quantum T -systems. In Section 5 we prove the existence of a quantum cluster algebra A t with t-commutations relations coherent with the framework of pq, tq-characters. Then, in Section 6 we prove that this quantum cluster algebra is isomorphic to the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional U q pĝq-modules; in this process, we established an algorithm to compute pq, tq-characters of Kirillov-Reshetikhin modules. Section 7 is devoted to the category O`, and to the proof of the inclusion Conjecture of [Bit19b]. Finally, the explicit computation mentioned just above is done in Section 8.
2 Cartan data and quantum Cartan data 2.1 Root data Let us fix some notations for the rest of the paper. Let the g be a simple Lie algebra of rank n and of type A,D or E. This restriction is necessary as one of the main arguments of the proof is the quantum T -systems, which have only been proven for these types as yet. Let γ be the Dynkin diagram of g and let I :" t1, . . . , nu be the set of vertices of γ.
The Cartan matrix of g is the nˆn matrix C such that Let us denote by pα i q iPI the simple roots of g, pα _ i q iPI the simple coroots and pω i q iPI the fundamental weights. We will use the usual lattices Q " À iPI Zα i , Q`" À iPI Nα i and P " À iPI Zω i . Let P Q " P b Q, endowed with the partial ordering : ω ď ω 1 if and only if ω 1´ω P Q`.
The Dynkin diagram of g is numbered as in [Kac90], and let a 1 , a 2 , . . . , a n be the Kac labels (a 0 " 1).
Let h be the (dual) Coxeter number of g: (2.1)

Quantum Cartan matrix
Let z be an indeterminate.
Definition 2.2.1. The quantum Cartan matrix of g is the matrix Cpzq with entries, Remark 2.2.2. The evaluation Cp1q is the Cartan matrix of g. As detpCq ‰ 0, then detpCpzqq ‰ 0 and we can defineCpzq, the inverse of the matrix Cpzq. The entries of the matrixCpzq belong to Qpzq.

One can write
where A is the adjacency matrix of γ. Hence, Therefore, we can write the entries ofCpzq as power series in z. For all i, j P I, Example 2.2.3. (i) For g " sl 2 , one has We will need the following lemma (see [ Let us extend the functionsC ij to symmetrical functions on Z C i,j pmq :"C ij pmq`C ij p´mq pm P Zq, (2.4) with the usual conventionC ij pmq " 0 if m ď 0. Then Lemma 2.2.4 translates as:

Height function
As g is simply-laced, its Dynkin diagram γ is a bipartite graph. There is partition I " I 0 \I 1 such that every edge in γ connects a vertex of I 0 to a vertex of I 1 .
Definition 2.3.1. Define, for all i P I, The map ξ : I Ñ t0, 1u is called a height function on γ.
Remark 2.3.2. In more generality, every function ξ : I Ñ Z satisfying ξpjq " ξpiq˘1, when j " i is a height function on γ. It defines an orientation of the Dynkin diagram γ: Our particular choice of height function defines a sink-source orientation.
Example 2.3.3. If g if of type D 5 , then γ is and if we fix ξ 1 " 0, then From now on, we fix such a height function ξ. We will also use the notation: (2.7)
Finally, we recall a useful notation from [HL16a]. For pi, rq PÎ´, define k i,r :"´r`ξ i 2 . (2.11) The vertex pi, rq is the k i,r th vertex in its column in G´, starting at the top.

Finite-dimensional representations of quantum affine algebras
In this section, we recall the notations and different results regarding quantum affine algebras and finite-dimensional representations of quantum affine algebras.

Quantum affine algebra
Letĝ be the untwisted affine Lie algebra corresponding to g.
Fix an nonzero complex number q, which is not a root of unity, and h P C such that q " e h . Then for all r P Q, q r :" e rh . Since q is not a root of unity, for r, s P Q, we have q r " q s if and only if r " s.
Let U q pĝq be the quantum enveloping algebra of the Lie algebraĝ (see [CP95a]), it is a C-Hopf algebra.

Finite-dimensional representations
Let C be the category of all (type 1) finite-dimensional U q pĝq-modules. As U q pĝq is a Hopf algebra, C is a tensor category. The simple modules in C have been classified by ), in terms of Drinfeld polynomials.
The simple finite-dimensional U q pĝq-modules are indexed by the monomials in the infinite set of variables pY i,a q iPI,aPCˆ, called dominant monomials ( [FR99]). For such a monomial m, let Lpmq denote the corresponding simple U q pĝq-module.
We define the following sets of dominant monomials: i,q r | pi, rq PÎ, n i,r P Z ě0 , n i,r " 0 except for a finite number of pi, rq i,q r | pi, rq PÎ´, n i,r P Z ě0 , n i,r " 0 except for a finite number of pi, rq + .
Definition 3.2.1. Let C Z be the full subcategory of C of objects whose composition factors are of the form Lpmq, with m P M.
Let CŹ be the full subcategory of C of objects whose composition factors are of the form Lpmq, with m P M´.
The category C Z is a tensor category and CŹ is a monoidal category ([HL10, 5.2.4] and [HL16a, Proposition 3.10]).
Remark 3.2.2. Every simple object in C can be written as a tensor product of simple objects which are essentially in C Z (see [HL10, Section 3.7]). Thus, the description of the simple objects of C reduces to the description of the simple objects of C Z .
Let us introduce some particular irreducible finite-dimensional representations. Note that fundamental module are particular KR-modules, for k " 1, m piq 1,r " Y i,r .

q-characters and truncated q-characters
Frenkel and Reshetikhin introduced in [FR99] an injective ring morphism, called the qcharacter morphism, on the Grothendieck ring K 0 pC q of the category C : Moreover, the q-character χ q pV q of a U q pĝq-module V gives information about the decomposition into Jordan subspaces for the action of a large commutative subalgebra of U q pĝq.
Here, as we restrict ourselves to the study of the category C Z , the q-character will only involve variables Y˘1 i,q r , for pi, rq PÎ. Hence, for simplicity of notation we denote them by: Y i,r :" Y i,q r . The q-character we are interested in is the injective ring morphism: We use the usual notation [FR99], for all pi, rq PĴ (see (2.10)). Note that A i,r is a Laurent monomial in the variables Y j,s , with pj, sq PÎ. The monomials A i,r are analogs of the simple roots.
where M p is a monomial in the variables A´1 i,r , with pi, rq PĴ.
Let us recall Nakajima's partial order on monomials. For m and m 1 Laurent monomials in Y, m ď m 1 ðñ m 1 m´1 is a product of A i,r , with pi, rq PÎ. (3.7) Remark 3.3.2. Note that Proposition 3.3.1 can be translated as follows: for all dominant monomials m, the monomials occurring in the q-character of the finite-dimensional irreducible representation Lpmq are lower than m, for Nakajima's partial order.
Proposition 3.3.4. [HL16a, Proposition 3.10] The assignment rLpmqs Þ Ñ χq pLpmqq extends to an injective ring homomorphism As such, all simple modules in CŹ are identified with their isoclasses through the truncated q-character morphism.

Cluster algebra structure
One of the main ingredient we want to use in this work is the cluster algebra structure of the Grothendieck ring of the category CŹ .
Consider the cluster algebra A :" Apu u u, G´q, with initial seed pu u u, G´q, where • u u u are initial cluster variables indexed byÎ´, u u u " • G´is the semi-infinite quiver with vertex setÎ´defined in the previous section.
Consider the identification, for all pi, rq PÎ´, (3.10) This identification makes sense, as these monomials are algebraically independent. From the Laurent phenomenon, we know that all cluster variables of A are Laurent polynomials in the variables u i,r . Thus, via the identification (3.10), A is seen as a subring of Z " Y˘1 i,r | pi, rq PÎ´ı.
Theorem 3.4.1. [HL16a, Theorem 5.1] The injective ring homomorphism χq is an isomorphism between the Grothendieck ring of the category CŹ and the cluster algebra A, after identification (3.10): χq Moreover, truncated q-characters of Kirillov-Reshetikhin modules can be obtained as cluster variables, via the identification of initial seed (3.10), it is the main result of [HL16a].

Quantum Grothendieck rings
We will recall in this section the definition of the quantum Grothendieck ring of the category C Z , introduce that of the category CŹ , and study those rings.
Let t be an indeterminate. The quantum Grothendieck rings of the categories C Z and CŹ are non-commutative t-deformations of the Grothendieck rings.

Quantum torus
Let Y t be the Zrt˘1s-algebra generated by the variables Y˘1 i,r , for pi, rq PÎ, and the tcommutations relations: where N i,j : Z Ñ Z is the antisymmetrical map: using the notations from Section 2.2. .
Remark 4.1.1. Here we work with the quantum torus of [Her04] and [HL15], which is slightly different from the original quantum torus used to define the quantum Grothendieck ring in [Nak04] and [VV03].
Example 4.1.2. If we continue Example 2.2.3, for g " sl 2 , in this case,Î " p1, 2Zq, for r P Z, one has Y 1,2r˚Y1,2s " t 2p´1q s´r Y 1,2s˚Y1,2r , @s ą r ą 0. The Zptq-algebra Y t is viewed as a quantum torus of infinite rank. We extend this quantum torus by adjoining a fixed square root t 1{2 of t: Let Yt be the quantum torus defined exactly the same way, except by only taking as generators the Y˘1 i,r , for pi, rq PÎ´.
Let us denote by π : Y t Ñ Yt , (4.5) the projection of Y t onto Yt , Remark 4.1.3. Even if Yt is an infinite rank quantum torus, it can be seen as a limit of finite rank quantum tori. As finite rank quantum tori are of polynomial growth, they are Ore domains (see [IM94]). Moreover, the Ore condition being local (any pair of elements of Yt belongs to some sufficiently larger finite rank quantum torus), Yt is an Ore domain. Hence we can consider its skew field of fractions F t .

Commutative monomials
For a family of integers with finitely many non-zero components pu i,r q pi,rqPÎ , define the commutative monomial where on the right-hand side an order onÎ is chosen so as to give meaning to the sum, and the product˚is ordered by it (notice that the result does not depend on the order chosen). The commutative monomials form a basis of the Zrt 1{2 s-vector space Y t . The non-commutative product of two commutative monomials m 1 and m 2 in Y t is given by: where m 1 m 2 denotes the commutative product of the monomials, and We define the quantum Grothendieck ring K t pC Z q of the category C Z as in [HL15, Section 5.4] (see Remark 4.1.1 for original references). For all pi, rq PĴ, let A i,r denote the commutative monomial in Y t defined as in (3.5): For all i P I, define K i,t the subring of Y t generated by the (4.10) In [Her03], the K i,t are defined as kernels of t-deformed screening operators, motivated by the results in [FR99]. Let us detail this, as it will be important in the proof of the main result. For all i P I, define the free Y t -modules Then let Y t,i be the quotient of Y l t,i by the left-Y t -module generated by the elements Lemma 4.3.1. For all i P I, the module Y t,i is free.
Proof. The elements Q i,r are linearly independent and for all r 0 such that pi, r 0 q PÎ fixed, the set tQ i,r , S i,r 0 | pi, rq P Iu forms a basis of Y l t,i . Hence Y t,i is a quotient of a free module by a submodule generated by elements of a basis, thus it is free. (4.11) which is a derivation and such that From [Nak01] and [Her04] we know that for all dominant monomials m P M, there exists a unique element F t pmq P K t pC Z q such that m occurs in F t pmq with multiplicity 1 and no other dominant monomial occurs in F t pmq. Thus, all elements of K t pC Z q are characterized by the coefficients of their dominant monomials.
The F t pmq linearly generate K t pC Z q. (4.14) The rLpY i,r qs t generate K t pCŹ q algebraically.

The pq, tq-characters
For a dominant monomial m P M, write it as a commutative monomial in Y t : where αpmq P 1 2 Z is fixed such that m occurs with multiplicity one in the expansion of rM pmqs t on the basis of the commutative monomials of Y t , and the product Ð Ý is taken with decreasing r P Z.
In particular, from (4.14), for all pi, rq PÎ, rLpY i,r qs t " rM pY i,r qs t . (4.17) One has, for all m P M, This result is a direct consequence of the definition of rM pmqs t , as it is satisfied for the fundamental modules rLpY i,r qs t " F t pY i,r q. Thus rM pmqs t is called the pq, tq-character of the standard module M pmq.
As in [Nak04], we consider the Z-algebra anti-automorphism of Y t defined by: This map is called the bar-involution.
Theorem 4.4.1. [Nak04] There exists a unique family trLpmqs t u mPM of elements of K t pC Z q such that, for all m P M, • rLpmqs t " rLpmqs t , • rLpmqs t P rM pmqs t`ř m 1 ăm t´1Zrt´1srM pm 1 qs t , where m 1 ă m for Nakajima's partial order (3.7).
The following Theorem extends (4.18), but more importantly gives an algorithm, similar to the Kazhdan-Lusztig algorithm, to compute the pq, tq-characters (and thus the q-characters) of the simple modules.
Theorem 4.4.2. [Nak04, Corollary 3.6] The evaluation at t " 1 of the pq, tq-characters recovers the q-characters. For all m P M, Moreover, the coefficients of the expansion of rLpmqs t as a linear combination of Laurent monomials in the variables pY i,r q pi,rqPÎ belong to Nrt˘1s.
Note that the positivity result of this Theorem has only been proven for ADE types as yet.
4.5 Truncated pq, tq-characters and quantum Grothendieck ring K t pCŹ q As in Section 3.3, one can define truncated versions of the pq, tq-characters.
For all dominant monomials m in M´, let rLpmqst be the Laurent polynomial obtained from rLpmqs t by removing any term in which a variable Y i,r , with pi, rq PÎzÎ´occurs: where π is the projection defined in (4.5).
Define K t pCŹ q as the Zrt 1{2 s-submodule of Yt generated by the truncated pq, tqcharacters rLpmqst of the simple finite-dimensional modules Lpmq in the category CŹ .
Lemma 4.5.1. The quantum Grothendieck ring K t pCŹ q is actually a subalgebra of Yt . Moreover, it is algebraically generated by the truncated pq, tq-characters of the fundamental modules: K t pCŹ q " Proof. For every dominant monomials m 1 , m 2 P M, one can write: Hence the image of (4.22) by the projection π of (4.5) is: Thus K t pCŹ q is stable by products. By definition the truncated pq, tq-characters of the fundamental modules LpY i,r q, for pi, rq PÎ´belong to K t pCŹ q.
Reciprocally, the pq, tq-characters of the fundamental modules LpY i,r q, for all pi, rq PÎ, algebraically generate the quantum Grothendieck ring K t pC Z q (see remark 4.3.2). Hence the truncated pq, tq-characters rLpY i,r qst , for all pi, rq PÎ, algebraically generate K t pCŹ q.
From Proposition 3.3.1 and Theorem 4.4.1, for all dominant monomials m P M, the pq, tq-character of the simple representation Lpmq is of the form where M p is a monomial in the variables pA´1 i,r q i,rqPĴ , with coefficients in Zrt˘1s. Thus, Hence, K t pCŹ q is algebraically generated by the rLpY i,r qst , with pi, rq PÎ´.
K t pCŹ q is a t-deformed version of the Grothendieck ring of the category CŹ , in the sense that the evaluation rLpmqst t"1 Ý Ý Ñ χq pLpmqq extends to a ring homomorphism where K 0 pCŹ q is identified with its image under the truncated q-character (3.9), which is an injective map.

Quantum T -systems
The pq, tq-characters of the Kirillov-Reshetikhin modules satisfy some algebraic relations called quantum T -systems. Those are t-deformed versions of the Moreover, the tensor product of the KR-modules W piq k´1,r`2 b W piq k`1,r is irreducible (this result is proved in [Cha02] and also by explicit computation of its pq, tq-character in [Nak03a]). Thus their respective pq, tq-characters t-commute (see [HL15, Corollary 5.5]). As their dominant monomials commute, these pq, tq-characters in fact commute and their product can be written as a commutative product, as in Section 4.2.
By the same arguments, for j " i, the pq, tq-characters rW pjq k,r`1 s t commute so the order of the factors in˚j "i in (4.24) does not matter.
By taking the image of (4.24) through the projection π of (4.5), one obtains the following relation in K t pCŹ q. For all pi, rq PÎ and k P Z ą0 , where αpi, kq and γpi, kq are defined in (4.25). Note that in (4.27), the products appearing on the left-hand side are commutative products, which are well-defined from Remark 4.6.2. Hence the change of notations since (4.24).

Quantum cluster algebra structure
We define in this section the quantum cluster algebra structure built within the quantum torus Yt .

A compatible pair
For all pi, rq PÎ´, the variable u i,r , written as in (3.10), can be seen as commutative monomial in Yt . Define They satisfy the following t-commutation relations. For all ppi, rq, pj, sqq P pÎ´q 2 , Let B´be theÎ´ˆÎ´-matrix encoding the quiver G´, for all ppi, rq, pj, sqq P pÎ´q 2 : B´ppi, rq, pj, sqq " |t arrows pi, rq Ñ pj, sq in G´u|´|t arrows pj, sq Ñ pi, rq in G´u|.
More precisely, we prove the following.
Proposition 5.1.1. For all ppi, rq, pj, sqq P pÎ´q 2 , Remark 5.1.2. In [BZ05], by definition a pair of JˆJ-matrices pΛ, Bq forms a compatible pair if T BL is a diagonal matrix with positive integer coefficients. But as explained in [Bit19b], quantum cluster algebras can be built exactly the same way given as data a pair pΛ, Bq such that T BL is a diagonal matrix with integer coefficients with constant signs.
Proof. Fix ppi, rq, pj, sqq P pÎ´q 2 , there are different cases to consider.
• If r ď´2, one has: One has L ppi, r´2q, pj, sqq´L ppi, r`2q, pj, sqq "´C ij ps´r´1q´C ij ps´r`1q where ξ : I Ñ t0, 1u is the height function on the Dynkin diagram of g fixed in Section 2.3.
On the other hand, for all k " i, one has L ppk, r`1q, pj, sqq´L ppk, r´1q, pj, sqq " C kj ps´rq´C kj p´r`2´ξ j q.

The quantum cluster algebra A t
Definition 5.2.1. Let T be the based quantum torus with generators t u i,r | pi, rq PÎ´u satisfying the quasi-commutation relations (5.1): u i,r˚uj,s " t Lppi,rq,pj,sqq u j,s˚ui,r .
Let F be the skew-field of fractions of T .
As pL, B´q forms a compatible pair, it defines a quantum seed in F. Let S be the mutation equivalence class of the quantum seed pL, B´q.
Definition 5.2.2. Let A t be the quantum cluster algebra defined by the quantum seed S, as in [BZ05].
By definition, A t is a Zrt˘1 {2 s-subalgebra of F. However, by the quantum Laurent phenomenon, A t is actually a Zrt˘1 {2 s-subalgebra of the quantum torus T .
where the variables U i,r are defined in (3.10), is an isomorphism of quantum tori.
Proof. First of all, this map is well-defined because the variables u i,r t-commute exactly as the variables U i,r , by definition of the matrix L (5.1). Secondly, this map is invertible, with inverse: η´1 : .
With this lemma, we know that the quantum cluster algebra A t belongs to the quantum torus Yt . The following result is the main result of this paper, it extends Theorem 3.4.1 to the quantum setting.
Theorem 5.2.4. The image of the quantum cluster algebra A t by the injective ring morphism η is the quantum Grothendieck ring of the category CŹ , (5.10) Moreover, the truncated pq, tq-characters of the Kirillov-Reshetikhin modules which are in CŹ are obtained as quantum cluster variables.
The proof of this Theorem will be developed in the following section. It is mainly based on Proposition 6.3.1.
Remark 5.2.5. In [Qin17,Theorem 8.4.3], for certain subcategories of C generated by a finite number of fundamental modules, Qin proved that there existed an isomorphism between the quantum Grothendieck ring of the category and a quantum cluster algebra, which identifies classes of Kirillov-Reshetikhin modules to cluster variables. It is our understanding that this identification coincides with the truncated pq, tq-characters in our work. Here, the isomorphism is given explicitly, and we obtain directly the truncated pq, tq-characters.

A note on the bar-involution
The bar-involution , as defined in (4.19), is a Z-algebra anti-automorphism of the quantum torus Y t . The commutative monomials are invariant under this involution.
On the other hand, the quantum cluster algebra A t is also equipped with a Z-linear bar-involution morphism on its quantum torus T (see [BZ05, Section 6]), which satisfies As noted in [Bit19b, Section 7.1], these definitions are compatible. In our case, they define exactly the same involution on Yt ; the following diagram is commutative: From [BZ05, Remark 6.4], all cluster variables are invariant under the bar-involution. Thus, the images of the quantum cluster variables in A t are bar-invariant elements of Yt .
We will use the terminology "commutative products", as in Section 4.2 for bar-invariant elements of the quantum torus T .

A sequence of vertices
In [HL16a] Hernandez and Leclerc exhibited a particular sequence of mutations in the cluster algebra Apu u u, G´q (see Section 3.4) in order to obtain the truncated q-characters of all the KR-modules, up to a shift of spectral parameter.
The key idea we used was that at each step of this sequence, the exchange relation was a T -system equation.
We will recall this sequence of mutations and show that, if applied to the quantum cluster algebra A t , the quantum exchange relations at each step are in fact quantum Tsystems relations, as in (4.6).
Recall the height function ξ : I Ñ t0, 1u fixed on the Dynkin diagram of g in Section 2.3. First, fix an order on the columns of G´: such that if k ď l then ξ i k ď ξ i l (select first the vertices i such that ξ i " 0 then the others). Then, the sequence S is defined by reading each column, from top to bottom, in this order.
We fix the following order on the columns: 1, 3, 3. Then the sequence S is S " p1, 0q, p1,´2q, p1,´4q, . . . , p3, 0q, p3,´2q, p3,´4q, . . . , p2,´1q, p2,´3q, p2,´5q 6.3 Truncated pq, tq-characters as quantum cluster variables As in [HL16a], let µ S be the sequence of quantum cluster mutations in A t indexed by the sequence of vertices S . For all m ě 1, let u pmq i,r be the quantum cluster variable obtained at vertex pi, rq after applying m times the sequence of mutations µ S to the quantum cluster algebra A t with initial seed t u i,r | pi, rq PÎ´u. By the quantum Laurent phenomenon, the u where k i,r is defined in (2.11).
In particular, if 2m ě h, this truncated pq, tq-character is equal to its pq, tq-character and w pmq i,r " rW piq k i,r ,r´2m s t . (6.6) Remark 6.3.2. The sequence of vertices S is infinite. However, in order to compute one fixed truncated pq, tq-character, one only has to compute a finite number of mutations in the infinite sequence µ S . In [Bit19b, Section 7.2], the exact finite sequence needed to compute the pq, tq-characters of the fundamental representations V i,r is given explicitly, we will also recall it in Section 7.
Proof. We prove this Proposition by induction on m, the number of times the mutation sequence µ S is applied on the initial quantum cluster variables t u i,r | pi, rq PÎ´u. The base step is given noting, as in [HL16a], that the images by the isomorphism η of the initial quantum cluster variables are indeed truncated pq, tq-characters. For all pi, rq PÎ´, Let m ě 0 and pi, rq PÎ´. Supposed we have applied m times the mutation sequence µ S , and a pm`1qth time on all vertices preceding pi, rq in the sequence S , and that all those previous vertices satisfy (6.5).
We want to write the quantum exchange relation corresponding to the mutation at vertex pi, rq. From the proof of Theorem 3.1 in [HL16a], we know the shape of the quiver just before this mutation (A t and Apu u u, G´q are defined on the same initial quiver and the mutation process on the quiver is the same for classical and quantum cluster algebras).
As explained in [HL16a, Section 3.2.3], for a general simply laced Lie algebra g, the mutation process takes place at vertices pi, rq having two (or one if r "´ξ i ) in-going arrows from pi, r˘2q and outgoing arrows to vertices pj, sq, with j " i. Thus the effect of the mutation sequence µ S on two fixed columns of the quiver is the same as the effect of an iteration of the mutation sequence on the corresponding quiver of rank 2.
Let us recall the mutation process on the quiver when g is of type A 2 . (1,0) where u pmq i,r`2 " 1 if r`2 ě 0 and ǫ i is defined in (2.7), and both terms are commutative products. This relation can also be written: where α, β P 1 2 Z. If we apply η to (6.9), and use the induction hypothesis, we get the following relation in Yt : Whereas, the corresponding (truncated) quantum T -system relation (4.27) is where α 1 , β 1 P 1 2 Z are given in (4.25). Note that one has indeed k i,r`2 " k i,r´1 , k i,r´2 " k i,r`1 , and k j,r´ǫ i " k i,r , for j " i. (6.12) Let k " k i,r and r 1 " r´2m. Let us precise how to obtain the coefficients α and β. From (6.8) and (6.9), α and β are such that the terms t α rW piq k´1,r 1 st rW piq k`1,r 1´2 st˚´rW piq k,r 1 st¯´1 , (6.13) and t β ź j"i rW pjq k,r 1´1 st˚´rW piq k,r 1 st¯´1 , (6.14) are bar-invariant. Thus, if one takes only the dominant monomials of (6.13) and (6.14), they are bar-invariant: This enables us to compute α and β.
Hence β " γpi, kq " 1 2´C ii p2k´1q`C ii p2k`1q¯. Thus α " α 1 and β " β 1 , and This concludes the induction. Finally, from [FM01, Corollary 6.14], we know that for all pi, rq PÎ´, the monomials m occurring in the q-character χ q pW piq k,r q of the KR-module W piq k,r are products of Y˘1 j,r`s , with 0 ď s ď 2k`h. Moreover, from Theorem 4.4.1, if one writes the pq, tq-character rW piq k,r s t of this KR-module as a linear combination of Laurent monomials in the variables Y i,s , with coefficients in Zrt˘1s, all monomials which occur in this expansion also occur in its q-character. Thus, if r`2k`h ď 0, the truncated pq, tq-character rW piq k,r st is equal to the pq, tq-character rW piq k,r s t . In particular, for m ě h, w pmq i,r " rW piq k i,r ,r´2m s t .

Proof of Theorem 5.2.4
We can now prove Theorem 5.2.4. This proof is a quantum analog of the proof of [HL16a, Theorem 5.1]. Naturally, there are technical difficulties brought forth by the non-commutative quantum tori structure. For example, in our situation, the quantum cluster algebra A t is isomorphic to the truncated quantum Grothendieck ring K t pCŹ q only via the isomorphism of quantum tori η (5.9). In particular, we need the following result.
Lemma 6.4.1. The identification (6.17) extends to a well-defined injective Zrt˘1 {2 s-algebras morphism where F t is the skew-field of fractions of Y t (see Remark 4.1.3). Moreover, the restriction of η 1 to the quantum cluster algebra A t has its image in the quantum torus Y t and the Zrt˘1 {2 s-algebra morphisms η, η 1 and π satisfy the following commutative diagram: Proof. From Proposition 6.3.1, for all pi, rq PÎ´, the full pq, tq-character rW piq k i,r ,r´2h s t is obtained as the image of the cluster variable sitting at vertex pi, rq after applying h times the mutation sequence S (which is locally a finite sequence of mutations): ηpu phq i,r q " rW piq k i,r ,r´2h s t . (6.19) In particular, for any two vertices pi, rq, pj, sq, the variables u phq i,r and u phq j,s belong to a common cluster and t-commute. Thus the pq, tq-characters rW piq k i,r ,r´2h s t t-commute. As the quantum torus Y t is invariant by shift of quantum parameters (Y j,s Þ Ñ Y j,s`2 ), the pq, tqcharacters rW piq k i,r ,r s t also t-commute for the product˚. Their t-commutation relations are determined by their dominant monomials, which are ηpu i,r q. Thus the rW piq k i,r ,r s t satisfy exactly the same t-commutations relations are the u i,r . This proves the first part of the lemma.
Let X be a cluster variable of A t obtained from the initial seed u u u " tu i,r u via finite sequence of mutations σ. We want to show that η 1 pXq P Y t . As the sequence of mutations σ is finite, it will only involve a finite number of cluster variables. Now apply h times the mutation sequence µ S to the initial seed so as to replace each cluster variable considered by u phq i,r (again, we only need a finite number of mutations). Let us summarize: η 1 pu i,r q " rW piq k i,r ,r s t , ηpu phq i,r q " rW piq k i,r ,r´2h s t .
Let X 1 be the cluster variable obtained by applying to this new seed the sequence of mutations σ. By construction, ηpX 1 q is equal to η 1 pXq, up to the downward shift of spectral parameters by 2h: every variable Y˘1 j,s is replaced by Y˘1 j,s´2h . In particular, η 1 pXq P Y t .
Next, the commutation of diagram (6.18) is verified as it is satisfied on the initial seed u u u " tu i,r u.
Let R t be the image of the quantum cluster algebra A t R t :" ηpA t q P Yt . (6.20) The inclusion K t pCŹ q Ă R t is essentially contained in Proposition 6.3.1. For the reverse inclusion, the main idea is to use the characterization of the quantum Grothendieck ring as the intersection of kernel of operators, called deformed screening operators. We show by induction on the length of a sequence of mutations that the images of all cluster variables belong to those kernels. The images of the initial cluster variables u i,r clearly belong to the quantum Grothendieck ring, and the screening operators being derivations, the exchange relations force the newly created cluster variables to be in the intersection of the kernels too. let us prove this in details.
Proof. Recall from Lemma 4.5.1 that the quantum Grothendieck ring K t pCŹ q is algebraically generated by the truncated pq, tq-characters of the fundamental modules: By Proposition 6.3.1, for all pi, rq PÎ´, (6.21) Which proves the first inclusion: We prove the reverse inclusion as explained just above. As explained in Section 4.3, Hernandez proved in [Her03] that for all i P I there exists operators S i,t : Y t Ñ Y i,t , where Y i,t is a Y t -module, which are Zrt˘1s-linear and derivations, such that č iPI kerpS i,t q " K t pC Z q. (6.23) Notice that these operators characterize the quantum Grothendieck ring K t pC Z q and not K t pCŹ q. Hence the need for Lemma 6.4.1.
Let us prove by induction that all cluster variables Z in A t satisfy η 1 pZq P K t pC Z q. Let Z be a quantum cluster variable in A t . If Z belongs to the initial cluster variables, Z " u i,r and η 1 pZq " η 1 pu i,r q " rW k i,r ,r s t P K t pC Z q. (6.24) If not, then by induction on the length of the sequence µ, one can assume that Z is obtained via a quantum exchange relation where Z 1 is a quantum cluster variable of A t , M 1 and M 2 are quantum cluster monomials of A t and η 1 pZ 1 q, η 1 pM 1 q, η 1 pM 2 q P K t pC q.
Apply η 1 to (6.25): η 1 pZq˚η 1 pZ 1 q " t α η 1 pM 1 q`t β η 1 pM 2 q. (6.26) For all i P I, apply the derivation S i,t : However, by hypothesis, Moreover, η 1 pZ 1 q ‰ 0 and the images of the screening operator is in a free module over Y t by Lemma 4.3.1. Thus S i,t pη 1 pZqq " 0, for all i P I. Hence η 1 pZq P K t pCŹ q, which concludes the induction. We have proven Then, by the commutation of the diagram (6.18) in Lemma 6.4.1, Which concludes the proof of the theorem.

Application to the proof of an inclusion conjecture
In this section, we use the quantum cluster algebra structure of the Grothendieck ring CŹ to prove that the quantum Grothendieck ring K t pOZ q defined in [Bit19b] contains K t pCŹ q.
In other words, we prove Conjecture 1 in [Bit19b]. The result was already proven in that paper in type A, but the core argument used was different.

The quantum Grothendieck ring K t pOZ q
The quantum Grothendieck ring K t pOZ q is defined as a quantum cluster algebra on the full infinite quiver Γ, of which the semi-infinite quiver G´is a subquiver. Let us recall some notations. For all i, j P I, F ij : Z Ñ Z is a anti-symmetrical map such that, for all m ě 0, ij pm´2k`1q. (7.1) Let T t be the quantum torus defined as the Zrt˘1 {2 s-algebra generated by the variables zȋ ,r , for pi, rq PÎ, with a non-commutative product˚, and the t-commutations relations z i,r˚zj,s " t F ij ps´rq z j,s˚zi,r ,´pi, rq, pj, sq PÎ¯.
Recall also from [Bit19b, Proposition 5.2.2] the inclusion of quantum tori J (with a slight shift of parameters on the z i,r ): Let Λ be the infinite skew-symmetricÎˆÎ-matrix: Λ pi,rq,pj,sq " F ij ps´rq,´pi, rq, pj, sq PÎ¯.
From [Bit19b], the quiver Γ and the skew-symmetric matrix Λ form a compatible pair. Let A t pΓ, Λq be the associated quantum cluster algebra. Then, as in [Bit19b, Definition 6.3.5], K t pOZ q :" A t pΓ, ΛqbE, (7.5) where E is a commutative ring and the completion allows for certain countable sums.

Intermediate quantum cluster algebras
The general idea is to see the quantum cluster algebra A t as a "sub-quantum cluster algebra" of A t pΓ, Λq (this term is not well-defined). However, as in Section 6.4 and contrary to the aforementioned proof, as we are dealing with quantum cluster algebras in our setting, this is not done trivially. Mainly, one notices that the map is not a well-defined inclusion of quantum tori, as the generators u i,r and z i,r do not satisfy the same t-commutation relations.
First, consider the subquiver Γ´of Γ of index set Iď 2 :"Î X tpi, rq | i P I, r ď 2u, (7.6) such that the vertices pi, rq, with r ą 0 are frozen. To summarize, for ξ i " 0 and j " i:  [GG18,Theorem 4.5] the inclusion of seeds Γ´, Λ Ă Γ, Λ induces an inclusion of the quantum cluster algebra A t pΓ´, Λq into the quantum cluster algebra of A t pΓ, Λq.
Now we need to link the quantum cluster algebras A t and A t pΓ´, Λq.
In order to do that, we use a result from [GL14], which deals with graded cluster algebras.
Definition 7.2.1. [GL14] A quantum cluster seed pB, Λq is graded if there exists an integer column vector G such that, for all mutable indices k, the kth row of B, B j satisfies B j G " 0.
Then, the G-degree of the initial variables are set by the vector G: for all cluster variables X k in the initial clusterX, deg G pXq " G i .
The grading condition is equivalent to the following, for all mutable variables X k , the sum of the degrees of all variables with arrows to X k is equal to the sum of the degrees of all variables with arrows coming from X k , i.e. exchange relations are homogeneous. Hence, each cluster variable has a well-defined degree.
Let us start by considering the quantum cluster algebra A t pΓ´, Lq, built on the quiver Γ´, with coefficients 1 on the frozen vertices. This quantum cluster algebra is clearly isomorphic to A t , and is a graded quantum cluster algebra of grading G " 0.
Next, for all i P I, we apply the process of [GL14, Theorem 4.6] to add coefficients f i , while twisting the t-commutation relations.
Let u i pj, sq " δ i,j ,´pj, sq PÎď 2¯( 7.7) and t i pj, sq "´F ij p2´s´ξ i q,´pj, sq PÎď 2¯, (7.8) with ξ : I Ñ t0, 1u the height function, fixed in Section 2.3. Then Lemma 7.2.2. For all i P I, u i and t i are gradings for the ice quiver Γ´.
•L encodes the t-commutations relations, such that, for all pi, rq PÎ´, and j P I, f j˚ui,r " t t j pi,rq u i,r˚fj , (7.10) and the f j pairwise commute.
•G is a multi-grading, i.e. instead of being an integer column vector, each entry inG is in the lattice Z I . It is defined by, for all pi, rq PÎď 2 , Gpũ i,r q " e i P Z I , (7.11) This is indeed the construction of [GL14], with the initial grading on A t pΓ´, Lq being G " 0. The new quantum cluster algebra is denoted by A u,t t pΓ´, Lq, to show that it is a twisted version of A t pΓ´, Lq.
Proposition 7.2.3. The quantum cluster algebra A t pũ u u,B,Lq is isomorphic to the quantum cluster algebra A t pΓ´, Λq.
First, notice that, for all i, j P I, m P Z, This result is proven in [Bit19b], in the course of the proof of Proposition 5.2.2. Thus, for all pi, rq, pj, sq PÎ´, ij p´ξ j´r`2 q " F ij ps´rq´t i pj, sq`t j pi, rq.
And of course, for all i P I, pj, sq PÎ´, Λ ppi,´ξ i`2 q, pj, sqq " F ij ps`ξ i´2 q " t i pj, sq.
Thus, one had indeed,L " Λ |Îď 2 . (7.14) From now on, we will use the notations z z z " tz i,r , f j u pi,rqPÎ´,jPI for the initial clusters variables of both A t pΓ´, Λq and A u,t t pΓ´, Lq. Remark 7.2.4.
• This result is natural, if we look at what the different initial cluster variables mean in terms of ℓ-weights. From (3.10), for all pi, rq PÎ´, the cluster variable u i,r can be identified with the (commutative) dominant monomial U i,r " ś kě0 r`2kď0 Y i,r`2k . Whereas, the quantum tori Y t and T t are compared via the inclusion J of (7.3) J : Y i,r Þ ÝÑ z i,r pz i,r`2 q´1 , @pi, rq PÎ.
Thus, the link between the variables u i,r and z i,r is the following: with the previous convention f i " z i,´ξ i`2 . More precisely, there are different maps of quantum tori: where identification (7.15) is the resulting dotted map, which we will denote by ρ: The quantum cluster algebra A u,t t pΓ´, Lq was built with this identification in mind.
• This process could also be seen as a quantum version of the multi-grading homogenization process of the seed pu u u, G´q, as in [Gra15,Lemma 7.1], where the multigrading is defined by (7.11).

Inclusion of quantum Grothendieck rings
In the section, we prove that the quantum Grothendieck ring K t pOZ q, or more precisely, the quantum cluster algebra A t pΓ, Λq contains the quantum Grothendieck ring K t pC Z q, which is the statement of Conjecture 1 in [Bit19b]. Recalled that in [Bit19b], the ring K t pOZ q was defined as a completion of the quantum cluster algebra A t pΓ, Λq, but the aim was to see it as a quantum Grothendieck ring for the category of representations OZ from [HJ12] and [HL16b]. As this category contains the category C Z , it was expected for the quantum Grothendieck ring K t pOZ q to contain K t pC Z q.
In order to prove this result we will actually prove Conjecture 2 of the same paper, which is a stronger result. We state it as follows.
Theorem 7.3.1. The pq, tq-characters of all fundamental representations in C Z are obtained as quantum cluster variables in the quantum cluster algebra A t pΓ, Λq.
More precisely, we show that for all i P I, there exists a specific finite sequence of mutations S i in A t pΓ, Λq such that, if applied to the initial seed tz j,s u pj,sqPÎ , the cluster variable sitting at vertex pi,´ξ i q is the image by J (of (7.3)) of the pq, tq-character of the fundamental module rLpY i,´ξ i´2 h 1 qs t , where with h the Coxeter number of the simple Lie algebra g. Let us define the sequence S i . Let pi 1 , i 2 , . . . , j n q be an ordering of the columns of Γ as in (6.2), such that i 1 " i (take first all columns j such that ξ j " ξ i ). The sequence S i is a sequence of vertices of Γ, and more precisely of Γ´, defined as follows. First read all vertices pi 1 , rq for´2h 1`2 ď r ď 0, from top to bottom, then all pi 2 , rq, with´2h 1 ď r ď 0, and so on, then read again all vertices pi 1 , rq for´2h 1`4 ď r ď 0, and continue browsing the columns successively, until at the last step you only read the vertex pi,´ξ i q.
Note that applying this sequence S i of mutations on the quivers Γ´or G´has exactly the same effect on the cluster variable sitting at vertex pi,´ξ i q than applying h 1 times the infinite sequence S from Section 6.2.
Moreover,χ i,r 0 " J prLpY i,r 0 qs t q . (7.20) Proof. The cluster variableχ i,r 0 is a variable of the quantum cluster algebra A t pΓ´, Λq, which is isomorphic to A u,t With notations from Section 6.3, u ph 1 q i,´ξ i is the cluster variable of A t pΓ´, Lq obtained at vertex pi,´ξ i q after applying the mutations of the sequence S i . Also from [GL14,Corollary 4.7], we know that there exists integers a j P Z such that written as a commutative product (bothχ i,r 0 and u ph 1 q i,´ξ i are bar-invariant), with ρ defined in (7.17). The term u ph 1 q i,´ξ i is a Laurent polynomial in the variables u j,s , which satisfy ρpu j,s q " z j,s pf j q´1 from (7.15). Thus expression (7.21) is a way of writingχ i,r 0 as a Laurent polynomial in the initial variables tz j,s , f k u. However, one can writeχ i,r 0 " N {D, where N is the Laurent polynomial in the cluster variables tz j,s , f k u, with coefficients in Zrt˘1 {2 s and not divisible by any of the f k , and D is a monomial in the non-frozen variables tz j,s u. Thus ś jPI f a j j is the smallest monomial such that contains only non-negative powers of the frozen variables f k . Moreover, from Proposition 6.3.1, However, all Laurent monomials occurring in rLpY i,r 0 qs t already occurred in the qcharacter χ q pLpY i,r 0 qq, as the pq, tq-character rLpY i,r 0 qs t has positive coefficients. Indeed, the pq, tq-characters of fundamental modules have been explicitly computed and have been found to have non-negative coefficients (in [Nak03b] for types A and D and [Nak10] for type E).
From [FM01], all monomials in χ q pLpY i,r 0 qq are products of Y˘1 j,s , with s ď r 0`h , but by definition of r 0 , r 0`h ď 0, and the term with the highest quantum parameter being the anti-dominant monomial Y´1 i,r 0`h , where : I Ñ I is the involutive map such that ω 0 pα j q "´α j , with ω 0 the longest element of the Weyl group of g (no relation with the bar-involution of Section 6.1).
Consider the change of variables, for pj, sq PÎ´, y j,s " η´1pY j,s q " Thus ρpy j,s q " All monomials occurring in rLpY i,r 0 qs t are commutative monomials in the variables Y˘1 j,s , with s ď r 0`h ď 0. Moreover, the only monomials in which the variables Y˘1 j,s , with s`2 ě 0, occur are the anti-dominant monomial Y´1 i,r 0`h , and any possible monomial in which some variable Y j,r 0`h´1 occurs. But for such a monomial m, the variable Y˘1 j,s with the highest s in m occurs with a negative power in m (the monomial m is "right-negative", as from [FM01, Lemma 6.5]), thus the variable Y j,r 0`h´1 also occurs with a negative power. The image by ρ˝η´1 of any monomial in the variables Y˘1 j,s , with s`2 ď 0 is a monomial in the variables tz˘1 j,s u (without frozen variables). Thus, the image ρ´u ph 1 q i,´ξ i¯" ρ˝η´1 prLpY i,r 0 qs t q is a Laurent polynomial with only positive powers of the variables f j .
Remark 7.3.4. At some point in the proof we used the fact that the pq, tq-characters of the fundamental modules had non-negative coefficients. Note that this part of the proof could easily be extended to non-simply laced types, as the pq, tq-characters of their fundamental representations have also be explicitly computed, and also have non-negative coefficients (in types B and C, the pq, tq-characters of all fundamental representations are equal to their respective q-character, and all coefficients are actually equal to 1 [Her05, Proposition 7.2], and see [Her04] for type G 2 and [Her05] for type F 4 ).
Corollary 7.3.5. For all pi, rq PÎ there exists a quantum cluster variableχ i,r in the quantum cluster algebra A t pΓ, Λq such that Proof. For all pi, rq PÎ, letχ i,r be the cluster variable of the quantum cluster algebra A t pΓ, Λq obtained at vertex pi, r`2h 1 q after applying the sequence of mutations S i , but starting at vertex pi, r`2h 1 q instead of pi,´ξ i q.
Consider the change of variables in A t pΓ, Λq: s : z j,s Þ ÝÑ z j,s`r 0´r , @pj, sq PÎ. (7.25) The quantum cluster algebra A t pΓ, Λq is invariant under this shift s, and this change of variables is clearly invertible (s´1pz j,s q " z j,s`r´r 0 ). One has spz i,r q " z i,r 0 , and spz i,r`2h 1 q " z i,´ξ i , thus s pχ i,r q "χ i,r 0 , and from Proposition 7.3.3, χ i,r " s´1 pχ i,r 0 q " s´1 pJ prLpY i,r 0 qs t qq .
However, from the definition of the map J in (7.3), the shift s also acts as a change of variables in the quantum torus Y t , s : Y j,u Þ ÝÑ Y j,u`r 0´r . Hence, Thus we have proven Theorem 7.3.1.
• When Conjecture 2 was formulated in [Bit19b], a recent positivity result of Davison [Dav18] was mentioned there. This work proves the so-called "positivity conjecture" for quantum cluster algebras, which states that the coefficients of the Laurent polynomials into which the cluster variables decompose from the Laurent phenomenon are in fact non-negative. This is an important result, but also a difficult one, and it is not actually needed in order to obtain our result.
• One can note from this proof that we know a close bound on the number of mutations needed in order to compute the pq, tq-character of a fundamental module. For g a simple-laced simple Lie algebra of rank n and of (dual) Coxeter number h, if h 1 " rh{2s, then the number of steps is lower than n h 1 ph 1`1 q 2 . (7.26) We have chosen to go into details on the number of steps required for this process because it made sense from an algorithmic point of view to know its complexity.
One can compare this algorithm to Frenkel-Mukhin [FM01] to compute q-characters. As explained in [Nak10], when trying to compute q-characters of fundamental representations of large dimension (for example, in type E 8 , the q-character of the 5th fundamental representation has approximately 6.4ˆ2 30 monomials), one encounters memory issues. Indeed, this algorithm has to keep track of all the previously computed terms. This advantage of the cluster algebra approach is that one only had to keep the seed in memory.
One can notice that the required number of steps is indeed lower than 24, which was the bound given in (7.26).
Let us give explicitly the mutations on the quiver Γ´(which encodes more than G´), as well as the quantum cluster variables obtained at (almost every) step. We give the quantum cluster variables as Laurent polynomials in the variables tz i,r , f j u pi,rqPÎ,jPI , as well as in the form (7.21). For completeness, we also compute the multi-degrees of the quantum cluster variables.