Extended T-System of Type G₂

We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type G₂ extending the celebrated T-system relations of type G₂. We show that these relations can be used to compute classes of certain irreducible modules,...

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Datum:	2013
Hauptverfasser:	Li, J., Mukhin, E.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2013
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1493482019-02-22T01:24:16Z Extended T-System of Type G₂ Li, J. Mukhin, E. We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type G₂ extending the celebrated T-system relations of type G₂. We show that these relations can be used to compute classes of certain irreducible modules, including classes of all minimal affinizations of type G₂. We use this result to obtain explicit formulas for dimensions of all participating modules. 2013 Article Extended T-System of Type G₂ / J. Li, E. Mukhin // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 81R50; 82B23 DOI: http://dx.doi.org/10.3842/SIGMA.2013.054 http://dspace.nbuv.gov.ua/handle/123456789/149348 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type G₂ extending the celebrated T-system relations of type G₂. We show that these relations can be used to compute classes of certain irreducible modules, including classes of all minimal affinizations of type G₂. We use this result to obtain explicit formulas for dimensions of all participating modules.
format	Article
author	Li, J. Mukhin, E.
spellingShingle	Li, J. Mukhin, E. Extended T-System of Type G₂ Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Li, J. Mukhin, E.
author_sort	Li, J.
title	Extended T-System of Type G₂
title_short	Extended T-System of Type G₂
title_full	Extended T-System of Type G₂
title_fullStr	Extended T-System of Type G₂
title_full_unstemmed	Extended T-System of Type G₂
title_sort	extended t-system of type g₂
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149348
citation_txt	Extended T-System of Type G₂ / J. Li, E. Mukhin // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 22 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT lij extendedtsystemoftypeg2 AT mukhine extendedtsystemoftypeg2
first_indexed	2025-07-12T21:54:05Z
last_indexed	2025-07-12T21:54:05Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 054, 28 pages Extended T -System of Type G2 ? Jian-Rong LI † and Evgeny MUKHIN ‡ † Department of Mathematics, Lanzhou University, Lanzhou 730000, P.R. China E-mail: lijr@lzu.edu.cn, lijr07@gmail.com URL: http://scholar.google.com/citations?user=v_0AZ7oAAAAJ&hl=en ‡ Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA E-mail: mukhin@math.iupui.edu URL: http://www.math.iupui.edu/~mukhin/ Received April 03, 2013, in final form August 16, 2013; Published online August 22, 2013 http://dx.doi.org/10.3842/SIGMA.2013.054 Abstract. We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type G2 extending the celebrated T -system relations of type G2. We show that these relations can be used to compute classes of certain irreducible modules, including classes of all minimal affinizations of type G2. We use this result to obtain explicit formulas for dimensions of all participating modules. Key words: quantum affine algebra of type G2; minimal affinizations; extended T -systems; q-characters; Frenkel–Mukhin algorithm 2010 Mathematics Subject Classification: 17B37; 81R50; 82B23 1 Introduction Kirillov–Reshetikhin modules are simplest examples of irreducible finite-dimensional modules over quantum affine algebras, and the T -system is a famous family of short exact sequences of tensor products of Kirillov–Reshetikhin modules, see [10, 15, 16, 20]. There are numerous applications of the T -systems in representation theory, combinatorics and integrable systems, see the survey [17]. Minimal affinizations of quantum affine algebras form an important family of irreducible modules which contains the Kirillov–Reshetikhin modules, see [3]. A procedure to extend the T -system to a larger set of relations to include the minimal affinization was described in [18], where it was conjectured to work in all types. In [18] this procedure was carried out in types A and B. In this paper, we show the existence of the extended T -system for type G2. We work with the quantum affine algebra Uqĝ of type G2. The irreducible finite-dimensional modules of quantum affine algebras are parameterized by the highest l-weights or Drinfeld polynomials. Let T be an irreducible Uqĝ-module such that zeros of all Drinfeld polynomials belong to a lattice aqZ for some a ∈ C×. Following [18], we define the left, right, and bottom modules, denoted by L, R, B respectively. The Drinfeld polynomials of left, right, and bottom modules are obtained by stripping the rightmost, leftmost, and both left- and rightmost zeros of the union of zeros of the Drinfeld polynomials of the top module T . Then the relations of the extended T -system have the form [L][R] = [T ][B] + [S], where [·] denotes the equivalence class of a Uqĝ-module in the Grothendieck ring of the category of ?This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html mailto:lijr@lzu.edu.cn mailto:lijr07@gmail.com http://scholar.google.com/citations?user=v_0AZ7oAAAAJ&hl=en mailto:mukhin@math.iupui.edu http://www.math.iupui.edu/~mukhin/ http://dx.doi.org/10.3842/SIGMA.2013.054 http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html 2 J.R. Li and E. Mukhin finite-dimensional representations of Uqĝ. Moreover, in all cases the modules T ⊗ B and S are irreducible. We start with minimal affinizations as the top modules T , then the left, right and bottom modules are minimal affinizations as well. We compute S and decompose it as a product of irreducible modules which we call sources. It turns out that the sources are not always minimal affinizations. Therefore, we follow up with taking the sources as top modules and compute new left, right, bottom modules, and sources. Then we use all new modules obtained on a previous step as top modules and so on. We end up with several families of modules which we denote by B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` , B̃(s) k,`, C̃ (s) k,` , D̃ (s) k,`, Ẽ (s) k,` , F̃ (s) k,` , where s ∈ Z, k, ` ∈ Z≥0. This is the minimal set of modules which contains all minimal affinizations ( these are modules B(s) k,`, B̃ (s) k,` ) and which is closed under our set of relations. Namely, if any of the above modules is chosen as a top module then the left, right, bottom modules and all sources belong to this set as well, see Theorems 3.4, 7.4. The spirit of the proof Theorems 3.4, 7.4 follows the works in [10, 18, 21]. We show that the extended T -system allows us to compute the modules B(s) k,`, C (s) k,` , D (s) k,`, E(s) k,` , F (s) k,` recursively in terms of fundamental modules, see Proposition 3.6. We use this to compute the dimensions of all participating modules, in particular, we give explicit formulas for dimensions of all minimal affinizations of type G2, see Theorem 8.1. We hope further, that one can use use the extended T -system to obtain the decomposition of all participating modules as the Uqg-modules. Let us point out some similarities and differences with types A and B. The type A, the extended T -system is closed within the class of minimal affinizations, meaning that all sources are minimal affinizations as well. In type B, the extended T -system is not closed within the class of minimal affinizations, but it is closed in the class of so called snake modules, see [18]. For the proofs and computations it is important that all modules participating in extended T -systems of types A and B are thin and special, moreover their q-characters are known explicitly in terms of skew Young tableaux in type A, and in terms of path models in type B, see [6, 18, 19, 22]. In general the modules of the extended T -system of type G2 are not thin and at the moment there is no combinatorial description of their q-characters. However, all modules turn out to be either special or anti-special. Therefore we are able to use the FM algorithm, see [8], to compute the sufficient information about q-characters in order to complete the proofs. Note, that a priori it is was not obvious that the extended T -system will be closed within special or anti-special modules. Moreover, since the q-characters of G2 modules are not known explicitly, the property of being special or anti-special had to be established in each case, see Theorems 3.3, 7.2. Note that in general the minimal affinizations of types C, D, E, F are neither special nor anti-special, therefore the methods of this paper cannot be applied in those cases. There is a remarkable conjecture on the cluster algebra relations in the category of finite- dimensional representations of quantum affine algebras of type A, D, E, see [12]. Taking into account the work of [13, 14], one could expect that the conjecture of [12] can be formulated for other types as well, in particular for type G2. We expect that the extended T -system is a part of cluster algebra relations. The paper is organized as follows. In Section 2, we give some background material. In Sec- tion 3, we define the modules B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` and state our main result, Theorem 3.4. In Section 4, we prove that the modules B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` are special. In Section 5, we prove Theorem 3.4. In Section 6, we prove that the module T ⊗B is irreducible for each relation in the extended T -system. In Section 7, we deduce the extended T -system for the modules B̃(s) k,`, C̃(s) k,` , D̃ (s) k,`, Ẽ (s) k,` , F̃ (s) k,` . In Section 8, we compute the dimensions of the modules in the extended T -systems. Extended T -System of Type G2 3 2 Background 2.1 Cartan data Let g be a complex simple Lie algebra of type G2 and h a Cartan subalgebra of g. Let I = {1, 2}. We choose simple roots α1, α2 and scalar product (·, ·) such that (α1, α1) = 2, (α1, α2) = −3, (α2, α2) = 6. Let {α∨1 , α∨2 } and {ω1, ω2} be the sets of simple coroots and fundamental weights respectively. Let C = (Cij)i,j∈I denote the Cartan matrix, where Cij = 2(αi,αj) (αi,αi) . Let r1 = 1, r2 = 3, D = diag(r1, r2) and B = DC. Then C = ( 2 −3 −1 2 ) , B = ( 2 −3 −3 6 ) . Let Q (resp. Q+) and P (resp. P+) denote the Z-span (resp. Z≥0-span) of the simple roots and fundamental weights respectively. Let ≤ be the partial order on P in which λ ≤ λ′ if and only if λ′ − λ ∈ Q+. Let ĝ denote the untwisted affine algebra corresponding to g. Fix a q ∈ C×, not a root of unity. Let qi = qri , i = 1, 2. Define the q-numbers, q-factorial and q-binomial: [n]q := qn − q−n q − q−1 , [n]q! := [n]q[n− 1]q · · · [1]q, [ n m ] q := [n]q! [n−m]q![m]q! . 2.2 Quantum affine algebra The quantum affine algebra Uqĝ in Drinfeld’s new realization, see [7], is generated by x±i,n (i ∈ I, n ∈ Z), k±1 i (i ∈ I), hi,n (i ∈ I, n ∈ Z\{0}) and central elements c±1/2, subject to the following relations: kikj = kjki, kihj,n = hj,nki, kik −1 i = k−1 i ki = 1, kix ± j,nk −1 i = q±Bijx±j,n,[ hi,n, x ± j,m ] = ± 1 n [nBij ]qc ∓\|n\|/2x±j,n+m, x±i,n+1x ± j,m − q ±Bijx±j,mx ± i,n+1 = q±Bijx±i,nx ± j,m+1 − x ± j,m+1x ± i,n, [hi,n, hj,m] = δn,−m 1 n [nBij ]q cn − c−n q − q−1 , [x+ i,n, x − j,m] = δij c(n−m)/2φ+ i,n+m − c−(n−m)/2φ−i,n+m qi − q−1 i , ∑ π∈Σs s∑ k=0 (−1)k [ s k ] qi x±i,nπ(1) · · ·x ± i,nπ(k) x±j,mx ± i,nπ(k+1) · · ·x±i,nπ(s) = 0, s = 1− Cij , for all sequences of integers n1, . . . , ns, and i 6= j, where Σs is the symmetric groups on s letters, φ±i,n = 0 (n < 0) and φ±i,n’s (n ≥ 0) are determined by the formula φ±i (u) := ∞∑ n=0 φ±i,±nu ±n = k±1 i exp ( ± ( q − q−1 ) ∞∑ m=1 hi,±mu ±m ) . (2.1) There exist a coproduct, counit and antipode making Uqĝ into a Hopf algebra. 4 J.R. Li and E. Mukhin The quantum affine algebra Uqĝ contains two standard quantum affine algebras of type A1. The first one is Uq1(ŝl2) generated by x±1,n (n ∈ Z), k±1 1 , h1,n (n ∈ Z\{0}) and central ele- ments c±1/2. The second one is Uq2(ŝl2) generated by x±2,n (n ∈ Z), k±1 2 , h2,n (n ∈ Z\{0}) and central elements c±1/2. The subalgebra of Uqĝ generated by (k±i )i∈I , (x±i,0)i∈I is a Hopf subalgebra of Uqĝ and is isomorphic as a Hopf algebra to Uqg, the quantized enveloping algebra of g. In this way, Uqĝ- modules restrict to Uqg-modules. 2.3 Finite-dimensional representations and q-characters In this section, we recall the standard facts about finite-dimensional representations of Uqĝ and q-characters of these representations, see [2, 4, 8, 9, 18]. A representation V of Uqĝ is of type 1 if c±1/2 acts as the identity on V and V = ⊕ λ∈P Vλ, Vλ = { v ∈ V : kiv = q(αi,λ)v } . (2.2) In the following, all representations will be assumed to be finite-dimensional and of type 1 without further comment. The decomposition (2.2) of a finite-dimensional representation V into its Uqg-weight spaces can be refined by decomposing it into the Jordan subspaces of the mutually commuting operators φ±i,±r, see [9]: V = ⊕ γ Vγ , γ = ( γ±i,±r ) i∈I, r∈Z≥0 , γ±i,±r ∈ C, (2.3) where Vγ = { v ∈ V : ∃ k ∈ N,∀ i ∈ I,m ≥ 0, ( φ±i,±m − γ ± i,±m )k v = 0 } . If dim(Vγ) > 0, then γ is called an l-weight of V . For every finite dimensional representation of Uqĝ, the l-weights are known, see [9], to be of the form γ±i (u) := ∞∑ r=0 γ±i,±ru ±r = qdegQi−degRi i Qi(uq −1 i )Ri(uqi) Qi(uqi)Ri(uq −1 i ) , (2.4) where the right hand side is to be treated as a formal series in positive (resp. negative) integer powers of u, and Qi, Ri are polynomials of the form Qi(u) = ∏ a∈C× (1− ua)wi,a , Ri(u) = ∏ a∈C× (1− ua)xi,a , (2.5) for some wi,a, xi,a ∈ Z≥0, i ∈ I, a ∈ C×. Let P denote the free abelian multiplicative group of monomials in infinitely many formal variables (Yi,a)i∈I, a∈C× . There is a bijection γ from P to the set of l-weights of finite-dimensional modules such that for the monomial m =∏ i∈I, a∈C× Y wi,a−xi,a i,a , the l-weight γ(m) is given by (2.4), (2.5). Let ZP = Z [ Y ±1 i,a ] i∈I, a∈C× be the group ring of P. For χ ∈ ZP, we write m ∈ P if the coefficient of m in χ is non-zero. The q-character of a Uqĝ-module V is given by χq(V ) = ∑ m∈P dim(Vm)m ∈ ZP, where Vm = Vγ(m). Extended T -System of Type G2 5 Let Rep(Uqĝ) be the Grothendieck ring of finite-dimensional representations of Uqĝ and [V ] ∈ Rep(Uqĝ) the class of a finite-dimensional Uqĝ-module V . The q-character map defines an injective ring homomorphism, see [9], χq : Rep(Uqĝ)→ ZP. For any finite-dimensional representation V of Uqĝ, denote by M (V ) the set of all monomials in χq(V ). For each j ∈ I, a monomial m = ∏ i∈I, a∈C× Y ui,a i,a , where ui,a are some integers, is said to be j-dominant (resp. j-anti-dominant) if and only if uj,a ≥ 0 (resp. uj,a ≤ 0) for all a ∈ C×. A monomial is called dominant (resp. anti-dominant) if and only if it is j-dominant (resp. j-anti-dominant) for all j ∈ I. Let P+ ⊂ P denote the set of all dominant monomials. Let V be a representation of Uqĝ and m ∈M (V ) a monomial. A non-zero vector v ∈ Vm is called a highest l-weight vector with highest l-weight γ(m) if x+ i,r · v = 0, φ±i,±t · v = γ(m)±i,±tv, ∀ i ∈ I, r ∈ Z, t ∈ Z≥0. The module V is called a highest l-weight representation if V = Uqĝ ·v for some highest l-weight vector v ∈ V . It is known, see [2, 4], that for each m+ ∈ P+ there is a unique finite-dimensional irreducible representation, denoted L(m+), of Uqĝ that is highest l-weight representation with highest l- weight γ(m+), and moreover every finite-dimensional irreducible Uqĝ-module is of this form for some m+ ∈ P+. Also, if m+, m′+ ∈ P+ and m+ 6= m′+, then L(m+) 6∼= L(m′+). For m+ ∈ P+, we use χq(m+) to denote χq(L(m+)). The following lemma is well-known. Lemma 2.1. Let m1, m2 be two monomials. Then L(m1m2) is a sub-quotient of L(m1)⊗L(m2). In particular, M (L(m1m2)) ⊆M (L(m1))M (L(m2)). For b ∈ C×, define the shift of spectral parameter map τb : ZP → ZP to be a homomorphism of rings sending Y ±1 i,a to Y ±1 i,ab. Let m1, m2 ∈ P+. If τb(m1) = m2, then τbχq(m1) = χq(m2). (2.6) Let m+ be a dominant l-weight. We call the polynomial χq(m+) special if it contains exactly one dominant monomial. A finite-dimensional Uqĝ-module V is said to be special if and only if M (V ) contains exactly one dominant monomial. It is called anti-special if and only if M (V ) contains exactly one anti- dominant monomial. It is called thin if and only if no l-weight space of V has dimension greater than 1. We also call a polynomial in ZP special, antispecial, or thin if this polynomial contains a unique dominant monomial, a unique anti-dominant monomial, or if all coefficients are zero and one respectively. A finite-dimensional Uqĝ-module is said to be prime if and only if it is not isomorphic to a tensor product of two non-trivial Uqĝ-modules, see [5]. Clearly, if a module is special or anti-special, then it is irreducible. Define Ai,a ∈ P, i ∈ I, a ∈ C×, by A1,a = Y1,aqY1,aq−1Y −1 2,a , A2,a = Y2,aq3Y2,aq−3Y −1 1,aq−2Y −1 1,a Y −1 1,aq2 . Let Q be the subgroup of P generated by Ai,a, i ∈ I, a ∈ C×. Let Q± be the monoids generated by A±1 i,a , i ∈ I, a ∈ C×. There is a partial order ≤ on P in which m ≤ m′ if and only if m′m−1 ∈ Q+. (2.7) For all m+ ∈ P+, M (L(m+)) ⊂ m+Q−, see [8]. 6 J.R. Li and E. Mukhin A monomial m is called right negative if and only if ∀ a ∈ C× and ∀ i ∈ I, we have the following property: if the power of Yi,a is non-zero and the power of Yj,aqk is zero for all j ∈ I, k ∈ Z>0, then the power of Yi,a is negative. For i ∈ I, a ∈ C×, A−1 i,a is right-negative. A product of right-negative monomials is right-negative. If m is right-negative, then m′ ≤ m implies that m′ is right-negative. 2.4 Minimal affinizations of Uqg-modules Let λ = kω1 + `ω2. A simple Uqĝ-module L(m+) is called a minimal affinization of V (λ) if and only if m+ is one of the following monomials( `−1∏ i=0 Y2,aq6i )( k−1∏ i=0 Y1,aq6`+2i+1 ) , ( k−1∏ i=0 Y1,aq2i )( `−1∏ i=0 Y2,aq2k+6i+5 ) , for some a ∈ C×, see [3]. In particular, when k = 0 or ` = 0, the minimal affinization L(m+) is called a Kirillov–Reshetikhin module. Let L(m+) be a Kirillov–Reshetikhin module. It is shown in [10] that any non-highest monomial in M (L(m+)) is right-negative and in particular L(m+) is special. 2.5 q-characters of Uqŝl2-modules and the FM algorithm The q-characters of Uq ŝl2-modules are well-understood, see [1, 9]. We recall the results here. Let W (a) k be the irreducible representation Uq ŝl2 with highest weight monomial X (a) k = k−1∏ i=0 Yaqk−2i−1 , where Ya = Y1,a. Then the q-character of W (a) k is given by χq ( W (a) k ) = X (a) k k∑ i=0 i−1∏ j=0 A−1 aqk−2j , where Aa = Yaq−1Yaq. For a ∈ C×, k ∈ Z≥1, the set Σ (a) k = {aqk−2i−1}i=0,...,k−1 is called a string. Two strings Σ (a) k and Σ (a′) k′ are said to be in general position if the union Σ (a) k ∪Σ (a′) k′ is not a string or Σ (a) k ⊂ Σ (a′) k′ or Σ (a′) k′ ⊂ Σ (a) k . Denote by L(m+) the irreducible Uq ŝl2-module with highest weight monomial m+. Let m+ 6= 1 and ∈ Z[Ya]a∈C× be a dominant monomial. Then m+ can be uniquely (up to permuta- tion) written in the form m+ = s∏ i=1  ∏ b∈Σ (ai) ki Yb  , where s is an integer, Σ (ai) ki , i = 1, . . . , s, are strings which are pairwise in general position and L(m+) = s⊗ i=1 W (ai) ki , χq(m+) = s∏ i=1 χq ( W (ai) ki ) . Extended T -System of Type G2 7 For j ∈ I, let βj : Z [ Y ±1 i,a ] i∈I;a∈C× → Z [ Y ±1 a ] a∈C× be the ring homomorphism which sends, for all a ∈ C×, Yk,a 7→ 1 for k 6= j and Yj,a 7→ Ya. Let V be a Uqĝ-module. Then βi(χq(V )), i = 1, 2, is the q-character of V considered as a Uqi(ŝl2)-module. In some situation, we can use the q-characters of Uq ŝl2-modules to compute the q-characters of Uqĝ-modules for arbitrary g, see [8, Section 5]. The corresponding algorithm is called the FM algorithm. The FM algorithm recursively computes the minimal possible q-character which contains m+ and is consistent when restricted to Uqi(ŝl2), i = 1, 2. Although the FM algorithm does not give the q-character of a Uqĝ-module in general, the FM algorithm works for a large family of Uqĝ-modules. For example, if a module L(m+) is special, then the FM algorithm applied to m+, produces the correct q-character χq(m+), see [8]. 2.6 Truncated q-characters We use the truncated q-characters [12, 18]. Given a set of monomials R ⊂ P, let ZR ⊂ ZP denote the Z-module of formal linear combinations of elements of R with integer coefficients. Define truncR : P → R; m 7→ { m if m ∈ R, 0 if m 6∈ R, and extend truncR as a Z-module map ZP → ZR. Given a subset U ⊂ I ×C×, let QU be the subgroups of Q generated by Ai,a with (i, a) ∈ U . Let Q±U be the monoid generated by A±1 i,a with (i, a) ∈ U . We call truncm+Q−U χq(m+) the q-character of L(m+) truncated to U . If U = I × C×, then truncm+Q−U χq(m+) is the ordinary q-character of L(m+). The main idea of using the truncated q-characters is the following. Given m+, one chooses R in such a way that the dropped monomials are all right-negative and the truncated q-character is much smaller than the full q-character. The advantage is that the truncated q-character is much easier to compute and to describe in a combinatorial way. At the same time, if the truncating set R can be used for both m+ 1 and m+ 2 , then the same R works for the tensor product L(m+ 1 )⊗ L(m+ 2 ). Moreover, the product of truncated characters of L(m+ 1 ) and L(m+ 2 ) contains all dominant monomials of the tensor product L(m+ 1 )⊗L(m+ 2 ) and can be used to find the decomposition of it into irreducible components in the Grothendieck ring. We compute the truncated q-characters using the following theorem. Theorem 2.2 ([18, Theorem 2.1]). Let U ⊂ I × C× and m+ ∈ P+. Suppose that M ⊂ P is a finite set of distinct monomials such that (i) M⊂ m+Q−U , (ii) P+ ∩M = {m+}, (iii) for all m ∈M and all (i, a) ∈ U , if mA−1 i,a 6∈ M, then mA−1 i,aAj,b 6∈ M unless (j, b) = (i, a), (iv) for all m ∈ M and all i ∈ I, there exists a unique i-dominant monomial M ∈ M such that truncβi(MQ−U ) χq(βi(M)) = ∑ m′∈mQ{i}×C× βi(m ′). 8 J.R. Li and E. Mukhin Then truncm+Q−U χq(m+) = ∑ m∈M m. Here by χq(βi(M)) we mean the q-character of the irreducible Uqi(ŝl2)-module with highest weight monomial βi(M) and by truncβi(MQ − U ) we mean keeping only the monomials of χq(βi(M)) in the set βi(MQ−U ). 3 Main results 3.1 First examples Without loss of generality, we fix a ∈ C× and consider modules V with M (V ) ⊂ Z[Yi,aqk ]i∈I, k∈Z. In the following, for simplicity we write is, i −1 s (s ∈ Z) instead of Yi,aqs , Y −1 i,aqs respectively. The q-characters of fundamental modules are easy to compute by using the FM algorithm. Lemma 3.1. The fundamental q-characters for Uqĝ of type G2 are given by χq(10) = 10 + 1−1 2 21 + 14162−1 7 + 141−1 8 + 1−1 6 1−1 8 25 + 1102−1 11 + 1−1 12 , χq(20) = 20 + 1113152−1 6 + 11131−1 7 + 111−1 5 1−1 7 24 + 1−1 3 1−1 5 1−1 7 2224 + 11192−1 10 + 242−1 8 + 1−1 3 19222−1 10 + 1517192−1 8 2−1 10 + 111−1 11 + 1−1 3 1−1 11 22 + 15171−1 11 2−1 8 + 151−1 9 1−1 11 + 1−1 7 1−1 9 1−1 11 26 + 2−1 12 . For s ∈ Z, χq(1s) and χq(2s) are obtained by shift all indices by s in χq(10) and χq(20) respectively. It is convenient to keep in mind the following lemma. Lemma 3.2. If b ∈ Z\{±2,±8,±12}, then L(101b) = L(10)⊗ L(1b), dimL(101b) = 49. If b ∈ Z\{±6,±8,±10,±12}, then L(202b) = L(20)⊗ L(2b), dimL(202b) = 225. If b ∈ Z\{±7,±11}, then L(102b) = L(10)⊗ L(2b), L(201b) = L(20)⊗ L(1b), dimL(102b) = dimL(201b) = 105. In addition, we have dimL(1012) = 34, dimL(1018) = 42, dimL(10112) = 48, dimL(2026) = 92, dimL(2028) = 210, dimL(20210) = 183, dimL(20212) = 224, dimL(1027) = dimL(2017) = 71, dimL(10211) = dimL(20111) = 98. Proof. By Lemma 3.1, the tensor products in the first three cases of the lemma are special. Therefore the tensor products are irreducible. Hence the first three cases of the lemma are true. The last part of the lemma can be proved using the methods of Section 5. In fact some of the dimensions follow from Theorem 8.1. We do not use this lemma in the proofs. Therefore we omit the details of the proof. � Extended T -System of Type G2 9 3.2 Definition of the modules B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` For s ∈ Z, k, ` ∈ Z≥0, define the following monomials. B (s) k,` = ( k−1∏ i=0 2s+6i )( `−1∏ i=0 1s+6k+2i+1 ) , C (s) k,` = ( k−1∏ i=0 2s+6i )( `−1∏ i=0 2s+6k+6i+4 ) , D (s) k,` = ( k−1∏ i=0 2s+6i ) 1s+6k+1 ( `−1∏ i=0 2s+6k+6i+8 ) , F (s) k,` = ( k−1∏ i=0 1s+2i )( `−1∏ i=0 1s+2k+2i+6 ) , E (s) k,` = ( k−1∏ i=0 1s+2i )b `−1 2 c∏ i=0 2s+2k+6i+3  b `−2 2 c∏ i=0 2s+2k+6i+5  . Note that, in particular, for k ∈ Z≥0, s ∈ Z, we have the following trivial relations B(s) k,0 = C(s) k,0 = C(s−4) 0,k , D(s) k,0 = B(s) k,1, E(s) k,0 = B(s−1) 0,k = F (s−6) 0,k = F (s) k,0. (3.1) Denote by B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` the irreducible finite-dimensional highest l-weight Uqĝ- modules with highest l-weight B (s) k,` , C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` respectively. Note that B(s) k,`, D (s) 0,` , D (s) k,0 are minimal affinizations. The modules B(s) 0,` , C (s) 0,` , F (s) 0,` , B(s) k,0, C(s) k,0, E(s) k,0, F (s) k,0 are Kirillov–Reshetikhin modules. Our first result is Theorem 3.3. The modules B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` , s ∈ Z, k, ` ∈ Z≥0, are special. In particular, the FM algorithm works for all these modules. We prove Theorem 3.3 in Section 4. Note that the case of B(s) k,` has been proved in Theorem 3.8 of [11]. In general, the modules in Theorem 3.3 are not thin. For example, χq(1012) has monomial 14161−1 8 1−1 10 with coefficient 2. 3.3 Extended T -system It is known that Kirillov–Reshetikhin modules B(s) k,0, B(s) 0,` satisfy the following T -system relations, see [16],[ B(s) 0,` ][ B(s+2) 0,` ] = [ B(s) 0,`+1 ][ B(s+2) 0,`−1 ] + [ B(s+1) b `+2 3 c,0 ][ B(s+3) b `+1 3 c,0 ][ B(s+5) b `3c,0 ] , (3.2)[ B(s) k,0 ][ B(s+6) k,0 ] = [ B(s) k+1,0 ][ B(s+6) k−1,0 ] + [ B(s+1) 0,3k ] , (3.3) where s ∈ Z, k, ` ∈ Z≥1. Our main result is Theorem 3.4. For s ∈ Z and k, ` ∈ Z≥1, t ∈ Z≥2, we have the following relations in Rep(Uqĝ):[ B(s) k,`−1 ][ B(s+6) k−1,` ] = [ B(s) k,` ][ B(s+6) k−1,`−1 ] + [ E(s+1) 3k−1,d 2`−2 3 e ][ B(s+6k+6) b `−1 3 c,0 ] , (3.4)[ E(s) 0,` ] = [ B(s+3) b `+1 2 c,0 ][ B(s+5) b `2c,0 ] , (3.5)[ E(s) 1,` ] = [ D(s−1) 0,b `2c ][ B(s+5) b `+1 2 c,0 ] , (3.6)[ E(s) t,`−1 ][ E(s+2) t−1,` ] = [ E(s) t,` ][ E(s+2) t−1,`−1 ] 10 J.R. Li and E. Mukhin +  [ D(s+1) r,p−1 ][ B(s+3) r+p,0 ][ B(s+5) r,3p−2 ] , if t = 3r + 2, ` = 2p− 1,[ B(s+1) r+p+1,0 ][ C(s+3) r,p ][ B(s+5) r,3p−1 ] , if t = 3r + 2, ` = 2p,[ B(s+1) r+1,3p−2 ][ D(s+3) r,p−1 ][ B(s+5) r+p,0 ] , if t = 3r + 3, ` = 2p− 1,[ B(s+1) r+1,3p−1 ][ B(s+3) r+p+1,0 ][ C(s+5) r,p ] , if t = 3r + 3, ` = 2p,[ B(s+1) r+p+1,0 ][ B(s+3) r+1,3p−2 ][ D(s+5) r,p−1 ] , if t = 3r + 4, ` = 2p− 1,[ C(s+1) r+1,p ][ B(s+3) r+1,3p−1 ][ B(s+5) r+p+1,0 ] , if t = 3r + 4, ` = 2p, (3.7) [ C(s) k,`−1 ][ C(s+6) k−1,` ] = [ C(s) k,` ][ C(s+6) k−1,`−1 ] + [ F (s+1) 3k−2,3`−2 ] , (3.8)[ D(s) 0,`−1 ][ B(s+8) `,0 ] = [ D(s) 0,` ][ B(s+8) `−1,0 ] + [ B(s+4) 0,3`−1 ] , (3.9)[ D(s) k,`−1 ][ D(s+6) k−1,` ] = [ D(s) k,` ][ D(s+6) k−1,`−1 ] + [ F (s+1) 3k−1,3`−1 ] , (3.10)[ F (s) k,`−1 ][ F (s+2) k−1,` ] = [ F (s) k,` ][ F (s+2) k−1,`−1 ] (3.11) +  [ B(s+1) r,0 ][ D(s+3) r,b `3c ][ C(s+5) r,b `+1 3 c ][ B(s+2k+11) b `−1 3 c,0 ] , if k = 3r + 1,[ C(s+1) r+1,b `+1 3 c ][ B(s+3) r,0 ][ D(s+5) r,b `3c ][ B(s+2k+11) b `−1 3 c,0 ] , if k = 3r + 2,[ D(s+1) r+1,b `3c ][ C(s+3) r+1,b `+1 3 c ][ B(s+5) r,0 ][ B(s+2k+11) b `−1 3 c,0 ] , if k = 3r + 3. We prove Theorem 3.4 in Section 5. Note that since D (s) k,0 = B (s) k,1, equations for D(s) k,0 are included in the equations for B(s) k,1. All relations except (3.5), (3.6) in Theorem 3.4 are written in the form [L][R] = [T ][B] + [S], where L, R, T , B are irreducible modules which we call left, right, top and bottom modules and S is a tensor product of some irreducible modules. We call the factors of S sources. Moreover, we have the following theorem. Theorem 3.5. For each relation in Theorem 3.4, all summands on the right hand side, T ⊗ B and S, are irreducible. We will prove Theorem 3.5 in Section 6. Recall that the q-characters of modules for different s are related by the simple shift of indexes, see (2.6). We have the following proposition. Proposition 3.6. Given χq(1s), χq(2s), one can obtain the q-characters of B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` , s ∈ Z, k, ` ∈ Z≥0, recursively, by using (3.1), and computing the q-character of the top module through the q-characters of other modules in relations in Theorem 3.4. Proof. Claim 1. Let n, m be positive integers. Then the q-characters χq ( B(s) k,` ) , k ≤ n, ` ≤ m, χq ( C(s) k,` ) , k ≤ n− 1, ` ≤ ⌈ 2m+ 1 6 ⌉ , χq ( D(s) k,` ) , k ≤ n− 1, ` ≤ ⌈ 2m+ 1 6 ⌉ , χq ( E(s) k,` ) , k ≤ 3n− 1, ` ≤ ⌈ 2m− 2 3 ⌉ , χq ( F (s) k,` ) , k ≤ 3n− 4, ` ≤ m+ 2, can be computed recursively starting from χq(10), χq(20). Extended T -System of Type G2 11 We use induction on n, m to prove Claim 1. For simplicity, we do not write the upper- subscripts “(s)” in the remaining part of the proof. We know that, see [10], the q-characters of Kirillov–Reshetikhin modules can be computed from χq(10), χq(20). When n = 0, m = 1, Claim 1 is clearly true. It is clear that χq(D0,1) can be computed using (3.9). Therefore Claim 1 holds for n = 1, m = 0, Suppose that for n ≤ n1 and m ≤ m1, Claim 1 is true. Let n = n1 + 1, m = m1. We need to show that Claim 1 is true. Then we need to show that χq(Bn1+1,`), ` ≤ m1, χq(Cn1,`), ` ≤ ⌈ 2m1 + 1 6 ⌉ , χq(Dn1,`), ` ≤ ⌈ 2m1 + 1 6 ⌉ , χq(Ek,`), k = 3n1, 3n1 + 1, 3n1 + 2, ` ≤ ⌈ 2m1 − 2 3 ⌉ , χq(Fk,`), k = 3n1 − 3, 3n1 − 2, 3n1 − 1, ` ≤ m1 + 2, can be computed. We compute the following modules χq(F3n1−3,`), ` ≤ m1 + 2, χq(F3n1−2,`), ` ≤ m1 + 2, χq(Cn1,`), ` ≤ ⌊ m1 + 3 3 ⌋ , χq(F3n1−1,`), ` ≤ m1 + 2, χq(Dn1,`), ` ≤ ⌈ 2m1 + 1 6 ⌉ , χq(Cn1,`), ` ≤ ⌈ 2m1 + 1 6 ⌉ , χq(E3n1,`), ` ≤ ⌈ 2m1 − 2 3 ⌉ , χq(E3n1,`), ` ≤ ⌈ 2m1 − 2 3 ⌉ , χq(E3n1+1,`), ` ≤ ⌈ 2m1 − 2 3 ⌉ , χq(E3n1+2,`), ` ≤ ⌈ 2m1 − 2 3 ⌉ , χq(Bn1+1,`), ` ≤ m1 in the order as shown. At each step, we consider the module that we want to compute as a top module and use the corresponding relation in Theorem 3.4 and known q-characters. For example, we consider the first set of modules χq(F3n1−3,`), ` ≤ m1+2. Since ⌊ m1+3 3 ⌋ ≤ ⌈ 2m1+1 6 ⌉ , χq(Cn1−1,`), ` ≤ ⌊ m1+3 3 ⌋ , is known by induction hypothesis. Similarly, χq(Dn1−1,`), ` ≤ ⌊ m1+2 3 ⌋ is known. Therefore χq(F3n1−3,`), ` ≤ m1 + 2, is computed using the last equation of (3.11). Similarly, we show that Claim 1 holds for n = n1, m = m1 + 1. Therefore Claim 1 is true for all n ≥ 1, m ≥ 1. � 4 Proof of Theorem 3.3 In this section, we will show that the modules B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` are special. Since B(s) 0,` , C (s) 0,` , F (s) 0,` , B(s) k,0, C(s) k,0, E(s) k,0, F (s) k,0 are Kirillov–Reshetikhin modules, they are special. 4.1 The case of C(s) k,` Let m+ = C (s) k,` with k, ` ∈ Z≥1. Without loss of generality, we can assume that s = 6. Then m+ = (26212 · · · 26k)(26k+1026k+16 · · · 26k+6`+4). Case 1. k = 1. Let U = I × {aqs : s ∈ Z, s < 6`+ 13}. Clearly, all monomials in χq(m+)− truncm+Q−U χq(m+) are right-negative. Therefore it is sufficient to show that truncm+Q−U χq(m+) is special. 12 J.R. Li and E. Mukhin Let M be the finite set consisting of the following monomials m0 = m+, m1 = m0A −1 2,9, m2 = m1A −1 1,12, m3 = m2A −1 1,10, m4 = m3A −1 1,8, m5 = m4A −1 2,11. It is clear that M satisfies the conditions in Theorem 2.2. Therefore truncm+Q−U χq(m+) = ∑ m∈M m and truncm+Q−U χq(m+) is special. Case 2. k > 1. Since the conditions of Theorem 2.2 do not apply to this case, we use another technique to show that L(m+) is special. We embed L(m+) into two different tensor products. In both tensor products, each factor is special. Therefore we can use the FM algorithm to compute the q-characters of the factors. We classify the dominant monomials in the first tensor product and show that the only dominant monomial in the first tensor product which occurs in the second tensor product is m+ which proves that L(m+) is special. The first tensor product is L(m′1)⊗ L(m′2), where m′1 = 26212 · · · 26k, m′2 = 26k+1026k+16 · · · 26k+6`+4. We use the FM algorithm to compute χq(m ′ 1), χq(m ′ 2) and classify all dominant monomials in χq(m ′ 1)χq(m ′ 2). Let m = m1m2 be a dominant monomial, where mi ∈ χq(m′i), i = 1, 2. If m2 6= m′2, then m is a right negative monomial therefore m is not dominant. Hence m2 = m′2. If m1 6= m′1, then m1 is right negative. Since m is dominant, each factor with a neg- ative power in m1 needs to be canceled by a factor in m′2. All possible cancellations can- cel 26k+10 in m′2. We have M(L(m′1)) ⊂ M(χq(26212 · · · 26k−6)χq(26k)). Only monomials in χq(26k) can cancel 26k+10. These monomials are 16k+116k+92−1 6k+10, 1−1 6k+316k+926k+22−1 6k+10, and 16k+516k+716k+92−1 6k+82−1 6k+10. Therefore m1 is in one of the following polynomials χq(26212 · · · 26k−6)16k+116k+92−1 6k+10, (4.1) χq(26212 · · · 26k−6)1−1 6k+316k+926k+22−1 6k+10, (4.2) χq(26212 · · · 26k−6)16k+516k+716k+92−1 6k+82−1 6k+10. (4.3) Subcase 2.1. Let m1 be in (4.1). If m1 = 26212 · · · 26k−616k+116k+92−1 6k+10, then m = m1m2 = 26212 · · · 26k−616k+116k+926k+16 · · · 26k+6`+4 (4.4) is dominant. Suppose that m1 6= 26212 · · · 26k−616k+116k+92−1 6k+10. Then m1 = n116k+116k+92−1 6k+10, where n1 is a non-highest monomial in χq(26212 · · · 26k−6). Since n1 is right negative, 16k+1 or 16k+9 should cancel a factor of n1 with a negative power. Using the FM algorithm, we see that there exists a factor 12 6k−1 or 12 6k+7 in a monomial in χq(26212 · · · 26k−6)16k+116k+92−1 6k+10. By Lemma 3.1, neither 12 6k−1 nor 12 6k+7 appear. This is a contradiction. Subcase 2.2. Let m1 be in (4.2). If m1 = 26212 · · · 26k−61−1 6k+316k+926k+22−1 6k+10, then m = m1m2 is not dominant. Suppose that m1 6= 26212 · · · 26k−61−1 6k+316k+926k+22−1 6k+10. Then m1 = n11−1 6k+316k+926k+22−1 6k+10, where n1 is a non-highest monomial in χq(26212 · · · 26k−6). Since n1 is right negative, 16k+9 or 26k+2 should cancel a factor of n1 with a negative power. Using the Extended T -System of Type G2 13 FM algorithm, we see that there exists either a factor 12 6k+7 or a factor 22 6k−4 in a monomial in χq(26212 · · · 26k−6)16k+116k+92−1 6k+10. By Lemma 3.1, neither 12 6k+7 nor 22 6k−4 appear. This is a contradiction. Subcase 2.3. Let m1 be in (4.3). If m1 = 26212 · · · 26k−616k+516k+716k+92−1 6k+82−1 6k+10, then m = m1m2 is not dominant. Suppose that m1 6= 26212 · · · 26k−616k+516k+716k+92−1 6k+82−1 6k+10. Then we have m1 = n116k+516k+716k+92−1 6k+82−1 6k+10, where n1 is a non-highest monomial in χq(26212 · · · 26k−6). Since n1 is right negative, 16k+5 or 16k+7 or 16k+9 should cancel a factor of n1 with a negative power. Using the FM algorithm, we see that there exists a factor 16k+7 or 16k+5 or 12 6k+3 in a monomial in χq(26212 · · · 26k−6)16k+116k+92−1 6k+10. By Lemma 3.1, 16k+7, 16k+5, and 12 6k+3 do not appear. This is a contradiction. Therefore the only dominant monomials in χq(m ′ 1)χq(m ′ 2) are m+ and (4.4). The second tensor product is L(m′′1)⊗ L(m′′2), where m′′1 = 26212 · · · 26k−6, m′′2 = 26k26k+1026k+16 · · · 26k+6`+4. The monomial (4.4) is n = m+A −1 2,6k+3A −1 1,6k+6A −1 1,6k+4A −1 2,6k+7. (4.5) Since Ai,a, i ∈ I, a ∈ C× are algebraically independent, the expression (4.5) of n of the form m+ ∏ i∈I, a∈C× A −vi,a i,a , where vi,a are some integers, is unique. Suppose that the monomial n is in χq(m ′′ 1)χq(m ′′ 2). Then n = n1n2, where ni ∈ χq(m′′i ), i = 1, 2. By the expression (4.5), we have n1 = m′′1 and n2 = m′′2A −1 2,6k+3A −1 1,6k+6A −1 1,6k+4A −1 2,6k+7. By the FM algorithm, the monomial m′′2A −1 2,6k+3A −1 1,6k+6A −1 1,6k+4A −1 2,6k+7 is not in χq(m ′′ 2). This contradicts the fact that n2 ∈ χq(m′′2). Therefore n is not in χq(m ′′ 1)χq(m ′′ 2). 4.2 The case of B(s) k,` Let m+ = B (s) k,` with k, ` ∈ Z≥1. Without loss of generality, we can assume that s = 6. Then m+ = (26212 · · · 26k)(16k+716k+9 · · · 16k+2`+5). Let U = I × {aqs : s ∈ Z, s < 6k + 2`+ 6}. Clearly, all monomials in the polynomial χq(m+)− truncm+Q−U χq(m+) are right-negative. Therefore it is sufficient to show that the truncated q-character truncm+Q−U χq(m+) is special. Let M be the finite set consisting of the following monomials m0 = m+, m1 = m0A −1 2,6k+3, m2 = m1A −1 2,6k−3, . . . , mk = mk−1A −1 2,9. It is clear that M satisfies the conditions in Theorem 2.2. Therefore truncm+Q−U χq(m+) = ∑ m∈M m and truncm+Q−U χq(m+) is special. 14 J.R. Li and E. Mukhin 4.3 The case of D(s) k,` Let m+ = D (s) k,` with k, ` ∈ Z≥0. Without loss of generality, we can assume that s = 0. Then m+ = (2026 · · · 26k−6)16k+1(26k+826k+14 · · · 26k+6`+2). Case 1. k = 0. Let U = I × {aqs : s ∈ Z, s < 6` + 5}. Clearly, all monomials in χq(m+) − truncm+Q−U χq(m+) are right-negative. Therefore it is sufficient to show that truncm+Q−U χq(m+) is special. Let M = { m+,m+A −1 1,2 } . It is clear that M satisfies the conditions in Theorem 2.2. Therefore truncm+Q−U χq(m+) = ∑ m∈M m and truncm+Q−U χq(m+) is special. Case 2. k > 0. Let m′1 = 2026 · · · 26k−616k+1, m′2 = 26k+826k+14 · · · 26k+6`+2, m′′1 = 2026 · · · 26k−6, m′′2 = 16k+126k+826k+14 · · · 26k+6`+2. Then M (L(m+)) ⊂M (χq(m ′ 1)χq(m ′ 2)) ∩M (χq(m ′′ 1)χq(m ′′ 2)). By using similar arguments as the case of C(s) k,` , we show that the only dominant monomials in χq(m ′ 1)χq(m ′ 2) are m+ and n = 2026 · · · 26k−616k+516k+726k+1426k+20 · · · 26k+6`+2 = m+A −1 1,6k+2A −1 2,6k+5. Moreover, n is not in χq(m ′′ 1)χq(m ′′ 2). Therefore the only dominant monomial in χq(m+) is m+. 4.4 The case of E(s) k,` Let m+ = E (s) k,` with k, ` ∈ Z≥0. Without loss of generality, we can assume that s = 1. Suppose that ` = 2r + 1, r ≥ 0 and k = 3p, p ≥ 1. The cases of ` = 2r, r ≥ 1, or k = 0 or k = 3p + 1, p ≥ 0 or k = 3p+ 2, p ≥ 0 are similar. Then m = (1113 · · · 16p−1)(26p+426p+10 · · · 26p+6r−226p+6r+4)(26p+626p+12 · · · 26p+6r). Let U = I × {aqs : s ∈ Z, s < 6p+ 6r + 3}. Clearly, all monomials in the polynomial χq(m+)− truncm+Q−U χq(m+) are right-negative. Therefore it is sufficient to show that the truncated q-character truncm+Q−U χq(m+) is special. Let M be the finite set consisting of the following monomials m0 = m+, m1 = m0A −1 1,6p, m2 = m1A −1 1,6p−2, . . . , m3p = m3p−1A −1 1,2, m3p+1 = m3pA −1 2,6p−4, m3p+2 = m3p+1A −1 2,6p−10, . . . , m4p = m4p−2A −1 2,6. It is clear that M satisfies the conditions in Theorem 2.2. Therefore truncm+Q−U χq(m+) = ∑ m∈M m and truncm+Q−U χq(m+) is special. Extended T -System of Type G2 15 4.5 The case of F (s) k,` Let m+ = F (s) k,` with k, ` ∈ Z≥1. Without loss of generality, we can assume that s = 1. Then m+ = (1113 · · · 12k−1)(12k+712k+9 · · · 12k+2`+5). Case 1. k = 1. Let U = I × {aqs : s ∈ Z, s < 2` + 8}. Clearly, all monomials in χq(m+) − truncm+Q−U χq(m+) are right-negative. Therefore it is sufficient to show that truncm+Q−U χq(m+) is special. Let M be the finite set consisting of the following monomials m0 = m+, m1 = m0A −1 1,2, m2 = m1A −1 2,5. It is clear that M satisfies the conditions in Theorem 2.2. Therefore truncm+Q−U χq(m+) = ∑ m∈M m and truncm+Q−U χq(m+) is special. Case 2. k > 1. Let m′1 = 1113 · · · 12k−1, m′2 = 12k+712k+9 · · · 12k+2`+5, m′′1 = 1113 · · · 12k−3, m′′2 = 12k−112k+712k+9 · · · 12k+2`+5. Then M (L(m+)) ⊂M (χq(m ′ 1)χq(m ′ 2)) ∩M (χq(m ′′ 1)χq(m ′′ 2)). By using similar arguments as the case of C(s) k,` , we can show that the only dominant monomials in χq(m ′ 1)χq(m ′ 2) are m+ and n1 = 1113 · · · 12k−312k+312k+912k+11 · · · 12k+2`+5 = m+A −1 1,2kA −1 2,2k+3A −1 1,2k+6, n2 = 1113 · · · 12k−312k+712k+912k+1312k+15 · · · 12k+2`+5 = n1A −1 1,2k+4A −1 2,2k+7A −1 1,2k+10, n3 = 1113 · · · 12k−512k+712k+1312k+15 · · · 12k+2`+5 = n2A −1 1,2k−2A −1 2,2k+1A −1 1,2k+4A −1 1,2k+2A −1 2,2k+5A −1 1,2k+8, n4 = 1113 · · · 12k−712k+1312k+15 · · · 12k+2`+5 = n3A −1 1,2k−4A −1 2,2k−1A −1 1,2k+2A −1 1,2kA −1 2,2k+3A −1 1,2k+6. Moreover, n1, n2, n3, n4 are not in χq(m ′ 1)χq(m ′ 2). Therefore the only dominant monomial in χq(m+) is m+. 5 Proof of Theorem 3.4 We use the FM algorithm to classify dominant monomials in χq(L)χq(R), χq(T )χq(B), and χq(S). 5.1 Classif ication of dominant monomials in χq(L)χq(R) and χq(T )χq(B) Lemma 5.1. We have the following cases. (1) Let M = B (s) k,`−1B (s+6) k−1,` , k ≥ 1, ` ≥ 1. Then dominant monomials in χq ( B(s) k,`−1 ) χq ( B(s+6) k−1,` ) are M0 = M, M1 = MA−1 1,s+6k+2`−2, 16 J.R. Li and E. Mukhin M2 = M1A −1 1,s+6k+2`−4, . . . , M`−1 = M`−2A −1 1,s+6k+2, M` = M`−1A −1 2,s+6k−3A −1 1,s+6k, M`+1 = M`A −1 2,s+6k−9, M`+2 = M`+1A −1 2,s+6k−15, . . . , Mk+`−1 = Mk+`−2A −1 2,s+3. The dominant monomials in χq ( B(s) k,` ) χq ( B(s+6) k−1,`−1 ) are M0, . . . ,Mk+`−2. (2) Let M = C (s) k,`−1C (s+6) k−1,` , k ≥ 1, ` ≥ 1. Then dominant monomials in χq ( C(s) k,`−1 ) χq ( C(s+6) k−1,` ) are M0 = M, M1 = MA−1 2,s+6k+6`−5, M2 = M1A −1 2,s+6k+6`−11, . . . , M`−1 = M`−2A −1 2,s+6k+7, M` = M`−1A −1 2,s+6k−3A −1 1,s+6kA −1 1,s+6k−2A −1 2,s+6k+1, M`+1 = M`A −1 2,s+6k−9, M`+2 = M`+1A −1 2,s+6k−15, . . . , Mk+`−1 = Mk+`−2A −1 2,s+3. The dominant monomials in χq ( C(s) k,` ) χq ( C(s+6) k−1,`−1 ) are M0, . . . ,Mk+`−2. (3) Let M = D (s) 0,`−1B (s+8) `,0 , ` ≥ 1. Then dominant monomials in χq ( D(s) 0,`−1 ) χq ( B(s+8) `,0 ) are M0 = M, M1 = MA−1 2,s+6`−1, M2 = M1A −1 2,s+6`−7, . . . , M`−1 = M`−2A −1 2,s+11, M` = M`−1A −1 1,s+6k+2A −1 2,s+6k+5. The dominant monomials in χq(D(s) 0,`)χq(B (s+8) `−1,0 ) are M0, . . . ,M`−1. (4) Let M=D (s) k,`−1D (s+6) k−1,`, k ≥ 1, ` ≥ 1. Then dominant monomials in χq ( D(s) k,`−1 ) χq ( D(s+6) k−1,` ) are M0 = M, M1 = MA−1 2,s+6k+6`−1, M2 = M1A −1 2,s+6k+6`−7, . . . , M`−1 = M`−2A −1 2,s+6k+11, M` = M`−1A −1 1,s+6k+2A −1 2,s+6k+5, M`+1 = M`A −1 2,s+6k−3A −1 1,s+6k, M`+2 = M`+1A −1 2,s+6k−9, . . . , Mk+` = Mk+`−1A −1 2,s+3. The dominant monomials in χq ( D(s) k,` ) χq ( D(s+6) k−1,`−1 ) are M0, . . . ,Mk+`−1. (5) Let M = E (s) k,`−1E (s+2) k−1,` , k ≥ 1, ` ≥ 1. If ` = 2r + 1, then dominant monomials in χq ( E(s) k,`−1 ) χq ( E(s+2) k−1,` ) are M0 = M, M1 = MA−1 2,s+2k+3`−3, M2 = M1A −1 2,s+2k+3`−9, . . . , Mr = Mr−1A −1 2,s+2k+6, Mr+1 = MrA −1 1,s+2k−1A −1 1,s+2k−3A −1 2,s+2k, Mr+2 = Mr+1A −1 1,s+2k−5, Mr+3 = Mr+2A −1 1,s+2k−7, . . . , Mk+r−1 = Mk+r−2A −1 1,s+1. The dominant monomials in χq ( E(s) k,` ) χq ( E(s+2) k−1,`−1 ) are M0, . . . ,Mk+r−2. If ` = 2r, then dominant monomials in χq ( E(s) k,`−1 ) χq ( E(s+2) k−1,` ) are M0 = M, M1 = MA−1 2,s+2k+3`−4, M2 = M1A −1 2,s+2k+3`−10, . . . , Mr−1 = Mr−2A −1 2,s+2k+8, Extended T -System of Type G2 17 Mr = Mr−1A −1 1,s+2k−1A −1 2,s+2k+2, Mr+1 = MrA −1 1,s+2k−3, Mr+2 = Mr+1A −1 1,s+2k−5, . . . , Mk+r−1 = Mk+r−2A −1 1,s+1. The dominant monomials in χq ( E(s) k,` ) χq ( E(s+2) k−1,`−1 ) are M0, . . . ,Mk+r−2. (6) Let M = F (s) k,`−1F (s+2) k−1,` , k ≥ 1, ` ≥ 1. Then dominant monomials in χq ( F (s) k,`−1 ) χq ( F (s+2) k−1,` ) are M0 = M, M1 = MA−1 1,s+2k+2`+3, M2 = M1A −1 1,s+2k+2`+1, . . . , M`−1 = M`−2A −1 1,s+2k+7, M` = M`−1A −1 1,s+2k−1A −1 2,s+2k+2A −1 1,s+2k+5, M`+1 = M`A −1 1,s+2k−3, M`+2 = M`+1A −1 1,s+2k−5, . . . , Mk+`−1 = Mk+`−2A −1 1,s+1. The dominant monomials in χq ( F (s) k,` ) χq ( F (s+2) k−1,`−1 ) are M0, . . . ,Mk+`−2. In each case, for each i, the multiplicity of Mi in the corresponding product of q-characters is 1. Proof. We prove the case of χq ( C(s) k,`−1 ) χq ( C(s+6) k−1,` ) . The other cases are similar. Let m′1 = C (s) k,`−1, m′2 = C (s+6) k−1,` . Without loss of generality, we assume that s = 6. Then m′1 = (26212 · · · 26k)(26k+1026k+16 · · · 26k+6`−2), m′2 = (212 · · · 26k)(26k+1026k+16 · · · 26k+6`−226k+6`+4). Let m = m1m2 be a dominant monomial, where mi ∈ χq(m′i), i = 1, 2. Denote by m3 = 26k+1026k+16 · · · 26k+6`+4. If m2 ∈ χq(212 · · · 26k)(χq(m3)−m3), then m = m1m2 is right negative and hence m is not dominant. Therefore m2 ∈ χq(212 · · · 26k)m3. Suppose that m2 ∈ M (L(m′2)) ∩M (χq(212 · · · 26k−6)(χq(26k) − 26k)m3). By the FM al- gorithm for L(m′2) and Lemma 3.1, m2 must have a factor 2−1 6k+6 or 1−1 6k+7 or 2−1 6k+8. By Lemma 3.1, m1 does not have the factors 26k+6 and 26k+8. Therefore m2 cannot have fac- tors 2−1 6k+6 and 2−1 6k+8 since m = m1m2 is dominant. Hence 1−1 6k+7 is a factor of m2. Since m = m1m2 is dominant, we need to cancel 1−1 6k+7 using a factor in m1. By Lemma 3.1, the only possible way to cancel 1−1 6k+7 by m1 is to use the factor 16k+516k+716k+92−1 6k+82−1 6k+10 or 16k+516k+71−1 6k+112−1 6k+8 of m1 coming from χq(26k). Since 2−1 6k+8 cannot be canceled by any monomials in χq(26212 · · · 26k−6), we have the factor 2−1 6k+8 in m = m1m2 and hence m is not dominant. Therefore m2 ∈M (L(212 · · · 26k−6))26km3. By the FM algorithm, m2 = m′2. If m1 ∈ χq(26 · · · 26k26k+1026k+16 · · · 26k+6`−8)(χq(26k+6`−2) − 26k+6`−2 − 2−1 6k+6`+416k+6`−116k+6`+116k+6`+3), then m = m1m2 is right-negative and hence not dominant. Therefore m1 is in one of the following polynomials χq(26 · · · 26k26k+1026k+16 · · · 26k+6`−8)26k+6`−2, (5.1) χq(26 · · · 26k26k+1026k+16 · · · 26k+6`−8)2−1 6k+6`+416k+6`−116k+6`+116k+6`+3. (5.2) If m1 is in (5.1), then m1 = m′1. The dominant monomial we obtain is M0 = m′1m ′ 2. If m1 is the highest monomial in (5.2), then we obtain the dominant monomial M1 = m1m ′ 2. Suppose that m1 is in M (L(m′1)) ∩M (χq(26 · · · 26k26k+1026k+16 · · · 26k+6`−14)(χq(26k+6`−8)− 26k+6`−8) × 2−1 6k+6`+416k+6`−116k+6`+116k+6`+3). 18 J.R. Li and E. Mukhin By the FM algorithm for L(m′1), m1 ∈ χq(26 · · · 26k26k+1026k+16 · · · 26k+6`−14) × ( 2−1 6k+6`−216k+6`−716k+6`−516k+6`−3 )( 2−1 6k+6`+416k+6`−116k+6`+116k+6`+3 ) . We obtain the dominant monomial M2 = m1m ′ 2. Continue this procedure, we obtain dominant monomials M3, . . . ,M`−1 and the remaining dominant monomials are of the form m1m ′ 2, where m1 is a non-highest monomial in M (L(m′1)) ∩M (L(26 · · · 26k))2 −1 6k+162−1 6k+22 · · · 2 −1 6k+6`+416k+1116k+13 · · · 16k+6`+3. Suppose that m1 is a non-highest monomial in the above set. Since the non-highest monomials in χq(26 · · · 26k) are right-negative, we need cancellations of factors with negative powers of some monomial in χq(26 · · · 26k) with 26k+1016k+1116k+13 · · · 16k+6`+3. The only cancellation can happen is to cancel 26k+10 or 16k+11. Since 12 6k+9 does not appear in χq(26 · · · 26k), 16k+11 cannot be canceled. Therefore we need a cancellation with 26k+10. The only monomials in χq(26 · · · 26k) which can cancel 26k+10 is in one of the following polynomials χq(26 · · · 26k−6)16k+116k+92−1 6k+10, χq(26 · · · 26k−6)1−1 6k+316k+926k+22−1 6k+10, χq(26 · · · 26k−6)16k+516k+716k+92−1 6k+82−1 6k+10. Therefore m1 is in one of the following sets M (L(m′1)) ∩M (L(26 · · · 26k−6))16k+116k+92−1 6k+10 · · · 2 −1 6k+6`+416k+11 · · · 16k+6`+3, (5.3) M (L(m′1)) ∩M (L(26 · · · 26k−6)) × 1−1 6k+316k+926k+22−1 6k+10 · · · 2 −1 6k+6`+416k+11 · · · 16k+6`+3, (5.4) M (L(m′1)) ∩M (L(26 · · · 26k−6)) × 16k+516k+716k+92−1 6k+82−1 6k+10 · · · 2 −1 6k+6`+416k+11 · · · 16k+6`+3. (5.5) If m1 is in (5.4), then we need to cancel 1−1 6k+3. We have M (L(26 · · · 26k−6)) ⊂M (χq(26 · · · 26k−12)χq(26k−6)). By Lemma 3.1, only the monomials 16k−516k+32−1 6k+4, 1−1 6k−316k+326k−42−1 6k+4, 16k−116k+116k+32−1 6k+22−1 6k+4 in χq(26k−6) can cancel 1−1 6k+3. But these monomials have the factor 2−1 6k+4 which cannot be canceled by any monomials in χq(26 · · · 26k−12) or by m′2. Hence m1 is not in (5.4). Ifm1 is in (5.5), then we need to cancel 2−1 6k+8. But 2−1 6k+8 cannot be canceled by any monomials in χq(26 · · · 26k−6) or by m′2. Therefore m1 is not in (5.5). Hence m1 is in (5.3). If m1 is the highest monomial in (5.3) with respect to ≤ defined in (2.7), then m1m ′ 2 = M`. Suppose that m1 a non-highest monomial in (5.3). By the FM algorithm, m1 must be in χq(26 · · · 26k−12)2−1 6k 16k−516k−316k−116k+116k+92−1 6k+10 · · · 2 −1 6k+6`+416k+11 · · · 16k+6`+3. If m1 is the highest monomial in the above set, then m1m ′ 2 = M`+1. Continue this procedure, we can show that the only remaining dominant monomials are M`+2, . . . ,Mk+`−1. It is clear that the multiplicity of Mi, i = 1, . . . , k + `− 1, in χq(m1)χq(m2) is 1. � Extended T -System of Type G2 19 5.2 Products of sources are special Lemma 5.2. Let [S] be the last summand in one of the relations (3.4)–(3.11). Then S is special. Proof. We give a proof for S in the last line of (3.7) and in the last line of (3.11). The other cases are similar. Let S1 = χq(C(s+1) r+1,p)χq(B(s+3) r+1,3p−1)χq(B(s+5) r+p+1,0). Let n1 = 2s+12s+7 · · · 2s+6r−52s+6r+1, n′1 = 2s+6r+112s+6r+17 · · · 2s+6r+6p+5, n2 = 2s+32s+9 · · · 2s+6r−32s+6r+3, n′2 = 1s+6r+101s+6r+12 · · · 1s+6r+6p+6, n3 = 2s+52s+11 · · · 2s+6r+6p+5. Then C (s+1) r+1,p = n1n ′ 1, B (s+3) r+1,3p−1 = n2n ′ 2, B (s+5) r+p+1,0 = n3. Let m′ = m1m2m3 be a dominant monomial, where m1 ∈M ( C(s+1) r+1,p ) , m2 ∈M ( B(s+3) r+1,3p−1 ) , m3 ∈M ( B(s+5) r+p+1,0 ) . If m3 6= B (s+5) r+p+1,0 or m1 ∈ χq(n1)(χq(n ′ 1) − n′1) or m2 ∈ χq(n2)(χq(n ′ 2) − n′2), then m′ is right-negative which contradicts the fact that m′ is dominant. Therefore m3 = B (s+5) r+p+1,0, m1 ∈ χq(n1)n′1, and m2 ∈ χq(n2)n′2. If m2 is in M (L(n2n ′ 2)) ∩M (χq(2s+32s+9 · · · 2s+6r−3)(χq(2s+6r+3)− 2s+6r+3)n′2), (5.6) then m2 ∈ χq(2s+32s+9 · · · 2s+6r−3)2−1 s+6r+91s+6k+41s+6k+61s+6k+8n ′ 2. By Lemma 3.1, the factor 2−1 s+6r+9 cannot be canceled by any monomial in either χq(n1) or χq(2s+32s+9 · · · 2s+6r−3). It is clear that 2−1 s+6r+9 cannot be canceled by n′1, n′2, n3. Therefore 2−1 s+6r+9 cannot be canceled. Hence m2 is not in (5.6). Thus m2 must be in M (L(n2n ′ 2)) ∩M (L(2s+32s+9 · · · 2s+6r−3))2s+6r+3n ′ 2. Therefore m2 = B (s+3) r+1,3p−1. Suppose that m1 6= C (s+1) r+1,p . Then m1 = m′1n ′ 1, where m′1 is a non-highest monomial in χq(n1). Since the non-highest monomials in χq(n1) are right-negative, we need a cancellation with n′1n ′ 2m3. The only cancellation can happen is to cancel 2s+6r+11 in n′1, or cancel one of 2s+6r+3, 1s+6r+10 in n2n ′ 2, or cancel one of 2s+6r+5, 2s+6r+11 in m3. By the FM algorithm, 2s+6r+11 cannot be canceled. By Lemma 3.1, 1s+6r+10, 2s+6r+3 and 2s+6r+5 cannot be canceled. This is a contradiction. Therefore m1 = C (s+1) r+1,p . Therefore the only dominant monomial in S1 is C (s+1) r+1,pB (s+3) r+1,3p−1B (s+5) r+p+1,0. Let S2 = χq ( D(s+1) r+1,b `3c ) χq ( C(s+3) r+1,b `+1 3 c ) χq ( B(s+5) r,0 ) χq ( B(s+6r+17) b `−1 3 c,0 ) , r ≥ 0, and ` = 3p, p ≥ 1. The cases of ` = 3p+ 1, p ≥ 0 and ` = 3p+ 2, p ≥ 0 are similar. Let n1 = 2s+12s+7 · · · 2s+6r+11s+6r+8, n′1 = 2s+6r+152s+6r+21 · · · 2s+6r+6p+9, n2 = 2s+32s+9 · · · 2s+6r−32s+6r+3, n′2 = 2s+6r+132s+6r+20 · · · 2s+6r+6p+7, n3 = 2s+52s+11 · · · 2s+6r−1, n4 = 2s+6r+172s+6r+23 · · · 2s+6r+6p+5. Then D (s+1) r+1,b `3c = n1n ′ 1, C (s+3) r+1,b `+1 3 c = n2n ′ 2, B (s+5) r,0 = n3, B (s+6r+17) b `−1 3 c,0 = n4. 20 J.R. Li and E. Mukhin Let m′ = m1m2m3m4 be a dominant monomial, where m1 ∈M ( D(s+1) r+1,b `3c ) , m2 ∈M ( C(s+3) r+1,b `+1 3 c ) , m3 ∈M ( B(s+5) r,0 ) , m4 ∈M ( B(s+6r+17) b `−1 3 c,0 ) . If m4 6= n4 or m1 ∈ χq(n1)(χq(n ′ 1) − n′1) or m2 ∈ χq(n2)(χq(n ′ 2) − n′2), then m′ is right- negative which contradicts the fact that m′ is dominant. Therefore m4 = n4, m1 ∈ χq(n1)n′1, and m2 ∈ χq(n2)n′2. If m1 ∈M (L(n1n ′ 1)) ∩M (χq(2s+12s+7 · · · 2s+6r+1)(χq(1s+6r+8)− 1s+6r+8)n′1), (5.7) then by the FM algorithm for L(n1n ′ 1), m1 ∈ χq(2s+12s+7 · · · 2s+6r+1)1−1 s+6r+102s+6r+9n ′ 1. It is clear that 1−1 s+6r+10 is not canceled by n′1, n′2, n4, and any monomial in χq(n3). By the FM algorithm for χq(n2n ′ 2), 1−1 s+6r+10 cannot be canceled by any monomial in χq(n2n ′ 2). Therefore, by Lemma 3.1, 1−1 s+6r+10 can only be canceled by one of the factors 1s+6r+21s+6r+102−1 s+6r+11, 1−1 s+6r+41s+6r+102s+6r+32−1 s+6r+11, 1s+6r+61s+6r+81s+6r+102−1 s+6r+92−1 s+6r+11 coming from χq(2s+6r+1), where 2s+6r+1 is in n1. But then 2−1 s+6r+11 cannot be canceled. This contradicts the fact that m′ is dominant. Hence m1 is not in (5.6). Thus m1 must be in M (L(n1n ′ 1)) ∩M (L(n1n ′ 1)) ∩M (L(2s+12s+7 · · · 2s+6r+1))1s+6r+8n ′ 1. If m1 is in M (L(n1n ′ 1)) ∩M (L(n1n ′ 1)) ∩M (χq(2s+1 × 2s+7 · · · 2s+6r−5)(χq(2s+6r+1)− 2s+6r+1)1s+6r+8n ′ 1). Then m1 ∈ χq(2s+12s+7 · · · 2s+6r−5)2−1 s+6r+71s+6r+21s+6r+41s+6r+61s+6r+8n ′ 1. The only possible way to cancel 2−1 s+6r+7 is to use one of the terms 1s+6r+41−1 s+6r+81−1 s+6r+102s+6r+7, 1−1 s+6r+61−1 s+6r+81−1 s+6r+102s+6r+52s+6r+7, 2s+6r+72−1 s+6r+11 in χq(2s+6r+3), where 2s+6r+3 is in n2. But then we have to cancel 1−1 s+6r+10 or 2−1 s+6r+11. But 1−1 s+6r+10 and 2−1 s+6r+11 cannot be canceled. This is a contradiction. Therefore m1 must be in M (L(n1n ′ 1)) ∩M (L(2s+12s+7 · · · 2s+6r−5))2s+6r+11s+6r+8n ′ 1. Hence m1 = n1n ′ 1. By the FM algorithm, when we compute the q-character for χq(n2n ′ 2), we can only choose one of the following terms 2s+6r+3, 1s+6r+41s+6r+61s+6r+82−1 s+6r+9, 1s+6r+41s+6r+61−1 s+6r+10, Extended T -System of Type G2 21 1s+6r+41−1 s+6r+81−1 s+6r+102s+6r+7, 1−1 s+6r+61−1 s+6r+81−1 s+6r+102s+6r+52s+6r+7, 2−1 s+6r+112s+6r+7 in χq(2s+6r+3). Since 2−1 s+6r+9, 1−1 s+6r+10, and 2−1 s+6r+11 cannot be canceled, we can only choose 2s+6r+3. Therefore m2 is in M (L(n2n ′ 2)) ∩M (L(2s+32s+9 · · · 2s+6r−3))2s+6r+3n ′ 2. Therefore m2 = n2n ′ 2. If m3 is in M (L(n3)) ∩M (χq(2s+52s+11 · · · 2s+6r−7)(χq(2s+6r−1)− 2s+6r−1)), then, by Lemma 3.1, m = m1m2m3m4 is non-dominant since m1 = n1n ′ 1,m2 = n2n ′ 2,m4 = n4. This contradicts the fact that m is dominant. Therefore m3 is in M (L(n3)) ∩M (L(2s+52s+11 · · · 2s+6r−7))2s+6r−1. Hence m3 = n3. Therefore the only dominant monomial in S2 is D (s+1) r+1,b `3c C (s+3) r+1,b `+1 3 c B (s+5) r,0 B (s+6r+17) b `−1 3 c,0 . � 5.3 Proof of Theorem 3.4 By Lemmas 5.1 and 5.2, the dominant monomials in the q-characters of the left hand side and of the right hand side of every relation in Theorem 3.4 are the same. The theorem follows. 6 Proof of Theorem 3.5 By Lemma 5.2, S is special and hence irreducible. Therefore we only have to show that T ⊗B is irreducible. It suffices to prove that for each non-highest dominant monomial M in T ⊗ B, we have M (L(M)) 6⊂M (T ⊗ B). The idea is similar as in [10, 18, 21]. Recall that the dominant monomials in T ⊗ B are described by Lemma 5.1. Lemma 6.1. We consider the same cases as in Lemma 5.1. In each case Mi are the dominant monomials described by that lemma. (1) For k ≥ 1, ` ≥ 1, let n1 = M1A −1 1,s+6k+2`−2, n2 = M2A −1 1,s+6k+2`−4, . . . , n`−1 = M`−1A −1 1,s+6k+2, n` = M`A −1 2,s+6k−3A −1 1,s+6k, n`+1 = M`+1A −1 2,s+6k−9, . . . , nk+`−2 = Mk+`−2A −1 2,s+9. Then for i = 1, . . . , k + `− 2, ni ∈ χq(Mi) and ni 6∈ χq ( B(s) k,` ) χq ( B(s+6) k−1,`−1 ) . (2) For k ≥ 1, ` ≥ 1, let n1 = M1A −1 2,s+6k+6`−5, n2 = M2A −1 2,s+6k+6`−11, . . . , n`−1 = M`−1A −1 2,s+6k+7, n` = M`A −1 2,s+6k−3A −1 1,s+6kA −1 1,s+6k−2A −1 2,s+6k+1, n`+1 = M`+1A −1 2,s+6k−9, . . . , nk+`−2 = Mk+`−2A −1 2,s+9. Then for i = 1, . . . , k + `− 2, ni ∈ χq(Mi) and ni 6∈ χq ( C(s) k,` ) χq ( C(s+6) k−1,`−1 ) . 22 J.R. Li and E. Mukhin (3) For ` ≥ 1, let n1 = M1A −1 2,s+6`−1, n2 = M2A −1 2,s+6`−7, . . . , n`−1 = M`−1A −1 2,s+11. Then for i = 1, . . . , `, ni ∈ χq(Mi) and ni 6∈ χq ( D(s) 0,` ) χq ( B(s+8) `−1,0 ) . (4) For k ≥ 1, ` ≥ 1, let n1 = M1A −1 2,s+6k+6`−1, n2 = M2A −1 2,s+6k+6`−7, . . . , n`−1 = M`−1A −1 2,s+6k+11, n` = M`A −1 1,s+6k+2A −1 2,s+6k+5, n`+1 = M`+1A −1 2,s+6k−3A −1 1,s+6k, n`+2 = M`+2A −1 2,s+6k−9, . . . , nk+`−1 = Mk+`−1A −1 2,s+9. Then for i = 1, . . . , k + `− 1, ni ∈ χq(Mi) and ni 6∈ χq ( D(s+6) k−1,`−1 ) χq ( D(s) k,` ) . (5) For k ≥ 0, ` = 2r + 1, r ≥ 0, let n1 = M1A −1 2,s+2k+3`−3, n2 = M2A −1 2,s+2k+3`−9, . . . , nr = MrA −1 2,s+2k+3, nr+1 = Mr+1A −1 1,s+2k−1A −1 1,s+2k−3A −1 2,s+2k, nr+2 = Mr+2A −1 1,s+2k−5, . . . , nk+r−2 = Mk+r−2A −1 1,s+3. Then for i = 1, . . . , r + k − 2, ni ∈ χq(Mi) and ni 6∈ χq ( E(s) k,` ) χq ( E(s+2) k−1,`−1 ) . For k ≥ 0, ` = 2r, r ≥ 1, let n1 = M1A −1 2,s+2k+3`−4, n2 = M2A −1 2,s+2k+3`−10, . . . , nr−1 = Mr−1A −1 2,s+2k+8, nr = MrA −1 1,s+2k−1A −1 2,s+2k+2, nr+1 = Mr+1A −1 1,s+2k−3, . . . , nk+r−2 = Mk+r−2A −1 1,s+3. Then for i = 1, . . . , r + k − 2, ni ∈ χq(Mi) and ni 6∈ χq ( E(s) k,` ) χq ( E(s+2) k−1,`−1 ) . (6) For k ≥ 1, ` ≥ 1, let n1 = M1A −1 1,s+2k+2`+3, n2 = M2A −1 1,s+2k+2`+1, . . . , n`−1 = M`−1A −1 1,s+2k+7, n` = M`A −1 1,s+2k−1A −1 2,s+2k+2A −1 1,s+2k+5, n`+1 = M`+1A −1 1,s+2k−3, . . . , nk+`−2 = Mk+`−2A −1 1,s+3. Then i = 1, . . . , k + `− 2, ni ∈ χq(Mi) and ni 6∈ χq ( F (s) k,` ) χq ( F (s+2) k−1,`−1 ) . Proof. We give a proof in the case of χq ( C(s) k,` ) χq ( C(s+6) k−1,`−1 ) . The other cases are similar. By definition, we have C (s) k,` = (2s2s+6 · · · 2s+6k−6)(2s+6k+42s+6k+10 · · · 2s+6k+6`−82s+6k+6`−2), C (s+6) k−1,`−1 = (2s+62s+12 · · · 2s+6k−6)(2s+6k+42s+6k+10 · · · 2s+6k+6`−8), M1 = C (s) k,`C (s+6) k−1,`−1A −1 2,s+6k+6`−5 = C (s) k,`C (s+6) k−1,`−12−1 s+6k+6`−82−1 s+6k+6`−21s+6k+6`−71s+6k+6`−51s+6k+6`−3. By Uq2(ŝl2) argument, it is clear that n1 = M1A −1 2,s+6k+6`−5 is in χq(M1). Extended T -System of Type G2 23 If n1 is in χq ( C(s) k,` ) χq ( C(s+6) k−1,`−1 ) , then C (s) k,`A −1 2,s+6k+6`−5 is in χq ( C(s) k,` ) which is impossible by the FM algorithm for C(s) k,` . Similarly, ni ∈ χq(Mi), i = 2, . . . , ` − 1, but n2, . . . , n`−1 are not in χq ( C(s) k,` ) χq ( C(s+6) k−1,`−1 ) . By definition, M` = (2s2s+6 · · · 2s+6k−6)(2s+62s+12 · · · 2s+6k−12)(1s+6k−51s+6k+31s+6k+5 · · · 1s+6k+6`−3). Let U = { (1, aqs+6k), (1, aqs+6k−3), (2, aqs+6k−2), (2, aqs+6k+1) } ⊂ I × C×. Let M be the finite set consisting of the following monomials m0 = M`, m1 = m0A −1 2,s+6k−3, m2 = m1A −1 1,s+6k, m3 = m2A −1 1,s+6k−2, m4 = m3A −1 2,s+6k+1. It is clear that M satisfies the conditions in Theorem 2.2. Therefore truncm+Q−U (χq(M`)) = ∑ m∈M m and hence n` = M`A −1 2,s+6k−3A −1 1,s+6kA −1 1,s+6k−2A −1 2,s+6k+1 is in χq(M`). If n` is in χq ( C(s) k,` ) χq ( C(s+6) k−1,`−1 ) , then C (s) k,`A −1 2,s+6k−3A −1 1,s+6kA −1 1,s+6k−2A −1 2,s+6k+1 is in χq ( C(s) k,` ) which is impossible by the FM algorithm for C(s) k,` . Similarly, we show that for i = `+ 1, . . . , k+ `− 2, ni ∈ χq(Mi) and ni 6∈ χq ( C(s) k,` ) χq ( C(s+6) k−1,`−1 ) . � 7 The second part of the extended T -system Let B̃ (s) k,` , C̃ (s) k,` , D̃ (s) k,`, Ẽ (s) k,` , F̃ (s) k,` be the monomials obtained from B (s) k,` , C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` by replacing ia with i−a, i = 1, 2. Namely, B̃ (s) k,` = ( `−1∏ i=0 1−s−6k−2i−1 )( k−1∏ i=0 2−s−6i ) , C̃ (s) k,` = ( `−1∏ i=0 2−s−6k−6i−4 )( k−1∏ i=0 2−s−6i ) , D̃ (s) k,` = ( `−1∏ i=0 2−s−6k−6i−8 ) 1−s−6k−1 ( k−1∏ i=0 2−s−6i ) , F̃ (s) k,` = ( `−1∏ i=0 1−s−2k−2i−6 )( k−1∏ i=0 1−s−2i ) , Ẽ (s) k,` = b `−2 2 c∏ i=0 2−s−2k−6i−5  b `−1 2 c∏ i=0 2−s−2k−6i−3 (k−1∏ i=0 1−s−2i ) . Note that, in particular, for k ∈ Z≥0, s ∈ Z, we have the following trivial relations B̃(s) k,0 = C̃(s) k,0 = C̃(s−4) 0,k , D̃(s) k,0 = B̃(s) k,1, Ẽ(s) k,0 = B̃(s−1) 0,k = F̃ (s−6) 0,k = F̃ (s) k,0. (7.1) We also have D(s) 0,k = B̃(−s−6k−2) k,1 , D̃(s) 0,k = B(−s−6k−2) k,1 , k ∈ Z≥0, s ∈ Z. Note that B̃(s) k,`, D̃ (s) 0,` , D̃ (s) k,0 are minimal affinizations. In general, the modules B̃(s) k,`, C̃ (s) k,` , D̃ (s) k,`, Ẽ(s) k,` , F̃ (s) k,` are not special. For example, we have the following proposition. 24 J.R. Li and E. Mukhin Proposition 7.1. The module B̃(0) 3,1 = L(101214211) is not special. Proof. Suppose that L(101214211) is special. Then the FM algorithm applies to L(101214211). Therefore, by the FM algorithm, the monomials 101214211, 10121−1 6 25211, 101−1 4 1−1 6 2325211, 1−1 2 1−1 4 1−1 6 212325211, 2−1 7 2325211, 2−1 7 2−1 9 14161825211, 2−1 7 14161−1 10 25211, 141−1 8 1−1 10 25211, 1−1 6 1−1 8 1−1 10 22 5211, 25 are in M (L(101214211)). Hence M (L(101214211)) has at least two dominant monomials 101214211 and 25. This contradicts the assumption that L(101214211) is special. � Theorem 7.2. The modules B̃(s) k,`, C̃ (s) k,` , D̃ (s) k,`, Ẽ (s) k,` , F̃ (s) k,` , s ∈ Z, k, l,∈ Z≥0 are anti-special. Proof. This theorem can be proved using the dual arguments of the proof of Theorem 3.3. � Lemma 7.3. Let ι : ZP → ZP be a homomorphism of rings such that Y1,aqs 7→ Y −1 1,aq12−s , Y2,aqs 7→ Y −1 2,aq12−s for all a ∈ C×, s ∈ Z. Then χq ( B̃(s) k,` ) = ι ( χq ( B(s) k,` )) , χq ( C̃(s) k,` ) = ι ( χq ( C(s) k,` )) , χq ( D̃(s) k,` ) = ι ( χq ( D(s) k,` )) , χq ( Ẽ(s) k,` ) = ι ( χq ( E(s) k,` )) , χq ( F̃ (s) k,` ) = ι ( χq ( F (s) k,` )) . Proof. Let m+ be one of B (s) k,` , C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` . Then χq(m̃+) can be computed by the FM algorithm starting from the lowest weight using Ai,a with i ∈ I, a ∈ C×. The procedure is dual to the computation of χq(m+) which starts from m+ using A−1 i,a with i ∈ I, a ∈ C×. The highest (resp. lowest) l-weight in χq(m+) is sent to the lowest (resp. highest) l-weight in χq(m̃+) by ι. � Note that Lemma 7.3 can also proved using the Cartan involution in [1]. The modules B̃(s) k,`, C̃ (s) k,` , D̃ (s) k,`, Ẽ (s) k,` , F̃ (s) k,` satisfy the same relations as in Theorem 3.4 but the roles of left and right modules are exchanged. More precisely, we have the following theorem. Theorem 7.4. For s ∈ Z, k, ` ∈ Z≥1, t ∈ Z≥2, we have the following relations in Rep(Uqĝ).[ B̃(s+6) k−1,` ][ B̃(s) k,`−1 ] = [ B̃(s) k,` ][ B̃(s+6) k−1,`−1 ] + [ Ẽ(s+1) 3k−1,d 2`−2 3 e ][ B̃(s+6k+6) b `−1 3 c,0 ] ,[ Ẽ(s) 0,` ] = [ B̃(s+3) b `+1 2 c,0 ][ B̃(s+5) b `2c,0 ] , [ Ẽ(s) 1,` ] = [ D̃(s−1) 0,b `2c ][ B̃(s+5) b `+1 2 c,0 ] ,[ Ẽ(s+2) t−1,` ][ Ẽ(s) t,`−1 ] = [ Ẽ(s) t,` ][ Ẽ(s+2) t−1,`−1 ] +  [ D̃(s+1) r,p−1 ][ B̃(s+3) r+p,0 ][ B̃(s+5) r,3p−2 ] , if t = 3r + 2, ` = 2p− 1,[ B̃(s+1) r+p+1,0 ][ C̃(s+3) r,p ][ B̃(s+5) r,3p−1 ] , if t = 3r + 2, ` = 2p,[ B̃(s+1) r+1,3p−2 ][ D̃(s+3) r,p−1 ][ B̃(s+5) r+p,0 ] , if t = 3r + 3, ` = 2p− 1,[ B̃(s+1) r+1,3p−1 ][ B̃(s+3) r+p+1,0 ][ C̃(s+5) r,p ] , if t = 3r + 3, ` = 2p,[ B̃(s+1) r+p+1,0 ][ B̃(s+3) r+1,3p−2 ][ D̃(s+5) r,p−1 ] , if t = 3r + 4, ` = 2p− 1,[ C̃(s+1) r+1,p ][ B̃(s+3) r+1,3p−1 ][ B̃(s+5) r+p+1,0 ] , if t = 3r + 4, ` = 2p,[ C̃(s+6) k−1,` ][ C̃(s) k,`−1 ] = [ C̃(s) k,` ][ C̃(s+6) k−1,`−1 ] + [ F̃ (s+1) 3k−2,3`−2 ] , Extended T -System of Type G2 25[ B̃(s+8) `,0 ][ D̃(s) 0,`−1 ] = [ D̃(s) 0,` ][ B̃(s+8) `−1,0 ] + [ B̃(s+4) 0,3`−1 ] ,[ D̃(s+6) k−1,` ][ D̃(s) k,`−1 ] = [ D̃(s) k,` ][ D̃(s+6) k−1,`−1 ] + [ F̃ (s+1) 3k−1,3`−1 ] ,[ F̃ (s+2) k−1,` ][ F̃ (s) k,`−1 ] = [ F̃ (s) k,` ][ F̃ (s+2) k−1,`−1 ] +  [ B̃(s+1) r,0 ][ D̃(s+3) r,b `3c ][ C̃(s+5) r,b `+1 3 c ][ B̃(s+2k+11) b `−1 3 c,0 ] , if k = 3r + 1,[ C̃(s+1) r+1,b `+1 3 c ][ B̃(s+3) r,0 ][ D̃(s+5) r,b `3c ][ B̃(s+2k+11) b `−1 3 c,0 ] , if k = 3r + 2,[ D̃(s+1) r+1,b `3c ][ C̃(s+3) r+1,b `+1 3 c ][ B̃(s+5) r,0 ][ B̃(s+2k+11) b `−1 3 c,0 ] , if k = 3r + 3. Moreover, the modules corresponding to each summand on the right hand side of the above relations are all irreducible. Proof. The theorem follows from the relations in Theorem 3.4, Theorem 3.5, and Lem- ma 7.3. � The following proposition is similar to Proposition 3.6. Proposition 7.5. Given χq(1s), χq(2s), one can obtain the q-characters of B̃(s) k,`, C̃ (s) k,` , D̃ (s) k,`, Ẽ (s) k,` , F̃ (s) k,` , s ∈ Z, k, ` ∈ Z≥0, recursively, by using (7.1), and computing the q-character of the top module through the q-characters of other modules in relations in Theorem 7.4. 8 Dimensions In this section, we give dimension formulas for the modules B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` , B̃ (s) k,`, C̃ (s) k,` , D̃(s) k,`, Ẽ (s) k,` , F̃ (s) k,` . Note that dimensions do not depend on the upper index s. Note also that dimM = dim M̃ for each M = B(s) k,`, C (s) k,` , D (s) k,`, E (s) k,` , F (s) k,` . Theorem 8.1. Let s ∈ Z, k, ` ∈ Z≥0. Then dimB(s) k,3` = (`+ 2)(`+ 1)(1 + k)(k + 3 + `)(k + 2 + `) × ( 54`3k3 + 243`2k3 + 363`k3 + 180k3 + 2784`2k2 + 1080k2 + 162`4k2 + 2880`k2 + 1134`3k2 + 162`5k + 1539`4k + 5490`3k + 9132`2k + 7057`k + 2040k + 54`6 + 648`5 + 3069`4 + 7272`3 + 8977`2 + 5380`+ 1200 ) /14400, dimB(s) k,3`+1 = (`+ 3)(`+ 2)(`+ 1)(1 + k)(k + 2 + `)(k + 4 + `)(k + 3 + `) × ( 171`k2 + 120k2 + 54`2k2 + 600k + 621`2k + 108`3k + 1116`k + 54`4 + 450`3 + 1341`2 + 1665`+ 700 ) /14400, dimB(s) k,3`+2 = (`+ 3)(`+ 2)(`+ 1)(1 + k)(k + 4 + `)(k + 3 + `)(2 + k + `) × ( 300k2 + 261`k2 + 54`2k2 + 891`2k + 2376`k + 2040k + 108`3k + 54`4 + 630`3 + 2691`2 + 4995`+ 3400 ) /14400, dim C(s) k,` = (`+ 2)(`+ 1)(k + 2)(k + 1)(k + 3 + `)(k + 2 + `) × ( 3k2 + 3`k2 + 12k + 15`k + 3`2k + 3`2 + 12`+ 10 ) /240, dimD(s) k,` = (`+ 2)(`+ 1)(k + 2)(k + 1)(k + 3 + `)(k + 4 + `) × ( 3`k2 + 6k2 + 3`2k + 30k + 21`k + 6`2 + 30`+ 35 ) /240, 26 J.R. Li and E. Mukhin dim E(s) 3k,2` = (`+ 2)(`+ 1)(k + 1)(k + `+ 1)(k + `+ 2)2(k + `+ 3)2 × ( 27k4`2 + 81k4`+ 54k4 + 81k3`3 + 468k3`2 + 825k3` + 432k3 + 81k2`4 + 711k2`3 + 2184k2`2 + 2754k2`+ 1179k2 + 27k`5 + 342k`4 + 1593k`3 + 3438k`2 + 3435k`+ 1260k + 18`5 + 180`4 + 696`3 + 1296`2 + 1160`+ 400 ) /28800, dim E(s) 3k,2`+1 = (`+ 3)(`+ 2)(`+ 1)(k + 1)(k + `+ 4)(k + `+ 2)2(k + `+ 3)2 × ( 27k4`+ 54k4 + 81k3`2 + 414k3`+ 510k3 + 81k2`3 + 684k2`2 + 1842k2`+ 1611k2 + 27k`4 + 342k`3 + 1512k`2 + 2808k`+ 1875k + 18`4 + 180`3 + 642`2 + 960`+ 500 ) /28800, dim E(s) 3k+1,2` = (`+ 2)(`+ 1)(k + 1)(k + `+ 4)(k + `+ 2)2(k + `+ 3)2 × ( 27k4`2 + 81k4`+ 54k4 + 81k3`3 + 477k3`2 + 852k3` + 450k3 + 81k2`4 + 747k2`3 + 2373k2`2 + 3069k2`+ 1341k2 + 27k`5 + 387k`4 + 1935k`3 + 4353k`2 + 4461k`+ 1665k + 36`5 + 360`4 + 1374`3 + 2490`2 + 2140`+ 700 ) /28800, dim E(s) 3k+1,2`+1 = (`+ 3)(`+ 2)(`+ 1)(k + 1)(k + `+ 2)(k + `+ 3)2(k + `+ 4)2 × ( 27k4`+ 54k4 + 81k3`2 + 450k3`+ 582k3 + 81k2`3 + 774k2`2 + 2310k2`+ 2193k2 + 27k`4 + 414k`3 + 2124k`2 + 4488k` + 3375k + 36`4 + 396`3 + 1590`2 + 2760`+ 1750 ) /28800, dim E(s) 3k+2,2` = (`+ 2)(`+ 1)(k + 2)(k + 1)(k + `+ 4)(k + `+ 2)(k + `+ 3)2 × ( 27k4`2 + 81k4`+ 54k4 + 108k3`3 + 648k3`2 + 1176k3` + 630k3 + 162k2`4 + 1458k2`3 + 4629k2`2 + 6057k2` + 2691k2 + 108k`5 + 1296k`4 + 5946k`3 + 12942k`2 + 13230k`+ 4995k + 27`6 + 405`5 + 2439`4 + 7515`3 + 12429`2 + 10395`+ 3400 ) /28800, dim E(s) 3k+2,2`+1 = (`+ 3)(`+ 2)(`+ 1)(k + 2)(k + 1)(k + `+ 5)(k + `+ 2) × ( k + `+ 3)2(k + `+ 4)2(9k2`+ 18k2 + 18k`2 + 99k`+ 128k + 9`3 + 81`2 + 237`+ 225 ) /9600, dimF (s) 3k,3` = (`+ 2)2(`+ 1)2(k + 2)2(k + 1)2(k + `+ 3)2 × ( 27k4`2 + 81k4`+ 54k4 + 54k3`3 + 405k3`2 + 801k3`+ 432k3 + 27k2`4 + 405k2`3 + 1746k2`2 + 2646k2`+ 1179k2 + 81k`4 + 801k`3 + 2646k`2 + 3342k`+ 1260k + 54`4 + 432`3 + 1179`2 + 1260`+ 400 ) /57600, dimF (s) 3k+1,3` = (`+ 2)2(`+ 1)2(k + 3)(k + 1)(k + 2)2(k + `+ 4)(k + `+ 3) × ( 27k4`2 + 81k4`+ 54k4 + 54k3`3 + 414k3`2 + 828k3`+ 450k3 + 27k2`4 + 414k2`3 + 1854k2`2 + 2907k2`+ 1341k2 + 81k`4 + 864k`3 + 3063k`2 + 4116k`+ 1665k + 54`4 + 498`3 + 1563`2 + 1905`+ 700 ) /57600, dimF (s) 3k+2,3` = (`+ 2)2(`+ 1)2(k + 3)(k + 1)(k + 2)2(k + `+ 4)(k + `+ 3) × ( 27k4`2 + 81k4`+ 54k4 + 54k3`3 + 504k3`2 + 1098k3`+ 630k3 + 27k2`4 + 558k2`3 + 3042k2`2 + 5355k2`+ 2691k2 + 135k`4 + 1764k`3 + 7395k`2 Extended T -System of Type G2 27 + 11190k`+ 4995k + 162`4 + 1734`3 + 6249`2 + 8475`+ 3400 ) /57600, dimF (s) 3k,3`+1 = (`+ 3)(`+ 1)(`+ 2)2(k + 2)2(k + 1)2(k + `+ 4)(k + `+ 3) × ( 27k4`2 + 81k4`+ 54k4 + 54k3`3 + 414k3`2 + 864k3`+ 498k3 + 27k2`4 + 414k2`3 + 1854k2`2 + 3063k2`+ 1563k2 + 81k`4 + 828k`3 + 2907k`2 + 4116k`+ 1905k + 54`4 + 450`3 + 1341`2 + 1665`+ 700 ) /57600, dimF (s) 3k+1,3`+1 = (`+ 3)(`+ 1)(`+ 2)2(k + 3)(k + 1)(k + 2)2(k + `+ 3)(k + `+ 4) × ( 27k4`2+ 81k4`+ 54k4 + 54k3`3+ 450k3`2 + 972k3`+ 570k3+ 27k2`4 + 450k2`3 + 2214k2`2 + 3891k2`+ 2061k2 + 81k`4 + 972k`3 + 3891k`2 + 6060k`+ 2985k + 54`4 + 570`3 + 2061`2 + 2985`+ 1400 ) /57600, dimF (s) 3k+2,3`+1 = (`+ 3)(`+ 1)(`+ 2)2(k + 3)(k + 1)(k + 2)2(k + `+ 3)(k + `+ 5) × ( k + `+ 4)2(9k2`2 + 27k2`+ 18k2 + 45k`2 + 135k` + 88k + 54`2 + 164`+ 105 ) /19200, dimF (s) 3k,3`+2 = (`+ 3)(`+ 1)(`+ 2)2(k + 2)2(k + 1)2(k + `+ 4)(k + `+ 3) ( 27k4`2 + 135k4`+ 162k4 + 54k3`3 + 558k3`2 + 1764k3`+ 1734k3 + 27k2`4 + 504k2`3 + 3042k2`2 + 7395k2`+ 6249k2 + 81k`4 + 1098k`3 + 5355k`2 + 11190k`+ 8475k + 54`4 + 630`3 + 2691`2 + 4995`+ 3400 ) /57600, dimF (s) 3k+1,3`+2 = (`+ 3)(`+ 1)(`+ 2)2(k + 3)(k + 1)(k + 2)2(k + `+ 3)(k + `+ 5) × ( k + `+ 4)2(9k2`2 + 45k2`+ 54k2 + 27k`2 + 135k` + 164k + 18`2 + 88`+ 105 ) /19200, dimF (s) 3k+2,3`+2 = (`+ 3)(`+ 1)(`+ 2)2(k + 3)(k + 1)(k + 2)2(k + `+ 4)(k + `+ 5) × ( 27k4`2 + 135k4`+ 162k4 + 54k3`3 + 630k3`2 + 2124k3`+ 2166k3 + 27k2`4 + 630k2`3 + 4374k2`2 + 11661k2`+ 10473k2 + 135k`4 + 2124k`3 + 11661k`2 + 26748k`+ 21759k + 162`4 + 2166`3 + 10473`2 + 21759`+ 16400 ) /57600. Proof. We check the initial conditions, namely dimensions of B(s) 0,1, B(s) 1,0. We check the dimen- sions are compatible with relations (3.1), (3.2), (3.3). We directly check that the formulas satisfy the relations in Theorems 3.4. For the checks we employed the computer algebra system Maple. The theorem follows since the solution of the extended T -system is unique, see Proposi- tion 3.6. � Acknowledgements We would like to thank D. Hernandez, B. Leclerc, T. Nakanishi, C.A.S. Young for helpful discussions. JL would like to thank IUPUI Department of Mathematical Sciences for hospitality during his visit when this work was carried out. JL is partially supported by a CSC scholarship, the Natural Science Foundation of Gansu Province (No. 1107RJZA218), and the Fundamental Research Funds for the Central Universities (No. lzujbky-2012-12) from China. The research of EM is supported by the NSF, grant number DMS-0900984. 28 J.R. Li and E. 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Math. 496 (1998), 181–212, q-alg/9502008. http://dx.doi.org/10.1007/BF02102063 http://dx.doi.org/10.1007/BF00750760 http://arxiv.org/abs/hep-th/9410036 http://arxiv.org/abs/hep-th/9411145 http://dx.doi.org/10.1215/S0012-7094-87-05423-8 http://dx.doi.org/10.1007/s002200000323 http://arxiv.org/abs/math.QA/9911112 http://dx.doi.org/10.1090/conm/248/03823 http://arxiv.org/abs/math.QA/9810055 http://dx.doi.org/10.1515/CRELLE.2006.052 http://arxiv.org/abs/math.QA/0501202 http://dx.doi.org/10.1007/s00220-007-0332-1 http://arxiv.org/abs/math.QA/0607527 http://dx.doi.org/10.1215/00127094-2010-040 http://arxiv.org/abs/0903.1452 http://dx.doi.org/10.4171/PRIMS/95 http://arxiv.org/abs/1001.1880 http://dx.doi.org/10.4171/PRIMS/96 http://arxiv.org/abs/1001.1881 http://dx.doi.org/10.1007/BF02342935 http://dx.doi.org/10.1142/S0217751X94002119 http://arxiv.org/abs/hep-th/9309137 http://dx.doi.org/10.1088/1751-8113/44/10/103001 http://dx.doi.org/10.1088/1751-8113/44/10/103001 http://arxiv.org/abs/1010.1344 http://dx.doi.org/10.1007/s00029-011-0083-x http://arxiv.org/abs/1104.3094 http://dx.doi.org/10.1016/j.aim.2012.06.012 http://arxiv.org/abs/1103.5873 http://dx.doi.org/10.1090/S1088-4165-03-00164-X http://dx.doi.org/10.1090/S1088-4165-03-00164-X http://arxiv.org/abs/math.QA/0204185 http://dx.doi.org/10.4007/annals.2004.160.1057 http://arxiv.org/abs/math.QA/0105173 http://dx.doi.org/10.1515/crll.1998.029 http://arxiv.org/abs/q-alg/9502008 1 Introduction 2 Background 2.1 Cartan data 2.2 Quantum affine algebra 2.3 Finite-dimensional representations and q-characters 2.4 Minimal affinizations of Uqg-modules 2.5 q-characters of Uq2-modules and the FM algorithm 2.6 Truncated q-characters 3 Main results 3.1 First examples 3.2 Definition of the modules Bk, (s), Ck, (s), Dk, (s), Ek, (s), Fk, (s) 3.3 Extended T-system 4 Proof of Theorem 3.3 4.1 The case of Ck, (s) 4.2 The case of Bk, (s) 4.3 The case of Dk, (s) 4.4 The case of Ek, (s) 4.5 The case of Fk, (s) 5 Proof of Theorem 3.4 5.1 Classification of dominant monomials in q(L)q(R) and q(T)q(B) 5.2 Products of sources are special 5.3 Proof of Theorem 3.4 6 Proof of Theorem 3.5 7 The second part of the extended T-system 8 Dimensions References

Extended T-System of Type G₂

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