Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A

Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A

Let G be a simply connected simple algebraic group over C, B and B− be two opposite Borel subgroups in G and W be the Weyl group. For u, v∈W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorphic to an upper cluster algebra A¯(i)C and the generalized min...

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Дата:2015
Автори: Kanakubo, Y., Nakashima, T.
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Опубліковано: Інститут математики НАН України 2015
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A / Y. Kanakubo, T. Nakashima // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 11 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1470102019-02-13T01:24:47Z Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A Kanakubo, Y. Nakashima, T. Let G be a simply connected simple algebraic group over C, B and B− be two opposite Borel subgroups in G and W be the Weyl group. For u, v∈W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorphic to an upper cluster algebra A¯(i)C and the generalized minors {Δ(k;i)} are the cluster variables belonging to a given initial seed in C[Gu,v] [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. In the case G=SLr₊₁(C), v=e and some special u∈W, we shall describe the generalized minors {Δ(k;i)} as summations of monomial realizations of certain Demazure crystals. 2015 Article Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A / Y. Kanakubo, T. Nakashima // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 11 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13F60; 81R50; 17B37 DOI:10.3842/SIGMA.2015.033 http://dspace.nbuv.gov.ua/handle/123456789/147010 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 Let G be a simply connected simple algebraic group over C, B and B− be two opposite Borel subgroups in G and W be the Weyl group. For u, v∈W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorphic to an upper cluster algebra A¯(i)C and the generalized minors {Δ(k;i)} are the cluster variables belonging to a given initial seed in C[Gu,v] [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. In the case G=SLr₊₁(C), v=e and some special u∈W, we shall describe the generalized minors {Δ(k;i)} as summations of monomial realizations of certain Demazure crystals.
format Article
author Kanakubo, Y.
Nakashima, T.
spellingShingle Kanakubo, Y.
Nakashima, T.
Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Kanakubo, Y.
Nakashima, T.
author_sort Kanakubo, Y.
title Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
title_short Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
title_full Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
title_fullStr Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
title_full_unstemmed Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A
title_sort cluster variables on certain double bruhat cells of type (u,e) and monomial realizations of crystal bases of type a
publisher Інститут математики НАН України
publishDate 2015
url http://dspace.nbuv.gov.ua/handle/123456789/147010
citation_txt Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A / Y. Kanakubo, T. Nakashima // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 11 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT kanakuboy clustervariablesoncertaindoublebruhatcellsoftypeueandmonomialrealizationsofcrystalbasesoftypea
AT nakashimat clustervariablesoncertaindoublebruhatcellsoftypeueandmonomialrealizationsofcrystalbasesoftypea
first_indexed 2025-07-11T01:08:34Z
last_indexed 2025-07-11T01:08:34Z
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 033, 32 pages Cluster Variables on Certain Double Bruhat Cells of Type (u, e) and Monomial Realizations of Crystal Bases of Type A? Yuki KANAKUBO and Toshiki NAKASHIMA Division of Mathematics, Sophia University, Yonban-cho 4, Chiyoda-ku, Tokyo 102-0081, Japan E-mail: j chi sen you ky@sophia.ac.jp, toshiki@sophia.ac.jp Received October 01, 2014, in final form April 14, 2015; Published online April 23, 2015 http://dx.doi.org/10.3842/SIGMA.2015.033 Abstract. Let G be a simply connected simple algebraic group over C, B and B− be two opposite Borel subgroups in G and W be the Weyl group. For u, v ∈W , it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v = BuB∩B−vB− is isomorphic to an upper cluster algebra Ā(i)C and the generalized minors {∆(k; i)} are the cluster variables belonging to a given initial seed in C[Gu,v] [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1–52]. In the case G = SLr+1(C), v = e and some special u ∈ W , we shall describe the generalized minors {∆(k; i)} as summations of monomial realizations of certain Demazure crystals. Key words: cluster variables; double Bruhat cells; crystal bases; monomial realizations, generalized minors 2010 Mathematics Subject Classification: 13F60; 81R50; 17B37 1 Introduction As is well-known that theory of cluster algebras has been initiated by S. Fomin and A. Zelevinsky in the study of product expressions by q-commuting elements for upper global bases (= dual canonical bases). Crystal bases are obtained from global bases considering the parameter q at 0. Thus, we can guess that they should be deeply related each other at their origins. Let G be a simply connected simple algebraic group over C of rank r. Let B and B− be two opposite Borel subgroups in G, N ⊂ B and N− ⊂ B− their unipotent radicals, H := B ∩ B− a maximal torus, and W the associated Weyl group. In [1], it is shown that for u, v ∈ W the coordinate ring C[Gu,v] of double Bruhat cell Gu,v := BuB ∩ B−vB− has the structure of an upper cluster algebra. The initial cluster variables of this upper cluster algebras are given as certain generalized minors on Gu,v. In [11], the second author revealed the relations between some generalized minors and mono- mial realizations of crystal bases. A naive definition of monomial realizations of crystal bases is as follows (see Section 3 for the exact definitions): Let Y be the set of monomials in infinitely many variables (see Section 3, equation (3.2)). We shall define the crystal structures on Y as- sociated with certain set of integers p = (pi,j)1≤i 6=j≤r and a Cartan matrix. And we can obtain a crystal for an irreducible module as a connected component of Y. For example, for type A4 and pi,j = 1 if i < j and pi,j = 0 if i > j, we have the following crystal graph of the crystal B(Λ3), where Λ3 is the 3rd fundamental weight. The set of integers p gives the cyclic sequence of indices ?This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is available at http://www.emis.de/journals/SIGMA/LieTheory2014.html mailto:j_chi_sen_you_ky@sophia.ac.jp mailto:toshiki@sophia.ac.jp http://dx.doi.org/10.3842/SIGMA.2015.033 http://www.emis.de/journals/SIGMA/LieTheory2014.html 2 Y. Kanakubo and T. Nakashima . . . 123412341234 . . . and we associate variables {τj} as follows . . . 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 . . . . . . τ−4 τ−3 τ−2 τ−1 τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8 τ9 τ10 . . . Note that the skip between τ7 and τ8 or τ9 and τ10 means no corresponding variable appears in the following crystal graph τ−2 τ−1τ2 τ3 τ2 τ4 τ−1τ5 τ6 τ3τ5 τ4τ6 τ5 τ7 τ−1 τ8 τ3 τ4τ8 τ6 τ7τ8 1 τ9 , 3 // 4 // 2 �� 4 // 1 �� 4 // // 2 ��1 //3�� 2 // 3 �� 1 (1.1) where the highest weight monomial τ−2 has a weight Λ3 and the lowest weight monomial 1 τ9 has a weight −Λ2. As stated above, the initial cluster variables on Gu,v are expressed by generalized minors {∆(k; i) | 1 ≤ k ≤ l(u) + l(v)}, where i is the reduced expression of (u, v) ∈ W ×W . Now, as an example we consider the case G = SL5(C) as above. Let W = S5 = 〈si | 1 ≤ i ≤ 4〉 be the symmetric group and set u := s1s2s3s4s1s2s3s1s2s1, v := e, and set a reduced word i for u as i := (1, 2, 3, 4, 1, 2, 3, 1, 2, 1). We have for x = (xi,j) ∈ SL5(C) ∆(6; i)(x) = ∣∣∣∣x31 x32 x41 x42 ∣∣∣∣ = x31x42 − x32x41. (1.2) See Section 5 for the detailed explanation. Now, let us consider the generalized minors on Lu,v := NuN ∩B−vB− instead of Gu,v since their difference is, indeed, only the factor from the torus part. We call Lu,v reduced double Bruhat cell. For the above u, v, there exists a birational map xLi : (C×)10 ∼−→ Lu,v given by xLi (τ1, . . . , τ10) = x−1(τ1)x−2(τ2)x−3(τ3)x−4(τ4)x−1(τ5)x−2(τ6)x−3(τ7)x−1(τ8)x−2(τ9)x−1(τ10) =  1 τ1τ5τ8τ10 0 0 0 0 A τ1τ5τ8τ10 τ2τ6τ9 0 0 0 B C τ2τ6τ9 τ3τ7 0 0 D E τ3τ9 τ4 + τ6τ9 τ7 τ3τ7 τ4 0 1 τ10 τ9 τ7 τ4  , (1.3) where A = τ1τ5τ8 τ2τ6τ9 + τ1τ5 τ2τ6τ10 + τ1 τ2τ8τ10 + 1 τ5τ8τ10 , B = τ2τ6 τ3τ7 + τ2τ8 τ3τ9 + τ5τ8 τ6τ9 + τ2 τ3τ10 + τ5 τ6τ10 + 1 τ8τ10 , C = τ2τ6τ10 τ3τ7 + τ2τ8τ10 τ3τ9 + τ5τ8τ10 τ6τ9 , D = τ3 τ4 + τ6 τ7 + τ8 τ9 + 1 τ10 , E = τ3τ10 τ4 + τ6τ10 τ7 + τ8τ10 τ9 , Cluster Variables and Monomial Realizations of Crystal Bases 3 x−i(t) = ith  . . . t−1 0 1 t . . .  . Therefore, by (1.2) and (1.3) we find ∆L(6; i)(τ) := ( ∆(6; i) ◦ xLi ) (τ1, . . . , τ10) = ∣∣∣∣B C D E ∣∣∣∣ = τ2 τ4 + τ3τ5 τ4τ6 + τ5 τ7 + τ3 τ4τ8 + τ6 τ7τ8 + 1 τ9 . (1.4) Now, observing the crystal graph (1.1) and the Laurent polynomial (1.4), we realize that each term in (1.4) appears in (1.1) and they constitute so-called lower Demazure crystal associated with the element u≤6 ∈ S5 [7]. Those facts motivate us to find a new linkage between the cluster variables on Lu,v ⊂ Gu,v and the monomial realizations of crystals. In this paper, we shall treat the case G = SLr+1(C), v = e and some special u ∈W = Sr+1. More precisely, we treat an element u ∈W whose reduced word (Definition 2.1) can be written as a left factor of the standard longest word (1, 2, 3, . . . , r, 1, 2, 3, . . . , (r − 1), . . . , 1, 2, 1): u = s1s2 · · · srs1 · · · sr−1 · · · s1 · · · sr−m+2s1 · · · sin , where n := l(u) is the length of u and 1 ≤ in ≤ r−m+1. And we treat (reduced) double Bruhat cells of the form Gu,e := BuB∩B− and Lu,e := NuN∩B−, where B (resp. B−) is the subgroup of upper (resp. lower) triangular matrices in G = SLr+1(C). Then generalized minors are a part of classical minors (Definition 4.10). This case matches well to the Demazure crystals. In fact, we shall describe generalized minors in terms of summations over certain monomial realizations of Demazure crystals in the main result Theorem 5.6. For example, (1.4) shows that the generalized minor ∆L(6; i)(τ) is described in terms of summation over certain monomial realization of the Demazure crystal B−u≤6(−Λ2). In forthcoming paper, we shall treat more general setting, like as, the Weyl group element v ∈ W is non-identity or type C. In these cases, the generalized minors are described also by monomial realizations of crystals. 2 Factorization theorem for type A In this section, we shall introduce (reduced) double Bruhat cells Gu,v, Lu,v, and their properties in the case G = SLr+1(C), v = e and some special u ∈ W . In [2, 3], these properties had been proven for simply connected, connected, semisimple complex algebraic groups and arbitrary u, v ∈W . For l ∈ Z>0, we set [1, l] := {1, 2, 3, . . . , l}. 2.1 Double Bruhat cells Let G = SLr+1(C) be the simple complex algebraic group of type Ar, B and B− be two opposite Borel subgroups in G, that is, B (resp. B−) is the subgroup of upper (resp. lower) triangular matrices in G = SLr+1(C). Let N ⊂ B and N− ⊂ B− be their unipotent radicals, H := B ∩B− a maximal torus, and W := NormG(H)/H the Weyl group. In this case, Weyl group W is isomorphic to the symmetric group Sr+1. 4 Y. Kanakubo and T. Nakashima We have two kinds of Bruhat decompositions of G as follows G = ∐ u∈W BuB = ∐ u∈W B−uB−. Then, for u, v ∈W , we define the double Bruhat cell Gu,v as follows Gu,v := BuB ∩B−vB−. This is biregularly isomorphic to a Zariski open subset of an affine space of dimension r+ l(u)+ l(v) [3, Theorem 1.1]. We also define the reduced double Bruhat cell Lu,v as follows Lu,v := NuN ∩B−vB− ⊂ Gu,v. As is similar to the case Gu,v, Lu,v is biregularly isomorphic to a Zariski open subset of an affine space of dimension l(u) + l(v) [2, Proposition 4.4]. Definition 2.1. Let u = si1 · · · sin be a reduced expression of u ∈ W , i1, . . . , in ∈ [1, r]. Then the finite sequence i := (i1, . . . , in) is called reduced word i for u. In this paper, we treat (reduced) double Bruhat cells of the form Gu,e := BuB ∩ B− and Lu,e := NuN ∩ B−, where u ∈ W is an element whose reduced word can be written as a left factor of (1, 2, 3, . . . , r, 1, 2, 3, . . . , (r − 1), . . . , 1, 2, 1): u = s1s2 · · · srs1 · · · sr−1 · · · s1 · · · sr−m+2s1 · · · sin , (2.1) where n := l(u) is the length of u and 1 ≤ in ≤ r −m+ 1. Let i be a reduced word of u: i = (1, . . . , r︸ ︷︷ ︸ 1st cycle , 1, . . . , (r − 1)︸ ︷︷ ︸ 2nd cycle , . . . , 1, . . . , (r −m+ 2)︸ ︷︷ ︸ (m− 1)th cycle , 1, . . . , in︸ ︷︷ ︸ mth cycle ). (2.2) Note that (1, 2, 3, . . . , r, 1, 2, 3, . . . , (r − 1), . . . , 1, 2, 1) is a reduced word of the longest element in W . 2.2 Factorization theorem for type A In this subsection, we shall introduce the isomorphisms between double Bruhat cell Gu,e and H × (C×)l(u), and between Lu,e and (C×)l(u). As in the previous section, we consider the case G := SLr+1(C). We set g := Lie(G) with the Cartan decomposition g = n− ⊕ h ⊕ n. Let ei, fi (i ∈ [1, r]) be the generators of n, n−. For i ∈ [1, r] and t ∈ C, we set xi(t) := exp(tei), yi := exp(tfi). Let ϕi : SL2(C) → G be the canonical embedding corresponding to each simple root αi. Then we have xi(t) = ϕi ( 1 t 0 1 ) , yi(t) = ϕi ( 1 0 t 1 ) . We can express xi(t), yi(t), as the following matrices xi(t) = ith  . . . 1 t 0 1 . . .  , yi(t) = ith  . . . 1 0 t 1 . . .  . (2.3) Cluster Variables and Monomial Realizations of Crystal Bases 5 For a reduced word i = (i1, i2, . . . , in), we define a map xGi : H × Cn → G as xGi (a; t1, . . . tn) := a · yi1(t1)yi2(t2) · · · yin(tn). Theorem 2.2 ([3, Theorem 1.2]). We set u ∈ W and its reduced word i as in (2.1) and (2.2). The map xGi defined above can be restricted to a biregular isomorphism between H × (C×)l(u) and a Zariski open subset of Gu,e. Next, for i ∈ [1, r] and t ∈ C×, we define as follows α∨i (t) := ϕi ( t 0 0 t−1 ) , x−i(t) := yi(t)α ∨ i (t−1) = ϕi ( t−1 0 1 t ) . We can express x−i(t) and α∨i (t) as the following matrices x−i(t) = ith  . . . t−1 0 1 t . . .  , α∨i (t) = diag(1, . . . , 1, i ∨ t, i+1 ∨ t−1, 1, . . . , 1). (2.4) For i = (i1, . . . , in) (i1, . . . , in ∈ [1, r]), we define a map xLi : Cn → G as xLi (t1, . . . , tn) := x−i1(t1) · · ·x−in(tn). We have the following theorem which is similar to the previous one. Theorem 2.3 ([2, Proposition 4.5]). We set u ∈W and its reduced word i as in (2.1) and (2.2). The map xLi defined above can be restricted to a biregular isomorphism between (C×)l(u) and a Zariski open subset of Lu,e. Finally, we define a map x̄Gi : H × (C×)n → Gu,v as x̄Gi (a; t1, . . . , tn) = axLi (t1, . . . , tn). Proposition 2.4. In the above setting, the map x̄Gi is a biregular isomorphism between H × (C×)n and a Zariski open subset of Gu,e. Proof. We set l0 := 0, l1 := r, l2 := r + (r − 1), . . . , lm := r + (r − 1) + · · ·+ (r −m+ 1). We define a map φ : H × (C×)n → H × (C×)n, t = (a; t1, . . . , tn) 7→ (a(t); τ1(t), . . . , τn(t)) as a(t) = a · α∨1 (t1)−1 · · ·α∨r (tr) −1︸ ︷︷ ︸ 1st cycle · · ·α∨1 (tlm−1+1)−1 · · ·α∨in(tlm−1+in)−1︸ ︷︷ ︸ mth cycle , τls+j(t) = (tls+1+j−1tls+2+j−1 · · · tlm−1+j−1)(tls+j+1tls+1+j+1 · · · tlm−1+j+1) tls+j(tls+1+j · · · tlm−1+j)2 , (2.5) where in (2.5), if i does not include j (resp. j + 1, j − 1) in ζth cycle then we set tlζ−1+j = 1 (resp. tlζ−1+j+1 = 1, tlζ−1+j−1 = 1). This is a biregular isomorphism. Let us prove x̄Gi (a; t1, . . . , tn) = ( xGi ◦ φ ) (a; t1, . . . , tn), which implies that x̄Gi : H × (C×)n → Gu,e is a biregular isomorphism by Theorem 2.2. 6 Y. Kanakubo and T. Nakashima First, we can verify the following relations by the explicit forms (2.3), (2.4) and direct calcu- lations: α∨i (c)−1yj(t) =  yi(c 2t)α∨i (c)−1 if i = j, yj(c −1t)α∨i (c)−1 if |i− j| = 1, yj(t)α ∨ i (c)−1 otherwise, (2.6) for 1 ≤ i, j ≤ r and c, t ∈ C×. On the other hand, we obtain (xGi ◦ φ)(a; t1, . . . , tn) = a× α∨1 (t1)−1 · · ·α∨r (tr) −1 · · ·α∨1 (tlm−1+1)−1 · · ·α∨in(tlm−1+in)−1 × y1(τ1(t))y2(τ2(t)) · · · yr(τr(t)) · · · y1(τlm−1+1(t)) · · · yin(τlm−1+in(t)). (2.7) For each s and j, let us move α∨j (tls+j) −1, α∨j+1(tls+j+1)−1, . . . , α∨in(tlm−1+in)−1 to the right of yj(τls+j(t)) by using the relations (2.6). For example, α∨j (tls+j) −1 · · ·α∨j−1(tlm−1+j−1)−1α∨j (tlm−1+j) −1 × α∨j+1(tlm−1+j+1)−1 · · ·α∨in(tlm−1+in)−1yj(τls+j(t)) = α∨j (tls+j) −1 · · · yj ( t2lm−1+j tlm−1+j−1tlm−1+j+1 τls+j(t) ) α∨j−1(tlm−1+j−1)−1 × α∨j (tlm−1+j) −1α∨j+1(tlm−1+j+1)−1 · · ·α∨in(tlm−1+in)−1. Repeating this argument, we have = yj ( (tls+j · · · tlm−1+j) 2 (tls+j−1 · · · tlm−1+j−1)(tls+j+1 · · · tlm−1+j+1) τls+j(t) ) α∨j (tls+j) −1 · · ·α∨in(tlm−1+in)−1. Note that (tls+j ...tlm−1+j )2 (tls+j−1···tlm−1+j−1)(tls+j+1···tlm−1+j+1)τls+j(t) = tls+j . By (2.7), we have( xGi ◦ φ ) (a; t1, . . . , tn) = a · y1(t1)α∨1 (t1)−1 · · · yr(tr)α∨r (tr) −1 · · · × y1(tlm−1+1)α∨1 (tlm−1+1)−1 · · · yin(tlm−1+in)α∨in(tlm−1+in)−1 = a · x−1(t1) · · ·x−r(tr) · · ·x−1(tlm−1+1) · · ·x−in(tlm−1+in) = x̄Gi (a; t1, . . . , tn). � 3 Monomial realizations of crystal bases In this section, we review the monomial realizations of crystals [6, 8, 10]. Let I := {1, 2, . . . , r} be a finite index set. 3.1 Monomial realizations of crystal bases for type A Definition 3.1. Let A = (aij)i,j∈I be the Cartan matrix of type Ar: A = (aij)i,j∈I is defined as aij =  2 if i = j, −1 if |i− j| = 1, 0 otherwise. (3.1) Let Π = {αi | i ∈ I} (resp. Π∨ = {hi | i ∈ I}) be the set of simple roots (resp. co-roots), and P be the weight lattice. A crystal associated with the Cartan matrix A is a set B together with the maps wt : B → P , ẽi, f̃i : B ∪ {0} → B ∪ {0} and εi, ϕi : B → Z ∪ {−∞}, i ∈ I, satisfying the following properties: For b ∈ B, i ∈ I, Cluster Variables and Monomial Realizations of Crystal Bases 7 (i) ϕi(b)− εi(b) = 〈hi, wt(b)〉, (ii) wt(ẽib) = wt(b) + αi, if ẽib ∈ B, (iii) wt(f̃ib) = wt(b)− αi, if f̃ib ∈ B, (iv) εi(ẽib) = εi(b)− 1, ϕi(ẽib) = ϕi(b) + 1 if ẽib ∈ B, (v) εi(f̃ib) = εi(b) + 1, ϕi(f̃ib) = ϕi(b)− 1 if f̃ib ∈ B, (vi) f̃ib = b′ ⇔ b = ẽib ′, if b, b′ ∈ B, (vii) ϕi(b) = −∞, b ∈ B, ⇒ ẽib = f̃ib = 0. Let Uq(g) be the universal enveloping algebra associated with the Cartan matrix A in (3.1), and g = slr+1(C). Let B+(λ) (resp. B−(λ)) be the crystal base of the Uq(g)-highest (resp. lowest) weight module [5, 9]. Note that B+(λ) = B−(w0λ), where w0 is the longest element of W . In particular, in the case λ = MΛd, M ∈ Z>0, we have B+(MΛd) = B−(−MΛr−d+1). Let us introduce monomial realizations which realize each element of B±(λ) as a certain Laurent monomial. First, we define a set of integers p = (pj,i)j,i∈I, j 6=i such that pj,i = { 1 if j < i, 0 if i < j. Second, for doubly-indexed variables {Ys,i | i ∈ I, s ∈ Z}, we define the set of monomials Y := Y = ∏ s∈Z, i∈I Y ζs,i s,i ∣∣∣∣∣ ζs,i ∈ Z, ζs,i = 0 except for finitely many (s, i)  . (3.2) Finally, we define maps wt : Y → P , εi, ϕi : Y → Z, i ∈ I. For Y = ∏ s∈Z, i∈I Y ζs,i s,i ∈ Y, wt(Y ) := ∑ i,s ζs,iΛi, ϕi(Y ) := max ∑ k≤s ζk,i | s ∈ Z  , εi(Y ) := ϕi(Y )− wt(Y )(hi). We set As,i := Ys,iYs+1,i ∏ j 6=i Y aj,i s+pj,i,j =  Ys,1Ys+1,1 Ys,2 if i = 1, Ys,iYs+1,i Ys,i+1Ys+1,i−1 if 2 ≤ i ≤ r − 1, Ys,rYs+1,r Ys+1,r−1 if i = r, (3.3) and define the Kashiwara operators as follows f̃iY = { A−1 nfi ,i Y if ϕi(Y ) > 0, 0 if ϕi(Y ) = 0, ẽiY = { Anei ,iY if εi(Y ) > 0, 0 if εi(Y ) = 0, where nfi := min n ∣∣∣∣∣ϕi(Y ) = ∑ k≤n ζk,i  , nei := max n ∣∣∣∣∣ϕi(Y ) = ∑ k≤n ζk,i  . (3.4) Then the following theorem holds: 8 Y. Kanakubo and T. Nakashima Theorem 3.2 ([8, 10]). (i) For the set p = (pj,i) as above, (Y,wt, ϕi, εi, f̃i, ẽi)i∈I is a crystal. When we emphasize p, we write Y as Y(p). (ii) If a monomial Y ∈ Y(p) satisfies εi(Y ) = 0 (resp. ϕi(Y ) = 0) for all i ∈ I, then the connected component containing Y is isomorphic to B+(wt(Y )) (resp. B−(wt(Y ))). Definition 3.3. Let Y ∈ Y(p) be a monomial and let B be the unique connected component in Y(p) including Y . Suppose that λ is the highest (resp. lowest) weight of B. We denote the embedding µY : B+(λ) ↪→ B ⊂ Y(p) (resp. µY : B−(λ) ↪→ B). Note that if Y and Y ′ are in the same component then µY = µY ′ . Remark 3.4. The actions of ẽi and f̃i on Y are determined by wt(Y )(hi), ϕi(Y ) and εi(Y ), which are determined by the factors Y ±1 s,i , s ∈ Z. Thus, when we consider the actions of ẽi and f̃i, we need to see the factors {Y ±1 s,i }s∈Z only. Example 3.5. For λ = βΛd (resp. λ = −βΛd), β ∈ Z>0, d ∈ I, we can embed B+(λ) (resp. B−(λ)) in Y as a crystal by vλ 7→ Yβ+γ,dYβ−1+γ,d · · ·Y1+γ,d, ( resp. vλ 7→ 1 Yβ+γ,dYβ−1+γ,d · · ·Y1+γ,d ) , where vλ is the highest (resp. lowest) weight vector of B+(λ) (resp. B−(λ)), and γ is an ar- bitrary integer. For Y + := Yβ+γ,dYβ−1+γ,d · · ·Y1+γ,d (resp. Y − := 1 Yβ+γ,dYβ−1+γ,d···Y1+γ,d ), µY + (resp. µY −) denotes the embedding in Definition 3.3. Then Y + (resp. Y −) is the highest (resp. lowest) weight vector in µY +(B+(λ)) (resp. µY −(B−(λ))). We set l0 := 0, l1 := r, l2 := r + (r − 1), . . . , ls := r + (r − 1) + · · · + (r − s + 1), . . . , lr := r + (r − 1) + · · ·+ 2 + 1 and changing the variables Ys,j to τls+j , 1 ≤ j ≤ r − s. For s < 0, we transform the variables Ys,j to τ−(r+1−j), 1 ≤ j ≤ r, . . . r 1 . . . r − 1 r 1 . . . r − 1 r 1 2 . . . . . . τ−1 τl0+1 . . . τl0+r−1 τl0+r τl1+1 . . . τl1+r−1 τl2+1 τl2+2 . . . Remark 3.6. In the above setting, the variables {Ys,j | r − s < j} do not correspond to any variables in τ . As we have seen in (1.1), these variables do not appear in the crystal base which we treat in this paper. In other words, we only need variables associated with j = (1, . . . , r, 1, . . . , r − 1, . . . , 1, 2, 1), which coincides with a specific reduced word of the longest element of W . Remark 3.7. For the variables τls+0, τls+r+1 (0 ≤ s ≤ m− 1) we understand τls+0 = τls+r+1 = 1. For example, if i = 1 then τls+i−1 = 1. Cluster Variables and Monomial Realizations of Crystal Bases 9 Example 3.8. Let us consider the action of ẽ1 on the monomial 1 τlr−1+1 . Following the method in Section 3.1, we have wt( 1 τlr−1+1 ) = −Λ1, ϕ1( 1 τlr−1+1 ) = 0, ε1( 1 τlr−1+1 ) = ϕ1( 1 τlr−1+1 ) − wt( 1 τlr−1+1 )(h1) = 1, and ne1 = r − 2. Thus, since we have Ar−2,1 = τlr−2+1τlr−1+1τ a2,1 lr−2+p2,1+2 = τlr−2+1τlr−1+1 τlr−2+2 , we get ẽ1 1 τlr−1+1 = Ar−2,1 1 τlr−1+1 = τlr−2+1 τlr−2+2 . Similarly, we have ẽ2ẽ1 1 τlr−1+1 = Ar−3,2 τlr−2+1 τlr−2+2 = τlr−3+2 τlr−3+3 , Ar−3,2 = τlr−3+2τlr−2+2 τlr−3+3τlr−2+1 , ẽ3ẽ2ẽ1 1 τlr−1+1 = Ar−4,3 τlr−3+2 τlr−3+3 = τlr−4+3 τlr−4+4 , Ar−4,3 = τlr−4+3τlr−3+3 τlr−4+4τlr−3+2 . Applying ẽi repeatedly, we obtain ẽk · · · ẽ2ẽ1 1 τlr−1+1 = Ar−1−k,kẽk−1 · · · ẽ2ẽ1 1 τlr−1+1 = τlr−1−k+k τlr−1−k+k+1 , k = 1, . . . , r, where, Ar−1−k,k = τlr−1−k+kτlr−k+k τlr−1−k+k+1τlr−k+k−1 , τl−1+r = r, τl−1+r+1 := 1. For i ∈ I, we have ϕi( 1 τlr−1+1 ) = 0. Hence f̃i( 1 τlr−1+1 ) = 0. Example 3.9. For a given i ∈ I and Y = ∏ s∈Z τ ζs,i ls+i , we define νY (n) := ∑ s≤n ζs,i. For j ∈ Z>0, we set Y = 1 τlq1+iτlq2+i · · · τlqj+i , 0 ≤ q1 < q2 < · · · < qj ≤ r − 1. First, let us calculate nei (3.4). We obtain wt(Y ) = −jΛi and νY (n) = 0 for n < 0, νY (0) = νY (1) = · · · = νY (q1 − 1) = 0, νY (q1) = νY (q1 + 1) = · · · = νY (q2 − 1) = −1, νY (q2) = νY (q2 + 1) = · · · = νY (q3 − 1) = −2, νY (q3) = · · · = νY (q4 − 1) = −3, . . . . Thus, we get ϕi(Y ) = max{νY (n) |n ∈ Z} = 0 and nei = max{n | νY (n) = 0} = q1 − 1. Next, since wt(Y )(hi) = −j, we have εi(Y ) = ϕi(Y )− wt(Y )(hi) = j > 0. Therefore, ẽiY = Aq1−1,iY = τlq1−1+i τlq1−1+i+1τlq1+i−1τlq2+i · · · τlqj+i , Aq1−1,i = τlq1−1+iτlq1+i τlq1−1+i+1τlq1+i−1 . Similarly, for k = 1, 2, . . . , j, we get ẽki Y = Aqk−1,i · · ·Aq2−1,iAq1−1,iY = k∏ s=1 ( τlqk−1+i τlqk−1+i+1τlqk+i−1 ) 1 τlqk+1 +i · · · τlqj+i . 10 Y. Kanakubo and T. Nakashima 3.2 Demazure crystal For w ∈ W , let us define an upper Demazure crystal B+ w (λ). This is a subset of the crystal B+(λ) defined as follows. Definition 3.10. Let uλ be the highest weight vector of B+(λ). For the identity element e of W , we set B+ e (λ) := {uλ}. For w ∈W , if siw < w, B+ w (λ) := { f̃ki b | k ≥ 0, b ∈ B+ siw(λ), ẽib = 0 } \ {0}. Similarly, we define a lower Demazure crystal B−w (λ) as follows. Definition 3.11. Let vλ be the lowest weight vector of B−(λ). We set B−e (λ) := {vλ}. For w ∈W , if siw < w, B−w (λ) := { ẽki b | k ≥ 0, b ∈ B−siw(λ), f̃ib = 0 } \ {0}. Theorem 3.12 ([7]). For w ∈W , let w = si1 · · · sin be an arbitrary reduced expression. Let uλ (resp. vλ′) be the highest (resp. lowest) weight vector of B+(λ) (resp. B−(λ′)). Then B+ w (λ) = { f̃ a(1) i1 · · · f̃a(n) in uλ | a(1), . . . , a(n) ∈ Z≥0 } \ {0}, B−w (λ′) = { ẽ a(1) i1 · · · ẽa(n) in vλ′ | a(1), . . . , a(n) ∈ Z≥0 } \ {0}. Let P+ be the set of dominant weights. We set P− := −P+. Definition 3.13. Let Y(p) be the monomial realization of crystal associated with p = (pi,j). Suppose that Y ∈ Y(p) be a highest (resp. lowest) monomial with a weight λ ∈ P±. Thus, Y is included in µY (B±w (λ)). Let us define the Demazure polynomial D±w [λ, Y ;C] associated with a monomial Y , w ∈W and coefficients C = (c(b))b∈B±w (λ) (c(b) ∈ Z>0), D±w [λ, Y ;C] := ∑ b∈B±w (λ) c(b)µY (b). Remark 3.14. In this paper, we only treat the case that the coefficients c(b) are equal to 1 for all b ∈ B−w (λ) (see Theorem 5.6). But when G 6= SLr+1(C) (for example, G = Sp2r(C)), we need to treat the case c(b) is not necessary equal to 1 for some b ∈ B−w (λ). Therefore, we need non-trivial coefficients c(b) ∈ Z>0 in Definition 3.13. 4 Cluster algebras and generalized minors In this section, we shall review the notions of cluster algebras. For all definitions in this section, see, e.g., [1, 4]. We set [1, l] := {1, 2, . . . , l} and [−1,−l] := {−1,−2, . . . ,−l} for l ∈ Z>0. For n,m ∈ Z>0, let x1, . . . , xn, xn+1, . . . , xn+m be variables and P be a free multiplicative abelian group generated by xn+1, . . . , xn+m. We set ZP := Z[x±1 n+1, . . . , x ±1 n+m]. Let K := { gh | g, h ∈ ZP, h 6= 0} be the field of fractions of ZP, and F := K(x1, . . . , xn) be the field of rational functions. 4.1 Cluster algebras of geometric type Definition 4.1. We set n-tuple of variables x = (x1, . . . , xn). Let B̃ = (bij)1≤i≤n, 1≤j≤n+m be n × (n + m) integer matrix whose principal part B := (bij)1≤i,j≤n is sign skew symmetric. Then a pair Σ = (x, B̃) is called a seed, x a cluster and x1, . . . , xn cluster variables. For a seed Σ = (x, B̃), principal part B of B̃ is called the exchange matrix. Cluster Variables and Monomial Realizations of Crystal Bases 11 Definition 4.2. For a seed Σ = (x, B̃ = (bij)), an adjacent cluster in direction k ∈ [1, n] is defined by xk = (x \ {xk}) ∪ {x′k}, where x′k is the new cluster variable defined by the exchange relation xkx ′ k = ∏ 1≤i≤n+m, bki>0 xbkii + ∏ 1≤i≤n+m, bki<0 x−bkii . Definition 4.3. Let A = (aij), A ′ = (a′ij) be two matrices of the same size. We say that A′ is obtained from A by the matrix mutation in direction k, and denote A′ = µk(A) if a′ij = −aij if i = k or j = k, aij + |aik|akj + aik|akj | 2 otherwise. For A, A′, if there exists a finite sequence (k1, . . . , ks), ki ∈ [1, n], such that A′ = µk1 · · ·µks(A), we say A is mutation equivalent to A′, and denote A ∼= A′. Next proposition can be easily verified by the definition of µk: Proposition 4.4 ([4, Proposition 3.6]). Let A be a skew symmetrizable matrix. Then any matrix that is mutation equivalent to A is sign skew symmetric. For a seed Σ = (x, B̃), we say that the seed Σ′ = (x′, B̃′) is adjacent to Σ if x′ is adjacent to x in direction k and B̃′ = µk(B̃). Two seeds Σ and Σ0 are mutation equivalent if one of them can be obtained from another seed by a sequence of pairwise adjacent seeds and we denote Σ ∼ Σ0. Now let us define a cluster algebra of geometric type. Definition 4.5. Let B̃ be a skew symmetrizable matrix, and Σ = (x, B̃) a seed. We set A := Z[xn+1, . . . , xn+m]. The cluster algebra (of geometric type) A = A(Σ) over A associated with seed Σ is defined as the A-subalgebra of F generated by all cluster variables in all seeds which are mutation equivalent to Σ. For a seed Σ, we define ZP-subalgebra U(Σ) of F by U(Σ) := ZP [ x±1 ] ∩ ZP [ x±1 1 ] ∩ · · · ∩ ZP [ x±1 n ] . Here, ZP[x±1] is the Laurent polynomial ring in x. Definition 4.6. Let Σ0 = (x, B̃) be a seed such that B̃ is skew symmetrizable. We define an upper cluster algebra A = A(Σ0) as the intersection of the subalgebras U(Σ) for all seeds Σ ∼ Σ0. Following the inclusion relation holds [1]: A(Σ) ⊂ A(Σ). 4.2 Cluster algebras on double Bruhat cells of type A As in Section 2, let G = SLr+1(C) be the simple algebraic group of type Ar and W = Sr+1 be its Weyl group. We set u ∈W and its reduced word i as in (2.1) and (2.2): u = s1s2 · · · sr︸ ︷︷ ︸ 1st cycle s1 · · · sr−1︸ ︷︷ ︸ 2nd cycle · · · s1 · · · sr−m+2︸ ︷︷ ︸ (m− 1)th cycle s1 · · · sin︸ ︷︷ ︸ mth cycle , (4.1) 12 Y. Kanakubo and T. Nakashima i = (1, . . . , r︸ ︷︷ ︸ 1st cycle , 1, . . . , (r − 1)︸ ︷︷ ︸ 2nd cycle , . . . , 1, . . . , (r −m+ 2)︸ ︷︷ ︸ (m− 1)th cycle , 1, . . . , in︸ ︷︷ ︸ mth cycle ). (4.2) We shall constitute the upper cluster algebra A(i) from i. Let ik, k ∈ [1, l(u)], be the kth index of i from the left. At first, we define a set e(i) as e(i) := [−1,−r] ∪ {k | there exist some l > k such that ik = il}. Next, let us define a matrix B̃ = B̃(i). Definition 4.7. Let B̃(i) be an integer matrix with rows labeled by all the indices in [−1,−r]∪ [1, l(u)] and columns labeled by all the indices in e(i). For k ∈ [−1,−r] ∪ [1, l(u)] and l ∈ e(i), an entry bkl of B̃(i) is determined as follows bkl =  − sgn((k − l) · ip) if p = q, − sgn((k − l) · ip · a|ik||il|) if p < q and sgn(ip · iq)(k − l)(k+ − l+) > 0, 0 otherwise. Proposition 4.8 ([1, Proposition 2.6]). The matrix B̃(i) is skew symmetrizable. By Proposition 4.4, Definition 4.6 and Proposition 4.8, we can construct the upper cluster algebra from B̃(i): Definition 4.9. We denote this upper cluster algebra by A(i). Now, we set Ā(i)C := Ā(i)⊗C and FC := F⊗C. It is known that the coordinate ring C[Gu,e] of the double Bruhat cell is isomorphic to Ā(i)C (Theorem 4.11). To describe this isomorphism explicitly, we need generalized minors. For k ∈ [1, l(u)], let ik be the kth index of i (4.2) from the left, and we suppose that it belongs to the m′th cycle. We set u≤k = u≤k(i) := s1s2 · · · sr︸ ︷︷ ︸ 1st cycle s1 · · · sr−1︸ ︷︷ ︸ 2nd cycle · · · s1 · · · sik︸ ︷︷ ︸ m′th cycle . (4.3) For k ∈ [−1,−r], we set u≤k := e and ik := k. In the case G = SLr+1(C), the generalized minors are nothing but the ordinary minors of a matrix: Definition 4.10 ([1]). For x ∈ G = SLr+1(C) and k ∈ [−1,−r] ∪ [1, l(u)], we define the generalized minor ∆(k; i)(x) as the minor of x whose rows (resp. columns) are labeled by the elements of the set u≤k([1, |ik|]) (resp. [1, |ik|]). Finally, we set F (i) := {∆(k; i) | k ∈ [−1,−r] ∪ [1, l(u)]}. It is known that the set F (i) is an algebraically independent generating set for the field of rational functions C(Gu,e) [3, Theorem 1.12]. Then, we have the following theorem. Theorem 4.11 ([1, Theorem 2.10]). The isomorphism of fields ϕ : FC → C(Gu,e) defined by ϕ(xk) = ∆(k; i), k ∈ [−1,−r]∪[1, l(u)], restricts to an isomorphism of algebras Ā(i)C → C[Gu,e]. Cluster Variables and Monomial Realizations of Crystal Bases 13 5 Generalized minors and crystals In the rest of the paper, we consider the case G = SLr+1(C), and let u ∈ W and its reduced word i as in (4.1) and (4.2): u = s1s2 · · · srs1 · · · sr−1 · · · s1 · · · sr−m+2s1 · · · sin , (5.1) i = (1, . . . , r︸ ︷︷ ︸ 1st cycle , 1, . . . , (r − 1)︸ ︷︷ ︸ 2nd cycle , . . . , 1, . . . , (r −m+ 2)︸ ︷︷ ︸ (m− 1)th cycle , 1, . . . , in︸ ︷︷ ︸ mth cycle ), (5.2) that is, i is the left factor of (1, 2, 3, . . . , r, 1, 2, 3, . . . , (r− 1), . . . , 1, 2, 1). Let ik be the kth index of i from the left, and belong to m′th cycle. As we shall show in Lemma 5.4, we may assume in = ik. By Theorem 4.11, we can regard C[Gu,e] as an upper cluster algebra and {∆(k; i)} as its cluster variables belonging to a given initial seed. Each ∆(k; i) is a regular function on Gu,e. On the other hand, by Proposition 2.4 (resp. Theorem 2.3), we can consider ∆(k; i) as a function on H × (C×)l(u) (resp. (C×)l(u)). Then we change the variables of {∆(k; i)} as follows: Definition 5.1. For a ∈ H and t, τ ∈ (C×)l(u) we set ∆G(k; i)(a, t) := ( ∆(k; i) ◦ x̄Gi ) (a, t), ∆L(k; i)(τ) := ( ∆(k; i) ◦ xLi ) (τ), where t = (t1, . . . , tl(u)), τ = (τ1, . . . , τl(u)). We will describe the function ∆L(k; i)(τ) by using monomial realizations of Demazure crys- tals. 5.1 Generalized minor ∆G(k; i)(a, t) In this subsection, we shall prove that ∆G(k; i)(a, t) is immediately obtained from ∆L(k; i): Proposition 5.2. We set d := ik. For a = diag(a1, . . . , ar+1) ∈ H, ∆G(k; i)(a, t) = am′+1 · · · am′+d∆L(k; i)(t). This proposition follows from the following lemma: Lemma 5.3. In the above setting, ∆G(k; i)(a, t) (resp. ∆L(k; i)(τ)) is given as a minor whose row are labeled by the set {m′ + 1, . . . ,m′ + d} and column are labeled by the set {1, . . . , d} of the matrix a x−1(t1)x−2(t2) · · ·x−r(tr)︸ ︷︷ ︸ 1st cycle x−1(tl1+1) · · ·x−(r−1)(tl1+r−1)︸ ︷︷ ︸ 2nd cycle · · · × x−1(tlm−1+1) · · ·x−in(tlm−1+in)︸ ︷︷ ︸ mth cycle , resp. x−1(τ1)x−2(τ2) · · ·x−r(τr)︸ ︷︷ ︸ 1st cycle x−1(τl1+1) · · ·x−(r−1)(τl1+r−1)︸ ︷︷ ︸ 2nd cycle · · · × x−1(τlm−1+1) · · ·x−in(τlm−1+in)︸ ︷︷ ︸ mth cycle . (5.3) 14 Y. Kanakubo and T. Nakashima Proof. Let us prove this lemma for ∆L(k; i)(τ) since the case for ∆G(k; i)(a, t) is proven simi- larly. By the definition (4.3) of u≤k and ik = d, we have u≤k[1, |ik|] = u≤k{1, . . . , d} = s1 · · · sr︸ ︷︷ ︸ 1st cycle · · · s1 · · · sr−m′+2︸ ︷︷ ︸ (m′ − 1)th cycle s1 · · · sd︸ ︷︷ ︸ m′th cycle {1, . . . , d} = {m′ + 1, . . . ,m′ + d}. Hence, by Theorem 2.3 and Definition 4.10, ∆L(k; i)(τ) is given as a minor whose row (resp. column) are labeled by the set {m′ + 1, . . . ,m′ + d} (resp. {1, . . . , d}) of the matrix x−1(τ1)x−2(τ2) · · ·x−r(τr)︸ ︷︷ ︸ 1st cycle x−1(τl1+1) · · ·x−(r−1)(τl1+r−1)︸ ︷︷ ︸ 2nd cycle · · · × x−1(τlm−1+1) · · ·x−in(τlm−1+in)︸ ︷︷ ︸ mth cycle , which implies the desired result. � In the rest of the paper, we will treat ∆L(k; i)(τ) only by Proposition 5.2. 5.2 Generalized minor ∆L(k; i)(τ ) Lemma 5.4. Let u and i = (1, . . . , in) be as in the form (5.1) and (5.2) respectively. For in+1 ∈ [1, r] and u′ := usin+1 ∈W (l(u′) > l(u)) we set the reduced word for u′ as i′ := (1, . . . , in, in+1). We also set τ = (τ1, . . . , τn) and τ ′ = (τ1, . . . , τn, τn+1). For an integer k, 1 ≤ k ≤ n, if d := ik 6= in+1, then ∆L(k; i′)(τ ′) does not depend on τn+1, so we can regard it as a function on H × (C×)n. Furthermore, we have ∆L(k; i)(τ) = ∆L(k; i′)(τ ′). Proof. We denote the matrix (5.3) by T = (Ti,j)i,j∈[1,r+1]: T = (Ti,j)i,j∈[1,r+1] := x−1(τ1)x−2(τ2) · · ·x−in(τn). We also define the submatrix Tu of the matrix T whose rows (resp. columns) are labeled by the set {m′ + 1, . . . ,m′ + d} (resp. {1, . . . , d}), that is Tu := Tm′+1,1 . . . Tm′+1,d ... ... ... Tm′+d,1 . . . Tm′+d,d  . By Lemma 5.3, ∆L(k; i)(τ) is the determinant of Tu. Similarly, we define the submatrix T ′u′ of the matrix T ′ := T · x−in+1(τn+1) whose rows (resp. columns) are labeled by the set {m′ + 1, . . . ,m′ + d} (resp. {1, . . . , d}). By Lemma 5.3, ∆L(k; i′)(τ ′) is the determinant of T ′u′ . Using the explicit form of x−i(t) in (2.4), we have the following relation: If in+1 < ik = d, then T ′u′ = Tm′+1,1 . . . τ−1 n+1Tm′+1,in+1 + Tm′+1,in+1+1 τn+1Tm′+1,in+1+1 . . . Tm′+1,d ... ... ... ... ... ... Tm′+d,1 . . . τ−1 n+1Tm′+1,in+1 + Tm′+1,in+1+1 τn+1Tm′+1,in+1+1 . . . Tm′+d,d  . If in+1 > ik = d, then T ′u′ = Tu. Therefore, it is clear that for both cases, detT ′u′ = detTu, which means ∆L(k; i′)(τ ′) = ∆L(k; i)(τ). � Cluster Variables and Monomial Realizations of Crystal Bases 15 By this lemma, when we calculate ∆L(k; i)(τ), we may assume that in = ik without loss of generality. We set li = i∑ k=1 (r − k + 1)(1 ≤ i ≤ m− 1), and change the variables {Ym,j} to {τlm+j} as in Section 3.1. Remark 5.5. For i = (1, . . . , r︸ ︷︷ ︸ 1st cycle , 1, . . . , (r − 1)︸ ︷︷ ︸ 2nd cycle , . . . , 1, . . . , (r −m+ 2)︸ ︷︷ ︸ (m− 1)th cycle , 1, . . . , in︸ ︷︷ ︸ mth cycle ), (li + j)th index of i from the left is j which belongs to (i+ 1)th cycle (1 ≤ j ≤ r − i− 1). The following theorem is our main result. We describe ∆L(k; i)(τ) as a Demazure polyno- mial D−w in Definition 3.13. Theorem 5.6. In the above setting, we set d := ik = in and suppose that ik belongs to m′th cycle in i (5.2). We also set Y := 1 τlm−1+dτlm−2+d · · · τlm′+d . Then we have ∆L(k; i)(τ) = ∑ x∈B−u≤k ((m′−m)Λd) µY (x) = D−u≤k [(m′ −m)Λd, Y ; 1], where µY (x) is an embedding of x ∈ B−u≤k((m′ − m)Λd) as in Example 3.5, and 1 means all c(b) ≡ 1 (Definition 3.13). We shall prove this theorem in Section 6. In particular, we can explicitly write down ∆L(k; i)(τ) in the case ik = 1. Theorem 5.7. If ik = 1 and ik is in the m′th cycle in i (5.2), then we have ∆L(k; i)(τ) = ∑ 0≤j1<···<jm′≤m−1 j1−1∏ i=0 1 τlm−1−i+1 j2−1∏ i=j1+1 τlm−1−i+1 τlm−1−i+2 · · · m−1∏ i=jm′+1 τlm−1−i+m′ τlm−1−i+m′+1 = D−u≤k [(m′ −m)Λ1, 1 τlm−1+1τlm−2+1 · · · τlm′+1+1τlm′+1 ; 1]. We will prove this theorem in Section 6.5. 6 The proof of Theorems 5.6 and 5.7 In this section, we shall give the proof of Theorems 5.6 and 5.7. In Sections 6.1–6.4, we will prove Theorem 5.6. In Section 6.5, we prove Theorem 5.7. We use the same notation as in Section 5.2: li = i∑ k=1 (r − k + 1), 1 ≤ i ≤ m− 1. 16 Y. Kanakubo and T. Nakashima 6.1 The set Xd(m,m ′) of paths In this subsection, we shall introduce a set Xd(m,m ′) of “paths” which correspond to the terms of ∆L(k; i)(τ). Let m, m′ and d be the positive integers as in Section 5.2. Definition 6.1. Let us define the directed graph (Vd, Ed) as follows. We define the set Vd = Vd(m,m ′) of vertices as Vd(m,m ′) := {( m− s; a(s) 1 , a (s) 2 , . . . , a (s) d ) ∣∣ 0 ≤ s ≤ m, a(s) i ∈ Z, 1 ≤ a(s) 1 < a (s) 2 < · · · < a (s) d ≤ d+ s } . Note that a (0) 1 = 1, a (0) 2 = 2, . . . , a (0) d = d by 1 ≤ a(0) 1 < a (0) 2 < · · · < a (0) d ≤ d. And we define the set Ed = Ed(m,m ′) of directed edges as Ed(m,m ′) := {( m− s; a(s) 1 , . . . , a (s) d ) → ( m− s− 1; a (s+1) 1 , . . . , a (s+1) d ) ∣∣ 1 ≤ s ≤ m, 1 ≤ i ≤ d, a(s+1) i = a (s) i or a (s) i + 1 } . Definition 6.2. Let Xd(m,m ′) be the set of directed paths from (m; 1, . . . , d) to (0;m′ + 1, m′ + 2, . . . ,m′ + d) in (Vd, Ed). In other word, any path p ∈ Xd(m,m ′) p = ( m; a (0) 1 , . . . , a (0) d ) → ( m− 1; a (1) 1 , . . . , a (1) d ) → ( m− 2; a (2) 1 , . . . , a (2) d ) → · · · → ( 1; a (m−1) 1 , . . . , a (m−1) d ) → ( 0; a (m) 1 , . . . , a (m) d ) are characterized by the following conditions (i) a (s) i ∈ Z≥1, (ii) a (s) 1 < a (s) 2 < · · · < a (s) d , (iii) a (s+1) i = a (s) i or a (s) i + 1, (iv) a (0) i = i, a (m) i = m′ + i. Remark 6.3. By Definition 6.2(ii), (iii) and (iv), we have a (s) i ≤ a (s) d ≤ a (m) d = m′ + d for any 1 ≤ i ≤ d and 0 ≤ s ≤ m. Let us define a Laurent monomial associated with each path in Xd(m,m ′). Definition 6.4. Let p ∈ Xd(m,m ′) be a path p = ( m; a (0) 1 , . . . , a (0) d ) → ( m− 1; a (1) 1 , . . . , a (1) d ) → ( m− 2; a (2) 1 , . . . , a (2) d ) → · · · → ( 1; a (m−1) 1 , . . . , a (m−1) d ) → ( 0; a (m) 1 , . . . , a (m) d ) . (i) For each 0 ≤ s ≤ m, we define the label of the edge ( m−s; a(s) 1 , a (s) 2 , . . . , a (s) d ) → ( m−s−1; a (s+1) 1 , a (s+1) 2 , . . . , a (s+1) d ) as the Laurent monomial d∏ i=1 τ lm−s−1+a (s+1) i −1 τ lm−s−1+a (s) i . (6.1) Cluster Variables and Monomial Realizations of Crystal Bases 17 (ii) And we define the label Q(p) of the path p as the product of them Q(p) := m−1∏ s=0 ( d∏ i=1 τ lm−s−1+a (s+1) i −1 τ lm−s−1+a (s) i ) . Example 6.5. Let m = 3, m′ = 2, d = 2. We can describe X2(3, 2) and its labels as follows (3; 1, 2) (2; 1, 2) (2; 1, 3) (2; 2, 3) (1; 2, 3) (1; 2, 4) (1; 3, 4) (0; 3, 4) 1 τl2+2 �� 1 τl2+1 �� 1 '' 1 '' τl1+2 τl1+3 �� 1 '' τl1+1 τl1+3 �� τl1+1 τl1+2 �� 1 '' 1 '' τl0+3 τl0+4 �� τl0+2 τl0+4 �� Each edge has the label written on the left side of it. The paths in X2(3, 2) are as follows p1 = (3; 1, 2)→ (2; 1, 2)→ (1; 2, 3)→ (0; 3, 4), p2 = (3; 1, 2)→ (2; 1, 3)→ (1; 2, 3)→ (0; 3, 4), p3 = (3; 1, 2)→ (2; 1, 3)→ (1; 2, 4)→ (0; 3, 4), p4 = (3; 1, 2)→ (2; 2, 3)→ (1; 2, 3)→ (0; 3, 4), p5 = (3; 1, 2)→ (2; 2, 3)→ (1; 2, 4)→ (0; 3, 4), p6 = (3; 1, 2)→ (2; 2, 3)→ (1; 3, 4)→ (0; 3, 4). We have Q(p1) = 1 τl2+2 , Q(p2) = τl1+2 τl2+1τl1+3 , Q(p3) = τl0+3 τl2+1τl0+4 , Q(p4) = τl1+1 τl1+3 , Q(p5) = τl1+1τl0+3 τl1+2τl0+4 , Q(p6) = τl0+2 τl0+4 . Definition 6.6. For each path p ∈ Xd(m,m ′) p = ( m; a (0) 1 , . . . , a (0) d ) → ( m− 1; a (1) 1 , . . . , a (1) d ) → ( m− 2; a (2) 1 , . . . , a (2) d ) → · · · → ( 1; a (m−1) 1 , . . . , a (m−1) d ) → ( 0; a (m) 1 , . . . , a (m) d ) and i ∈ {1, . . . , d}, we call the following sequence a (0) i → a (1) i → a (2) i → · · · → a (m) i an i-sequence of p. 18 Y. Kanakubo and T. Nakashima Example 6.7. In the setting of Example 6.5, p1 := (3; 1, 2) → (2; 1, 2) → (1; 2, 3) → (0; 3, 4). Then, 1-sequence of p1 is 1→ 1→ 2→ 3 and 2-sequence of p1 is 2→ 2→ 3→ 4. 6.2 One-to-one correspondence between paths in Xd(m,m ′) and terms of ∆L(k; i)(τ ) Proposition 6.8. We use the setting (5.1), (5.2) and the notations in Section 5. Then, we have the following ∆L(k; i)(τ) = ∑ p∈Xd(m,m′) Q(p). To prove this proposition, we need the following preparations. For s (0 ≤ s ≤ m − 1), we define a matrix x(s)(τ) as x(s)(τ) := x−1(τ1)x−2(τ2) · · ·x−r(τr)︸ ︷︷ ︸ 1st cycle x−1(τl1+1) · · ·x−(r−1)(τl1+r−1)︸ ︷︷ ︸ 2nd cycle · · · × x−1(τls−1+1) · · ·x−(r−s+1)(τls−1+r−s+1)︸ ︷︷ ︸ sth cycle , where we understand x(0)(τ) means identity matrix. We also define x(m)(τ) := x−1(τ1)x−2(τ2) · · ·x−r(τr)︸ ︷︷ ︸ 1st cycle x−1(τl1+1) · · ·x−(r−1)(τl1+r−1)︸ ︷︷ ︸ 2nd cycle · · · × x−1(τlm−1+1) · · ·x−in(τlm−1+in)︸ ︷︷ ︸ mth cycle , which is equal to the matrix in (5.3). For x(s)(τ) = (x (s) i,j )i,j∈[1,r+1], we define the d dimensional column vector D(s; p) D(s; p) :=  x (s) m′+1,p ... x (s) m′+d,p  ∈ Cd. Note that, by the explicit form of x−i(τ) in (2.4), multiplying x−i(τ) from the right gives an elementary transformation of a matrix. Therefore, we get D(s; p) =  1 τls−1+p D(s− 1; p) +D(s− 1; p+ 1) if p = 1, τls−1+p−1 τls−1+p D(s− 1; p) +D(s− 1; p+ 1) if p > 1, (6.2) for 1 ≤ s ≤ m. For 0 ≤ s ≤ m and 1 ≤ i1 < · · · < id ≤ r, we set (s; i1, i2, . . . , id) := det(D(s; i1), D(s; i2), . . . , D(s; id)), (6.3) which coincides with the notation for a vertex in Xd(m,m ′) since later we identify each vertex with the minor above. Cluster Variables and Monomial Realizations of Crystal Bases 19 Proof of Proposition 6.8. We shall prove the proposition in the following three steps. Step 1. ∆L(k; i)(τ) = det(D(m; 1), D(m; 2), . . . , D(m; d)). It is followed from Lemma 5.3 that ∆L(k; i)(τ) is given as a minor whose row (resp. col- umn) are labeled by the set {m′ + 1, . . . ,m′ + d} (resp. {1, . . . , d}) of the matrix x(m)(τ). Thus, by using above notation, we have ∆L(k; i)(τ) = det(t(D(m; 1), D(m; 2), . . . , D(m; d))) = det(D(m; 1), D(m; 2), . . . , D(m; d)). Step 2. Calculation of det(D(m; 1), D(m; 2), . . . , D(m; d)) and labeled graph. By using (6.2), let us calculate ∆L(k; i)(τ) = det(D(m; 1), D(m; 2), . . . , D(m; d)) explicitly. We have (m; 1, 2, . . . , d) = det(D(m; 1), D(m; 2), . . . , D(m; d)) = det ( 1 τlm−1+1 D(m− 1; 1) +D(m− 1; 2), . . . , τlm−1+d−1 τlm−1+d D(m− 1; d) +D(m− 1; d+ 1) ) , (6.4) which implies that in the notation (6.3), (m; 1, 2, . . . , d) is a linear combination of {(m − 1; a (1) 1 , a (1) 2 , . . . , a (1) d ) | 1 ≤ a (1) 1 < a (1) 2 < · · · < a (1) d ≤ d + 1, a (1) i = i or i + 1}. By (6.4), the coefficient of each (m− 1; a (1) 1 , a (1) 2 , . . . , a (1) d ) is d∏ i=1 τ lm−1+a (1) i −1 τlm−1+i , which coincides with the label of the edge connecting (m; 1, . . . , d) and (m−1; a (1) 1 , a (1) 2 , . . . , a (1) d ) defined in Definition 6.4, see (6.1). Using the formula (6.2) again, we see that each (m− 1; a (1) 1 , a (1) 2 , . . . , a (1) d ) is a linear combi- nation of {(m−2; a (2) 1 , a (2) 2 , . . . , a (2) d ) | 1 ≤ a(2) 1 < a (2) 2 < · · · < a (2) d ≤ d+ 2, a (2) i = a (1) i or a (1) i + 1} in the same way as (6.4). Then, the coefficient of (m− 2; a (2) 1 , a (2) 2 , . . . , a (2) d ) is d∏ i=1 τ lm−2+a (2) i −1 τ lm−2+a (1) i , which coincides with the label of the edge connecting (m − 1; a (1) 1 , a (1) 2 , . . . , a (1) d ) and (m − 2; a (2) 1 , a (2) 2 , . . . , a (2) d ). Thus (m; 1, 2, . . . , d) is a linear combination of (m − 2; a (2) 1 , a (2) 2 , . . . , a (2) d ), 1 ≤ a(2) 1 < a (2) 2 < · · · < a (2) d ≤ d+ 2, whose coefficient is ∑ a (1) 1 ,...,a (1) d ( d∏ i=1 τ lm−1+a (1) i −1 τlm−1+i × d∏ i=1 τ lm−2+a (2) i −1 τ lm−2+a (1) i ) , where the index a (1) i (1 ≤ i ≤ d) runs over {(a(1) i : i ∈ {1, . . . , d}) | a(2) i = a (1) i or a (1) i + 1, a (1) i = i or i+ 1, a (1) i < a (1) i+1}. Repeating this argument, we see that (m; 1, 2, . . . , d) is a linear combination of (0; a (m) 1 , a (m) 2 , . . . , a (m) d ), 1 ≤ a(m) 1 < a (m) 2 < · · · < a (m) d ≤ m+ d, whose coefficient is ∑ a (j) i , i∈{1,...,d}, 1≤j≤m−1 ( d∏ i=1 τ lm−1+a (1) i −1 τlm−1+i × d∏ i=1 τ lm−2+a (2) i −1 τ lm−2+a (1) i × · · · × d∏ i=1 τ l0+a (m) i −1 τ l0+a (m−1) i ) , (6.5) 20 Y. Kanakubo and T. Nakashima where the index a (j) i runs over {(a(j) i : i ∈ {1, . . . , d}, 1 ≤ j ≤ m − 1) | a(j+1) i = a (j) i or a (j) i + 1, a (j) i < a (j) i+1, a (1) i = i or i+ 1}. Step 3. One-to-one correspondence between paths and terms of ∆L(k; i)(τ). Let us recall that (0; a (m) 1 , a (m) 2 , . . . , a (m) d ) means a minor of the identity matrix, and (0; a (m) 1 , a (m) 2 , . . . , a (m) d ) = det(D(0; a (m) 1 ), D(0; a (m) 2 ), . . . , D(0; a (m) d )), and each D(0; i) is given as follows: D(0; i) =  x (0) m′+1,i ... x (0) m′+d,i  = δm′+1,i ... δm′+d,i  . By 1 ≤ a(m) 1 < a (m) 2 < · · · < a (m) d ≤ m+ d (Definition 6.2), we have ( 0; a (m) 1 , a (m) 2 , . . . , a (m) d ) = { 1 if a (m) 1 = m′ + 1, a (m) 2 = m′ + 2, . . . , a (m) d = m′ + d, 0 otherwise. Therefore, ∆L(k; i)(τ) = det(D(m; 1), D(m; 2), . . . , D(m; d)) is equal to the coefficient of (0;m′+ 1, m′ + 2, . . . ,m′ + d). By (6.5), setting a (0) i = i, a (m) i = m′ + i, we have ∆L(k; i)(τ) = ∑ a (j) i , i∈{1,...,d}, 1≤j≤m−1 ( d∏ i=1 τ lm−1+a (1) i −1 τ lm−1+a (0) i × d∏ i=1 τ lm−2+a (2) i −1 τ lm−2+a (1) i × · · · × d∏ i=1 τ l0+a (m) i −1 τ l0+a (m−1) i ) , (6.6) where the index a (j) i runs over {(a(j) i : i ∈ {1, . . . , d}, 1 ≤ j ≤ m − 1) | a(j+1) i = a (j) i or a (j) i + 1, a (j) i < a (j) i+1, 0 ≤ j ≤ m − 1}. Note that these conditions equal to the ones in Definition 6.2(ii) and (iii), and the conditions a (0) i = i, a (m) i = m′ + i are equal to the ones in Definition 6.2(iv). We see that the summand in (6.6) is equal to the label of the path( m; a (0) 1 , . . . , a (0) d ) → ( m− 1; a (1) 1 , . . . , a (1) d ) → ( m− 2; a (2) 1 , . . . , a (2) d ) → · · · → ( 1; a (m−1) 1 , . . . , a (m−1) d ) → ( 0; a (m) 1 , . . . , a (m) d ) ∈ Xd(m,m ′). Hence we obtain ∆L(k; i)(τ) = ∑ p∈Xd(m,m′) Q(p). � Example 6.9. We use the same notations m, m′ and d = ik at the beginning of Section 5. We set rank r = 4, u = s1s2s3s4s1s2s3s1s2s1 ∈ W and k = 6, which is the same setting as in introduction. Let i := (1, 2, 3, 4, 1, 2, 3, 1, 2, 1) be a reduced for u, and let i′ = (1, 2, 3, 4︸ ︷︷ ︸ 1st cycle , 1, 2, 3︸ ︷︷ ︸ 2nd cycle , 1, 2︸︷︷︸ 3rd cycle ) be a reduced word for us1. By Lemma 5.4, we get ∆L(6; i)(τ) = ∆L(6; i′)(τ). Since m = 3, m′ = 2 and d := ik = 2, it follows from Example 6.5 and Proposition 6.8 that ∆L(6; i)(τ) = ∆L(6; i′)(τ) = ∑ p∈X2(3,2) Q(p) Cluster Variables and Monomial Realizations of Crystal Bases 21 = 1 τl2+2 + τl1+2 τl2+1τl1+3 + τl0+3 τl2+1τl0+4 + τl1+1 τl1+3 + τl1+1τl0+3 τl1+2τl0+4 + τl0+2 τl0+4 = 1 τ9 + τ6 τ8τ7 + τ3 τ8τ4 + τ5 τ7 + τ5τ3 τ6τ4 + τ2 τ4 , which is equal to (1.4). 6.3 The explicit description of ∆L(k; i)(τ ) In Proposition 6.8, we had described the terms of ∆L(k; i)(τ) as the paths in Xd(m,m ′). In this subsection, we shall describe ∆L(k; i)(τ) explicitly by using some properties of paths. This description will be used in the proof of Theorem 5.6. First, we need to show some lemmas. Let us write a path p ∈ Xd(m,m ′) as follows p = ( m; a (0) 1 , . . . , a (0) d ) → ( m− 1; a (1) 1 , . . . , a (1) d ) → ( m− 2; a (2) 1 , . . . , a (2) d ) → · · · → ( 1; a (m−1) 1 , . . . , a (m−1) d ) → ( 0; a (m) 1 , . . . , a (m) d ) . (6.7) We begin with the following lemma: Lemma 6.10. For a path p (6.7) and i ∈ {1, . . . , d}, we have # { s | a(s) i = a (s+1) i , 0 ≤ s ≤ m− 1 } = m−m′. Proof. By Definition 6.2(iii) and (iv), we have i = a (0) i ≤ a (1) i ≤ · · · ≤ a (m) i = m′ + i, a (s+1) i = a (s) i or a (s) i + 1. Thus, we get # { s | a(s+1) i = a (s) i + 1, 0 ≤ s ≤ m− 1 } = m′, which implies that #{s | a(s) i = a (s+1) i , 0 ≤ s ≤ m− 1} = m−m′. � Definition 6.11. For a path p (6.7) and i ∈ {1, . . . , d}, we set {q(j) i }1≤j≤m−m′ , 0 ≤ q(1) i < · · · < q (m−m′) i ≤ m− 1, as{ q (1) i , q (2) i , . . . , q (m−m′) i } := { q | a(q) i = a (q+1) i , 0 ≤ q ≤ m− 1 } . (6.8) We also set k (j) i ∈ [1,m′ + d] (1 ≤ i ≤ d, 1 ≤ j ≤ m−m′) as k (j) i := a (q (j) i ) i . (6.9) Lemma 6.12. (i) For 1 ≤ i ≤ d and 1 ≤ j ≤ m−m′, q (j) i = k (j) i + j − i− 1. (ii) For 1 ≤ j ≤ m−m′ and 1 ≤ i ≤ d− 1, 1 ≤ k(j) i < k (j) i+1 ≤ m ′ + d, q (j) i ≤ q (j) i+1. 22 Y. Kanakubo and T. Nakashima Proof. (i) The definition of q (j) i in (6.8) means that the path p has the following i-sequence (Definition 6.6): a (0) i = i, a (1) i = i+ 1, a (2) i = i+ 2, . . . , a (q (1) i ) i = i+ q (1) i , a (q (1) i +1) i = i+ q (1) i , a (q (1) i +2) i = i+ q (1) i + 1, . . . , a (q (2) i ) i = i+ q (2) i − 1, a (q (2) i +1) i = i+ q (2) i − 1, a (q (2) i +2) i = i+ q (2) i , . . . , a (q (3) i ) i = i+ q (3) i − 2, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (6.10) a (q (j−1) i +1) i = i+ q (j−1) i − j + 2, a (q (j−1) i +2) i = i+ q (j−1) i − j + 3, . . . , a (q (j) i ) i = i+ q (j) i − j + 1, a (q (j) i +1) i = i+ q (j) i − j + 1, a (q (j) i +2) i = i+ q (j) i − j + 2, . . . . Hence we have k (j) i = a (q (j) i ) i = i+ q (j) i − j + 1, which implies q (j) i = k (j) i + j − i− 1. (ii) By Definition 6.2(iii), i = a (0) i ≤ a (1) i ≤ · · · ≤ a (q (j) i+1) i ≤ a(q (j) i+1+1) i , (6.11) a (ζ) i = a (ζ−1) i or a (ζ−1) i + 1, 1 ≤ ζ ≤ q(j) i+1 + 1. We obtain q (j) i+1 + 1− j ≥ # { ζ | a(ζ) i = a (ζ−1) i + 1, 1 ≤ ζ ≤ q(j) i+1 + 1 } , (6.12) otherwise, it follows from (6.11) and (i) that a (q (j) i+1+1) i > i+ q (j) i+1 + 1− j = ki+1− 1 = a (q (j) i+1) i+1 − 1, and hence a (q (j) i+1+1) i ≥ a(q (j) i+1) i+1 = a (q (j) i+1+1) i+1 , which contradicts Definition 6.2(ii). The inequality (6.12) means that j ≤ # { ζ | a(ζ) i = a (ζ−1) i , 1 ≤ ζ ≤ q(j) i+1 + 1 } . (6.13) On the other hand, using the list (6.10) (or the definition of q (j) i in (6.8)), we have j = # { ζ | a(ζ) i = a (ζ−1) i , 1 ≤ ζ ≤ q(j) i + 1 } . (6.14) Since a (q (j) i ) i = a (q (j) i +1) i , the equation (6.14) means j − 1 = # { ζ | a(ζ) i = a (ζ−1) i , 1 ≤ ζ ≤ q(j) i } . (6.15) Thus, by (6.13) and (6.15), we have q (j) i < q (j) i+1+1, and hence q (j) i ≤ q (j) i+1, which yields k (j) i < k (j) i+1 since k (j) i = i+ q (j) i − j+ 1 < i+ q (j) i+1− j+ 2 = (i+ 1) + q (j) i+1− j+ 1 = k (j) i+1. Remark 6.3 implies that k (j) i+1 := a (q (j) i+1) i+1 ≤ m′ + d. � For 1 ≤ i ≤ m and 1 ≤ j ≤ r, we set the Laurent monomials C(i, j) := τli+j−1 τli+j . (6.16) Cluster Variables and Monomial Realizations of Crystal Bases 23 Lemma 6.13. For a path p (6.7), we set q (j) i and k (j) i as in (6.8) and (6.9). Then we have Q(p) = d∏ i=1 m−m′∏ j=1 C ( m− q(j) i − 1, k (j) i ) . (6.17) Proof. Let us recall the definition of Q(p) (Definition 6.4(ii)): Q(p) = m−1∏ s=0 ( d∏ i=1 τ lm−s−1+a (s+1) i −1 τ lm−s−1+a (s) i ) . (6.18) For each i = 1, 2, . . . , d, the i-sequence i = a (0) i ≤ a (1) i ≤ a (2) i ≤ · · · ≤ a (m) i = m′ + i of p satisfies a (s+1) i = a (s) i or a (s) i + 1, 0 ≤ s ≤ m− 1, by Definition 6.2(iii). If a (s+1) i = a (s) i + 1, then τ lm−s−1+a (s+1) i −1 τ lm−s−1+a (s) i = 1. Therefore, it follows from the definition (6.8) of q (j) i and k (j) i := a (q (j) i ) i = a (q (j) i +1) i that m−1∏ s=0 τ lm−s−1+a (s+1) i −1 τ lm−s−1+a (s) i = m−m′∏ j=1 τ l m−q(j) i −1 +k (j) i −1 τ l m−q(j) i −1 +k (j) i = m−m′∏ j=1 C ( m− q(j) i − 1, k (j) i ) , which implies (6.17) by (6.18). � Let us describe ∆L(k; i)(τ) explicitly. Proposition 6.14. ∆L(k; i)(τ) = ∑ (∗) d∏ i=1 m−m′∏ j=1 C ( m−K(j) i − j + i,K (j) i ) . where (∗) is the conditions for K (j) i (1 ≤ i ≤ d, 1 ≤ j ≤ m −m′): 1 ≤ K (j) 1 < K (j) 2 < · · · < K (j) d ≤ m ′ + d, 1 ≤ j ≤ m−m′, i ≤ K(1) i ≤ K(2) i ≤ · · · ≤ K(m−m′) i ≤ m′ + i, 1 ≤ i ≤ d. Proof. Using Lemmas 6.12(i) and 6.13, we see that Q(p), p ∈ Xd(m,m ′), is described as Q(p) = d∏ i=1 m−m′∏ j=1 C ( m− k(j) i − j + i, k (j) i ) . with {k(j) i }1≤i≤d,1≤j≤m−m′ which satisfy the conditions in Lemma 6.12(ii), that is, 1 ≤ k (j) 1 < k (j) 2 < · · · < k (j) d ≤ m ′ + d. Furthermore, Definition 6.2(iii) and (iv) show that i = a (0) i ≤ a (1) i ≤ · · · ≤ a(m) i = m′+ i, which means that i ≤ k(1) i ≤ k (2) i ≤ · · · ≤ k (m−m′) i ≤ m′+ i for 1 ≤ i ≤ d by k (j) i := a (q (j) i ) i and q (j) i < q (j+1) i (see Definition 6.11). Thus, {k(j) i } satisfies the conditions (∗) in Proposition 6.14. 24 Y. Kanakubo and T. Nakashima Conversely, let {K(j) i }1≤i≤d,1≤j≤m−m′ the set of numbers which satisfies the conditions (∗) in Proposition 6.14: 1 ≤ K(j) 1 < K (j) 2 < · · · < K (j) d ≤ m ′ + d, 1 ≤ j ≤ m−m′, and i ≤ K(1) i ≤ · · · ≤ K(m−m′) i ≤ m′ + i, 1 ≤ i ≤ d. (6.19) We set Q (j) i := K (j) i + j − i− 1, (6.20) for 1 ≤ i ≤ d and 1 ≤ j ≤ m −m′. We need to show that there exists a path p ∈ Xd(m,m ′) such that Q(p) = d∏ i=1 m−m′∏ j=1 C ( m−Q(j) i − 1,K (j) i ) = d∏ i=1 m−m′∏ j=1 C ( m−K(j) i − j + i,K (j) i ) . (6.21) Since we supposed K (j) i < K (j) i+1, we can easily verify Q (j) i ≤ Q (j) i+1, (6.22) by (6.20). We claim that 0 ≤ Q (j) i ≤ m − 1 for 1 ≤ i ≤ d and 1 ≤ j ≤ m − m′. By the condition (6.19), we get i ≤ K(j) i . So it is clear that 0 ≤ Q(j) i . It follows from the condition (6.19) and (6.20) that Q (j) i = K (j) i + j − i− 1 ≤ m′ + i + j − i− 1 = m′ + j − 1 ≤ m− 1. Therefore, we have 0 ≤ Q(j) i ≤ m− 1 for all 1 ≤ i ≤ d and 1 ≤ j ≤ m−m′. We define a path p = (m; a (0) 1 , . . . , a (0) d ) → · · · → (0; a (m) 1 , . . . , a (m) d ) ∈ Xd(m,m ′) as follows. For i, 1 ≤ i ≤ d, we define the i-sequence (Definition 6.6) of p as a (0) i = i, a (1) i = i+ 1, a (2) i = i+ 2, . . . , a (Q (1) i ) i = i+Q (1) i , a (Q (1) i +1) i = i+Q (1) i , a (Q (1) i +2) i = i+Q (1) i + 1, . . . , a (Q (2) i ) i = i+Q (2) i − 1, a (Q (2) i +1) i = i+Q (2) i − 1, a (Q (2) i +2) i = i+Q (2) i , . . . , a (Q (3) i ) i = i+Q (3) i − 2, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (6.23) a (Q (m−m′−1) i +1) i = i+Q (m−m′−1) i −m+m′ + 2, . . . , a (Q (m−m′) i ) i = i+Q (m−m′) i −m+m′ + 1, a (Q (m−m′) i +1) i = i+Q (m−m′) i −m+m′ + 1, a (Q (m−m′) i +2) i = i+Q (m−m′) i −m+m′ + 2, a (Q (m−m′) i +3) i = i+Q (m−m′) i −m+m′ + 3, . . . , a (m) i = m′ + i. It is easy to see that a (Q (j) i ) i = K (j) i , 1 ≤ j ≤ m−m′, by (6.20) and (6.23). Clearly, the path p satisfies Definition 6.2(iii) and (iv). For 1 ≤ s ≤ m, we obtain a (s) i < a (s) i+1 by (6.22) and (6.23). Hence p is well-defined, and (6.21) is follows from Lemma 6.13 (see (6.8) and (6.9)). Thus, Proposition 6.14 follows from Proposition 6.8. � Cluster Variables and Monomial Realizations of Crystal Bases 25 Example 6.15. We use the same setting in Example 6.9. Since m = 3, m′ = 2 and d := ik = 2, it follows from Proposition 6.14 that ∆L(6; i)(τ) = ∑ 1≤K1<K2≤4 ( 2∏ i=1 C(i−Ki + 2,Ki) ) = C(2, 1)C(2, 2) + C(2, 1)C(1, 3) + C(2, 1)C(0, 4) + C(1, 2)C(1, 3) + C(1, 2)C(0, 4) + C(0, 3)C(0, 4) = 1 τl2+2 + τl1+2 τl2+1τl1+3 + τl0+3 τl2+1τl0+4 + τl1+1 τl1+3 + τl1+1τl0+3 τl1+2τl0+4 + τl0+2 τl0+4 , which is equal to the one in Example 6.9. 6.4 The completion of the proof of Theorem 5.6 In this subsection, we shall complete the proof of Theorem 5.6. Let us recall the definition (3.3) of As,i. Since we identify the variables {Ys,i} with {τls+i}, we have the following (see Remark 3.7) As,i = τls+iτls+1+i τls+i+1τls+1+i−1 , 0 ≤ s ≤ m− 1, 1 ≤ i ≤ r. Therefore, by the definition (6.16) of C(s, i), we have C(s, i) ·As−1,i = C(s− 1, i+ 1). (6.24) Lemma 6.16. For a path p ∈ Xd(m,m ′), we describe the monomial Q(p) as in Lemma 6.13 Q(p) = d∏ i=1 m−m′∏ j=1 C ( m−K(j) i − j + i,K (j) i ) , (6.25) where {K(j) i } satisfies the condition (∗) in Proposition 6.14. For s, 1 ≤ s < m′+d, if ẽsQ(p) 6= 0, then there exist i and j, 1 ≤ i ≤ d, 1 ≤ j ≤ m−m′, such that K (j) i = s and ẽsQ(p) = Q(p) ·A m−K(j) i −j+i−1,s . (6.26) Furthermore, there exists a path p′ ∈ Xd(m,m ′) such that ẽsQ(p) = Q(p′). (6.27) Proof. We suppose that the monomial Q(p) does not include factor as in the form τ−1 lt+s , 0 ≤ t ≤ m− 1, which means wt(Q(p))(hi) = ϕi(Q(p)). Hence εi(Q(p)) := ϕi(Q(p))− wt(Q(p))(hi) = 0, which contradicts the assumption ẽsQ(p) 6= 0. Hence the monomial Q(p) includes factor τ−1 lt+s , 0 ≤ t ≤ m− 1. We set the numbers 0 ≤ t1 < t2 < · · · < tξ ≤ m− 1 by {t1, t2, . . . , tξ} := { t |Q(p) includes factors τ−1 lt+s } , 1 ≤ ξ. The definition (6.16) of C(t, s) and (6.25) show that there exist 1 ≤ i1, . . . , iξ ≤ d and 1 ≤ j1, . . . , jξ ≤ m−m′ such that K (ja) ia = s and ta = m−K(ja) ia − ja + ia, a = 1, 2, . . . , ξ. (6.28) 26 Y. Kanakubo and T. Nakashima As in Example 3.9, for a given s ∈ I and monomial Y = ∏ q∈Z, i∈I τ ζq,i lq+i , we define νY (n) :=∑ q≤n ζq,s, n ∈ Z. We set Y := Q(p). We claim that nes ∈ {t1 − 1, . . . , tξ − 1} (3.4), otherwise, we get nes /∈ {t1 − 1, . . . , tξ − 1}. Then ϕs(Y ) = νY (nes) ≤ νY (nes + 1). If νY (nes) < νY (nes + 1), then it contradicts the definition of ϕs(Y ) := max{νY (n) |n ∈ Z}. (6.29) If νY (nes) = νY (nes + 1), then it contradicts the definition of nes := max{n |ϕs(Y ) = νY (n)}. Hence we obtain nes ∈ {t1 − 1, . . . , tξ − 1}, which implies there exists a, 1 ≤ a ≤ ξ, such that nes = ta − 1. By (6.28), we have nes = ta − 1 = m−K(ja) ia − ja + ia − 1. Therefore, we obtain ẽsY := Y ·Anes ,s = Y ·A m−K(ja) ia −ja+ia−1,s . Since Q(p) includes the factor τ−1 lta+s, the path p includes the following edge( ta + 1; a (m−ta−1) 1 , · · · , a(m−ta−1) q−1 , s qth , a (m−ta−1) q+1 , · · · , a(m−ta−1) d ) → ( ta; a (m−ta) 1 , · · · , a(m−ta) q−1 , s qth , a (m−ta) q+1 , · · · , a(m−ta) d ) → ( ta − 1; a (m−ta+1) 1 , · · · , a(m−ta+1) q−1 , a(m−ta+1) q , a (m−ta+1) q+1 , · · · , a(m−ta+1) d ) for some 1 ≤ q ≤ d. We claim that a (m−ta) q+1 > s+ 1, otherwise a (m−ta) q+1 = s+ 1 and a (m−ta−1) q+1 = s + 1 by Definition 6.2(ii) and (iii), which implies that Q(p) does not include the factor τ−1 lta+s by Definition 6.4(i). It contradicts the definition of ta. So we get a (m−ta) q+1 > s+ 1. (6.30) If ta = 0, then Definition 6.2(iv) means that s = m′ + q (by the assumption s < m′ + d, we get q < d), and a (m−ta) q+1 = m′ + q + 1 = s+ 1, which contradicts (6.30). So we have 1 ≤ ta. We also claim that a (m−ta+1) q > s, otherwise a (m−ta+1) q = s, which implies that Q(p) includes the factor τ−1 lta−1+s by a (m−ta+1) q+1 ≥ a(m−ta) q+1 > s+ 1. Therefore, ϕs(Y ) = νY (nes) = νY (ta− 1) < νY (ta− 2), which contradicts (6.29). So we get a (m−ta+1) q > s, which means a(m−ta+1) q = s+ 1, (6.31) by Definition 6.2(iii). Let p′ ∈ Xd(m,m ′) be the path obtained from p by replacing the vertex( ta; a (m−ta) 1 , . . . , a (m−ta) q−1 , s qth , a (m−ta) q+1 , . . . , a (m−ta) d ) by ( ta; a (m−ta) 1 , . . . , a (m−ta) q−1 , s+ 1 qth , a (m−ta) q+1 , . . . , a (m−ta) d ) . By (6.30) and (6.31), the path p′ is well-defined. Definition 6.4(i) shows that Q(p′) = Q(p) · τlta−1+sτlta+s τlta−1+s+1τlta+s−1 = Q(p) ·A m−K(ja) ia −ja+ia−1,s = ẽsQ(p) by (6.28). � Cluster Variables and Monomial Realizations of Crystal Bases 27 Next lemma shows that the coefficients c(b) in Theorem 5.6 are equal to 1 for all b ∈ B−u≤k((m′ −m)Λd) by Proposition 6.8. Lemma 6.17. For paths p, p′ ∈ Xd(m,m ′), if p 6= p′ then Q(p) 6= Q(p′). Proof. We suppose that Q(p) = Q(p′). Let us prove p = p′. We denote p and p′ by p = ( m; a (0) 1 , . . . , a (0) d ) → · · · → ( 0; a (m) 1 , . . . , a (m) d ) and p′ = ( m; b (0) 1 , . . . , b (0) d ) → · · · → ( 0; b (m) 1 , . . . , b (m) d ) . Since Q(p) is the product of the labels d∏ i=1 τ lm−s−1+a (s+1) i −1 τ lm−s−1+a (s) i of the edges (m−s; a(s) 1 , . . . , a (s) d )→ (m−s−1; a (s+1) 1 , . . . , a (s+1) d ), 0 ≤ s ≤ m−1, the assumption Q(p) = Q(p′) means that d∏ i=1 τ lm−s−1+a (s+1) i −1 τ lm−s−1+a (s) i = d∏ i=1 τ lm−s−1+b (s+1) i −1 τ lm−s−1+b (s) i (6.32) for all 0 ≤ s ≤ m − 1. For s = 0, by Definition 6.2(iv), we get a (0) i = b (0) i = i, 1 ≤ i ≤ d. It follows from a (1) 1 < a (1) 2 < · · · < a (1) d , b (1) 1 < b (1) 2 < · · · < b (1) d and (6.32) for s = 0 that a (1) i = b (1) i , 1 ≤ i ≤ d. Repeating this argument for s = 1, 2, 3, . . . ,m − 1, we get a (s) i = b (s) i , 1 ≤ i ≤ d, which means that p = p′. � Proof of Theorem 5.6. We set B := {Q(p) | p ∈ Xd(m,m ′)} =  d∏ i=1 m−m′∏ j=1 C ( R (j) i ,K (j) i ) ∣∣∣∣∣R(j) i := m−K(j) i − j + i  , where {K(j) i } satisfies the conditions 1 ≤ K(j) 1 < K (j) 2 < · · · < K (j) d ≤ m ′ + d, 1 ≤ j ≤ m−m′, i ≤ K(1) i ≤ · · · ≤ K(m−m′) i ≤ m′ + i, 1 ≤ i ≤ d. By Proposition 6.14 and Lemma 6.17, we need to show that B = µY (B−u≤k((m′ −m)Λd)), (6.33) where µY (x) is an embedding of x ∈ B−u≤k((m′ −m)Λd) in Theorem 5.6, and Y := 1 τlm−1+dτlm−2+d · · · τlm′+d , (6.34) which is the lowest weight vector in µY (B−u≤k((m′ −m)Λd)) (Theorem 3.2(ii)). 28 Y. Kanakubo and T. Nakashima First, let us prove the inclusion µY (B−u≤k((m′ −m)Λd)) ⊂ B. Using C(a, b), the monomial Y in (6.34) is described as follows C(m− 1, 1) · C(m− 1, 2) · · ·C(m− 1, d) Y = · C(m− 2, 1) · C(m− 2, 2) · · ·C(m− 2, d) ... ... ... · C(m′, 1) · C(m′, 2) · · · C(m′, d). Thus, we see that Y ∈ B. It follows from Theorem 3.12 and the definition (4.3) of u≤k that µY (B−u≤k((m′ −m)Λd)) = { ẽ (N1(m′)) 1 · · · ẽ(Nr(m′)) r︸ ︷︷ ︸ 1st cycle ẽ (N1(m′−1)) 1 · · · ẽ(Nr−1(m′−1)) r−1︸ ︷︷ ︸ 2nd cycle · · · ẽ(N1(2)) 1 · · · ẽ(Nr−m+2(2)) r−m+2︸ ︷︷ ︸ (m′ − 1)th cycle ẽ (N1(1)) 1 . . . ẽ (Nd(1)) d︸ ︷︷ ︸ m′th cycle ·Y |Ni(j) ∈ Z≥0 } \ {0}. By (6.27), for any monomial Z ∈ B and 1 ≤ s < m′ + d, we have ẽsZ ∈ B ∪ {0}. (6.35) For an arbitrary set {Ni(1)}i=1,...,d of non negative integers, the monomial ẽ (N1(1)) 1 · · · ẽ(Nd(1)) d ·Y does not include factors in the form C(∗, a), d+2 ≤ a ≤ r−m+2, by (6.24) and (6.26). Therefore, ẽa · ẽ(N1(1)) 1 · · · ẽ(Nd(1)) d · Y = 0 for d + 2 ≤ a ≤ r −m + 2 by (6.26). Similarly, for an arbitrary set {Ni(2)}i=1,...,r−m+2 of non negative integers, the monomial ẽ (N1(2)) 1 · · · ẽ(Nr−m+2(2)) r−m+2 ẽ (N1(1)) 1 · · · ẽ(Nd(1)) d · Y does not include factors in the form C(∗, a), d+ 3 ≤ a ≤ r−m+ 3, which means that ẽa · ẽ(N1(2)) 1 · · · ẽ(Nr−m+2(2)) r−m+2 ẽ (N1(1)) 1 · · · ẽ(Nd(1)) d · Y = 0. Repeating this argument, we obtain µY (B−u≤k((m′ −m)Λd)) = { ẽ (N1(m′)) 1 · · · ẽ(Nm′+d−1(m′)) m′+d−1︸ ︷︷ ︸ 1st cycle ẽ (N1(m′−1)) 1 · · · ẽ(Nm′+d−2(m′−1)) m′+d−2︸ ︷︷ ︸ 2nd cycle · · · ẽ(N1(2)) 1 · · · ẽ(Nd+1(2)) d+1︸ ︷︷ ︸ (m′ − 1)th cycle ẽ (N1(1)) 1 · · · ẽ(Nd(1)) d︸ ︷︷ ︸ m′th cycle ·Y |Ni(j) ∈ Z≥0 } \ {0}. Therefore, we get µY (B−u≤k((m′ −m)Λd)) ⊂ B by (6.35). Next, we shall prove B ⊂ µY (B−u≤k((m′ −m)Λd)). For this, we take an arbitrary element M := d∏ i=1 m−m′∏ j=1 C(R (j) i ,K (j) i ) ∈ B. We need to show that there exists a set {Ni(s)} of non negative integers such that ẽ (N1(m′)) 1 · · · ẽ(Nm′+d−1(m′)) m′+d−1︸ ︷︷ ︸ 1st cycle ẽ (N1(m′−1)) 1 · · · ẽ(Nm′+d−2(m′−1)) m′+d−2︸ ︷︷ ︸ 2nd cycle · · · ẽ(N1(2)) 1 · · · ẽ(Nd+1(2)) d+1︸ ︷︷ ︸ (m′ − 1)th cycle ẽ (N1(1)) 1 · · · ẽ(Nd(1)) d︸ ︷︷ ︸ m′th cycle ·Y = M. (6.36) Cluster Variables and Monomial Realizations of Crystal Bases 29 We set {Ni(s)} as follows (see Example 6.18). First, we set {Ni(1)}i=1,2,...,d as Nd(1) := # { 1 ≤ j ≤ m−m′ |K(j) d − d = m′ } , Nd−1(1) := # { 1 ≤ j ≤ m−m′ |K(j) d−1 − (d− 1) = m′ } , Nd−2(1) := # { 1 ≤ j ≤ m−m′ |K(j) d−2 − (d− 2) = m′ } , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · N1(1) := # { 1 ≤ j ≤ m−m′ |K(j) 1 − 1 = m′ } . Note that since K (j) 1 < · · · < K (j) d , we have N1(1) < · · · < Nd(1). As seen in Example 3.9, by applying ẽ (Nd(1)) d to Y , the factors C(m′, d), C(m′+ 1, d), . . . , C(m′+Nd(1)−1, d) in Y turn out C(m′ − 1, d + 1), C(m′, d + 1), . . . , C(m′ + Nd(1) − 2, d + 1) by Remark 3.4, (6.24) and (6.26). Then the monomial ẽ (Nd(1)) d · Y does not include factors in the form τlq+d−1, 0 ≤ q ≤ m − 1. Thus, we can use Example 3.9 again when we apply ẽ (Nd−1(1)) d−1 to (ẽ (Nd(1)) d ·Y ). Then the factors C(m′, d−1), C(m′+1, d−1), . . . , C(m′+Nd−1(1)−1, d−1) in (ẽ (Nd(1)) d ·Y ) turn out C(m′−1, d), C(m′, d), . . . , C(m′ + Nd−1(1) − 2, d), and the monomial ẽ (Nd−1(1)) d−1 ẽ (Nd(1)) d · Y does not include factors in the form τlq+d−2, 0 ≤ q ≤ m − 1. After all, by using Example 3.9 repeatedly, we see that the monomial ẽ (N1(1)) 1 · · · ẽ(Nd−1(1)) d−1 ẽ (Nd(1)) d · Y is obtained from Y by replacing C(m′, ζ), C(m′+1, ζ), . . . , C(m′+Nζ(1)−1, ζ) by C(m′−1, ζ+1), C(m′, ζ+1), . . . , C(m′+Nζ(1)−2, ζ+1), 1 ≤ ζ ≤ d. We set {Ni(s)}2≤s≤m′, i=1,2,...,d+s−1 as Nd+1(2) := Nd(1), Nd(2) := Nd−1(1) + # { 1 ≤ j ≤ m−m′ |K(j) d − d = m′ − 1 } , Nd−1(2) := Nd−2(1) + # { 1 ≤ j ≤ m−m′ |K(j) d−1 − (d− 1) = m′ − 1 } , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · N2(2) := N1(1) + # { 1 ≤ j ≤ m−m′ |K(j) 2 − 2 = m′ − 1 } , N1(2) := # { 1 ≤ j ≤ m−m′ |K(j) 1 − 1 = m′ − 1 } , Nd+2(3) := Nd+1(2), Nd+1(3) := Nd(2), Nd(3) := Nd−1(2) + # { 1 ≤ j ≤ m−m′ |K(j) d − d = m′ − 2 } , Nd−1(3) := Nd−2(2) + # { 1 ≤ j ≤ m−m′ |K(j) d−1 − (d− 1) = m′ − 2 } , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · N2(3) := N1(2) + # { 1 ≤ j ≤ m−m′ |K(j) 2 − 2 = m′ − 2 } , N1(3) := # { 1 ≤ j ≤ m−m′ |K(j) 1 − 1 = m′ − 2 } , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Nm′+d−1(m′) := Nm′+d−2(m′ − 1), Nm′+d−2(m′) := Nm′+d−3(m′ − 1), · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Nd+1(m′) := Nd(m ′ − 1), Nd(m ′) := Nd−1(m′ − 1) + # { 1 ≤ j ≤ m−m′ |K(j) d − d = 1 } , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 30 Y. Kanakubo and T. Nakashima N2(m′) := N1(m′ − 1) + # { 1 ≤ j ≤ m−m′ |K(j) 2 − 2 = 1 } , N1(m′) := # { 1 ≤ j ≤ m−m′ |K(j) 1 − 1 = 1 } . For example, ifK (m−m′) d −d = m′, then the factor C(m′, d) in Y is acted by ẽm′+d−1 · · · ẽd+2ẽd+1ẽd since Nm′+d−1(m′) = Nm′+d−2(m′ − 1) = Nm′+d−3(m′ − 2) = · · · = Nd(1), and we obtain ẽm′+d−1 · · · ẽd+2ẽd+1ẽd · C(m′, d) = C(0,m′ + d) = C ( R (m−m′) d ,K (m−m′) d ) , by (6.24) and (6.26). In general, if K (j) i − i = m′− ζ, 0 ≤ ζ ≤ m′−1, then the factor C(m− j, i) in Y is acted by ẽm′−ζ+i−1 · · · ẽi+2ẽi+1ẽi, and we obtain ẽm′−ζ+i−1 · · · ẽi+2ẽi+1ẽiC(m− j, i) = C(m−m′ − j + ζ,m′ − ζ + i) = C(R (j) i ,K (j) i ), which means (6.36). Therefore, we get (6.33). � Example 6.18. We set r = 9, m = 6, m′ = 3 and d = 4. Let us see that we can obtain C(6, 1) · C(6, 2) · C(5, 4) · C(4, 6) M := C(4, 2) · C(4, 3) · C(4, 4) · C(3, 6) C(1, 4) · C(1, 5) · C(1, 6) · C(1, 7) from C(6, 1) · C(6, 2) · C(6, 3) · C(6, 4) Y := C(5, 1) · C(5, 2) · C(5, 3) · C(5, 4) C(4, 1) · C(4, 2) · C(4, 3) · C(4, 4). by applying {ẽNi(s)i } in the proof of Theorem 5.6, that is ẽ (N1(3)) 1 ẽ (N2(3)) 2 ẽ (N3(3)) 3 ẽ (N4(3)) 4 ẽ (N5(3)) 5 ẽ (N6(3)) 6︸ ︷︷ ︸ 3rd cycle · ẽ(N1(2)) 1 ẽ (N2(2)) 2 ẽ (N3(2)) 3 ẽ (N4(2)) 4 ẽ (N5(2)) 5︸ ︷︷ ︸ 2nd cycle · ẽ(N1(1)) 1 ẽ (N2(1)) 2 ẽ (N3(1)) 3 ẽ (N4(1)) 4︸ ︷︷ ︸ 1st cycle ·Y = M. (6.37) We set M = 3∏ j=1 4∏ i=1 C(R (j) i ,K (j) i ), that is, K (1) 1 = 1, K (1) 2 = 2, K (1) 3 = 4, K (1) 4 = 6, K (2) 1 = 2, K (2) 2 = 3, . . . ,K (3) 4 = 7. To change the factor C(4, 4) of Y into C(1, 7) of M , we need to apply ẽ6ẽ5ẽ4 to C(4, 4). In contrast, to change the factors C(5, 4) and C(6, 4) of Y into C(3, 6) and C(4, 6), we need to apply ẽ5ẽ4 to C(5, 4) and C(6, 4). Thus, C(4, 4) must be changed to C(3, 5) by the action of ẽ (N4(1)) 4 in first cycle, and C(5, 4), C(6, 4) do not have to be changed at the first cycle. So we set N4(1) := 1 = #{j |K(j) 4 − 4 = 3}. Similarly, we set N3(1) = N2(1) = N1(1) = 1, and the factors C(4, 1), C(4, 2), C(4, 3) of Y is changed to C(3, 2), C(3, 3), C(3, 4) by the action of first cycle. Next, to obtain M , the factor C(3, 5) in ẽ (N1(1)) 1 ẽ (N2(1)) 2 ẽ (N3(1)) 3 ẽ (N4(1)) 4 · Y must be changed to C(2, 6) at the second cycle. Thus, we set N5(2) := 1 = N4(1). The factor C(3, 4) must be changed to C(2, 5) by the action of ẽ (N4(2)) 4 in second cycle, and the factors C(5, 4) and C(6, 4) must also be changed at the second cycle. Hence, we set N4(2) := 3 = N3(1)+#{j |K(j) 4 −4 = 2}. Similarly, we set N3(2) = 1, N2(2) = 1, N1(2) = 0, N6(3) = 1, N5(3) = 3, N4(3) = 1, N3(3) = 3, N2(3) = 1 and N1(3) = 1. Then we get (6.37). Cluster Variables and Monomial Realizations of Crystal Bases 31 Example 6.19. We use the same setting in Example 6.15: u = s1s2s3s4s1s2s3s1s2s1, k = 6, m = 3 and m′ = 2. We have u≤6 = s1s2s3s4s1s2. Let Y := 1 τl2+2 ∈ Y, which has weight −Λ2. By Theorem 5.6, we obtain ∆L(6; i)(τ) = ∑ x∈B−u≤6 (−Λ2) µY (x) = Y + ẽ2Y + ẽ1ẽ2Y + ẽ3ẽ2Y + ẽ3ẽ1ẽ2Y + ẽ2ẽ3ẽ1ẽ2Y = 1 τl2+2 + τl1+2 τl2+1τl1+3 + τl1+1 τl1+3 + τl0+3 τl2+1τl0+4 + τl1+1τl0+3 τl1+2τl0+4 + τl0+2 τl0+4 . 6.5 The proof of Theorem 5.7 Let us prove Theorem 5.7. Suppose that ik = d = 1. Proof of Theorem 5.7. By Proposition 6.8, we have ∆L(k; i)(τ) = ∑ p∈X1(m,m′) Q(p). (6.38) The set X1(m,m′) consists of paths p as in the form p = (m, 1)→ ( m− 1, a(1) ) → ( m− 2, a(2) ) → · · · → ( 1, a(m−1) ) → (0,m′ + 1) such that a(s+1) = a(s) or a(s) + 1, 0 ≤ s ≤ m− 1. Here, a(0) := 1, a(m) := m′ + 1. By Lemma 6.10, we obtain # { s | a(s+1) = a(s) + 1, 0 ≤ s ≤ m− 1 } = m′. Set {s | a(s+1) = a(s) + 1, 0 ≤ s ≤ m− 1} := {j1, . . . , jm′}, 0 ≤ j1 < · · · < jm′ ≤ m− 1. Then we have a(0) = 1, a(1) = 1, a(2) = 1, . . . , a(j1) = 1, a(j1+1) = 2, a(j1+2) = 2, a(j1+3) = 2, . . . , a(j2) = 2, a(j2+1) = 3, a(j2+2) = 3, a(j2+3) = 3, . . . , a(j3) = 3, a(j3+1) = 4, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (6.39) a(jν+2) = ν + 1, a(jν+3) = ν + 1, . . . , a(jν+1) = ν + 1, a(jν+1+1) = ν + 2, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · a(jm′+2) = m′ + 1, a(jm′+3) = m′ + 1, . . . , a(m) = m′ + 1. Therefore, Definition 6.4(ii) means Q(p) = j1−1∏ i=0 1 τlm−1−i+1 j2−1∏ i=j1+1 τlm−1−i+1 τlm−1−i+2 · · · m−1∏ i=jm′+1 τlm−1−i+m′ τlm−1−i+m′+1 . Conversely, for a given {j1, . . . , jm′}, 0 ≤ j1 < · · · < jm′ ≤ m− 1, we can constitute a path p as in (6.39). Hence, by (6.38), we proved our claim. � 32 Y. Kanakubo and T. 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Algebra 399 (2014), 712–769, arXiv:1301.7301. http://dx.doi.org/10.1215/S0012-7094-04-12611-9 http://dx.doi.org/10.1215/S0012-7094-04-12611-9 http://arxiv.org/abs/math.RT/0305434 http://dx.doi.org/10.1007/s002220000102 http://arxiv.org/abs/math.RT/9912012 http://dx.doi.org/10.1090/S0894-0347-99-00295-7 http://arxiv.org/abs/math.RT/9802056 http://dx.doi.org/10.1215/S0012-7094-91-06321-0 http://dx.doi.org/10.1090/conm/325/05669 http://arxiv.org/abs/math.QA/0202268 http://dx.doi.org/10.1006/jabr.1994.1114 http://dx.doi.org/10.1090/conm/325/05669 http://arxiv.org/abs/math.QA/0204184 http://dx.doi.org/10.1016/j.jalgebra.2013.09.052 http://arxiv.org/abs/1301.7301 1 Introduction 2 Factorization theorem for type A 2.1 Double Bruhat cells 2.2 Factorization theorem for type A 3 Monomial realizations of crystal bases 3.1 Monomial realizations of crystal bases for type A 3.2 Demazure crystal 4 Cluster algebras and generalized minors 4.1 Cluster algebras of geometric type 4.2 Cluster algebras on double Bruhat cells of type A 5 Generalized minors and crystals 5.1 Generalized minor G(k;i)(a,t) 5.2 Generalized minor L(k;i)() 6 The proof of Theorems 5.6 and 5.7 6.1 The set Xd(m,m') of paths 6.2 One-to-one correspondence between paths in Xd(m,m') and terms of L(k;i)() 6.3 The explicit description of L(k;i)() 6.4 The completion of the proof of Theorem 5.6 6.5 The proof of Theorem 5.7 References

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