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arXiv:math/0105146v1 [math.QA] 17 May 2001

¨ THE BETHE EQUATION AT q = 0, THE MOBIUS INVERSION FORMULA, AND WEIGHT MULTIPLICITIES: (r ) III. THE XN CASE

ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI Abstract. It is shown that the numbers of o?-diagonal solutions to the Uq (XN ) Bethe equation at q = 0 coincide with the coe?cients in the recently introduced canonical power series solution of the Q-system. Conjecturally the canonical solutions are characters of the KR (Kirillov-Reshetikhin) modules. This implies that the numbers of o?-diagonal solutions agree with the weight multiplicities, which is interpreted as a formal completeness (r ) of the Uq (XN ) Bethe ansatz at q = 0.

(r )

1. Introduction Enumerating the solutions to the Bethe equation began with the invention of the Bethe ansatz [Be], where Bethe himself obtained a counting formula for sl2 -invariant Heisenberg chain. His calculation is based on the string hypothesis and has been generalized to higher spins [K1], sln [K2] and a general classical simple Lie algebra Xn [KR]. These works (1) concern the rational Bethe equation [OW], or in other words, Uq (Xn ) Bethe equation at q = 1. On the other hand, a systematic count at q = 0 started rather recently [KN1, KN2]. The two approaches are contrastive in many respects. To explain them, recall the general (1) setting where integrable Hamiltonians associated with Uq (Xn ) act on a ?nite dimensional module called the quantum space. At q = 1, the Hamiltonians are invariant and the Bethe vectors are singular with respect to the classical subalgebra Xn , while for q = 1, such aspects are no longer valid in general. Consequently, by completeness at q = 1 (resp. q = 0) we mean that the number of solutions to the Bethe equation coincides with the multiplicity of irreducible Xn modules (resp. weight multiplicities) in the quantum space. In this paper we study the Bethe equation associated with the quantum a?ne algebra (r ) Uq (XN ) [RW] at q = 0. By extending the analyses of the nontwisted case [KN1, KN2], an explicit formula R(ν, N ) is derived for the number of o?-diagonal solutions of the string (r ) center equation. Moreover we relate the result to the Q-system for Uq (XN ) introduced in [KR, K3, HKOTT]. It is a (yet conjectural in general) family of character identities for the KR modules (De?nition 2.1). Our main ?nding is that R(ν, N ) is identi?ed with the coe?cients in the canonical solution of the Q-system obtained in [KNT]. Under the Kirillov-Reshetikhin conjecture [KR] (cf. Conjecture 3.4), it leads to a character formula for tensor products of KR modules, which may be viewed as a formal completeness at q = 0. (r ) The outline of the paper is as follows. In Section 2 we study the Uq (XN ) Bethe equation at q = 0. For a generic string solution, the string centers satisfy the key equation (2.18), which we call the string center equation (SCE). There is a one-to-one correspondence between the generic string solutions to the Bethe equation and the generic solutions

1

2

ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI

to the SCE (Theorem 2.10). We then enumerate the o?-diagonal solutions of the SCE, and obtain the formula R(ν, N ) in Theorem 2.13. In Section 3 we recall the Q-system (r ) for Uq (XN ). It corresponds to a special case (called KR-type) of a more general system considered in [KNT]. There, power series solutions are studied, and the notion of the canonical solution is introduced unifying the ideas in [K1, K2, HKOTY, KN2]. For the Q-system in question, we ?nd that the coe?cients in the canonical solution are described by R(ν, N ), the number of o?-diagonal solutions of the SCE obtained in Section 3 (Theorem 3.3). A consequence of this fact is stated also in the light of the Kirillov-Reshetikhin conjecture [KR, C, KNT]. We note that the canonical solution of the Q-system is expressed also as a ratio of two power series [KNT], which matches the enumeration at q = 1 [KR] for the nontwisted cases. In this paper we omit most of the proofs and calculations, which are parallel with those in [KN1, KN2, KNT].

2. Bethe equation at q = 0 2.1. Preliminary. Let g = XN be a ?nite-dimensional complex simple Lie algebra of rank N . We ?x a Dynkin diagram automorphism σ of g of order r = 1, 2, 3. The a?ne (r ) (1) (1) (1) (1) (1) Lie algebras of type XN = An (n ≥ 1), Bn (n ≥ 3), Cn (n ≥ 2), Dn (n ≥ 4), En (n = (1) (1) (2) (2) (2) (2) (3) 6, 7, 8), F4 , G2 , A2n (n ≥ 1), A2n?1 (n ≥ 2), Dn+1(n ≥ 2), E6 and D4 are realized as the canonical central extension of the loop algebras based on the pair (g, σ ). Let g0 be the ?nite-dimensional σ -invariant subalgebra of g; namely,

g Xn A2n A2n?1 Dn+1 E6 D4 r 1 2 2 2 2 3 g0 Xn Bn Cn Bn F4 G2 Let A′ = (A′ij ) (i, j ∈ I ) and A = (Aij ) (i, j ∈ Iσ ) be the Cartan matrices of g and g0 , respectively, where Iσ is the set of σ -orbits of I . We de?ne the numbers d′i , di , ?′i , ?i (i ∈ I ) as follows: d′i (i ∈ I ) are coprime positive integers such that (d′iA′ij ) is symmetric; di (i ∈ Iσ ) are coprime positive integers such that (diAij ) is symmetric, and we set di = dπ(i) (i ∈ I ), where π : I → Iσ is the canonical projection. ?′i = r if σ (i) = i, and 1 otherwise; (r ) (2) ?i = 2 if A′iσ(i) < 0, and 1 otherwise. Let κ0 = 2 if XN = A2n , and 1 otherwise. By the de?nition one has d′i = di and ?′i = 1 if r = 1; d′i = 1 if r > 1; ?i = 1 if XN = A2n . In this paper we let {1, 2, . . . , N } and {1, 2, . . . , n} label the sets I and Iσ , respectively, (r ) and enumerate the nodes of the Dynkin diagram of XN by Iσ ∪ {0} as speci?ed in Table 1. The diagrams (and the enumeration of the nodes for r > 1) coincide with TABLE (2) A?1-3 in [Kac], except the A2n case. We ?x an injection ι : Iσ → I such that π ? ι = idIσ and Aab < 0 ? A′ι(a)ι(b) < 0 for any a, b ∈ Iσ . To be speci?c, assume that the labeling of the nodes for the Dynkin diagram of g are given by dropping the 0-th ones from XN case in Table 1. Then we simply set ι(a) = a and regard ι as the embedding of the subset {1, . . . , n} ?→ {1, . . . , N }. The symbols d′a , ?′a and A′ab for a, b ∈ Iσ = {1, . . . , n} should

(1) (r ) (2)

BETHE EQUATION AT q = 0

3

Table 1. Dynkin diagrams for XN . The enumeration of the nodes with Iσ ∪ {0} = {0, 1, . . . , n} is speci?ed under or the right side of the nodes. In addition, the numbers da (a ∈ Iσ ) are attached above the nodes if and only if da = 1.

e8

(r )

A1 : An : (n ≥ 2)

(1) Bn : (1)

(1)

0

e <> e

1

E8 : F4 : G2 : A2 : A2n : (n ≥ 2) A2n?1 : (n ≥ 3) Dn+1 : (n ≥ 2)

e e

(2) (2) (2) (2) (1) (1)

(1)

e

e

e

e

e

e

e

e

0

e

1

2

3

4

5

6

7

e0 ! aa ! ! aa !! e e ea e

0

e

2 e 2 e> e e 1 2 3 4 3 e> e 1 2 1

1

2

e 0T 2 e 2e 2

n?1 n

0 0

(n ≥ 3) Cn : (n ≥ 2) Dn : (n ≥ 4)

(1) (1)

2 e 3

e

1 0

e> e

2 e> e n?1 n

e< 2 e n?1 n en e e

e> e

1

2

0

e> 2 e

1

2 e 2

2 e> e n?1 n

e< 2 e

e0 e e

e0 e e e

1

2

e0 e4

n?2 n?1

1

2

3 2 e 2 2

e< 2 e

n?1 n

0 0

e< 2 e

1

E6 :

(1)

e

e

e

1

2

3

5

e7

6

E6 : D4 :

(3)

(2)

e

e

1 1

3

2 e 4

2 e> e n?1 n

e

E7 :

(1)

e

e

e

e

e

e

e

0

e< 3 e

2

0

1

2

3

4

5

6

be interpreted accordingly. One can check κ0 ?′a d′a = ?a da ,

r

A′aσs (b) =

s=1

?′a Aab . ?a

We use the notation: (2.1)

(r )

H = {(a, m) | a ∈ Iσ , m ∈ Z≥1 }.

(r )

Let Uq (XN ) be the quantum a?ne algebra. The irreducible ?nite-dimensional Uq (XN )modules are parameterized by N -tuples of polynomials (Pi (u))i∈I (Drinfeld polynomials ) ′ with unit constant terms [CP1, CP2]. They satisfy the relation Pσ(i) (u) = Pi (ω ?i u),

4

ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI

√ where ω = exp(2π ?1/r ). Thus it is enough to specify (Pb (u))b∈Iσ . Following [KNT] we introduce De?nition 2.1. For each (a, m) ∈ H and ζ ∈ C× , let Wm (ζ ) be the ?nite-dimensional (r ) irreducible Uq (XN )-module whose Drinfeld polynomials Pb (u) (b = 1, . . . , n) are speci?ed as follows: Pb (u) = 1 for b = a, and

m (a)

P a (u ) = We call

(a) Wm (ζ )

k =1

(1 ? ζq ?ada (m+2?2k) u).

a KR (Kirillov-Reshetikhin) module.

2.2. The Uq (XN ) Bethe equation. Let

(a) (a) N = { N = (Nm )(a,m)∈H | Nm ∈ Z≥0 , (a) (a,m)∈H (a) Nm < ∞ }.

(r )

Given ν = (νm ) ∈ N , we de?ne a tensor product module: (2.2)

(a)

Wν =

(a,m)∈H

(a) (a) ?νm (Wm (ζm )) ,

(a )

where ζm ∈ C× . In the context of solvable lattice models [B], one can regard W ν as the quantum space on which the commuting family of transfer matrices act. Reshetikhin and (r ) Wiegmann [RW] wrote down the Uq (XN ) Bethe equation and conjectured its relevance to the spectrum of those transfer matrices. In our formulation, it is the simultaneous (a) equation on the complex variables xi (i ∈ {1, 2, . . . , Ma }, a ∈ Iσ ) having the form: (2.3)

r ∞

′ (a) ω (xi ) ?′a q mκ0 da δa,σs (a) ? 1 1 ′ (a) ω s (xi ) ?′a ? q mκ0 da δa,σs (a)

s

1

νm

(a )

r

Mb

s=1 m=1

=?

ω s (xi ) ?′a q

(a)

1

(a)

1

′ κ0 d′ a Aaσ s (b)

s=1 b∈Iσ j =1

For the nontwisted case r = 1, this reduces to eq.(2.3) in [KN2]. The both sides are (a) actually rational functions of (xi ). In the sequel we consider a polynomial version of (2.3) speci?ed as follows: (2.4)

∞ (a) Fi+

ω s (xi ) ?′a ? (xj ) ?b q

(b)

1 ′

? (xj ) ?b

(b)

1 ′

′ κ0 d′ a Aaσ s (b)

.

Fi+ Gi? = Fi? Gi+ , = (xi q kκ0 ?a da ? 1)νk ,

(a)

′ ′ (a )

(a)

(a)

(a)

(a)

(a)

′

′

(a )

k =1 ∞

Fi? =

n ? (a) Gi+

(a)

k =1

(xi ? q kκ0 ?a da )νk ,

Mb (a) ((xi )

?′ ab ?′ a

=

b=1 j =1 n ? Mb

q

′ ′ κ0 ?′ ab da Aab

?

(b) (xj )

?′ ab ?′ b

),

(a) Gi?

=

b=1 j =1

(a) ((xi )

?′ ab ?′ a

?

(b) (xj )

?′ ab ?′ b

q κ0 ?ab da Aab ),

′

′

′

BETHE EQUATION AT q = 0

(2) (r ) (2)

5

where ?′ab = max(?′a , ?′b ), and n ? = n except for n ? = n + 1 for A2n . When XN = A2n , we (n+1) (n) have set xj = ?xj and Mn+1 = Mn . Remark 2.2. Let Pm (u) denote the a-th Drinfeld polynomial of the KR module Wm (1). Then we have Fi?

(a) (a) (a) (a)

Fi+

=

(a,m)∈H

q

?2?a da xi ) ?a da m Pm (q

(a)

(a)

νm

(a )

Pm (xi )

(a)

(a)

.

In view of this, we expect without proof that the solutions of (2.3) determine the spectrum (a) of transfer matrices acting on (2.2) with the choice ζm = 1. We consider a class of solutions (xi ) of (2.4) such that xi = xi (q ) is meromorphic function of q around q = 0. For a meromorphic function f (q ) around q = 0, let ord(f ) be the order of the leading power of the Laurent expansion of f (q ) around q = 0, i.e., f (q ) = q ord(f ) (f 0 + f 1 q + · · · ), f 0 = 0,

(a) (a) (a)

?(q ) := f 0 + f 1 q + · · · be the normalized series. When f (q ) is identically zero, we and let f (a) set ord(f ) = ∞. For each N = (Nm ) ∈ N , we set (2.5)

(a) H ′ = H ′ (N ) := { (a, m) ∈ H | Nm > 0 },

where H is de?ned in (2.1). We have |H ′| < ∞. De?nition 2.3. Let (Ma )n a=1 be the one in the Bethe equation (2.4), and let N = (Nm ) ∈ (a) (a) ∞ N satisfy m=1 mNm = Ma . A meromorphic solution (xi ) of (2.4) around q = 0 is called a string solution of pattern N if (a) (a) (i) ord(Fi+ Gi? ) < ∞ for any (a, i). (a) (a) (ii) (xi ) can be arranged as (xmαi ) with (a, m) ∈ H ′ ,

(a) α = 1, . . . , Nm , (a)

i = 1, . . . , m

such that (a) (a) (a) dmαi := ord(xmαi ) = (m + 1 ? 2i)κ0 ?′a d′a . (a) (a)0 (a)0 (a)0 (a)0 (b) zmα := xmα1 = xmα2 = · · · = xmαm (= 0), where xmαi is the coe?cient of the (a) leading power of xmαi . (a) (a) For each (a, m, α), (xmαi )m i=1 is called an m-string of color a, and zmα is called the string (a) (a) center of the m-string (xmαi )m i=1 . Thus, Nm is the number of the m-strings of color a. For a string solution xmαi (q ) = q dmαi x ?mαi (q ) of pattern N , the Bethe equation (2.4) reads (2.6) Fmαi+ Gmαi? = Fmαi? Gmαi+ ,

(a) (a) (a) (a) (a)

(a )

(a)

6

ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI

∞ (a) Fmαi+

(2.7) (2.8)

=

k =1 ∞

(? xmαi q dmαi +kκ0 ?a da ? 1)νk ,

(a)

(a ) ′ ′ (a )

(a)

(a )

′

′

(a )

Fmαi? =

n ? (a) Gmαi+

(a)

k =1

(? xmαi q dmαi ? q kκ0 ?a da )νk ,

∞ Nk

(b)

k (a) ((? xmαi )

(2.9)

=

b=1 k =1 β =1 j =1 n ? ∞ Nk

(b)

?′ ab ?′ a

q

?′ ab d(a) +κ ?′ d′ A′ 0 ab a ab mαi ?′ a

? q

(b) (? xkβj )

?′ ab ?′ b

q

?′ ab d(b) kβj ?′ b

),

k (a) ((? xmαi )

(2.10)

(a) Gmαi?

=

b=1 k =1 β =1 j =1

?′ ab ?′ a

q

?′ ab d(a) mαi ?′ a

?

(b) (? xkβj )

?′ ab ?′ b

?′ ab d(b) +κ ?′ d′ A′ 0 ab a ab kβj ?′ b

),

where for XN = A2n , we have set x ?kβj = ?x ?kβj , dkβj = dkβj and Nk = Nk . According to the procedure similar to [KN2], we can take the q → 0 limit of (2.6) and obtain a key equation:

m m i=1

(r )

(2)

(n+1)

(n)

(n+1)

(n)

(n+1)

(n)

(2.11)

1 = (?1)

Fmαi+ Gmαi? Fmαi? Gmαi+

(a)0 (a)0

(a)0

(a)0

.

In order to estimate the order of the Bethe equation (2.6), we introduce

∞

ξmαi+ = κ0 ?′a d′a

k =1 ∞

(a)

νk min(m + 1 ? 2i + k, 0), νk min(m + 1 ? 2i, k ),

(b)

(a)

ξmαi? = κ0 ?′a d′a

k =1

(a)

(a)

(2.12)

(a) ηmαi+

n ?

∞ Nk

k

= κ0

b=1 k =1 β =1 j =1 n ? ∞ Nk

(b)

?′ab min(d′a (m + 1 ? 2i + A′ab ), d′b(k + 1 ? 2j )), ?′ab min(d′a (m + 1 ? 2i), d′b (k + 1 ? 2j + A′ba )).

k

(a) ηmαi?

= κ0

b=1 k =1 β =1 j =1

De?nition 2.4. A string solution (xmαi ) to (2.6) is called generic if ord(Fmαi± ) = ξmαi± , ord(Gmαi+ ) = ηmαi+ + ζmαi ,

(a) (a) (a) (a) (a) (a) (a) (a)

(a)

(2.13)

ord(Gmαi? ) = ηmαi? + ζmαi+1 ,

(a) (a)

(a)

(a)

(a)

where ζmαi := ord(? xmαi ? x ?mαi?1 ) for 2 ≤ i ≤ m, and ζmα1 = ζmα,m+1 = 0.

BETHE EQUATION AT q = 0

7

Given a quantum space data ν ∈ N and a string pattern N ∈ N , we set

∞

(2.14) (2.15) (2.16)

(a) γm

=

(a) γm (ν )

=

k =1

min(m, k )νk , Aab (b) min(d′a m, d′b k )Nk , ?a d′b A′ab (b) min(d′a m, d′b k )Nk . ′ db

(a)

(a) (a) (a) Pm = Pm (ν, N ) = γm ?

(b,k )∈H

? (a) = P ? (a) (ν, N ) = γ (a) ? P m m m

(b,k )∈H

(a) ?m The number P will appear only in the RHS of (2.18).

Lemma 2.5. We have (ξmαi+ + ηmαi? ) ? (ξmαi? + ηmαi+ ) ? (a) (a) (a) ′ ′ ? ??κ0 ?a da (Pm+1?2i + Nm+1?2i ) ? κ0 ?m+1?2i = 0 ? ? ′ ′ (a) (a) (a) κ0 ?a da (P2i?m?1 + N2i?m?1 ) + κ0 ?2i?m?1

(a) ?j (a) (a) (a) (a) +1 1 ≤ i < m2 +1 i = m2 m+1 < i ≤ m, 2

where ?j = 0 except for the following nontwisted cases: If there is a′ such that da > da′ = 1 and Aaa′ = 0, then = ?N2j (a′ ) (a′ ) (a′ ) ?(N3j ?1 + N3j + N3j +1 )

(a′ )

(a)

da = 2 d a = 3.

(a)

For a generic string solution, one can determine the order ζmαi from (2.6), (2.13) and (a) Lemma 2.5. Requiring that the resulting ζmαi should be positive and ?nite (cf. De?nition 2.3), one has Proposition 2.6. A necessary condition for the existence of a generic string solution of pattern N is

min(i?1,m+1?i)

(2.17)

k =1 (a)

d′a (Pm+1?2k + Nm+1?2k ) + ?m+1?2k

(a)

(a)

(a)

> 0,

for (a, m) ∈ H ′, 1 ≤ α ≤ Nm , 2 ≤ i ≤ m. For a generic string solution, (2.11) becomes an equation for the string centers (zmα ). We call it the string center equation (SCE). Proposition 2.7. Let (xmαi ) be a generic string solution of pattern N . Then its string (a) (a) centers (zmα ) satisfy the following equations ((a, m) ∈ H ′ , 1 ≤ α ≤ Nm ):

Nk

(b)

(a)

(a)

(2.18)

(b,k )∈H ′ β =1

(zkβ )Aamα,bkβ = (?1)Pm

(b)

(a ) ? ( a ) + Nm +1

,

(2.19)

(a) (a) Aamα,bkβ = δab δmk δαβ (Pm + Nm )+

Aba min(d′a m, d′b k ) ? δab δmk . ?b d′a

8

ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI

Note that all the quantities in (2.15), (2.16) and (2.19) are integers. As in [KN2], Proposition 2.7 is derived by explicitly evaluating the ratio (2.11) by Lemma 2.8. For a ∈ {1, 2, . . . , n} and b ∈ {1, 2, . . . , n ? }, we have

m (a)0 Fmαi? i=1 m k (a) ((? xmαi ) i=1 j =1

=

(?1)γm famα

?′ ab ?′ a

(a )

?=+ ? = ?, ?

(b) (? xkβj )

?′ ab ?′ b ?′ ab d(b) + 1 (1??)κ ?′ d′ A′ 0 ab a ab kβj 2 ?′ b

(a) (a) (zmα )γm famα

q

?′ ab d(a) + 1 (1+?)κ ?′ d′ A′ 0 ab a ab ?′ a mαi 2

q

)0

bkβ for some famα and gamα , where we have set zmα

? ?′ ab ′ ′ ? (b) ?′ A′ min(d′ bkβ a m,db k )/db ?δab δmk g (?(zkβ ) b ) ab amk = ′ ′ ′ ? (a) ??ab A′ min(d′ bkβ a m,db k )/db ?δab δmk (zmα ) ′a ab (?1)(m?1)δab δmk δαβ gamk

(n+1)

?=1 ? = ?1,

(a)

:= ?zmα .

(n)

bkβ The quantities famα and gamα depend on the string centers (zmα ), whose explicit formulae are available in [KN2] for nontwisted case. However we do not need them bkβ here. A string solution is generic if and only if famα = 0 and gamα = 0 for any (a) (b) a ∈ {1, . . . , n}, b ∈ {1, . . . , n ? }, m, k ∈ Z≥1 , 1 ≤ α ≤ Nm , 1 ≤ β ≤ Nk . These conditions are equivalent to (2.20) (a) (a) zmα = 1 if there is k ≥ 1 such that νk > 0 and k ∈ m , (a) (zmα )

?′ ab ′ ?a

=

(b) (zkβ )

?′ ab ?′ b

where m = {m ? 1, m ? 3, . . . , ?m + 1}. Apart from the exceptional case (a, m, α) = (b, k, β ), the condition (2.20) says that the two terms in each factor in (2.7) – (2.10) possess di?erent leading terms whenever their orders coincide. De?nition 2.9. A solution to the SCE (2.18) is called generic if it satis?es (2.20). Let A be the matrix with the entry Aamα,bkβ in (2.19). The main theorem in this subsection is Theorem 2.10. Suppose that N ∈ N satis?es the conditions (2.17) and det A = 0. Then, there is a one-to-one correspondence between generic string solutions of pattern N to the Bethe equation (2.6) and generic solutions to the SCE (2.18) of pattern N . Remark 2.11. Given the Bethe equation (2.3), the choice of Fi± and Gi± in (2.4) is not (a) the unique one. For example one may restrict the b-product in Gi± to those satisfying A′ab = 0. Such an ambiguity in?uences De?nition 2.3 (i), (2.7) – (2.10), (2.12), (2.20), hence De?nition 2.9. However, the ratio in (2.11) is left unchanged, and all the statements in Lemma 2.5, Propositions 2.6, 2.7 and Theorem 2.10 remain valid. 2.3. Counting of o?-diagonal solutions to SCE . For k ∈ C and j ∈ Z, we de?ne the binomial coe?cient by Γ (k + 1) k = . j Γ (k ? j + 1)Γ (j + 1)

(a) (a)

if (a, m, α) = (b, k, β ) and d′a A′ab ∈ {id′a ? jd′b | i ∈ m , j ∈ k },

BETHE EQUATION AT q = 0

9

For each ν , N ∈ N , we de?ne the number R(ν, N ) by (2.21) R(ν, N ) =

(a,m),(b,k )∈H Nk

(b)

det

Fam,bk ′

(a,m)∈H ′

1 Nm

(a)

Pm + Nm ? 1 , (a) Nm ? 1

(a)

(a)

(2.22)

Fam,bk =

β =1

(a) Aamα,bkβ = δab δmk Pm + (a) (a)

Aba (b) min(d′a m, d′b k )Nk , ′ ?b da

for N = 0. Here H ′ = H ′ (N ) and Pm = Pm (ν, N ) are given by (2.5) and (2.15). For N = 0, we set R(ν, 0) = 1 irrespective of ν . It is easy to see that R(ν, N ) is an integer. De?nition 2.12. A solution (zmα ) to the SCE is called o?-diagonal (diagonal ) if zmα = (a) zmβ only for α = β (otherwise). Our main result in this subsection is Theorem 2.13. Suppose Pm (ν, N ) ≥ 0 for any (a, m) ∈ H ′ . Then the number of o?(a) diagonal solutions to the SCE (2.18) of pattern N divided by (a,m)∈H ′ Nm ! is equal to R(ν, N ). The proof is due to the inclusion-exclusion principle and an explicit evaluation of the M¨ obius inversion formula similar to [KN1, KN2]. 3. R(ν, N ) and Q-system So much for the Bethe equation, we now turn to the Q-system. For a, b ∈ Iσ and m, k ∈ Z, set ? ? ? ?1b Aba δm,k r>1 ? ? ? ? ? db /da = 2 ??Aba (δm,2k?1 + 2δm,2k + δm,2k+1 ) Gam,bk = ?Aba (δm,3k?2 + 2δm,3k?1 + 3δm,3k db /da = 3 ? ? ? +2δm,3k+1 + δm,3k+2 ) ? ? ? ??A δ otherwise. ab da m,db k Let αa and Λa (a ∈ Iσ ) be the simple roots and the fundamental weights of g0 . We set xa = e?a Λa , ya = e?αa , which are related as

n (a) (a) (a)

(3.1)

ya =

b=1

xb

?Aba /?b

.

De?nition 3.1. The system of equations (Q0 (y ) = 1) (3.2)

(a) m (a) (Qm (y ))2 = Qm+1 (y )Qm?1 (y ) + ya (Qm (y ))2 (b,k )∈H (a) (a) b) Gam,bk (Q( m (y ))

(a)

for a family (Qm (y ))(a,m)∈H of power series of y = (ya )n a=1 with unit constant terms is called the Q-system.

(a)

10

ATSUO KUNIBA, TOMOKI NAKANISHI, AND ZENGO TSUBOI

m The factor ya in the RHS is absorbed away if (3.2) is written in terms of the combi( a ) m nation xa Qm (y ). The resulting form of the Q-system has originally appeared in [KR] (1) (1) (1) (1) (1) (1) (1) (An , Bn , Cn , Dn ), [K3] (E6,7,8 , F4 , G2 ) and [HKOTT] (twisted case). De?nition 3.1 corresponds to an in?nite Q-system in the terminology of [KNT]. Its solution is not unique in general. Following [KNT] we introduce

De?nition 3.2. A solution of (3.2) is canonical if the limit limm→∞ Qm (y ) exists in the ring C[[y ]] of formal power series of y = (ya )n a=1 with the standard topology. Theorem 3.3. ([KNT]) There exists a unique canonical solution (Qm (y ))(a,m)∈H of the Q-system (3.2). Moreover, for any ν ∈ N , it admits the formula:

(a) (Qm (y ))νm = Rν (y ), (a,m)∈H

(a )

(a)

(a)

where the power series R (y ) is de?ned by

n

ν

R ν (y ) =

N ∈N

R(ν, N )

a=1

ya

∞ m=0

mNm

(a )

in terms of the integer R(ν, N ) in (2.21). In the proof of the theorem [KNT], the expression R(ν, N ) emerges from a general argument on the Q-system, which is independent of the Bethe equation. Our main ?nding in this paper is that it coincides with the number of o?-diagonal solutions to the SCE obtained in Theorem 2.13. Let us state the consequence of this fact in the light of the Kirillov-Reshetikhin con(a) (x) denote the Laurent polynomial of x = (xa )n jecture. Let chm a=1 representing the (a) (a) ?m (a) g0 -character of the KR module Wm (ζ ). Then, Qm (y ) := xa chm (x)|x=x(y) , where x(y ) is the inverse map of (3.1), is a polynomial of y = (ya )n a=1 with the unit constant term. (a) (a) We call Qm (y ) the normalized g0 -character of Wm (ζ ). The normalized character of the g0 -module W ν in (2.2) is given by Qν (y ) =

(a) (a) (a,m)∈H (a) (Qm (y ))νm .

(a )

The Kirillov-Reshetikhin conjecture [KR] is formulated in [KNT] as Conjecture 3.4. Qm (y ) = Qm (y ) for any (a, m) ∈ H . Combining Theorem 3.3 and Conjecture 3.4, we relate the weight multiplicity in the tensor product of KR modules to the number of o?-diagonal solutions to the SCE: Corollary 3.5 (Formal completeness of the Bethe ansatz at q = 0). Under Conjecture 3.4 one has Qν (y ) = Rν (y ).

m Conjecture 3.4 implies that ( (a,m)∈H xmν )Rν (y (x)) is a Laurent polynomial invariant a under the Weyl group of g0 . In fact canonical solutions have also been obtained as (r ) linear combinations of characters of irreducible ?nite dimensional g0 -modules for XN = (1) (1) (1) (1) (r ) (2) (2) (2) (3) An , Bn , Cn , Dn [KR, HKOTY], and for XN = A2n?1 , A2n , Dn+1 , D4 [HKOTT]. For the current status of Conjecture 3.4, see section 5.7 of [KNT]. (a )

BETHE EQUATION AT q = 0

11

References

[B] R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London (1982). [Be] H. A. Bethe, Zur Theorie der Metalle, I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Physik 71 (1931) 205–231. [C] V. Chari, On the fermionic formula and the Kirillov-Reshetikhin conjecture, math.QA/0006090. [CP1] V. Chari and A. Pressley, Quantum a?ne algebras and their representations, Canadian Math. Soc. Conf. Proc. 16 (1995) 59–78. [CP2] V. Chari and A. Pressley, Twisted Quantum a?ne algebras, Commun. Math. Phys. 196 (1998) 461–476. [HKOTY] G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Remarks on fermionic formula, Contemporary Math. 248 (1999) 243–291. [HKOTT] G. Hatayama, A. Kuniba, M. Okado , T. Takagi and Z. Tsuboi, Paths, Crystals and Fermionic Formula, math.QA/0102113. [Kac] V. G. Kac, In?nite dimensional Lie algebras, 3rd edition, Cambridge Univ. Press, Cambridge (1990). [K1] A. N. Kirillov, Combinatorial identities and completeness of states for the Heisenberg magnet, J. Sov. Math. 30 (1985) 2298–3310. [K2] A. N. Kirillov, Completeness of states of the generalized Heisenberg magnet, J. Sov. Math. 36 (1987) 115–128. [K3] A. N. Kirillov, Identities for the Rogers dilogarithm function connected with simple Lie algebras, J. Sov. Math. 47 (1989) 2450–2459. [KR] A. N. Kirillov and N. Yu. Reshetikhin, Representations of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras, J. Sov. Math. 52 (1990) 3156–3164. [KN1] A. Kuniba and T. Nakanishi, The Bethe equation at q = 0, the M¨ obius inversion formula, and weight multiplicities: I. The sl(2) case, Prog. in Math. 191 (2000) 185–216. [KN2] A. Kuniba and T. Nakanishi, The Bethe equation at q = 0, the M¨ obius inversion formula, and weight multiplicities: II. The Xn case, math.QA/0008047, J. Alg. in press. [KNT] A. Kuniba, T. Nakanishi and Z. Tsuboi, The canonical solutions of the Q-systems and the KirillovReshetikhin conjecture, math.QA/0105145. [OW] E. Ogievetsky and P. Wiegmann, Factorized S -matrix and the Bethe ansatz for simple Lie groups, Phys. Lett. B 168 (1986) 360–366. [RW] N. Yu. Reshetikhin and P. Wiegmann, Towards the classi?cation of completely integrable quantum ?eld theories (the Bethe ansatz associated with Dynkin diagrams and their automorphisms), Phys. Lett. B 189 (1987) 125–131. Institute of Physics, University of Tokyo, Tokyo 153-8902, Japan E-mail address : atsuo@gokutan.c.u-tokyo.ac.jp Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan E-mail address : nakanisi@math.nagoya-u.ac.jp Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan E-mail address : tsuboi@gokutan.c.u-tokyo.ac.jp

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