Remarks on Fermionic Formula

arXiv:math/9812022v3 [math.QA] 30 Jul 1999 Remarks on Fermionic Formula G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada Abstract. Fermioni...

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arXiv:math/9812022v3 [math.QA] 30 Jul 1999

Remarks on Fermionic Formula G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada Abstract. Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae asso(1) ciated with general non-twisted quantum affine algebra Uq (Xn ) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary Xn .

1. Introduction Many important q-polynomials and q-series arising in representation theory and combinatorics can be expressed as sums of products of q-binomial or multinomial coefficients. Among them there are a class of formulae connected with the Bethe ansatz. (1) Consider the tensor product of the L-copies of the vector representation W1 ≃ C2 of sl(2): (1)

W = W1

(1)

⊗ · · · ⊗ W1 .

Decomposing W into irreducible components one gets M W = M (W, λ)V (λ), λ

where the multiplicity M (W, λ) of the λ-highest weight representation V (λ) is the well known Kostka number Kλ,(1L ) . In physics one regards W as a Hilbert space of a statistical mechanical model, spin 21 Heisenberg chain, and tries to diagonalize an sl(2)-linear operator on W called the row transfer matrix (RTM). The Bethe ansatz is a method to construct the eigenvectors of RTM that are sl(2)-highest 1 . Thus M is counted as the number of the solutions to the Bethe equation. Under the string hypothesis [Be] it leads to 1991 Mathematics Subject Classification. Primary 81R10, 81R50, 82B23; Secondary 05E15. 1 This is so for systems with Yangian symmetry as in the XXX Heisenberg chain. Those systems associated with trigonometric solutions to the Yang-Baxter equation can have a RTM that does not commute with Uq (sl(2)) nor sl(2). In such a case, the Bethe ansatz produces not only the highest weight vectors but also the other weight vectors. 1

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G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

Fermionic formula: M (W, λ) = M (W, λ, q = 1) where Y  p i + mi  X c({m}) q M (W, λ, q) = mi q {m}

c({m}) =

X

j≥1

min(i, j)mi mj − L

i,j≥1

pi = L − 2

X

X i≥1

min(i, j)mj .

mi +

L(L − 1) , 2

j≥1

P Here the {m} sum is taken over mi ≥ 0 (i ≥ 1), such that 2 i≥1 imi = L − k for λ corresponding to the (k + 1)-dimensional representation. The q-analogue M (W, λ, q)2 is counting the spectrum of the momentum (cyclic lattice shift operator) which is a specialization of the RTM. The formula involves (q-)binomial without signs reflecting the “fermionic” nature of the counting in the Bethe ansatz. The idea to apply the Bethe ansatz to representation theory and combinatorics has been initiated in [KR1, KR2] and has led to fruitful results. For example, M (W, λ, q) in the above turns out to be the Kostka-Foulkes polynomial Kλ,(1L ) (q) [Ma]. See [Ki3] and reference therein. On the other hand, there is another important idea from Baxter’s corner transfer matrix (CTM) method [ABF, B]. It indicates that in the limit L → ∞, the space W has a structure of an integrable highest weight module over the quantum 3 b affine algebra Uq (sl(2)) . To be more precise we invoke the crystal base theory (1) b of Kashiwara [Kas] and regard W1 as a Uq′ (sl(2))-module, which has a perfect crystal B. Then in the limit L → ∞, a subset of the semi-infinite tensor product · · · ⊗ B ⊗ B turns out to be isomorphic to the crystal base of the level 1 integrable b highest weight module over Uq (sl(2)) [KMN1]. Moreover B ⊗L with finite L is isomorphic to the crystal base of a Demazure module [KMOU]. By using the notion of path and energy function, one can define One dimensional sum (1dsum): X P L−1 q j=1 (L−j)H(bj ⊗bj+1 ) , X= {b}

where H(1 ⊗ 1) = H(2 ⊗ 1) = H(2 ⊗ 2) = 1,

H(1 ⊗ 2) = 0,

P and the {b} sum is taken over bj ∈ {1, 2} (1 ≤ j ≤ L), such that m j=1 (δbj ,1 − δbj ,2 ) ≥ 0 for 1 ≤ m ≤ L − 1 and = k for m = L. This is the branching coefficient of the (k + 1)-dimensional Uq (sl(2))-module in the Demazure module, which is graded by the null root q = e−δ . See X(B ⊗L , λ, q) in (3.1) for the general definition. When q = 1, both M and X give the multiplicity of the same representation in W . In q case, they are related to the spectrum of the RTM and CTM, respectively. Their spectra are different in general although they are expected to be the same in the “CFT” limit, which involves L → ∞ at least. It is therefore remarkable that X =M 2 M (W, λ, q) 3 Here

here is q L(L−1)/2 times that in (4.3). we have the q-deformed XXZ chain in mind rather than the XXX chain.

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is known to hold for finite L 4 . Note here that the Bethe ansatz play a complementary role with the CTM and the crystal base theory. The latter provides a conceptual definition of the 1dsum X, while the former offers a specific formula M which is fermionic. This kind of phenomena have been conjectured or proved for a variety of setting and a wide class of representations. There are immense literatures on this, see for example [Ber, BMS, BMSW, BLS, DF, DKKMM, FLW, FOW, FS, Ge, HKKOTY, KM, Ki3, KR1, KSS, NRT, OPW, SW, Wa] and the reference therein. Especially there are extensive results on fractional level case of (1) (1) (1) A1 cf. [BMS, FLW]. However, beyond the A1 or An case to which these literatures are mostly devoted, there are relatively few works concerning general (1) Xn especially at higher level, cf. [KR2, KNS, Y]. The aim of this paper is to treat all the non-twisted quantum affine algebra (1) Uq (Xn ) on an equal footing. We formulate a general X = M conjecture and discuss its several consequences and applications that generalize or unify many earlier results. Let us give an overview of them aligned in the sections 2 – 8 in the main text. In section 2, we introduce a conjectural family of crystals B r,s of the finite (r) (1) dimensional representation Ws of Uq′ (Xn ) (1 ≤ r ≤ n, s ∈ Z≥1 ). Its existence is suggested from the Bethe ansatz argument in [KR2] for the Yangian Y (Xn ) case. We propose a criterion telling whether B r,s is perfect or not according to r, s and the root data of Xn . When B r,s is perfect, the theory of [KMN1] applies and the semi-infinite tensor product · · · ⊗ B r,s ⊗ B r,s becomes isomorphic to the crystal (1) base of an integrable highest weight Uq (Xn )-module. When B r,s is non-perfect, r,s r,s we conjecture that · · · ⊗ B ⊗ B is isomorphic to the crystal base of a certain tensor product module. This conjecture, indicated again from the Bethe ansatz [Ku], has been proved in [HKKOT] in some cases. In section 3, the 1dsums, either classically restricted one X(B, λ, q) or level restricted one Xl (B, λ, q), are defined for arbitrary inhomogeneous B = B r1 ,s1 ⊗ · · · ⊗ B rL ,sL . By definition, the classically restricted one X(W, λ, q) at q = 1 (r ) (r ) coincides with the multiplicity [W : λ] of V (λ) in W = Ws1 1 ⊗ · · ·⊗ WsLL regarded as a Uq (Xn )-module. Relations of L → ∞ limit of the 1sums (homogeneous case) to the branching functions are also stated including the non-perfect case. We shall also formulate the conjecture X = M , where the definition of the fermionic form M is postponed to section 4. In section 4, we define the fermionic formula M (W, λ, q) and its restricted (r ) (r ) version Ml (W, q) for general inhomogeneous W = Ws1 1 ⊗ · · · ⊗ WsLL . We shall also introduce a modified one Nl (W, λ, q), for which we allow negative “vacancy (a) numbers” pi in the Bethe ansatz terminology. This is a noteworthy difference from M ’s. Nevertheless we conjecture that they are the same for λ in the dominant chamber, see Conjecture 4.3. The N∞ (W, λ, 1) has a nice behaviour under the Weyl group (cf. Remark 8.8, Conjecture 8.9) and plays an essential role in section 8. One of the important advantages of the fermionic formulae is that we can derive the spinon character formulae for the branching functions by taking the limit (r)⊗L , λ, q). We do this in section 5, generalizing the calculations in L → ∞ in M (Ws [HKKOTY]. In general, the spinon character formulae are related with the particle 4 For

q = 1 this was proved by Bethe [Be] already in 1931.

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G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

structure of the model, or the domain wall description of the paths [NY1, NY2, ANOT, BPS, BS, BLS]. For simply laced Xn , the spinon character formulae derived here are consistent with the conjectured particle structure of the model, i.e. for any level k, the particles are labeled by the fundamental representations [FT] (1) and their S-matrices are the tensor products of those for Uq (Xn ) and its RSOS analogues [R]. In section 6, a recursion relation satisfied by the fermionic formulae is proved. In section 7, we discuss closely related difference equations among the characters (r) (r) Qs = chWs [Ki1, KR2] (Q-system). We verify that the Q-system is indeed valid for An , Bn , Cn and Dn as announced in [KR2] and include a proof for Bn case. (r) In addition we establish some asymptotic property of the Qs -functions needed in section 8. Finally, in section 8, we formulate and prove Theorem 8.1 related to the combinatorial completeness of string hypothesis of the Bethe ansatz for arbitrary (r ) (r ) Xn and W = Ws1 1 ⊗ · · · ⊗ WsLL . By combinatorial completeness it is usually meant the equality [W : λ] = M (W, λ, q = 1), which is an indication (not a proof) that there are as many solutions to the Bethe equation as the Xn -highest weight vectors in W . Instead of this, what our Theorem 8.1 tells is that [W : λ] = N∞ (W, λ, q = 1) is reduced to checking the Q-system and the asymptotic property (r) of the Qs -functions 5 . This is a support of our X = M conjecture in a weak sense in that we restrict to q = 1 and need Conjecture 4.3 to replace M by N∞ . (X = [W : λ] holds by definition, cf. (3.3).) The heart of our proof of the Theorem is to derive an integral representation of the fermionic formulae by means of the Q-system. It is a generalization of the An case in [Ki2], where not [W : λ] = M but [W : λ] = N∞ was shown similarly. String hypothesis is an origin of the fermionic formulae. However we emphasize that our Theorem 8.1 stands totally independent of it and elucidates the following points which have not necessarily been recognized so evidently: (i) The combinatorial completeness is reduced to the Q-system and asymptotic property of the (r) Qs -functions for general Xn . (ii) The relevant fermionic form is not M (W, λ, 1) but N∞ (W, λ, 1) which contains contributions from negative vacancy numbers. (iii) Despite the significant difference in their definitions, M = N∞ is valid (Conjecture 4.3). At present no explanation is available for the remarkable coincidence of the two fermionic forms M and N∞ . Although it is not “physical” to allow the negative (a) vacancy numbers pi in the Bethe ansatz context, N∞ enjoys a nice symmetry under the Weyl group. Compare (4.7) and Conjecture 8.9. (r) In Appendix A, we list explicit examples of the fermionic formulae M (Ws , λ, q −1 ), which is a q-version of the list in [Kl]. Some of those data require a very long CPU time to be evaluated. In Appendix B we illustrate the calculation of the 1dsum and the fermionic (1) form with an example from C2 . In Appendix C we give the explicit form of the Q-system for non-simply laced Xn .

5 In section 7 we have been able to establish these conditions for A , B , C and D only. n n n n But Theorem 8.1 itself works also for E6,7,8 , F4 and G2 once these conditions are guaranteed.

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Acknowledgements The authors thank K. Aomoto, M. Kashiwara, A.N. Kirillov, Y. Koga, A. Lascoux, B. Leclerc, J. Lepowsky, T. Miwa, T. Nakanishi, M. Shimozono and J.-Y. Thibon for valuable discussions and interest in this work. A. K. and M.O. thank N. Jing and K.C. Misra, organizers of the “Conference on Affine and Quantum Affine Algebras and Related Topics” held at North Carolina State University, Raleigh during May 21-24, 1998, for the invitation and warm hospitality.

2. A conjectural family of crystals 2.1. Preliminaries. Let Xn denote one of the classical simple Lie algebra (1) An (n ≥ 1), Bn (n ≥ 2), Cn (n ≥ 2), Dn (n ≥ 4), E6,7,8 , F4 and G2 , and let Xn be the associated non-twisted affine Lie algebra. We recapitulate necessary facts and (1) notations concerning the quantum affine algebra Uq (Xn ). Let αi , hi , Λi (i ∈ I = (1) {0, 1, . . . , n}) be the simple roots, simple coroots, fundamental weights of Xn . We enumerate the vertices of the Dynkin diagram as in Table 1, which is the same with TABLE Fin and Aff1 in [Kac]. Let (·|·) be the standard bilinear form normalized by (αi |αi ) = 2 with αi a long root. We shall write |x|2 to mean (x|x). Let δ and L c denote the L null root ZΛi Zδ be the and the canonical central element, respectively. Let P = i∈I P weight lattice. We define the following subsets of P : P + = i∈I Z≥0 Λi , Pl+ = Pn Pn + {λ ∈ P + | hλ, ci = l}, P = i=1 ZΛi , P = i=1 Z≥0 Λi . Here Λi = Λi − hΛi , ciΛ0 is the classical part of Λi . This map is extended to a map on P so that it is (1) Z-linear and δ = 0. To consider finite dimensional Uq′ (Xn )-modules, the classical weight Llattice Pcl = P/Zδ is also needed. In this paper we canonically identify Pcl with i∈I ZΛi ⊂ P . For a precise treatment, see section 3.1 of [KMN1]. We let Ln Q = i=1 Zαi denote the classical root lattice. For later convenience we introduce a few more notations concerning the classical Lie algebra Xn .

(2.1)

ta =

2 ∈ {1, 2, 3}, (αa |αa )

  1 An , Dn , E6,7,8 t = max1≤a≤n ta = 2 Bn , Cn , F4 ,   3 G2

−1 (2.2) Cab = ta (αa |αb ), Cab = ta (Λa |Λb ) 1 ≤ a, b ≤ n, n X Cba Λb . (2.3) αa = b=1

C and C −1 are the Cartan and the inverse Cartan matrices. We shall write a ∼ b to mean that Cab < 0. The following explicit form will be of later use. (2.4)

(n+1)

(Λa |Λb ) = Ka,b  2  3   2 1

3 6 4 2

(An ), 2 4 3 3 2

min(a, b) min(a, b) (Bn ), (Cn ), t t 2  a b 1   2  2 1  (G2 ), 3  (F4 ), 1 32 2 1

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G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

Table 1. Dynkin diagrams An :

e 1

e 2

e e n−1 n

Bn :

e 1

e 2

e> e n−1 n

Cn :

e 1

e 2

e< e n−1 n

(1)

A1 :

(1)

An : (n ≥ 2)

e<> e 0 1 e0 ! aa ! ! aa ! e! e ea e 1 2 n−1 n e0

(1)

Bn : (n ≥ 3)

en Dn :

e 1

e 2

e e n−2 n−1

e 1

e 2

e 3

e 4

e 5

e 1

e 2

e 3

e 4

e 5

e> e n−1 n

e> e 0 1

e 2

e< e n−1 n en

e0

(1)

Dn : (n ≥ 4)

e7 E7 :

e 3

(1)

Cn : (n ≥ 2)

e6 E6 :

e 2

e 1

e 1

e 2

e 6

e e n−2 n−1 e0 e6

e8 E8 :

e 1

e 2

e 3

e 4

F4 :

e 1

e> e 2 3

e 4

G2 :

e> e 1 2

e 5

(1)

e 6

e 7

E6 :

e 1

e 2

e 3

e 4

e 5

e7 (1)

E7 :

e 2

e 3

e 4

(1)

e 0

e 1

e 2

e 3

e 4

(1)

e 0

e 1

e> e 2 3

e 4

(1)

e 0

e> e 1 2

F4 : G2 :

(n+1)

e 1

e 5

e 6

e8 E8 :

where Ka,b

e 0

e 5

e 6

e 7

is defined by

ij . l To each non-simply laced algebra Xn = Bn , Cn , F4 and G2 , we associate the two algebras Zn and Yn such that Zn ⊂ Xn ⊂ Yn . (2.5)

(l)

(l)

Ki,j = Kl−i,l−j = min(i, j) −

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Table 2. Xn Yn Zn

Bn Dn+1 A1

Cn A2n−1 An−1

F4 E6 A2

G2 B3 A1

Here Zn is obtained from Xn by removing those vertices a with ta = 1 in the Dynkin diagram. The embedding Xn ֒→ Yn is well known. Let D = (Dab )1≤a,b≤n,ta =tb =t be the Cartan matrix of the subalgebra Zn . Then its inverse has the matrix elements 1 Bn ,  2 δan δbn   K (n) C n, −1 a,b (2.6) Dab = (3)  K F4 ,    1 a−2,b−2 G2 . 2 δa2 δb2 Given a positive integer l we consider the following subsets of Z × Z:

(2.7)

Hl = {(a, j) | 1 ≤ a ≤ n, 1 ≤ j ≤ ta l},

H l = {(a, j) | 1 ≤ a ≤ n, 1 ≤ j ≤ ta l − 1}, ta Hl [i] = {(a, j) | 1 ≤ a ≤ n, (i − 1) < j ≤ ta l} 1 ≤ i ≤ tl + 1. (2.9) t Thus Hl [tl + 1] = ∅ and Hl [1] = Hl . One can check (2.8)

(2.10)

Hl [i] = Hl [i + 1] ⊔ {(a,

ta ta i) | 1 ≤ a ≤ n, i ∈ Z}. t t (1)

(1)

2.2. Terminology for crystals. A crystal base B of a Uq (Xn ) (Uq′ (Xn ), Uq (Xn ))module can be regarded as a set of basis vectors of the module at q = 0. On B one still has Chevalley-like generators e˜i , f˜i : B −→ B ⊔ {0}, which are sometimes called Kashiwara operators. For b ∈ B we set εi (b) = max{k ≥ 0 | e˜ki b 6= 0}, ϕi (b) = max{k ≥ 0 | f˜ik b 6= 0}. If B1 and B2 are crystals, the crystal structure on the tensor product B1 ⊗ B2 is given by ( e˜i b1 ⊗ b2 if ϕi (b1 ) ≥ εi (b2 ), e˜i (b1 ⊗ b2 ) = (2.11) b1 ⊗ e˜i b2 otherwise, ( f˜i b1 ⊗ b2 if ϕi (b1 ) > εi (b2 ), f˜i (b1 ⊗ b2 ) = (2.12) b1 ⊗ f˜i b2 otherwise. We mainly use two categories of crystals. The first one contains the crystal (1) base B(λ) of the irreducible integrable Uq (Xn )-module L(λ) with highest weight + λ ∈ P . B(λ) is a P -weighted crystal. The other one contains a crystal base B (1) of a finite-dimensional Uq′ (Xn )-module. As opposed to B(λ), B is a finite set. It is PclP -weighted. We shall Pcall it a finite crystal. For a finite crystal B, we set ε(b) = i εi (b)Λi , ϕ(b) = i ϕi (b)Λi , wt b = ϕ(b) − ε(b), and introduce the level of B by lev B = min{hc, ε(b)i | b ∈ B}.

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We further set Bmin = {b ∈ B | hc, ε(b)i = lev B}, and call an element of Bmin (1) minimal. A Uq′ (Xn )-module can be viewed as a Uq (Xn )-module. For the latter the irreducible representation with highest weight λ ∈ P

+

will be denoted by V (λ).

(r)

2.3. Family Ws . We shall present a conjectural family of finite-dimensional having crystal bases. This is a revelation of the Bethe Ansatz.

(1) Uq′ (Xn )-modules

Conjecture 2.1. For each (r, s) (1 ≤ r ≤ n, s ≥ 1), there exists an irreducible (1) (r) finite-dimensional Uq′ (Xn )-module Ws having the following features: l m (r) (1) Ws has a crystal base B r,s . B r,s is a finite crystal of level tsr 6 . Moreover, it is perfect if tsr is an integer, and not perfect if not an integer. (r)

(2) As a Uq (Xn )-module, Ws decomposes itself into M Ws(r) = M (Ws(r) , λ, q = 1) V (λ), λ∈P

+

(r)

where M (Ws , λ, q) is defined in (4.3). (r) (r) (r) (3) Set Qs = ch Ws , then Qs satisfies the Q-system (7.1). (r)

From (4.3) the decomposition (2) of Ws Ws(r)

has the form

= V (sΛr ) ⊕ · · · ,

where · · · contains irreducible modules with highest weights strictly lower than (r) sΛr only. For Xn = An , Ws = V (sΛr ), i.e. representation corresponding to a rectangular Young diagram. In the other cases, we need the · · · part in general. (See [CP1, KR2] for the Yangian case.) A list of such decompositions is available in Appendix A (by setting q = 1). The notion of perfect crystals is introduced in [KMN1]. From a perfect crystal B we can construct a set of paths, which is isomorphic to the crystal of an (1) irreducible integrable Uq (Xn )-module. In [HKKOT] the definition of a set of paths is generalized to non-perfect crystals. Also in this case, the set of paths is isomorphic to the crystal of an integrable module, but not necessarily irreducible as we will see next. (r)

(r)

Remark 2.2. For v ∈ C× , let Ws (v) denote the pull back of Ws by the Hopf algebra automorphism τv defined on the Drinfel’d generators [CP2]. Then (r) (1) we expect that Ws (v) is the irreducible Uq′ (Xn )-module characterized by the δa,r  s−3 −s+1 s−1 up Drinfel’d polynomial Pa (u) = (1 − q ta uv)(1 − q ta uv) · · · (1 − q ta uv) to a normalization of v. Remark 2.3. To our knowledge, the following crystals have been shown to be perfect for the non-twisted case so far. [KMN2] [KKM] [Ko] [Ya] 6 The

An : B r,s , Bn : B 1,s , Cn : B n,s , Dn : B 1,s B n−1,s B n,s , Cn : B 1,2s , Bn : B r,tr , Cn : B r,2 (r 6= n), Dn : B r,1 (r 6= n − 1, n), G2 : B 1,s .

symbol ⌈x⌉ denotes the smallest integer not less than x.

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We give some examples of B r,s . Example 2.4. For Xn = An , B 1,s is isomorphic to B(sΛ1 ) as a crystal for Uq (An ). As a set, B 1,s = {(x1 , . . . , xn+1 ) ∈ Zn+1 | xi ≥ 0,

n+1 X

xi = s}.

i=1

The actions of e˜i , f˜i and ε, ϕ are defined as follows (cf. e.g. [HKKOTY]): For b = (x1 , . . . , xn+1 ) ∈ B 1,s , e˜0 b = (x1 − 1, x2 , . . . , xn+1 + 1), f˜0 b = (x1 + 1, x2 , . . . , xn+1 − 1), e˜i b = (x1 , . . . , xi + 1, xi+1 − 1, . . . , xn+1 ) for i = 1, . . . , n, f˜i b = (x1 , . . . , xi − 1, xi+1 + 1, . . . , xn+1 ) for i = 1, . . . , n, where the RHS is regarded as 0 if it is not an element of B 1,s , and ε(b) =

n X

xi+1 Λi , ϕ(b) = xn+1 Λ0 +

n X

xi Λi .

i=1

i=0

In this case, we have (B 1,s )min = B 1,s . B 1,s is perfect of level s. Example 2.5. For Xn = Cn , B 1,s is isomorphic to B(sΛ1 ) ⊕ B((s − 2)Λ1 ) ⊕ · · · ⊕ (B(Λ1 ) or B(0)) as a crystal for Uq (Cn ). As a set, B 1,s = {(x1 , . . . , xn , x ¯n , . . . , x ¯1 ) ∈ Z2n | xi , x ¯i ≥ 0, n X

(xi + x ¯i ) ≤ s,

i=1

n X

(xi + x ¯i ) ≡ s

(mod 2)}.

i=1

The actions of e˜i , f˜i and ε, ϕ are defined as follows (cf. [KKM, HKKOT]): For b = (x1 , . . . , xn , x ¯n , . . . , x ¯1 ) ∈ B 1,s ,   ¯2 , x¯1 ) if x1 ≥ x ¯1 + 2, (x1 − 2, x2 , . . . , x e˜0 b = (x1 − 1, x2 , . . . , x ¯2 , x¯1 + 1) if x1 = x ¯1 + 1,   (x1 , x2 , . . . , x ¯2 , x¯1 + 2) if x1 ≤ x ¯1 ,   ¯2 , x¯1 ) if x1 ≥ x ¯1 , (x1 + 2, x2 , . . . , x f˜0 b = (x1 + 1, x2 , . . . , x ¯2 , x¯1 − 1) if x1 = x ¯1 − 1,   (x1 , x2 , . . . , x ¯2 , x¯1 − 2) if x1 ≤ x ¯1 − 2, ( (x1 , . . . , xi + 1, xi+1 − 1, . . . , x ¯1 ) if xi+1 > x¯i+1 e˜i b = for i = 1, . . . , n − 1, (x1 , . . . , x¯i+1 + 1, x ¯i − 1, . . . , x ¯1 ) if xi+1 ≤ x¯i+1 ( (x1 , . . . , xi − 1, xi+1 + 1, . . . , x ¯1 ) if xi+1 ≥ x¯i+1 ˜ for i = 1, . . . , n − 1, fi b = (x1 , . . . , x¯i+1 − 1, x ¯i + 1, . . . , x ¯1 ) if xi+1 < x¯i+1 e˜n b = (x1 , . . . , xn + 1, x ¯n − 1, . . . , x ¯1 ), f˜n b = (x1 , . . . , xn − 1, x ¯n + 1, . . . , x ¯1 ),

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G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

where the RHS is regarded as 0 if it is not an element of B 1,s , and P   n−1 X s − ni=1 (xi + x¯i ) (¯ xi + (xi+1 − x ¯i+1 )+ ) Λi + x ¯n Λn , ε(b) = + (x1 − x ¯1 )+ Λ0 + 2 i=1 P   n−1 X s − ni=1 (xi + x¯i ) (xi + (¯ xi+1 − xi+1 )+ ) Λi + xn Λn , ϕ(b) = + (¯ x1 − x1 )+ Λ0 + 2 i=1 where (x)+ := max(x, 0). In this case, we have  1,s   {(x1 , . . . , xn , xn , . . . , x1 ) ∈ B } if s is even,  B 1,s min = {(x1 , . . . , xk , . . . , xn , xn , . . . , xk ± 1, . . . , x1 ) ∈ B 1,s | k = 1, . . . , n}    if s is odd.

B 1,s is perfect (level s2 ) if s is even, and non-perfect (level

s+1 2 )

if s is odd.

Example 2.6. For Xn = Cn , B r,1 is isomorphic to B(Λr ) as a crystal for Uq (Cn ). As a set, B r,1 = {(x1 , . . . , xn , x ¯n , . . . , x ¯1 ) ∈ Z2n | xi , x ¯i = 0 or 1, n X

(xi + x¯i ) = r, if xk = x ¯k = 1 then

i=1

The actions of e˜i and f˜i are defined as follows (cf. [AK]): For b = (x1 , . . . , xn , x ¯n , . . . , x ¯1 ) ∈ B r,1 , e˜0 b = (x1 − 1, x2 , . . . , x ¯2 , x ¯1 + 1), f˜0 b = (x1 + 1, x2 , . . . , x ¯2 , x ¯1 − 1),  (x1 , . . . , x ¯i+1 + 1, x ¯i − 1, . . . , x¯1 )      if (¯ xi+1 , x ¯i ) = (0, 1) and (xi , xi+1 ) 6= (1, 0)    e˜i b = (x1 , . . . , xi + 1, xi+1 − 1, . . . , x¯1 )    if (¯ xi+1 , x ¯i ) 6= (0, 1) and (xi , xi+1 ) = (0, 1)      0 otherwise   (x1 , . . . , xi − 1, xi+1 + 1, . . . , x¯1 )    if (xi , xi+1 ) = (1, 0) and (¯ xi+1 , x ¯i ) 6= (0, 1)    f˜i b = (x1 , . . . , x ¯i+1 − 1, x ¯i + 1, . . . , x¯1 )    if (xi , xi+1 ) 6= (1, 0) and (¯ xi+1 , x ¯i ) = (1, 0)      0 otherwise

k X

(xi + x ¯i ) ≤ k}.

i=1

for i = 1, . . . , n − 1,

for i = 1, . . . , n − 1,

e˜n b = (x1 , . . . , xn + 1, x ¯n − 1, . . . , x ¯1 ), f˜n b = (x1 , . . . , xn − 1, x ¯n + 1, . . . , x ¯1 ),

where the RHS is regarded as 0 if it is not an element of B r,1 . B r,1 is not perfect except when r = n. The level of B r,1 is 1.

REMARKS ON FERMIONIC FORMULA

11

2.4. Set of paths. Let B be a finite crystal of level k. From B we construct a subset of · · · ⊗ B ⊗ · · · ⊗ B called a set of paths. See also [HKKOT]. First we fix a reference path p = · · · ⊗ bj ⊗ · · · ⊗ b2 ⊗ b1 . For any j, ε(bj ) should have level k, and satisfy (2.13)

ϕ(bj+1 ) = ε(bj ).

Set (2.14)

P(p, B) = {p = · · · ⊗ bj ⊗ · · · ⊗ b2 ⊗ b1 | bj ∈ B, bJ = bJ for J ≫ 1}.

An element of P(p, B) is called a path. To a path we can associate a weight in P . For this purpose we consider the energy function H : B × B → Z, which is determined up to constant difference by the following rule: H(˜ ei (b1 ⊗ b2 )) = H(b1 ⊗ b2 ) + 1 if i = 0, ϕ0 (b1 ) ≥ ε0 (b2 ), = H(b1 ⊗ b2 ) − 1 if i = 0, ϕ0 (b1 ) < ε0 (b2 ), = H(b1 ⊗ b2 ) if i 6= 0. Using this function we define the energy E(p) and weight wt p of p ∈ P(p, B) by E(p) =

∞ X

j(H(bj+1 ⊗ bj ) − H(bj+1 ⊗ bj )),

j=1

wt p = ϕ(b1 ) +

∞ X

(wt bj − wt bj ) − E(p)δ.

j=1

Notice that they have L finite values. To calculate the weight of an element of B, we identify Pcl with i∈I ZΛi . Let us start with examining P(p, B) when B is perfect of level k. Take λ ∈ Pk+ and choose a unique b1 such that ϕ(b1 ) = λ. Then all bj is uniquely fixed from (2.13). We denote this reference path by p(λ) . It is known in [KMN1] that we have the following isomorphism of P -weighted crystals. P(p(λ) , B) ≃ B(λ). Of course, the actions of e˜i , f˜i on the left hand side obey the tensor-product rule of crystals (2.11,2.12). “Signature rule” is a good way to calculate the actions on multiple tensor products. See [KMOU] for that. Remark 2.7. The set of paths P(p, B) admits a natural filtration P0 (p, B) ⊂ P1 (p, B) ⊂ · · · ⊂ PL (p, B) ⊂ · · · ⊂ P(p, B), PL (p, B) = {p ∈ P(p, B) | bj = bj for j > L}. If B is perfect and λ = kΛ0 , the finite subset PL (p(λ) , B) can be identified with the crystal base corresponding to a Demazure module Lw (λ) for a special affine Weyl group element w. See [KMOU, KMOTU1] for details. (λ)

Example 2.8. We give examples of p(λ) = · · · ⊗ bj i=0 λi Λi . For Xn = An and B = B 1,s (cf. Example 2.4),

Pn

(λ)

bj

(λ)

⊗ · · · ⊗ b1 . Set λ =

= (λ(j) , λ(j+1) , · · · , λ(j+n) ) ∈ B 1,s .

Here (a) denotes the integer such that (a) ≡ a (mod n + 1), 0 ≤ (a) ≤ n.

12

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

For Xn = Cn and B = B 1,s (s: even) (cf. Example 2.5), (λ)

bj

= (λ1 , · · · , λn , λn , · · · , λ1 ) ∈ B 1,s .

Next we consider the non-perfect case. Assume B is non-perfect and its level is k. Then even if we fix b1 , the reference path is not unique. There are actually infinitely many reference paths. Among them, we conjecture the existence of p(λ,µ) which is friendly to representation theory in the following sense. For B = B r,s set (1) + B † = B r,1 . For our Xn recall Yn in Table 2. We take λ from Pk−1 of Xn , but (1)

µ from P1+ of Yn . Then there exist reference paths p(λ,µ) for B and p†(µ) for B † and the following isomorphism. P(p(λ,µ) , B) ≃ B(λ) ⊗ P(p†(µ) , B † ). (1)

Moreover, at the character level, P(p†(µ) , B † ) can be identified with the Yn (1) (1) module LY (µ) considered as an Xn -module through the natural embedding Xn ֒→ (1) (1) ′ 1,s Yn . This is proven for Uq (Cn )-crystal B (s: odd) in [HKKOT]. (1)

Example 2.9. In this Uq′ (Cn )-crystal B 1,s (s: odd) case, p(λ,µ) = · · · ⊗ (λ,µ)

(λ,µ)

†(µ)

†(µ)

are given as follows. Set ⊗ · · · ⊗ b1 , p†(µ) = · · · ⊗ bj ⊗ · · · ⊗ b1 Pn (1) (1) + λ = i=0 λi Λi ∈ P s−1 of Cn and take µ = Λ0 of A2n−1 .

bj

2

(λ,Λ0 )

bj

†(Λ0 )

bj

 (λ1 , · · · , λn , λn , · · · , λi + 1, · · · , λ1 ) if j ≡ i (mod 2n)     for some i (1 ≤ i ≤ n) =  (λ , · · · , λ + 1, · · · , λ , λ , · · · , λ ) if j ≡ 1 − i (mod 2n) 1 i n n 1    for some i (1 ≤ i ≤ n), (0,Λ0 )

= bj

(i.e. set formally λ = 0 in the above).

Thus we have the isomorphism P(p(λ,Λ0 ) , B 1,s ) ≃ B(λ) ⊗ P(p†(Λ0 ) , B 1,1 ), (1)

and the character of P(p†(Λ0 ) , B 1,1 ) agrees with that of the A2n−1 -module LY (Λ0 ) (1) considered as a Cn -module. 3. One dimensional sums r,s

Fix any B = B and let k be the level of B. Take b0 ∈ B such that ϕ(b0 ) = + kΛ0 . In all known cases, such b0 is unique for B. For λ ∈ P , we define X∗ P L−1 (3.1) q j=0 (L−j)H(bj ⊗bj+1 ) , X(B ⊗L , λ, q) = where the sum and

P∗

is taken over all b1 ⊗ · · · ⊗ bL ∈ B ⊗L satisfying e˜i (b1 ⊗ · · · ⊗ bL ) = 0 ∀i 6= 0.

(3.2)

Note that b0 is fixed. (3.2) can be rewritten as local conditions. hhi ,wt b1 +···+wt bj−1 i+1

e˜i

bj = 0

∀i 6= 0, 1 ≤ j ≤ L.

PL

j=1

wt bj = λ

REMARKS ON FERMIONIC FORMULA

13

(3.1) is introduced in [KMOTU2] in more general setting (b0 not limited to the above one) and called “classically restricted 1dsum”. It is easy to see from the definition that X(B ⊗L , λ, 1) = [W ⊗L : λ],

(3.3)

(1)

where W is the corresponding finite-dimensional Uq′ (Xn )-module of B and [M : µ] = dim Chv ∈ M | wt v = µ, ei v = 0 ∀i 6= 0i.

(3.4)

+

If B is perfect of level k, it is known [KMOTU2] that for λ ∈ P , X (3.5) [L(kΛ0 ) : λ + kΛ0 − iδ] q i . lim q −cL X(B ⊗L , λ, q) = L→∞ L≡0 mod γ

i

PL−1

Here cL = j=0 (L − j)H(bj ⊗ bj+1 ), b0 , · · · , bL are the last L + 1 components of p(kΛ0 ) (read from left to right), and γ is the period of the sequence b1 , b2 , · · · . Note that when L ≡ 0 mod γ, b0 agrees with the fixed b0 . Note also that in [L(kΛ0 ) : µ], µ is a weight in P as opposed to λ in (3.3) in P . The right hand side of (3.5) is called a branching function. If B is non-perfect, the conjecture in section 2.4 implies X (3.6) [L((k − 1)Λ0 ) ⊗ LY (Λ0 ) : λ + kΛ0 − iδ] q i . lim q −cL X(B ⊗L , λ, q) = L→∞ L≡0 mod γ

i

(1)

Here LY (Λ0 ) is the integrable Yn -module with highest weight Λ0 regarded as an (1) Xn -module. We are to state a conjecture for the expression of X. l m Conjecture 3.1. Let B = B r,s , set k = tsr and take b0 , b♮ ∈ B such that ϕ(b0 ) = kΛ0 , wt b♮ = sΛr . Then we have

X(B ⊗L , λ, q) = q c M (Ws(r)⊗L , λ, q) with c =

L(L−1) H(b♮ 2

⊗ b♮ ) + LH(b0 ⊗ b♮ ), where M is defined in (4.3) – (4.6).

Surprisingly, this conjecture admits “inhomogeneous” version. Consider the tensor product of crystals B0 ⊗ B r1 ,s1 ⊗ · · · ⊗ B rL ,sL . Here B0 = B ri1 ,si1 ⊗ · · · ⊗ B rit ,sit and (ri1 , si1 ), · · · , (rit , sit ) are mutually distinct elements of {(r1 , s1 ), · · · , (rL , sL )}. To explain the corresponding classically restricted 1dsum, one has to redefine the energy function H due to inhomogeneity [NY3, HKKOTY]. Suppose B1 and B2 are finite crystals, and b1 ⊗ b2 ∈ B1 ⊗ B2 mapped to b′2 ⊗ b′1 under the isomorphism B1 ⊗ B2 ≃ B2 ⊗ B1 . The energy function H on B1 ⊗ B2 is defined up to constant difference by the following rule: H(˜ ei (b1 ⊗ b2 )) = H(b1 ⊗ b2 ) + 1 if i = 0, ϕ0 (b1 ) ≥ ε0 (b2 ), ϕ0 (b′2 ) ≥ ε0 (b′1 ), = H(b1 ⊗ b2 ) − 1 if i = 0, ϕ0 (b1 ) < ε0 (b2 ), ϕ0 (b′2 ) < ε0 (b′1 ), = H(b1 ⊗ b2 )

otherwise.

14

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA (i)

Next consider the tensor product of finite crystals B1 ⊗ · · · ⊗ BL . We define bj (i < j) by Bi ⊗ · · · ⊗ Bj−1 ⊗ Bj bi ⊗ · · · ⊗ bj−1 ⊗ bj

≃ Bi ⊗ · · · ⊗ Bj ⊗ Bj−1 (j−1) ⊗ b′j−1 7→ bi ⊗ · · · ⊗ bj ··· ···

≃ ··· 7→ · · ·

≃ Bj ⊗ Bi ⊗ · · · ⊗ Bj−1 (i) 7 → bj ⊗ b′i ⊗ · · · ⊗ b′j−1 ,

(i)

and set bi = bi . Take some b0 ∈ B0 . The classically restricted 1dsum is defined similarly. X∗ P (i+1) q 0≤i
Note that b0 ∈ B0 , bj ∈ B rj ,sj (1 ≤ j ≤ L) and X depends also on b0 . Of course, we have a similar formula to (3.3). Let b♮0 = b0 ∈ B0 and b♮j ∈ B rj ,sj be the highest weight element for 1 ≤ j ≤ L. Conjecture 3.2. Set k = lev B0 . There exists an element b0 ∈ B0 such that ϕ(b0 ) = kΛ0 , and we have X(B r1 ,s1 ⊗ · · · ⊗ B rL ,sL , λ, q) = q c M (Ws(r1 1 ) ⊗ · · · ⊗ Ws(rLL ) , λ, q) P with c = 0≤i
The authors are informed [KSS] that this conjecture is proved for Xn = An . In this case, if all rj = 1, the conjecture has already been proved combining the results of [KR1] and [NY3]. + Let us give the “level restricted” version of Conjecture 3.2. For λ ∈ P and a positive integer l, define X† P (i+1) q 0≤i
e˜i

(b1 ⊗ · · · ⊗ bL ) = 0 ∀i.

Note that the above condition is the same as (3.2) except i = 0. Thus it is clear that Xl = X for sufficiently large l. Conjecture 3.3. With the same assumptions and c as in Conjecture 3.2, we have Xl (B r1 ,s1 ⊗ · · · ⊗ B rL ,sL , 0, q) = q c Ml (Ws(r1 1 ) ⊗ · · · ⊗ Ws(rLL ) , q), where Ml is defined in (4.8) – (4.12). For Xn = An there are several proven cases. See e.g. [Ber, FOW, Wa]. Remark 3.4. In the homogeneous case, there is a similar result to (3.5). If B + is perfect of level k (≤ l), it is known that for λ ∈ P , (3.7) lim

L→∞ L≡0 mod γ

q −cL Xl (B ⊗L , λ, q) =

X i

[[L((l − k)Λ0 ) ⊗ L(kΛ0 ) : λ + lΛ0 − iδ]] q i ,

REMARKS ON FERMIONIC FORMULA

15

where [[M : µ]] = dim Chv ∈ M | wt v = µ, ei v = 0 ∀ii.

(3.8)

Note the difference between (3.4) and (3.8). The RHS of (3.7) is called a coset branching function. If B is non-perfect and level k, we conjecture (cf. [Ku], p230) lim

L→∞ L≡0 mod γ

X

q −cL Xl (B ⊗L , λ, q) =

[[L((l − k)Λ0 ) ⊗ L((k − 1)Λ0 ) ⊗ LY (Λ0 ) : λ + lΛ0 − iδ]] q i .

i

4. Fermionic forms For m ∈ Z≥0 and p ∈ Z, we define   (q p+1 )∞ (q m+1 )∞ p+m (4.1) , = m (q)∞ (q p+m+1 )∞ q ( )      p+m for p ≥ 0, p+m m (4.2) = q m   q 0 for p < 0, Q∞ j where (x)∞ = j=0 (1 − xq ). (4.2) is the usual q-binomial coefficient. (4.1) is an extended version, which vanishes only for −m ≤ p ≤ −1 and has non-zero value  −p − 1 (−q p+(m+1)/2 )m for p ≤ −m − 1. In the q → 1 limit they become m q       Γ(p + m + 1) p+m p+m p+m , = . = m m m Γ(p + 1)Γ(m + 1) 1 1

We shall also use the notation (q)k = (q)∞ /(q k+1 )∞ for k ∈ Z≥0 . (a) Nn N (a) (a) ⊗νj Given any {νj ∈ Z≥0 | j ≥ 1, 1 ≤ a ≤ n}, we set W = a=1 j≥1 Wj . (a)

For each 1 ≤ a ≤ n we shall always assume that νj ’s are non-zero only for finitely many j’s. Let λ ∈ P . We define the fermionic form M (W, λ, q) by " # (a) (a) Y X + m p i i q c({m}) (4.3) , M (W, λ, q) = (a) m i 1≤a≤n {m} q

i≥1

(4.4)

c({m}) =

1 2

X

(αa |αb )

1≤a,b≤n



n X

X

(a)

(b)

min(tb j, ta k)mj mk

j,k≥1

X

(a)

(a)

νj min(j, k)mk ,

a=1 j,k≥1 (a)

(4.5)

pi

=

X

(a)

νj min(i, j) −

where the sum (a) pi

{m}

(αa |αb )

b=1

j≥1

P

n X

is taken over

(a) {mi

X

k≥1

∈ Z≥0 | 1 ≤ a ≤ n, i ≥ 1} satisfying

≥ 0 for 1 ≤ a ≤ n, i ≥ 1, and

(4.6)

n X X a=1 i≥1

(a)

imi αa =

n X X a=1 i≥1

(a)

(b)

min(tb i, ta k)mk ,

iνi Λa − λ

for 1 ≤ a ≤ n.

16

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

By definition M (W, λ, q) = 0 if the RHS of (4.6) does not belong to More strictly we have

(4.7)

unless λ ∈

M (W, λ, q) = 0

n X X

(a)

iνi Λa −

n X

a=1

a=1 i≥1

Pn

a=1

Z≥0 αa .

 + Z≥0 αa ∩ P .

(a)

To see this, compare (4.5) and (4.6), which leads to p∞ = (ta αa |λ). Thus the above property (4.7) holds because of (4.2). It is easy to see M (W, λ, q) ∈ Z≥0 [q −1 ]. (a)

Note that pi

(a)

= − ∂c({m}) from (4.4) and (4.5), regarding all mi (a) ∂mi

as independent (a)

continuous variables. Similar relations are valid also between cl ({m}) and pi ’s in the later equations (4.9)-(4.10) and (4.14)-(4.15). To introduce a “restricted version” of M (W, λ, q), we fix a positive integer l. (a) N (a) ⊗νj we define For W = (a,j)∈Hl Wj (4.8)

Ml (W, q) =

X

q

(a,i)∈Hl

{m}

(4.9)

1 cl ({m}) = 2

Y

cl ({m})

X

"

(a)

(a)

pi

+ mi (a) mi

#

,

q (a)

(b)

(αa |αb )min(tb j, ta k)mj mk

(a,j),(b,k)∈Hl



X

ta l X

(a)

(a)

νj min(j, k)mk ,

(a,j)∈Hl k=1

(4.10)

(a)

pi

=

ta l X

(a)

νj min(j, i) −

j=1

X

(b)

(αa |αb )min(tb i, ta k)mk ,

(b,k)∈Hl

P (a) (a) where the sum {m} is taken over {mi ∈ Z≥0 | (a, i) ∈ Hl } satisfying pi ≥ 0 for (a, i) ∈ Hl , and (4.11)

X

(a)

i mi αa =

(a,i)∈Hl

X

(a)

iνi Λa

for 1 ≤ a ≤ n,

(a,i)∈Hl

or equivalently,

(4.12)

ta l X

k=1

(a)

kmk =

X

(b,i)∈Hl

(b)

−1 Cab iνi

for 1 ≤ a ≤ n.

REMARKS ON FERMIONIC FORMULA

17

(a)

(a)

Notice that (4.11) is equivalent to pta l = 0. Eliminating mta l (1 ≤ a ≤ n) from (4.12) one can rewrite (4.8) – (4.11) as follows. " # (a) (a) Y X p i + mi cl ({m}) q (4.13) Ml (W, q) = , (a) mi {m} q

(a,i)∈H l

(4.14)

cl ({m}) =

1 2

X

X

(a,j)∈H l (a)

pi

=

tX a l−1

tX a l−1

(t l) (a) (a) Ki,ja νi mj

Here the sum

(a, i) ∈ H l , and



|

i=1

(t l) (a)

Ki,ja νj

X



j=1

P

(b)

(a)

(a,j),(b,k)∈H l



(4.15)

(t t l)

a b (αa |αb )Ktb j,t mj mk ak

P

(a)

(a,j)∈Hl

jνj Λa |2

2l

(t t l)

,

(b)

a b (αa |αb )Ktb i,t mk . ak

(b,k)∈H l

{m} is taken (a) mta l defined

over

(a) {mi

(a)

∈ Z≥0 | (a, i) ∈ H l } satisfying pi

≥ 0 for

by (4.12) is a non-negative integer for 1 ≤ a ≤ n.

Remark 4.1. The two fermionic forms M and Ml are related by M (W, 0, q) = M∞ (W, q). In the RHS, the limit l → ∞ poses no subtlety.  ⊗νj(a) N 2 (a) Remark 4.2. For W = (a,i)∈Hl Wj , set M l (W, q) = q |Λ| /2l Ml (W, q), P (a) Λ = (a,j)∈Hl jνj Λa , which corresponds to dropping the last term in (4.14). For simply laced Xn , H l=1 (2.8) is empty and therefore from (4.13), M 1 (W, q) = 1

for An , Dn , E6,7,8 .

On the other hand for non-simply laced Xn , H 1 is bijective to H t of Zn given in Table 2. (t > 1 here is the one for Xn and not for Zn .) Moreover a simple manipulation tells that M 1 (W, q) for Xn = M t (W ′ , q) for Zn , where W ′ = those for Zn .) (1)′

Bn : ν1

 ⊗νj(a)′ (a)′ (a)′ here denote is specified by (Ht and Wj W j (a,i)∈Ht

N

(1)′

(n)

= ν1 , ν2

=

n−1 X

aν1 +

(a) ν2

δa,n−1 + 2

(a)

a=1

Cn :

(a)′ ν1 (a)′

F4 : νj

(1)′

G2 : νj

=

(a) ν1 , (a+2)

= νj

(2)

= νj

(a)′ ν2

=

n − 1 (n) (n) ν1 + nν2 , 2 n−1 X

(1)′

(1)′

+

(a) 2ν2 )

+

(n) nν1

a=1

1 ≤ a, j ≤ 2 except ν2

1 ≤ j ≤ 2, ν3

(a) a(ν1

(1)

(2)

(1)

(3)

!

1 ≤ a ≤ n − 1, (3)

(4)

(4)

= 3ν1 + 6ν1 + 4ν1 + 9ν2 + 2ν1 + 4ν2 , (2)

(2)

= 2ν1 + ν1 + 2ν2 . (a)′

In the above the condition (4.12) ∈ Z ensures ∀νj

∈ Z.

18

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

The above fermionic forms are defined in terms of the usual q-binomial (4.2). However we shall also concern those involving (4.1). Given l ∈ Z≥1 , λ = λ1 Λ1 + (a) N (a) ⊗νj · · · + λn Λn ∈ P and W = (a,j)∈Hl Wj , we introduce the third fermionic form ( ) (a) (a) Y X p i + mi cl ({m}) (4.16) , q Nl (W, λ, q) = (a) m i (a,i)∈H {m} q

l

(a)

where cl ({m}) and pi are defined by the same formulae as (4.9) and (4.10), reP (a) spectively. The sum {m} is taken over {mi ∈ Z≥0 | (a, i) ∈ Hl } not bounded (a)

by the restriction pi ≥ 0 for (a, i) ∈ Hl , but only subject to the condition X X (a) (a) iνi Λa − λ (4.17) i mi αa = for 1 ≤ a ≤ n. (a,i)∈Hl

(a,i)∈Hl

By introducing (a)

γj

(4.18)

=

ta l X

(a)

(a) µa = γ∞ − λa ,

νk min (j, k),

k=1

(a)

one can rewrite (4.17) and pi (4.10) as X (b) (4.19) Cab kmk , µa = (b,k)∈Hl

(4.20)

(a) pi

=

(a) γi

− µa +

n X b=1

Cab tb ta

X

i
  tb (b) k − i mk . ta (a)

Although Nl (W, λ, q) is defined for all λ ∈ P , it is vanishing unless νi ’s and λ are Pn −1 chosen so that b=1 Cab µb ∈ Z≥0 for all 1 ≤ a ≤ n. In contrast with M and Ml , the above fermionic form contains, in general, (a) (a) non-zero contributions with possibly negative signs from pi < −mi cases. See the remarks after (4.2). Nevertheless our computer experiments suggest Conjecture 4.3. (4.21)

M (W, λ, q) = N∞ (W, λ, q)

(4.22)

Ml (W, q) = Nl (W, 0, q).

+

for λ ∈ P ,

The conjecture means that those “unphysical” contributions in the Bethe ansatz context cancel out totally. So far we have checked it for most of the fundamental (a) (1) representations W = W1 . (The case Xn = A1 , νj = Lδj,1 is exceptional in that (1)

(1)

the “unphysical contributions” are absent also in Nl (W, λ, q) because p1 ≥ p2 ≥ + + (1) · · · ≥ pl = (α1 |λ) ≥ 0 for λ ∈ P .) On the contrary, for λ 6∈ P the two fermionic forms M and N∞ are significantly different. Compare (4.7) and Conjecture 8.9. One can define Ml (W, λ, q) by restricting the summands in Nl (W, λ, q) to those (a) satisfying ∀pi ≥ 0. However Ml (W, λ, q) 6= Nl (W, λ, q) in general for λ 6= 0 as opposed to (4.22).

REMARKS ON FERMIONIC FORMULA

19

5. Generalized spinon character formula (r)⊗L

In this section we concentrate on the fermionic form M (Ws , λ, q) for the (r)⊗L homogeneous quantum space W = Ws and its limit L → ∞ such that LsΛr ∈ Q. Due to (3.5)–(3.6) and Conjecture 3.1, it yields explicit formulae for the branching functions. The result turns out to be a generalization of the spinon character formulae [NY1, NY2, ANOT, BPS, BS, BLS]. Such a connection between the spinon character formula and the limit of the fermionic form was first (1) shown in [HKKOTY] for An . (a) First we seek the minimum point of the quadratic form c({m}) (4.4) for νj = Lδa,r δj,s . The solution of the linear equation

∂c({m}) (a) ∂mj

= 0 is given by

Lemma 5.1. The simultaneous equation n X X (b) (5.1) (αa |αb )min(tb j, ta k)mk,0 = Lδa,r min(s, j) b=1 k≥1

for 1 ≤ a ≤ n, j ≥ 1 is satisfied by

(a)

−1 δj, ta s mj,0 = LCra tr

for any Xn provided that ( Bn :

(a)

mj,0 =

s tr

∈ Z. In case

s tr

6∈ Z, (5.1) holds for

La s−1 s+1 2 (δj, 2 + δj, 2 ) L 4 (n − 1)(δj,s−1 + δj,s+1 )

+ 2δj,s



1 ≤ a ≤ n − 1, a = n,

ar Lar (δj, ta (s−1) + δj, ta (s+1) ) + L(min(a, r) − )δj, ta s , 2 2 2 2n n  L (a) −1 F4 : mj,0 = (2δr,3 + δr,4 ) Ca2 (δj, ta (s−1) + δj, ta (s+1) ) + δa,3 δj,s 2 2 3 L + (δr,3 + 2δr,4 )δa,4 δj,s , 3 where the inverse Cartan matrix of F4 is given by (2.2) and (2.4). ( L(2δj, s−1 + δj, s+2 ) s ∈ 3Z + 1, (1) 3 3 G2 : mj,0 = L(δj, s−2 + 2δj, s+1 ) s ∈ 3Z + 2, 3 3 ( 1 L(δj,s−1 + 2 δj,s + 21 δj,s+2 ) s ∈ 3Z + 1, (2) mj,0 = L( 12 δj,s−2 + 21 δj,s + δj,s+1 ) s ∈ 3Z + 2. Cn :

(a)

mj,0 =

Remark 5.2. In all the cases in Lemma 5.1, one has n X X (a) jmj,0 αa = LsΛr . a=1 j≥1

When s/tr ∈ Z,

(a) mi,0

−1 is an integer iff LsCra ∈ Z. When s/tr 6∈ Z, it is so iff

Bn :L ∈ 2Z (n : odd), Cn :L ∈ 2nZ, F4 :L ∈ 3Z, G2 :L ∈ 2Z.

L ∈ 4Z (n : even),

20

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

For such L we have LsΛr ∈ Q. (a)

Remark 5.3. mj,0 6= 0 iff  ta s  {(a, tr ) | 1 ≤ a ≤ n} for tr ∈ Z,  (a, j) ∈ {(a, ta (s−s0 ) ), (a, ta (s−s0 +tr ) ) | 1 ≤ a ≤ n} tr tr   ∪{(a, s) | 1 ≤ a ≤ n, t = t (> 1)} for s 6∈ Z, a r tr

where s0 ≡ s mod tr Z and 1 ≤ s0 ≤ tr − 1. This is essentially the same with (3.5b) in [Ku], which appeared in the Bethe ansatz analysis. It corresponds to the physical picture that the ground state in “regime III-like” region is a Dirac sea of (a) color a length j-strings for those (a, j) such that mj,0 → ∞ as L → ∞. The value (a)

mi,0 should coincide with the 0-th Fourier component of the ground state density function of the (a, j)-string. (r)⊗L

As in Lemma 5.1 the limit of M (Ws , λ, q) is considerably complicated if s/tr 6∈ Z. We therefore treat the case s/tr ∈ Z first. Theorem 5.4. Assume that s/tr ∈ Z. Then (a)

(5.2) lim q −c({mi,0 }) M (Ws(r)⊗L , λ, q) = (5.3)

W (ζ) =

X M (W (ζ), λ, q −1 )M ts (W (ζ), q −1 ) r ζ n O

(q)ζ1 · · · (q)ζn

(a)⊗ζa

W1

,

.

a=1

(a)

−1 Here the limit L → ∞ is taken so that LCra ∈ Z for 1 ≤ a ≤ n. c({mi,0 }) = P 2 −1 −L sCrr /2 ∈ Z. The sum is taken over ζ1 , . . . , ζn ∈ Z≥0 such that ζ = na=1 ζa Λa ∈ Q.

Proof. We start with the expression (4.3) – (4.6). The limit is to be expanded (a) (a) from the minimum mi = mi,0 of c({m}) determined in Lemma 5.1. The non-zero (a)

(a)

mi,0 ’s are proportional to L. Thus under the identification ζa = p sta , the factor tr " (a) # (a) psta /tr + msta /tr Qn −1 as L → ∞. in (4.3) gives rise to ((q)ζ1 · · · (q)ζn ) (a) a=1 msta /tr (a)

After the shift mi

q (a)

→ mi

(a)

pi

(5.4)

(a)

+ mi,0 , the relation (4.5) and its i =

=−

n X

(αa |αb )

b=1

n X

(5.5)

b=1

−1 Cab ζb = −

X

k≥1

X

sta tr

case become

(b)

min(tb i, ta k)mk ,

k≥1

min(

ta s (a) , k)mk . tr

(a)

Eliminating m ta s with (5.5), one rewrites (5.4) as tr

(a) pi

 Pn P  ζa − b=1 (αa |αb ) k> tb s min(tb (i − ttars ), ta (k − tr tb s ta tb s ta s =  K ( ttra s) ζa − Pn (αa |αb ) P tr −1 K ( tr ) m(b) b=1 k=1 tb i,ta k k −i 1, tr

(b) tb s tr ))mk

i>

ta s tr

1≤i<

ta s tr

.

REMARKS ON FERMIONIC FORMULA (a)

By setting pi

(a)

(a)

(a)

=m ˆ i− ta s , i >

= pˆi− ta s and mi tr

tr

(a)

(a)

ta s tr

21

case in the above becomes

ˆ i for M (W (ζ), λ, q −1 ). Similarly, by setting the relation (4.5) between pˆi and m (a) (a) (a) (a) pi = p˜ ta s −i and mi = m ˜ ta s −i , i < ttars case becomes the relation (4.15) between tr tr Q (a) (a) s ˜ i for M tr (W (ζ), q −1 ). Thus in the product 1≤a≤n,i≥1 of (4.3), we are p˜i and m Q Q to extract M (W (ζ), λ, q −1 ) from 1≤a≤n i> sta part and M tsr (W (ζ), q −1 ) from tr Q s . Actually, the decomposition H part. See (2.8) for the definition of (a,i)∈H s tr tr

(a)

(a)

(a)

(a)

−c({mi,0 }) + c({mi → mi + mi,0 }) X X (a) (a) (a) (a) m ˜ i p˜i − c tsr ({m}) =− ˆ − m ˆ i pˆi − c({m}) ˜ 1≤a≤n,i≥1

(a,i)∈H l

is valid among the quadratic forms (4.4) and (4.14). The first (resp. last) two (a) (a) ˜ i }) terms on the RHS yield the quadratic form in the variables {m ˆ i } (resp. {m for M (W (ζ), λ, q −1 ) (resp. M tsr (W (ζ), q −1 )). It remains to establish that the sum (a)

over {mj

(a)

| 1 ≤ a ≤ n, j ≥ 1} is properly translated into the sums over {m ˆj

(a) a ≤ n, j ≥ 1}, {ζa | 1 ≤ a ≤ n} and {m ˜ j | (a, j) P P P P (i) na=1 ζa Λa ∈ Q, (ii) na=1 ζa Λa − na=1 i≥1

|1≤

∈ H l }. For this it suffices to check (a)

(a)

˜ ta s im ˆ i αa = λ (see (4.6)). (iii) m tr

Pn P ttars (a) −1 ζb (see (4.12)) is a non-negative integer. km ˜ k = b=1 Cab determined from k=1 Pn Pn P (a) ta s (i) Due to (5.5), a=1 ζa Λa = − a=1 k≥1 min( tr , k)mk αa ∈ Q holds. (ii) (a)

(a)

(a)

Making the shift mi → mi + mi,0 in (4.6) and applying Remark 5.2, we have Pn P Pn (a) λ = − a=1 k≥1 kmk αa . Combining this with the above we have a=1 ζa Λa − P ttars Pn P (a) (a) ˜k = λ = − a=1 k> ta s (k − ttars )mk αa . (iii) Replace the RHS of k=1 k m tr   Pn P ttars −1 ta s Pn (a) (a) −1 −1 tr ˜ ta s = ta s ( tr − i)mi = i=1 b=1 Cab ζb with that of (5.5), leading to m b=1 Cab ζb − tr P (a) (a) ˜ ta s ≥ 0. Setting i = ta in (5.4) we − i≥1 mi ∈ Z. Thus we are left to verify m tr Pn P (a) (b) (a) −1 ˜ ta s ≥ 0 indeed have − k≥1 min(ta , k)mk = b=1 Cab ptb ≥ 0. This tells that m tr

holds if ta = 1. Henceforth we assume that ta > 1 and use (5.4) again for i = 1, P Pn (b) (a) (a) ˜ ta s , we which reads p1 = − b=1 Cba k≥1 mk . From the above expression of m

rewrite this as (5.6)

tr

X

1≤b≤n,tb >1

(b)

(a)

Cba m ˜ tb s = p1 − tr

X

1≤b≤n,tb =1

(b)

Cba m ˜ tb s , tr

which is valid for each 1 ≤ a ≤ n such that ta > 1 For tb = 1 we have already shown (b) that m ˜ tb s ≥ 0, and moreover Cba ≤ 0 because ta > 1 = tb . Therefore the RHS tr

of (5.6) is non-negative. The coefficients (Cba )ta ,tb >1 form a symmetric submatrix, which is the Cartan matrix of the algebra Zn in Table 2. (Simply laced Xn is −1 out of question here.) Thus all the elements Dab (2.6) of their inverse are positive, which completes the proof. The above proof is a straightforward generalization of the one for Xn = An given in [HKKOTY]. Next we proceed to the case s/tr ∈ Z + 21 .

22

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

Theorem 5.5. Let Xn = Bn , Cn , F4 and assume that tr = 2, s ∈ 2Z + 1. Then we have (5.7)

(5.8) (5.9)

(a)

lim q −c({mi,0 }) M (Ws(r)⊗L , λ, q) X q φ M (W (η), λ, q −1 )M s−1 (W (ξ), q −1 ) Q2 Q = , (q) (q) (q) ζ ξ η ξ,η,ζ a a a 1≤a≤n,ta =2 1≤a≤n W (η) =

φ=

n O

(a)⊗ηa

W1

,

W (ξ) =

n O

(a)⊗ξa

W1

,

a=1

a=1 n X

1 | (ηa − ξa )Λa |2 + 2 a=1

X

−1 Dab (ζa −

1≤a,b≤n ta =tb =2

ξa + ηa ξb + ηb )(ζb − ), 2 2

where D−1 has been defined in (2.6). The limit L → ∞ is taken under the condition in Remark 5.2. The sum runs over non-negative integers ηa , ξa (1 ≤ a ≤ n) and ζa (1 ≤ a ≤ n, ta = 2) satisfying the constraints Bn :

ξn , ηn , ζn +

Cn :

n X

n−1 X

a(ξa + ηa ) +

a=1

aξa ,

a=1

F4 :

n X

aηa ∈ 2Z,

a=1

n−1 X

n−1 (ξn + ηn ) ∈ 2Z, 2 n

aζa −

a=1

1X a(ξa + ηa ) ∈ nZ, 2 a=1

ξ3 + 2ξ4 + η3 + 2η4 + ζ3 + 2ζ4 ∈ 3Z.

Finally we consider s/tr ∈ Z ±

1 3

case.

Theorem 5.6. Let Xn = G2 and r = 2 hence tr = 3. Assume that s ∈ 3Z + 1 or 3Z + 2, for which s0 = 1 or 2, respectively as in Remark 5.3. Then we have (5.10)

(5.11) (5.12) (5.13)

(a)

lim q −c({mi,0 }) M (Ws(2)⊗L , λ, q) X q φ M (W (η), λ, q −1 )M s−s0 (W (ξ), q −1 )Φ 3  Q , = (q) (q) (q) ζ ξ η ξ,η,ζ,m a a 1≤a≤2 W (η) =

2 O

a=1 2 X

(a)⊗ηa

W1

,

W (ξ) =

2 O

(a)⊗ξa

W1

,

a=1

 2 1 m2 s0 η2 + (3 − s0 )ξ2 3 2 φ= | ζ− , (ηa − ξa )Λa | + + 2 a=1 2 8 3   1 (ζ + (2 − s0 )η2 + (s0 − 1)ξ2 ) 2 Φ= . m q

Here the limit L → ∞ is taken so that L ∈ 2Z as in Remark 5.2. The sum runs over non-negative integers ηa , ξa (1 ≤ a ≤ 2), ζ and m satisfying ζ + (2 − s0 )η2 + (s0 − 1)ξ2 , m +

(3 − s0 )ξ2 + s0 η2 + ζ ∈ 2Z. 2

Our proofs of Theorem 5.5 and 5.6 are essentially the same with that for Theorem 5.4. Here we shall only mention a few key points. From Lemma 5.1, the

REMARKS ON FERMIONIC FORMULA (a)

variables mj

23

with those (a, j) in Remark 5.3 are tending to infinity. Correspond(a)

(a)

(a)

ingly we have set ξa = p ta (s−s ) , ηa = p ta (s−s

and ζa = ps (ta = tr ). For # " (2) (2) ps+3−2s0 + ms+3−2s0 G2 , ζ2 is simply denoted by ζ and Φ is in fact equal to . (2) ms+3−2s0 q The congruence conditions among the variables ξ, η, ζ (and m for G2 ) are equivalent to the condition that the following quantities are integers for all 1 ≤ a ≤ n P P (a) (a) (a) −1 −1 ; b=1 Cab ηb , m ta (s−s ) , m ta (s−s +t ) , ms (ta = tr ). From the b=1 Cab ξb , 0

tr

0

tr

condition, φ ∈ Z follows easily.

0 +tr )

tr

0

tr

r

6. Recursion relation of fermionic forms The fermionic form obeys a recursion relation compatible with the Q-system discussed in the next section. Theorem 6.1. Fix 1 ≤ a0 ≤ n and j0 ∈ Z≥1 arbitrarily. Given any λ ∈ P (a) N N (a) and W = 1≤a≤n j≥1 (Wj )⊗νj , set (a )

(a )

(a )

(a )

+

W1 = Wj0 0 ⊗ Wj0 0 ⊗ W,

0 0 ⊗ W, ⊗ Wj0 −1 W2 = Wj0 +1

W3 =

O

−Ca0 b −1

b∼a0

O

(b)

W (C j −k)/C ⊗ W, [ ba0 0 a0 b ]

k=0

where the symbol [x] denotes the largest integer not exceeding x. Then we have M (W1 , λ, q) = M (W2 , λ, q) + q −θ M (W3 , λ, q),

(6.1)

Ml (W1 , q) = Ml (W2 , q) + q −θ Ml (W3 , q),

(6.2)

Nl (W1 , λ, q) = Nl (W2 , λ, q) + q −θ Nl (W3 , λ, q), X (a ) νk 0 min (j0 , k), θ = j0 +

(6.3) (6.4)

k≥1

(a)

where in (6.2) and (6.3) we assume 0 ≤ j ≤ ta l for all Wj ’s contained in W1 , W2 and W3 . (a0 )

When j0 = 1 we understand that W2 = W2

⊗ W and similarly for W3 .

Proof. Weprove (6.1).  (6.2) and(6.3) are of the prop  similar. By means   p+m p + m p + m p + m−1 −mp erties = q and = + m m m m q q q  q−1 p+m−1 qp , we rewrite (4.3) as m−1 q (6.5) M (W1 , λ, q

−1

)=

X

q

c({m})

{m}

"

×

Y

(a,i)6=(a0 ,j0 ) (a )

(a )

pj0 0 + mj0 0 − 1 (a ) mj0 0

" #

(a)

(a)

pi

+ mi (a) mi "

+q q

(a ) pj 0 0

#

q

(a )

(a )

pj0 0 + mj0 0 − 1 (a ) mj0 0 − 1

#  . q

24

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

Here c({m}) = (a)

1 2

P

P

1≤a,b≤n (αa |αb )

(a)

j,k≥1

(b)

min(tb j, ta k)mj mk , which does not (a)

contain νj ’s explicitly as opposed to (4.4). Let us write (4.5) as pi (W, {m}) to N (b) (a) ⊗νj(a) and {m} = {mk }. Similarly, exhibit the dependence on W = a,j (Wj ) (a)

(a)

(4.6) is denoted by (4.6)W . Then pi = pi (W1 , {m}) for all (a, i) in (6.5). Suppose (b)′ (b) {m} obey (4.6)W1 , and set mk = mk − δb,a0 δk,j0 . Then it is easy to see (a)

(a)

{m} obeys (4.6)W2 ,

pi (W2 , {m}) = pi (W1 , {m}) − δa,a0 δi,j0 , (a) pi (W3 , {m′ })

=

c({m}) =

(a) pi (W1 , {m}), (a ) c({m′ }) − pj0 0 (W1 , {m}) +

{m′ } obeys (4.6)W3 , θ.

Thus in the expansion (6.5), the first and the second terms on the RHS yield M (W2 , λ, q −1 ) and q θ M (W3 , λ, q −1 ), respectively up to boundary effects. By bound(a ) ary effects we mean that the above expansion should not be applied if pj0 0 (W1 , {m}) = (a )′

(a )

(a )

mj0 0 = 0, because it amounts to allowing pj0 0 (W2 , {m}) = −1 or mj0 0 = −1     −1 −1 (a ) in either interpretation = 1 or = 1. In fact pj0 0 (W1 , {m}) = 0 q −1 q (a )

mj0 0 = 0 does not hold for any summand in M (W1 , λ, q −1 ). This can be guaran(a)

teed by combining (4.5) for pj (a)

(a)

(a)

= pj (W1 , {m}) as (a)

(a)

pj+1 − 2pj + pj−1 = −νj +

X

Cab

b∼a

(b)

(a)

− 2δa,a0 δj,j0 + 2mj

! tX b −1 tb (b) (b) (b) (tb − i)(mtb j−i + mtb j+i ) , mt + ta tab j i=1 (a )

(a )

where we interpret mk = 0 whenever k 6∈ Z. Thus pj0 0 = mj0 0 = 0 contradicts (a)

(a)

(a, j) = (a0 , j0 ) case of the above since ∀mj , pj

≥ 0 and Cab < 0 for b ∼ a.

For Xn = An a similar observation has been made also in [SW]. 7. Q-system Let xa = eΛa be a complex variable. We shall use the notations eαa and eρ , etc Qn Qn + ba to stand for b=1 xC and b=1 xb , etc. For any λ ∈ P the character chV (λ) of b ±1 the associated irreducible finite dimensional Xn -module belongs to Z[x±1 1 , . . . , xn ]. (a) (a) Let {Qj | 1 ≤ a ≤ n, j ∈ Z≥0 } be the functions satisfying Q0 = 1 and (7.1)

(a)2

Qj

(a)

(a)

= Qj+1 Qj−1 +

ab −1 Y −CY

b∼a

k=0

(b)

Qh Cba j−k i

j ≥ 1.

Cab

We shall call (7.1) Q-system of type Xn . It first appeared in [KR2] for Xn = An , Bn , Cn and Dn and in [Ki1] for the exceptional case. For the simply laced case Xn = An , Dn , E6,7,8 , it has the simple form Y (b) (a) (a) (a)2 Qj , Qj = Qj+1 Qj−1 + b∼a

where the symbol b ∼ a is defined before (2.4). We have listed the explicit forms for the non-simply laced case in Appendix C.

REMARKS ON FERMIONIC FORMULA

25

The rest of this section is devoted to a solution of (7.1) for the classical algebras Xn = An , Bn , Cn and Dn . Let us introduce the functions (7.2)

(a)

An :χj = chV (jΛa ), (P chV (k1 Λ1 + k2 Λ2 + · · · + ka Λa ) 1 ≤ a ≤ n − 1, (a) Cn :χj = (7.3) chV (jΛn ) a = n, X (a) chV (ka0 Λa0 + ka0 +2 Λa0 +2 + · · · + ka Λa ) 1 ≤ a ≤ n′ , (7.4) Bn , Dn :χj = n′ = n for Bn , n − 2 for Dn ,

(a) χj =

chV (jΛa ) a = n − 1, n

a0 ≡ a mod 2,

a0 = 0 or 1,

only for Dn .

Here Λ0 = 0 and chV (λ) denotes the irreducible Xn character with highest weight λ. The sum in (7.3) is taken over non-negative integers k1 , . . . , ka that satisfy k1 + · · · + ka ≤ j, kb ≡ jδa,b mod 2 for all 1 ≤ b ≤ a. The sum in (7.4) extends over non-negative integers ka0 , ka0 +2 , . . . , ka obeying the constraint ta (ka0 +ka0 +2 +· · ·+ ka−2 ) + ka = j. If one depicts the highest weights in the sum (7.3) (resp. (7.4)) by the Young diagrams as usual, they correspond to those obtained from the a × j rectangular one by successively removing 1 × 2 (resp. 2 × 1) dominoes. (For Bn with a = n, one needs to say a bit more precisely, see Appendix A.) Now we have Theorem 7.1. Let Xn = An , Bn , Cn or Dn . Then, P (a) + dλ ch V (λ), dλ ∈ C, d (A) χj = jΛa = 1, dλ = 0 unless λ ∈ jΛa − λ∈P Pn 1 ≤ a ≤ n, j ∈ Z≥0 , b=1 Z≥0 αb , (a) (a) (B) Qj = χj solves the Q-system (7.1),  (a)  χj αn α1 (C) limj→∞ | > 1. = x−1 (a) a in the domain |e |, . . . , |e χj+1

(A) is trivially true. The functions and (7.2)-(7.4) were first introduced in [KR2], where the statement that they satisfy the Q-system was also announced. However as they did not present a proof, we prove the property (B) for the typical case Xn = Bn . (An case is too much straightforward.) The property (C) is needed in section 8, for which we also sketch a proof. Proof. (Theorem 7.1 (B) for Bn .) Give a partition λ = (λ1 , λ2 , . . . ) with any length l(λ), define (7.5)

Q(λ) = (1)

where we understand that Qi

det

(1)

1≤i,j≤l(λ)

(Qλi −i+j ),

= 0 for i < 0.

Lemma 7.2. For j ∈ Z≥0 the following solves the Q-system (see Appendix C for (n) (1) (1) Xn = Bn ) in the polynomial ring of infinitely many variables Q1 , Q1 , Q2 , . . . . (7.6) (7.7)

(a)

Qta j = Q((j a )) (n)

(n)

Q2j+1 = Q1

j X

1 ≤ a ≤ n, (−1)k Q((j n−1 , k)).

k=0

This follows as a corollary of Theorem 5.1 in [KOS] by dropping the spectral parameter dependence. (To get (7.7) one needs to expand the determinant therein.)

26

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

Thus it suffices to show (n)

(7.8)

RHS of (7.6), (7.7) under the substitution Q1

(n)

(1)

→ χ1 , Q j

(1)

→ χj (∀j)

= RHS of (7.4).

In the remainder of this section we always assume this substitution for Q(λ) in (7.5). A partition λ = (λ1 , λ2 , . . . ) is identified with the Young diagram in the usual way. We denote by λ′ its transpose, |λ| = λ1 + λ2 + · · · and 2λ = (2λ1 , 2λ2 , . . . ). Let sλ λ be the Schur function of gln associated with the partition λ. We denote by LRµ,ν P λ sλ . It is well known the Littlewood-Richardson coefficient, i.e., sµ sν = λ LRµ,ν λ that LRµ,ν = 0 unless |λ| = |µ| + |ν|. (cf. [Ma]) Lemma 7.3. For a partition λ with l(λ) ≤ n, Q(λ) decomposes into the irreducible Bn characters as X λ Q(λ) = (7.9) LR(2κ) ′ ,µ chV (µ). κ,µ

Here the κ, µ sums run over all partitions such that (2κ)′ , µ ⊂ λ. The partition µ = (µ1 , µ2 , . . . , µn ) is identified with the Bn highest weight (µ1 − µ2 )Λ1 + · · · + (µn−1 − µn )Λn−1 + 2µn Λn . (7.8) follows from Lemma 7.3. To see this for (7.6), one only has to notice that (j a ) any non-zero LR(2κ)′ ,µ is 1 exactly when µ is obtained from (j a ) by removing (2κ)′ made of 2 × 1 dominoes. A similar argument tells that j X

(−1)k Q((j n−1 , k)) =

k=0 (n)

ǫ1 ,... ,ǫn =±1,ν∈P

(−1)nj−|µ| chV (µ).

µ⊂(j n )

Combining this with χ1 chV (Λn )chV (µ) X =

X

= chV (Λn ) and

chV (ν = µ + +

ǫn−1 − ǫn ǫ1 − ǫn Λ1 + . . . + Λn−1 + ǫn Λn ) 2 2

(cf. (5.1.15) in [N]), one can verify (7.8) for (7.7). It remains to show Lemma 7.3. For this purpose we introduce generating functions of characters: X (1) 1 1+z = χj z j , := Qn −1 φ(z) (1 − y z)(1 − y z) i i i=1 j≥0

X 1 1 := = pj z j , 2 ψ(z) (1 − z )φ(z) j≥0

(1)

where yi = xtii /xi−1 (1 ≤ i ≤ n, x0 = 1). Here χj = chV (jΛ1 ) is an irreducible Bn character while pj is a reducible one in general.

REMARKS ON FERMIONIC FORMULA

27

Lemma 7.4. X 1 = Q(λ)sλ (z), φ(z1 ) · · · φ(zn ) λ P chV (λ)sλ (z) 1 = Q λ , φ(z1 ) · · · φ(zn ) 1≤i
(7.10) (7.11) (7.12)

where λ (resp. κ) sum extends over all the partitions with l(λ) ≤ n (resp. l((2κ)′ ) ≤ n). Q(λ) and chV (λ) are characters of Bn hence the functions of y1 , . . . , yn only. Thus Lemma 7.3 immediately follows from Lemma 7.4. (7.12) is eq.(11.9;2) in [L] or Example 5(b) on p77 of [Ma]. Let us verify (7.11). Below we write any n × n determinant det1≤i,j≤n (Ai,j ) simply as |Ai,j |. Due to factorization of the Q Q Vandermonde type determinant |zin−j − zin+j | = 1≤i≤j≤n (1 − zi zj ) 1≤i
|zin−j − zin+j | φ(z1 ) · · · φ(zn ) ψ(z1 ) · · · ψ(zn ) X X ǫ1 · · · ǫn sgn σ z1n−ǫ1 σ1 +j1 · · · znn−ǫn σn +jn pj1 · · · pjn .

i
− zi zj )(zi − zj )

=

σ∈Sn ,ǫ1 ,... ,ǫn =±1

j1 ,... ,jn ≥0

P Setting li = n − ǫi σi + ji this equals l1 ,... ,ln |pli −n+j − pli −n−j |z1l1 · · · znln . In view of the anti-symmetry under the interchange of l1 , . . . , ln , we set li = λi + n − i and rewrite this as a sum over partitions λ = (λ1 , . . . , λn ): X |pλi −i+j − pλi −i−j ||zjλi +n−i |. λ,l(λ)≤n

Q Thus (7.11) follows from the well known character formulae |zjλi +n−i | = sλ (z) i
where the summation is taken over the set of non-negative integers ka , ka−2 , . . . such that ka + ka−2 + · · · + k2 ≤ m (resp. ka + ka−2 + · · · + k1 = m). The Bn character with highest weight λ = (l1 − l2 − 1)Λ1 + · · · + (ln−1 − ln − 1)Λn−1 + (2ln − 1)Λn is X D · chV (λ) = sgn(σ)ǫ1 · · · ǫn yσǫ11l1 · · · yσǫnnln , σ∈Sn ,ǫ1 ,... ,ǫn =±1

28

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

where D is the Weyl denominator independent of λ. First let us prove the case a even, for which we have X D · χ(a) eρ (y1 y2 )k2 (y1 y2 y3 y4 )k4 · · · (y1 · · · ya )ka m = k

). ± (other terms given by yi → yσ±1 i

Owing to the formula (7.13) explained later, this expression can be recast into  ρ (a) D · χm = e C2 (y1 y2 )m + C4 (y1 y2 y3 y4 )m + · · · + Ca (y1 · · · ya )m  1 + (1 − y1 y2 ) · · · (1 − y1 · · · ya ) ), ± (other terms given by yi → yσ±1 i where the coefficients Ci ’s have no dependence on m. Recall that the assumption of the property (C), |eα1 |, . . . , |eαn | > 1, now reads as |y1 | > · · · > |yn | > 1. Therefore in the m → ∞ limit we have X m D · χ(a) (±)τ (Ca eρ ) + O(γ m )), |γ| < 1. m = (y1 · · · ya ) ( Here the sum is taken over those τ ∈ Sn such that yτ1 · · · yτa = y1 · · · ya . (Action of τ is the natural permutation of the y-variables.) As the leading term does not (a) (a) vanish we get χm /χm+1 → (y1 · · · ya )−1 = e−Λa in this limit. In the case a odd the same estimation is obtained by taking the k-sum via X eρ (y1 )k1 (y1 y2 y3 )k3 · · · (y1 · · · ya )ka k

o n = eρ C˜1 (y1 )m + C˜3 (y1 y2 y3 )m + · · · + C˜a (y1 · · · ya )m ,

where the coefficients C˜i ’s again have no dependence on m. The key here is an elementary identity s X X j ξim Qs ξ11 · · · ξsjs = (7.13) , k=1,(k6=i) (1 − ξk /ξi ) i=1 j

where the sum in the left is taken over the non-negative integers j1 , . . . , js such that j1 + · · · + js = m. (For even a we have used this with one of the variables equal to 1.) Finally when a = n, the factor (y1 · · · ya )ka in the above is replaced by (y1 · · · yn )kn /2 . The k-sum is taken under the condition kn + 2(kn−2 + · · · + k2 ) ≤ m (resp. kn + 2(kn−2 + · · · + k1 ) = m) if n is even (resp. odd). Following the similar argument to the above, one gets the leading m-dependence proportional to (y1 · · · yn )m/2 = emΛn , hence the desired result. 8. Combinatorial completeness for Xn 8.1. Main theorem. Our goal in this section is to prove Theorem 8.1. Suppose that a linear combination of characters X (a) Qj = dλ ch V (λ) 1 ≤ a ≤ n, j ∈ Z≥0 λ∈P

possesses the properties:

+

REMARKS ON FERMIONIC FORMULA

(A) dλ ∈ C, djΛa = 1, dλ = 0 unless λ ∈ jΛa − (a) {Qj }

Pn

b=1

29

Z≥0 αb ,

satisfies the Q-system (7.1),  (a)  Qj αn α1 (C) limj→∞ | > 1. = x−1 (a) a in the domain |e |, . . . , |e (B)

Qj+1

(a)

Then for arbitrary νj (8.1)

∈ Z≥0 ’s we have

n Y Y

a=1 j≥1

(a)

Qj

νj(a)

=

X

N∞ (W, λ, 1)ch V (λ)

λ

P P Nn N (a) (a) ⊗νj(a) . The sum λ runs over λ ∈ ( na=1 γ∞ Λa − for W = a=1 j≥1 (Wj ) Pn + (a) b=1 Z≥0 αb ) ∩ P . (See (4.18) for the definition of γ∞ .)

We shall denote the LHS of (8.1) by chW . Before entering the proof in section 8.2, several remarks are in order. (a) Suppose conversely that (8.1) holds for any νi ’s. Then, for the function P (a) λ N∞ (Wj , λ, 1)chV (λ), one can derive (A) and (B). The former is clear from (a)

the definition (4.16) – (4.17) of N∞ (Wj , λ, 1). The latter follows from Theorem 6.1 (6.3) with l → ∞, q = 1. (a) Theorem 8.1 asserts the uniqueness of {Qj } satisfying (A) – (C), but does not P (a) assure the existence, i.e. it does not guarantee the functions { λ N∞ (Wj , λ, 1)chV (λ)} fulfill (B) and (C). (They do (A).) In this paper we have verified the existence in (a) (a) Theorem 7.1 for Xn = An , Bn , Cn and Dn . Namely the choice Qj = χj in (7.2)–(7.4) indeed fulfills (A)–(C). For the exceptional Xn , our Theorem 8.1 also works once the conditions (A)–(C) are shown. By combinatorial completeness of the string hypothesis it is usually meant that (8.2)

chW =

X

M (W, λ, 1)chV (λ),

λ

where the sum ranges over the same domain as in (8.1). Note the gap between (8.2) and (8.1) has been left as Conjecture 4.3 (4.21) in general. Our proof of Theorem 8.1 is a generalization of [Ki2], where not (8.2) but (8.1) was shown similarly for Xn = An . So far the above mentioned gap has been filled only by (a) (a) combinatorial means for Xn = An , νj = νj δa,1 [KR1] and νj general [KSS]. Although it is not “physical” to allow negative vacancy numbers, the fermionic form N∞ (W, λ, q) is not less interesting than M (W, λ, q) in view of the remarkable symmetry in Remark 8.8 and Conjecture 8.9. Of course there is another gap between the combinatorial completeness and the completeness in the literal sense. To discuss it is beyond the scope of this paper. See for example [EKS], [JD], [LS], [TV] and the references therein. String hypothesis is an origin of the fermionic formulae. However it should be emphasized that our Theorem 8.1 stands totally independent of it; it is based purely on the Q-system (B) and the properties of the Q-functions (A) and (C). In a sense the Q-system encodes the essential aspects of the string hypothesis as far as the combinatoral completeness is concerned. 8.2. Proof of Theorem 8.1.

30

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

8.2.1. Definition of ψi . For 1 ≤ i ≤ tl + 1, let Zi denote the set of variables ta (a) (a) (8.3) Zi = {zj,i−1 | (a, j) ∈ Hl [i]} = {zj,i−1 | (i − 1) < j ≤ ta l, 1 ≤ a ≤ n}. t See (2.1) for the definition of t. By definition Ztl+1 = ∅. Suppose the variables are related by −Cba (j− tta i) n  Y (a) (a) (b) zj,i = zj,i−1 (8.4) (a, j) ∈ Hl [i + 1] 1 − z tb t

b=1

i,i−1

(b) = 0 unless ttb i ∈ Z, for which the power t i,i−1 (a) that zj,i ∈ Zi+1 in the LHS while all the variables

for 1 ≤ i ≤ tl − 1. We assume that z tb

Cba (j − tta i) is an integer. Note appearing in the RHS belong to Zi . Thus we have the relation C = C(Ztl+1 ) ⊂ C(Ztl ) ⊂ · · · ⊂ C(Z1 )

(8.5)

among the fields of rational functions in Zi . For 1 ≤ i ≤ tl + 1 we define the function ψi by ψtl+1 = 1 and −(γj(a) −µa +1) Y  (a) (8.6) 1 ≤ i ≤ tl, 1 − zj, t j−1 ψi = ta

(a,j)∈Hl [i]

(a)

and µa specified in (4.18) as integer parameters.

which involves γj

Proposition 8.2. (8.7)

ψi = ψi+1

n  Y

1−

a=1

ψi ∈ C(Zi )

(8.8) For

ta t i

(a) z ta i,i−1 t

−(γ (a) −µa +1) t ai t

1 ≤ i ≤ tl,

1 ≤ i ≤ tl + 1.

(a)

(a)

t

t

6∈ Z, γ ta i is not needed since z ta i,i−1 = 0.

Proof. (8.7) is immediate by applying the decomposition (2.10) to (8.6). We show (8.8) by induction with respect to i. It is trivially true for i = tl + 1. By virtue (a) of (8.5) and (8.7) we only have to check z ta i,i−1 ∈ Zi (8.3), which is obvious. t

8.2.2. Series expansion of ψi . Proposition 8.3. As a formal power series in the variables Zi , the function ψi (1 ≤ i ≤ tl + 1) has the expansion ) ( (a) (a) mj(a)  X Y p j + mj (a) (8.9) . z ψi = j,i−1 (a) mj {m} (a,j)∈H [i] 1

l

(a)

Here the sum ranges over {mj

(a)

∈ Z≥0 | (a, j) ∈ Hl [i]} and pj

is given by (4.20).

Proof. We again invoke induction with respect to i. The case i = tl + 1 is trivially valid. Suppose (8.9) is valid for ψi+1 . Substituting it into (8.7) we get

ψi =

n  Y

a=1

1−

(a) z ta i,i−1 t

−(γ (a) −µa +1) X t ai t

Y

{m} (a,j)∈Hl [i+1]

(

(a)

(a)

p j + mj (a) mj

)

1



(a)

zj,i

mj(a)

.

REMARKS ON FERMIONIC FORMULA

31

(a)

By further substituting (8.4) to express zj,i in terms of Zi variables, the RHS becomes ( ) n  (a) (a) mj(a)  −p(a) Y XY ta −1 + m p (a) (a) i j j t z 1 − z ta i,i−1 j,i−1 (a) t mj (a,j)∈H [i+1] {m} a=1 1

l

with the help of (4.20). Expanding the first factor by means of X  p+m  −p−1 (1 − z) = (8.10) z m for any p ∈ Z, m 1 m≥0

and using (2.10), we obtain (8.9). (a)

8.2.3. Specialization of ψ1 . ψ1 is a rational function of Z1 = {zj,0 | (a, j) ∈ Hl }. Now we begin specializing these variables to a certain combination of the Q-functions. (a)

(a)

Lemma 8.4. Suppose {Qj } satisfies the Q-system (7.1) with Q0 = 1. Then the recursion relation (8.4) has a solution (8.11)

(a)

zj,i =

Cba n  Y (b)j− tta (ib +1) (b)−j+ tta ib b Qib +1 b , Qib

b=1

ib =



tb i t



for (a, j) ∈ Hl [i + 1] and 0 ≤ i ≤ tl − 1. Proof. For tta i ∈ Z it is straightforward (though tedious) to check that (a) z ta i,i−1 given by (8.11) satisfies t

(a)

(a)

(8.12)

1−

(a) z ta i,i−1 t

=

Q ta i−1 Q ta i+1 t

t

(a)2

Q ta i t

by using the Q-system. Upon substituting (8.11) and (8.12), both sides of (8.4) can be cast into products of Q-functions, which turn out to be equal. When i = 0 (8.11) reads (8.13)

(a)

zj,0 =

n Y

(b)−jCba

Q1

1 ≤ j ≤ tl

b=1

for the Z1 variables. As said in (B) we assume the Q-system and set (8.14)

(1)

(n)

Ψ = ψ1 |(8.13) ∈ C(Q1 , . . . , Q1 ), (a)

which depends on the integer parameters νi and λa in (4.17) – (4.18) as well. Based on the results in sections 8.2.1 and 8.2.2, it is easy to express Ψ either as a product or a power series.

32

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

Proposition 8.5. Let chW = have (8.15)

Ψ = chW

n Y

a=1

(8.16)

Ψ=

{u}

Here the sum



(a)

(a)

(a)

X

Y

{m} (a,j)∈Hl

(

,

) 

p j + mj (a) mj

1



n Y

(a)−

Q1

a=1

Pn

b=1

Cab ub

.

P extends over {ua ∈ Z≥0 | 1 ≤ a ≤ n} and the sum {m} Pta l (a) ∈ Z≥0 | (a, j) ∈ Hl } obeying the constraint k=1 kmk = ua for

P

(a)

does over {mj 1 ≤ a ≤ n.



Qta l+1

!λa +1

(a)

Qta l

(a)−µa +1 Q1

a=1

X

 ν (a) (a) j Q be the LHS of (8.1). We j j≥1

Q

Qn

{u}

Proof. To show (8.15) apply (8.12) to (8.6) with i = 1 and use the identity

(a)

for 1 ≤ j ≤ ta l (γ0

(a)

(a)

(a)

(a)

− γj+1 = νj

−γj−1 + 2γj

= 0). The result reads Ψ = chW

n Y

(a)

(a)γta l+1 −µa +1

(a)−µa +1 Qta l

Q1

(a)γ

(a)

ta l Qta l+1

a=1 (a)

−µa +1

.

(a)

From (4.18) it follows that γta l+1 − µa + 1 = γta l − µa + 1 = λa + 1, hence (8.15). To Pta l (a) show (8.16) substitute (8.13) into (8.9) with i = 1 and set ua = k=1 kmk .

8.2.4. Integral representation of Nl (W, λ, 1). Anticipating the change of vari(1) (n) ables from {Q1 , . . . , Q1 } to {x1 , . . . , xn }, we first prepare Lemma 8.6. (a)

(8.17)

det1≤a,b≤n

∂Q1 ∂xb

!

=

Y

(1 − e−α ),

α>0

where the RHS denotes the product over all positive roots α of Xn . Proof. Under the action of the simple reflection wa = wαa (a = 1, . . . , n) in ¯ the Weyl group, the variables e−αa and xa = eΛa transforms as wa (e−αb ) = e−αb +Cab αa ,

wa (xb ) = xb e−δab αb .

Moreover wa (αa ) = −αa and other positive roots are transformed with each other. The RHS of the equation Y ∆= (1 − e−α ), α>0

has an expansion ∆ = 1 + O(e−α ) and has a transformation property such as, wa (∆) = (−eαa )∆. On the other hand, the Jacobian J in the LHS (1)

J=

(n)

∂(Q1 , . . . , Q1 ) ∂(x1 , . . . , xn )

REMARKS ON FERMIONIC FORMULA

33

transforms in the same manner as ∆, i.e. wa (J) = −eαa J, since the Laurent (a) polynomials Q1 (a = 1, . . . , n) are Weyl group invariants and the volume form transforms as follows wa (dx1 ∧ · · · ∧ dxn ) = dx1 ∧ · · · ∧ dxa−1 ∧ d(wa (xa )) ∧ dxa+1 ∧ · · · ∧ dxn Y 1 ba x−C ) ∧ dxa+1 · · · ∧ dxn = dx1 ∧ · · · dxa−1 ∧ (− 2 dxa b xa b6=a

= −e−αa dx1 ∧ · · · ∧ dxn .

(a)

Hence, the ratio J/∆ is a Weyl group invariant. Since the polynomials Q1 have (a) the expansion Q1 = xa (1 + O(e−α )) owing to (A), the invariant J/∆ has an expansion J/∆ = 1 + O(e−α ) and hence it must be 1. (a) N (a) ⊗νj and λ = λ1 Λ1 + · · · + Proposition 8.7. Suppose W = (a,j)∈Hl Wj P −1 λn Λn ∈ P are chosen so that µa in (4.18) has the property nb=1 Cab µb ∈ Z≥0 . Then we have (8.18) !λa +1 I I (a) n (1) (n) Y Qta l dQ1 ∧ · · · ∧ dQ1 chW Nl (W, λ, 1) = · · · (a) (2πi)n Qta l+1 a=1 ! !λa +1 I I (a) n Y Y Qta l dx1 ∧ · · · ∧ dxn ρ −α (1 − e ) chW = ··· (8.19) . e (a) (2πi)n x1 · · · xn Q α>0 a=1

ta l+1

(1)

(n)

Here the integrand in (8.18) (resp. (8.19) ) is regarded as an element in C(Q1 , . . . , Q1 ) Q (b) (resp. C(x1 , . . . ,Q xn )) and the integration contours encircle ∞ in the domain b=1 |Q1 |Cba ≫ 1 (resp. |eαa | = b=1 |xb |Cba ≫ 1) for 1 ≤ a ≤ n.

Proof. By comparing (8.16) and (4.16) – (4.19) with q = 1, we find that Qn (a)−µa in Ψ. The integral (8.18) is picking the Nl (W, λ, 1) = coefficient of a=1 Q1 coefficient up as a multi-dimensional residue from Ψ in (8.15). In view of (8.16) Q (b) the cycles of the integrals are to be taken in the domain b=1 |Q1 |Cba ≫ 1. It (a) follows from this inequality that |Q1 | ≫ 1 for all 1 ≤ a ≤ n. (8.19) can be derived from (8.18) by changing variables with the aid of Lemma 8.6. When ∀|eαa | ≫ 1, (a) ∀|eΛa | ≫ 1 also holds. Thus the domain of {Q1 }-integrals matches that for {xa }(a) integrals because Q1 = xa (1 + O(e−αa )) owing to (A). Note that such a domain indeed exists, for example, the vicinity of |xa | = ey(Λa |ρ) with y ≫ 0.

8.2.5. Fermionic form N∞ (W, λ, 1) as branching coefficient. Define the branching coefficient [W : ω] ∈ C by X ch W = (8.20) [W : ω] ch V (ω), ω

Pn Pn (a) where, due to (A), the sum is actually limited to ω ∈ ( a=1 γ∞ Λa − a=1 Z≥0 αa )∩ + P . Now we are ready to finish the proof of Theorem 8.1, namely, to establish (8.21)

[W : λ] = N∞ (W, λ, 1)

+

for λ ∈ P .

34

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

Consider the limit l → ∞ in (8.19). Thanks to the assumption (C), we may replace  (a) λa +1 Q Qn Qta l by na=1 xa−λa −1 = e−λ−ρ , leading to (a) a=1 Qta l+1

(8.22) N∞ (W, λ, 1) =

I

···

I

dx1 ∧ · · · ∧ dxn (2πi)n x1 · · · xn

e

ρ

Y

(1 − e

−α

α>0

!

)

e−λ−ρ ch W.

Substitute (8.20) into the RHS and express chV (ω) via the Weyl-Kac character formula. The result reads (8.23) I I X dx1 ∧ · · · ∧ dxn −λ−ρ X e N∞ (W, λ, 1) = det w ew(ω+ρ) , [W : ω] · · · nx · · · x (2πi) 1 n + w ω∈P

P

+

where w denotes the sum over the Weyl group of Xn . Since λ, ω ∈ P , the integral equals δw,1 δω,λ . Thus we have (8.21), and thereby complete the proof of Theorem 8.1. Remark 8.8. From (8.23) it follows that N∞ (W, w(λ + ρ) − ρ, 1) = det w N∞ (W, λ, 1) for any λ ∈ P and any Weyl group element w. In particular, N∞ (W, λ, 1) = 0 if there is a root α such that hα, λ+ρi = 0. In fact our computer experiments indicate Conjecture 8.9. N∞ (W, w(λ + ρ) − ρ, q) = det w N∞ (W, λ, q)

for any λ ∈ P .

Remark 8.10. The integral representations (8.18), (8.19) and (8.22) of the (a) fermionic forms are valid for arbitrary νj ∈ C and λ ∈ CΛ1 + · · · + CΛn as long P (a) (a) as µ = a,j jνj Λa − λ is kept in Zα1 + · · · + Zαn . In such a case pi ∈ C in general, and (8.22) for example should read as ( ) (a) (a) X Y p i + mi (a) mi {m} 1≤a≤n 1

i≥1

=

I

···

I

Y

dx1 ∧ · · · ∧ dxn (2πi)n x1 · · · xn

(1 − e

α>0

−α

!

)



n Y ν (a) Y (a) j e−jΛa Qj ,

a=1 j≥1

P P (a) ∈ Z≥0 such that na=1 j≥1 jmj αa = where the sum in LHS is taken over all µ. In particular, the above holds as 0 = 0 when µ 6∈ Z≥0 α1 + · · · + Z≥0 αn . (a) mj

(r)

Appendix A. List of M (Ws , λ, q −1 ) Consider a formal linear combination X M (Ws(r) , λ, q −1 )V (λ). Ws(r) = λ∈P

+

(r)

(r)

(r)

In this section, we give a list of Ws for Xn = An , Bn , Cn , Dn and W1 ( W2 (r) or a conjecture for Ws in some cases) for Xn = E6,7,8 , F4 , G2 . Some conjectures here have arisen from the calculation using the efficient algorithm in [Kl]. When

REMARKS ON FERMIONIC FORMULA

35

q = 1 it reduces to (7.2) – (7.4) for An , Bn , Cn , Dn and those in [Kl] for E6,7,8 . Xn = An : Ws(r) = V (sΛr ). Xn = Bn : Ws(r) =

X

q (Λn |sΛr −λ) V (λ) ,

λ

P

+

where the sum λ is taken over λ ∈ {kr0 Λr0 +kr0 +2 Λr0 +2 +· · ·+kr Λr ∈ P | tr (kr0 + kr0 +2 + · · · + kr−2 ) + kr = s} with Λ0 = 0 and r0 ≡ r (mod 2), r0 = 0 or 1. In the (r) computation of M (Ws , kr0 Λr0 + kr0 +2 Λr0 +2 + · · · + kr Λr , q −1 ), the only choice (a) (a) of {mj } such that ∀pi ≥ 0 is the following: if r = 2u is even, then  (2a−1) Pmin(a−1,u) Pmin(a,u)  = b=1 δj,lb + b=1 δj,lb (a ≥ 1, 2a − 1 ≤ n − 1, j ≥ 1) mj P (2a) min(a,u) , mj = 2 b=1 δj,lb (a ≥ 1, 2a ≤ n − 1, j ≥ 1)   (n) Pu (j ≥ 1) mj = b=1 δj,2lb and if r = 2u + 1 is odd, then  (1)  mj = 0    m(2a) = Pmin(a−1,u) δ + Pmin(a,u) δ j,l j,lb j b=1 b=1 Pmin(a,u) b (2a+1)  = 2 m δ j,l  b j b=1   m(n) = Pu δ j,2l b j b=1

where lb =

1 tr (s

(j ≥ 1) (a ≥ 1, 2a ≤ n − 1, j ≥ 1) , (a ≥ 1, 2a + 1 ≤ n − 1, j ≥ 1) (j ≥ 1)

− kr ) − (kr−2 + kr−4 + · · · + k2b+r0 ) for the both cases.

Xn = Cn : Ws(r) =

(P

q (Λn |sΛr −λ) V (λ) V (sΛn ) λ

(1 ≤ r ≤ n − 1) , (r = n)

P + where the sum λ is taken over λ ∈ {k1 Λ1 + · · · + kr Λr ∈ P | k1 + · · · + kr ≤ (r) s, ka ≡ sδar (mod 2)}. In the computation of M (Ws , k1 Λ1 + · · · + kr Λr , q −1 ), (a) (a) the only choice of {mj } such that ∀pi ≥ 0 is the following: ( Pmin(a,r) (a) mj = b=1 δj,2lb (1 ≤ a ≤ n − 1, j ≥ 1) , P (n) (j ≥ 1) mj = rb=1 δj,lb where lb = 21 (s − (kb + kb+1 + · · · + kr )).

Xn = Dn : Ws(r)

=

(P

q (Λn |sΛr −λ) V (λ) V (sΛr ) λ

(1 ≤ r ≤ n − 2) , (r = n − 1, n)

P + where the sum λ is taken over λ ∈ {kr0 Λr0 + kr0 +2 Λr0 +2 + · · · + kr Λr ∈ P | kr0 + kr0 +2 + · · · + kr = s} with Λ0 = 0 and r0 ≡ r (mod 2), r0 = 0 or 1. In the (r) computation of M (Ws , kr0 Λr0 + kr0 +2 Λr0 +2 + · · · + kr Λr , q −1 ), the only choice

36

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA (a)

(a)

of {mj } such that ∀pi ≥ 0 is the following: if r = 2u is even, then  (2a−1) Pmin(a−1,u) Pmin(a,u)  = b=1 δj,lb + b=1 δj,lb mj Pmin(a,u) (2a) δj,l mj = 2 b=1  P b  (n−1) (n) = mj = ub=1 δj,lb mj and if r = 2u + 1 is odd, then  (1)  mj = 0    m(2a) = Pmin(a−1,u) δ + Pmin(a,u) δ j,l j,lb j b=1 b=1 Pmin(a,u) b (2a+1)  = 2 b=1 mj δj,lb    m(n−1) = m(n) = Pu δ j j b=1 j,lb

(a ≥ 1, 2a − 1 ≤ n − 2, j ≥ 1) , (a ≥ 1, 2a ≤ n − 2, j ≥ 1) (j ≥ 1)

(j ≥ 1) (a ≥ 1, 2a ≤ n − 2, j ≥ 1) , (a ≥ 1, 2a + 1 ≤ n − 2, j ≥ 1) (j ≥ 1)

where lb = s − (kr + kr−2 + · · · + k2b+r0 ) for the both cases. In Cn (1 ≤ r ≤ n − 1) (resp. Bn (1 ≤ r ≤ n − 1), Dn (1 ≤ r ≤ n − 2)) case, (Λn |sΛr − λ) is the number of 1 × 2 (resp. 2 × 1) dominoes to be removed from r × s rectangle Young diagram to obtain λ. In Bn , r = n case, we consider Young diagrams consisting of size 1 × 12 elementary pieces. Then the highest weights +

(n)

λ = kn Λn + kn−2 Λn−2 + · · · ∈ P occurring in Ws correspond to those diagrams obtained from n × 2s shape by removing 2 × 1 blocks (made of 4 elementary pieces). The quantity (Λn |sΛn − λ) is the number of the removed 2 × 1 blocks to obtain λ. Xn = E6 : (1)

=V (Λ1 ),

(2)

=V (Λ2 ) + q V (Λ5 ),

W1

W1 W1

 =q 3 V (0) + q V (Λ1 + Λ5 ) + V (Λ3 ) + q + q 2 V (Λ6 ),

(5) W1

=V (Λ5 ),

(6) W1

=q V (0) + V (Λ6 ).

(3) W1 (4)

=q V (Λ1 ) + V (Λ4 ),

(r)

In addition we have a conjecture for Ws : Ws(1) =V (sΛ1 ), Ws(2) =

s X

k=0

Ws(3)

=

q k V ((s − k)Λ2 + kΛ5 ), X

min (1 + j3 , 1 + s − j1 − 2j2 − j3 − j4 ) q 3s−2j1 −4j2 −3j3 −2j4

j1 +2j2 +j3 +j4 ≤s j1 ,j2 ,j3 ,j4 ∈Z≥0

× Ws(4) =

   j4 + 1 V j1 (Λ1 + Λ5 ) + j2 (Λ2 + Λ4 ) + j3 Λ3 + j4 Λ6 , 1 q

s X

q k V (kΛ1 + (s − k)Λ4 ),

k=0

Ws(5)

=V (sΛ5 ),

REMARKS ON FERMIONIC FORMULA

Ws(6) =

s X

37

q s−k V (kΛ6 ).

k=0 (3)

The conjecture for Ws

has been checked for 1 ≤ s ≤ 7.

Xn = E7 : (1)

W1

(2)

W1

(3)

W1

(3)

W2

(4)

W1

=q V (0) + V (Λ1 ),  =q 3 V (0) + q + q 2 V (Λ1 ) + V (Λ2 ) + q V (Λ5 ),   = q 4 + q 6 V (0) + q 2 V (2Λ1 ) + q 3 V (2Λ6 ) + 2 q 3 + q 4 + q 5 V (Λ1 ) + q V (Λ1 + Λ5 )    + q + q 2 + q 3 V (Λ2 ) + V (Λ3 ) + 2 q 2 + q 3 + q 4 V (Λ5 ) + q + q 2 V (Λ6 + Λ7 ),   = q 8 + q 10 + q 12 V (0) + 2 q 7 + q 8 + 3 q 9 + q 10 + q 11 V (Λ1 )   + 4 q 6 + 2 q 7 + 4 q 8 + q 9 + q 10 V (2 Λ1 ) + 2 q 5 + q 6 + q 7 V (3 Λ1 )  + q 4 V (4 Λ1 ) + q 5 + 2 q 6 + 5 q 7 + 3 q 8 + 2 q 9 V (Λ2 )   + q 2 + q 3 + 3 q 4 + q 5 + q 6 V (2 Λ2 ) + 2 q 4 + 5 q 5 + 5 q 6 + 3 q 7 + q 8 V (Λ1 + Λ2 )   + q 3 + q 4 + q 5 V (2 Λ1 + Λ2 ) + 2 q 4 + 3 q 5 + 7 q 6 + 2 q 7 + q 8 V (Λ3 )  + V (2 Λ3 ) + 3 q 3 + 3 q 4 + 3 q 5 V (Λ1 + Λ3 ) + q 2 V (2 Λ1 + Λ3 )   + q + q 2 + q 3 V (Λ2 + Λ3 ) + q 3 V (2 Λ4 ) + 2 q 6 + q 7 + 4 q 8 + q 9 + q 10 V (Λ5 )   + 3 q 4 + 2 q 5 + 4 q 6 + q 7 + q 8 V (2 Λ5 ) + 5 q 5 + 6 q 6 + 7 q 7 + 2 q 8 + q 9 V (Λ1 + Λ5 )  + 4 q 4 + 2 q 5 + 2 q 6 V (2 Λ1 + Λ5 ) + q 3 V (3 Λ1 + Λ5 )  + 2 q 3 + 5 q 4 + 7 q 5 + 3 q 6 + q 7 V (Λ2 + Λ5 )   + q 2 + 2 q 3 + q 4 V (Λ1 + Λ2 + Λ5 ) + 2 q 2 + 2 q 3 + 2 q 4 V (Λ3 + Λ5 )  + q V (Λ1 + Λ3 + Λ5 ) + 2 q 3 + q 4 + q 5 V (Λ1 + 2 Λ5 ) + q 2 V (2 Λ1 + 2 Λ5 )   + q 7 + q 9 V (2 Λ6 ) + q 6 V (4 Λ6 ) + q 4 + 5 q 5 + 4 q 6 + 2 q 7 V (Λ4 + Λ6 )  + 2 q 3 + 3 q 4 + q 5 V (Λ1 + Λ4 + Λ6 ) + q 2 V (Λ2 + Λ4 + Λ6 )  + 3 q 6 + q 7 + q 8 V (Λ1 + 2 Λ6 ) + q 5 V (2 Λ1 + 2 Λ6 )  + 2 q 4 + 2 q 5 + q 6 V (Λ2 + 2 Λ6 ) + 2 q 3 V (Λ3 + 2 Λ6 )  + 2 q 5 + q 6 + q 7 V (Λ5 + 2 Λ6 ) + q 4 V (Λ1 + Λ5 + 2 Λ6 )   + 2 q 5 + q 6 + q 7 V (2 Λ7 ) + q 3 + 4 q 4 + 3 q 5 + q 6 V (Λ4 + Λ7 )   + q 2 + q 3 V (Λ1 + Λ4 + Λ7 ) + q 5 + 3 q 6 + 4 q 7 + 2 q 8 V (Λ6 + Λ7 )   + 3 q 4 + 6 q 5 + 4 q 6 + q 7 V (Λ1 + Λ6 + Λ7 ) + q 3 + q 4 V (2 Λ1 + Λ6 + Λ7 )   + q 2 + 3 q 3 + 3 q 4 + q 5 V (Λ2 + Λ6 + Λ7 ) + q + q 2 V (Λ3 + Λ6 + Λ7 )  + 2 q 3 + 4 q 4 + 3 q 5 + q 6 V (Λ5 + Λ6 + Λ7 )   + q 2 + q 3 V (Λ1 + Λ5 + Λ6 + Λ7 ) + q 4 + q 5 V (3 Λ6 + Λ7 )  + q 4 V (Λ1 + 2 Λ7 ) + q 3 V (Λ5 + 2 Λ7 ) + q 2 + q 3 + q 4 V (2 Λ6 + 2 Λ7 ),   =q V (Λ1 + Λ6 ) + V (Λ4 ) + q 2 + q 3 V (Λ6 ) + q + q 2 V (Λ7 ),

38

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA (5)

=q 2 V (0) + q V (Λ1 ) + V (Λ5 ),

W1

(6)

=V (Λ6 ),

(7) W1

=q V (Λ6 ) + V (Λ7 ).

W1

(r)

In addition we have a conjecture for Ws (r = 1, 2, 4, 5, 6, 7): Ws(1) =

s X

k=0

Ws(2)

=

q s−k V (kΛ1 ), X

min (1 + j2 , 1 + s − j1 − j2 − 2j3 − j4 ) q 3s−2j1 −3j2 −4j3 −2j4

j1 +j2 +2j3 +j4 ≤s j1 ,j2 ,j3 ,j4 ∈Z≥0

Ws(4)

   j1 + 1 × V j1 Λ1 + j2 Λ2 + j3 Λ3 + j4 Λ5 , 1 q X (jΛ3 + 1)q c({j}) = {j}

      j +1 jΛ7 + 1 jΛ1 +Λ5 + 1 × Λ6 V 1 1 1 q q q

X

λ∈T1 ∪T2 ∪T3

!

jλ λ ,

P where the sum {j} is taken over {jλ ∈ Z≥0 | λ ∈ T1 ∪T2 ∪T3 } under the conditions P P P (1) λ∈T1 jλ +2 λ∈T2 jλ +3 λ∈T3 jλ = s , (2) jΛ5 jΛ1 +Λ6 = 0 , (3) jΛ2 +Λ5 jΛ1 +Λ4 = 0 . Here T1 = { Λ1 + Λ6 , Λ4 , Λ6 , Λ7 }, T2 = { Λ2 , Λ3 , Λ5 , Λ1 + Λ5 , Λ2 + Λ5 }, T3 = { Λ1 + Λ4 , Λ2 + Λ4 } and c({j}) = 3s − 2jΛ1 +Λ6 − 3jΛ4 − jΛ6 − 2jΛ7 − 2jΛ2 − 3jΛ3 − jΛ5 − 3jΛ1 +Λ5 − 4jΛ2 +Λ5 − 3jΛ1 +Λ4 − 4jΛ2 +Λ4 . X Ws(5) = q 2s−2k−j V (jΛ1 + kΛ5 ), j+k≤s j,k∈Z≥0

Ws(6) =V (sΛ6 ), Ws(7) =

s X

k=0

 q k V kΛ6 + (s − k)Λ7 . (2)

The conjecture for Ws

(4)

(Ws ) has been checked for 1 ≤ s ≤ 7 (1 ≤ s ≤ 8).

Xn = E8 : (1)

W1

(2)

W1

(3)

W1

(4)

W1

=q V (0) + V (Λ1 ),  =q 3 V (0) + q + q 2 V (Λ1 ) + V (Λ2 ) + q V (Λ7 ),   = q 4 + q 6 V (0) + q 2 V (2Λ1 ) + 2 q 3 + q 4 + q 5 V (Λ1 ) + q V (Λ1 + Λ7 )    + q + q 2 + q 3 V (Λ2 ) + V (Λ3 ) + 2 q 2 + q 3 + q 4 V (Λ7 ) + q + q 2 V (Λ8 ),   = q 6 + q 7 + q 8 + q 10 V (0) + q 3 + 2 q 4 + q 5 + q 6 V (2Λ1 ) + q 2 V (2Λ7 )   + q 4 + 3 q 5 + 3 q 6 + 2 q 7 + q 8 + q 9 V (Λ1 ) + q 2 + q 3 V (Λ1 + Λ2 )   + 2 q 2 + 3 q 3 + 2 q 4 + q 5 V (Λ1 + Λ7 ) + q + q 2 V (Λ1 + Λ8 )

REMARKS ON FERMIONIC FORMULA

(5)

W1

(6)

W1

(7)

W1

(8)

W1

 + 3 q 3 + 3 q 4 + 3 q 5 + q 6 + q 7 V (Λ2 ) + q V (Λ2 + Λ7 )   + q + 2 q 2 + q 3 + q 4 V (Λ3 ) + V (Λ4 ) + q + q 2 + q 3 V (Λ6 )   + q 3 + 4 q 4 + 3 q 5 + 2 q 6 + q 7 + q 8 V (Λ7 ) + 2 q 2 + 3 q 3 + 2 q 4 + q 5 + q 6 V (Λ8 ),  = q 7 + 3 q 9 + q 10 + 2 q 11 + q 12 + q 13 + q 15 V (0)  + 5 q 5 + 4 q 6 + 6 q 7 + 3 q 8 + 3 q 9 + q 10 + q 11 V (2Λ1 )  + 2 q 3 + q 4 + q 5 V (2Λ1 + Λ7 ) + q 3 V (2Λ2 )   + 3 q 3 + 2 q 4 + 3 q 5 + q 6 + q 7 V (2Λ7 ) + q 4 + q 6 V (3Λ1 )  + 4 q 6 + 5 q 7 + 8 q 8 + 5 q 9 + 5 q 10 + 3 q 11 + 2 q 12 + q 13 + q 14 V (Λ1 )  + q 2 V (Λ1 + 2Λ7 ) + 2 q 3 + 5 q 4 + 5 q 5 + 3 q 6 + 2 q 7 + q 8 V (Λ1 + Λ2 )   + 2 q 2 + q 3 + q 4 V (Λ1 + Λ3 ) + q + q 2 + q 3 V (Λ1 + Λ6 )  + 2 q 3 + 9 q 4 + 10 q 5 + 10 q 6 + 6 q 7 + 4 q 8 + 2 q 9 + q 10 V (Λ1 + Λ7 )  + 2 q 2 + 5 q 3 + 5 q 4 + 3 q 5 + 2 q 6 + q 7 V (Λ1 + Λ8 )  + 3 q 4 + 6 q 5 + 11 q 6 + 8 q 7 + 7 q 8 + 4 q 9 + 3 q 10 + q 11 + q 12 V (Λ2 )   + 3 q 2 + 4 q 3 + 4 q 4 + 2 q 5 + q 6 V (Λ2 + Λ7 ) + q + q 2 V (Λ2 + Λ8 )  + 5 q 3 + 5 q 4 + 7 q 5 + 4 q 6 + 3 q 7 + q 8 + q 9 V (Λ3 ) + q V (Λ3 + Λ7 )  + q + 2 q 2 + 2 q 3 + q 4 + q 5 V (Λ4 ) + V (Λ5 )  + 2 q 2 + 4 q 3 + 6 q 4 + 4 q 5 + 3 q 6 + q 7 + q 8 V (Λ6 )  + 5 q 5 + 6 q 6 + 9 q 7 + 6 q 8 + 6 q 9 + 3 q 10 + 2 q 11 + q 12 + q 13 V (Λ7 )  + q + 2 q 2 + 2 q 3 + q 4 V (Λ7 + Λ8 )  + q 3 + 5 q 4 + 8 q 5 + 7 q 6 + 6 q 7 + 4 q 8 + 2 q 9 + q 10 + q 11 V (Λ8 ),    = q 5 + q 7 V (0) + q 3 V (2Λ1 ) + q 3 + 2 q 4 + q 5 + q 6 V (Λ1 ) + q + q 2 V (Λ1 + Λ7 )   + 2 q 2 + q 3 + q 4 V (Λ2 ) + q V (Λ3 ) + V (Λ6 ) + q 2 + 2 q 3 + q 4 + q 5 V (Λ7 )  + q + q 2 + q 3 V (Λ8 ), =q 2 V (0) + q V (Λ1 ) + V (Λ7 ),   =q 4 V (0) + q 2 + q 3 V (Λ1 ) + q V (Λ2 ) + q + q 2 V (Λ7 ) + V (Λ8 ). (r)

In addition we have a conjecture for Ws (r = 1, 2, 7): Ws(1) =

s X

k=0

Ws(2)

=

q s−k V (kΛ1 ), X

j1 +j2 +2j3 +j4 ≤s j1 ,j2 ,j3 ,j4 ∈Z≥0

Ws(7)

39

min (1 + j2 , 1 + s − j1 − j2 − 2j3 − j4 ) q 3s−2j1 −3j2 −4j3 −2j4

   j1 + 1 × V j1 Λ1 + j2 Λ2 + j3 Λ3 + j4 Λ7 , 1 q X 2s−2k−j = q V (jΛ1 + kΛ7 ). j+k≤s j,k∈Z≥0

40

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA (2)

The conjecture for Ws

has been checked for 1 ≤ s ≤ 7.

Xn = F4 : (1)

W1

(2)

W1

(3)

W1

(3) W2

(4)

W1

=q V (0) + V (Λ1 ),  =q 3 V (0) + q + q 2 V (Λ1 ) + V (Λ2 ) + q V (2Λ4 ),

=V (Λ3 ) + q V (Λ4 ),   = q 4 + q 6 V (0) + 2 q 3 + q 4 + q 5 V (Λ1 ) + q 2 V (2 Λ1 )   + q + q 2 + q 3 V (Λ2 ) + q 3 V (Λ3 ) + V (2 Λ3 ) + 2 q 2 + q 3 + q 4 V (2 Λ4 )  + q + q 2 V (Λ3 + Λ4 ) + q V (Λ1 + 2 Λ4 ), =V (Λ4 ).

(r)

In addition we have a conjecture for Ws (r = 1, 2, 4): Ws(1) =

s X

k=0

Ws(2)

=

q s−k V (kΛ1 ), X

min (1 + j2 , 1 + s − j1 − j2 − 2j3 − j4 ) q 3s−2j1 −3j2 −4j3 −2j4

j1 +j2 +2j3 +j4 ≤s j1 ,j2 ,j3 ,j4 ∈Z≥0

   j1 + 1 × V j1 Λ1 + j2 Λ2 + 2j3 Λ3 + 2j4 Λ4 , 1 q

Ws(4) =

[s/2] k XX k=0 j=0

 q 2k−j V jΛ1 + (s − 2k)Λ4 . (2)

The conjecture for Ws

has been checked for 1 ≤ s ≤ 5.

Xn = G2 : (1)

=q V (0) + V (Λ1 ),

(2)

=V (Λ2 ).

W1

W1

(r)

In addition we have a conjecture for Ws : Ws(1) =

s X

q s−k V (kΛ1 )

k=0

Ws(2) =

        s + k − 2x x − 2k x − 2k , + +1 min 3 3 3 k=0 x=2k   k+1 × q x−k V (kΛ1 + (s − x − k)Λ2 ). 1 q

[s/3] s−k X X

The last conjecture has been checked for 1 ≤ s ≤ 16.

REMARKS ON FERMIONIC FORMULA

41

Appendix B. Example of one dimensional sums We present an example of the one dimensional configuration sums (1dsums) and (1) (1) the fermionic forms for C2 . First we consider the 1dsums. Let us take a Uq′ (C2 )crystal B := B 1,2 ⊗(B 2,1 )⊗3 ⊗(B 1,1 )⊗2 . As Uq (C2 )-crystals, B 1,2 ≃ B(2Λ1 )⊕B(0), B 2,1 ≃ B(Λ2 ) and B 1,1 ≃ B(Λ1 ). We use the parametrization of the Uq (C2 )-crystal B(sΛr ) given in [N]. We denote the element in B(0) by φ, and the elements in B(Λ2 ) by a, b, c, d, e for short, where

a=

1 1 2 , c= , , b= 2 2 2

d=

2 , e= 2 . 1 1

The tableau in this appendix with the content 1x1 2x2 ¯2x¯2 ¯1x¯1 corresponds to (x1 , x2 , x ¯2 , x ¯1 ) in Example 2.5 or Example 2.6. In Table 3, all the classically restricted paths in B with zero classical weight, namely the λ = 0 paths, are listed. The value of ε0 means, for instance, (˜ e0 )2 p 6= 0 and (˜ e0 )3 p = 0 if ε0 (p) = 2. Here it stands for the restriction level. In view of Conjecture 3.2 we have to find suitable b0 from B0 = B 1,2 ⊗ B 2,1 ⊗ B 1,1 . In this particular example, we further expect that we can take B0 = B 1,2 and b0 = φ (note that ϕ(φ) = Λ0 ). We also have b♮0 = φ, b♮1 = 11, b♮2 = b♮3 = b♮4 = a, b♮5 = b♮6 = 1. From these data, one can write down the expressions of the 1dsums Xl (B, λ = 0) over the level l = 1, 2 restricted paths, and X(B, λ = 0) over the classically restricted paths,

X1 (B, 0, q −1 ) = q 15 , X2 (B, 0, q −1 ) = q 8 + 2q 9 + 2q 10 + 3q 11 + 2q 12 + q 13 + q 15 , X(B, 0, q −1 ) = q 6 + 2q 7 + 2q 8 + 3q 9 + 2q 10 + 3q 11 + 2q 12 + q 13 + q 15 . (B.1)

P We illustrate the calculation of the “relative” energy E(p) = 0≤i
E(p) =

X

1≤i
(i+1)

H(bi ⊗ bj

) + H(φ ⊗ b1 ) − H(φ ⊗ 11).

42

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

We show the computation of the relative energy of the path 11 ⊗ a ⊗ d ⊗ e ⊗ 2 ⊗ 1. The necessary data are shown at the end of this appendix. It proceeds as 0

11 ⊗ (a) 7→ (a) ⊗ 11, 1

0

11 ⊗ a ⊗ (d) 7→ 11 ⊗ (a) ⊗ d 7→ (a) ⊗ 11 ⊗ d, 1

11 ⊗ a ⊗ d ⊗ (e) 7→ 11 ⊗ a ⊗ (d) ⊗ e 1

0

7→ 11 ⊗ (a) ⊗ d ⊗ e 7→ (a) ⊗ 11 ⊗ d ⊗ e, 0

11 ⊗ a ⊗ d ⊗ e ⊗ (2) 7→ 11 ⊗ a ⊗ d ⊗ (2) ⊗ e 0

7→ 11 ⊗ a ⊗ (1) ⊗ c ⊗ e 1

7→ 11 ⊗ (1) ⊗ d ⊗ c ⊗ e 0

7→ (1) ⊗ 11 ⊗ d ⊗ c ⊗ e, 1

11 ⊗ a ⊗ d ⊗ e ⊗ 2 ⊗ (1) 7→ 11 ⊗ a ⊗ d ⊗ e ⊗ (2) ⊗ 1 0

0

1

0

7→ · · · 7→ · · · 7→ · · · 7→ (1) ⊗ 11 ⊗ d ⊗ c ⊗ e ⊗ 1. (i+1)

The parenthesized elements denote bj (i+1) bi ⊗ bj

(i) bj

⊗ b′i by the 7→ (i+1) ) are −H(bi ⊗ bj

. They are to be moved to the left as

isomorphism of the crystals. The values of the energy

shown above the arrows, which amount to 6. We also function show the computation of the relative energy of the path φ ⊗ a ⊗ e ⊗ a ⊗ 2 ⊗ 1; 1

φ ⊗ (a) 7→ (a) ⊗ 22, 2

1

φ ⊗ a ⊗ (e) 7→ φ ⊗ (a) ⊗ e 7→ (a) ⊗ 22 ⊗ e, 0

φ ⊗ a ⊗ e ⊗ (a) 7→ φ ⊗ a ⊗ (e) ⊗ a 2

1

7→ φ ⊗ (a) ⊗ e ⊗ a 7→ (a) ⊗ 22 ⊗ e ⊗ a, 1

φ ⊗ a ⊗ e ⊗ a ⊗ (2) 7→ φ ⊗ a ⊗ e ⊗ (1) ⊗ c 0

7→ φ ⊗ a ⊗ (2) ⊗ c ⊗ c 1

7→ φ ⊗ (1) ⊗ c ⊗ c ⊗ c 1

7→ (1) ⊗ 11 ⊗ c ⊗ c ⊗ c, 1

φ ⊗ a ⊗ e ⊗ a ⊗ 2 ⊗ (1) 7→ φ ⊗ a ⊗ e ⊗ a ⊗ (2) ⊗ 1 1

0

1

1

7→ · · · 7→ · · · 7→ · · · 7→ (1) ⊗ 11 ⊗ c ⊗ c ⊗ c ⊗ 1. Here the energy amounts to 14. One should add to it another 1, since −(H(φ ⊗ φ) − H(φ ⊗ 11)) = 1 (see Table 7). (1) (2) (1) Now we consider the fermionic forms. Let W := W2 ⊗ (W1 )⊗3 ⊗ (W1 )⊗2 . Let us compute the level l = 1, 2 restricted versions of the fermionic form Ml (W, q −1 ) and M (W, λ = 0, q −1 ). First recall that {(αa |αb )} =



1 −1 −1 2



,

C

−1

=



1 1 2

1 1



,

t1 = 2,

t2 = 1.

REMARKS ON FERMIONIC FORMULA

43

Table 3. λ = 0 paths on B := B 1,2 ⊗ (B 2,1 )⊗3 ⊗ (B 1,1 )⊗2 path −E 11 ⊗ a ⊗ c ⊗ e ⊗ 1 ⊗ 1 7 11 ⊗ a ⊗ e ⊗ c ⊗ 1 ⊗ 1 8 11 ⊗ a ⊗ d ⊗ e ⊗ 2 ⊗ 1 6 11 ⊗ a ⊗ e ⊗ d ⊗ 2 ⊗ 1 7 11 ⊗ c ⊗ a ⊗ e ⊗ 1 ⊗ 1 9 11 ⊗ c ⊗ c ⊗ c ⊗ 1 ⊗ 1 12 11 ⊗ c ⊗ c ⊗ d ⊗ 2 ⊗ 1 11 11 ⊗ c ⊗ d ⊗ b ⊗ 1 ⊗ 1 9 11 ⊗ c ⊗ d ⊗ e ⊗ 1 ⊗ 1 11 11 ⊗ d ⊗ a ⊗ e ⊗ 2 ⊗ 1 10 11 ⊗ d ⊗ b ⊗ c ⊗ 1 ⊗ 1 8 11 ⊗ d ⊗ b ⊗ d ⊗ 2 ⊗ 1 9 11 ⊗ d ⊗ e ⊗ a ⊗ 2 ⊗ 1 10 φ ⊗ a ⊗ a ⊗ e ⊗ 2 ⊗ 1 11 φ⊗a⊗b⊗c⊗1⊗1 13 φ⊗a⊗b⊗d⊗2⊗1 12 φ ⊗ a ⊗ e ⊗ a ⊗ 2 ⊗ 1 15

ε0

3

2

1

From these data the constraint (4.12) on the summation variables m’s for W reads 2l X i=1

(1)

imi

= 7,

l X

(2)

imi

= 5.

i=1

Among the 105 possible configurations of the m’s, only those 6 listed in Table 4 have non-zero contributions to the fermionic forms. Consequently, M1 (W, q −1 ) = q 15 , M2 (W, q −1 ) = q 8 + 2q 9 + 2q 10 + 3q 11 + 2q 12 + q 13 + q 15 , M (W, λ = 0, q −1 ) = q 6 + 2q 7 + 2q 8 + 3q 9 + 2q 10 + 3q 11 + 2q 12 + q 13 + q 15 . They are exactly the same as (B.1) calculated as the 1dsums. Table 4. Relevant configurations to the fermionic forms m(1) , m(2) (1, 0, 0, 0, 0, 1), (0, 1, 1) (1, 0, 0, 0, 0, 1), (2, 0, 1) (0, 0, 1, 1), (1, 2) (1, 1, 0, 1), (1, 2) (1, 1, 0, 1), (3, 1) (1, 3), (5)

p(1) , p(2) contribution (1, 2, 2, 2, 1, 0), (2, 0, 0) q6 + q7 7 (2, 4, 3, 2, 1, 0), (0, 0, 0) q + q8 + q9 (2, 2, 0, 0), (1, 0) q8 + q9 9 (0, 0, 0, 0), (2, 0) q + q 10 + q 11 10 (1, 2, 1, 0), (0, 0) q + 2q 11 + 2q 12 + q 13 (0, 0), (0) q 15

Below we list the energy function and the isomorphism (combinatorial R(1) matrix) of the Uq′ (C2 ) crystals used in calculating the 1dsums. The data on B 1,1 ⊗ B 1,1 , B 2,1 ⊗ B 2,1 and B 2,1 ⊗ B 1,1 are quoted from [Y]. Here we added

44

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

the cases including B 1,2 . For B 1,1 ⊗ B 1,1 , B 2,1 ⊗ B 2,1 and B 1,2 ⊗ B 1,2 cases, the isomorphism is the trivial (identity) map. The blanks in the tables signify that the energy is 0. Table 5. −H(b1 ⊗ b2 ) on B 1,1 ⊗ B 1,1 b1

b2 2 2 1 1 1

1

1 2 2 1

1 1 1 1

Table 6. −H(b1 ⊗ b2 ) on B 2,1 ⊗ B 2,1 b1

a

a b c d e

b 1

b2 c 1 1 1

d e 1 2 1 1 1 1 1

Table 7. −H(b1 ⊗ b2 ) on B 1,2 ⊗ B 1,2 b1 11 12 12 11 22 22 21 22 21 11 φ

b2 11 12 12 11 22 22 21 22 21 11 1 1 2 2 2 2 2 2 2 1 1 2 1 2 2 2 2 2 1 1 2 1 1 2 1 2 2 1 1 2 1 1 2 1 2 2 1 1 2 1 2 2 2 1 1 2 1 2 2 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1

1

1

1

1

1

1

1

1

1

φ 1 1 1 1 1 1 1 1 1 1 2

REMARKS ON FERMIONIC FORMULA

45

Table 8. Isomorphism and −H(b1 ⊗ b2 ) on B 2,1 ⊗ B 1,1 b1 a b c d e

b2 1 1⊗a 1⊗b 2⊗b 2⊗c 2⊗c

2 2⊗a 2⊗a 1⊗a 2⊗d 2⊗d

b2 2 1⊗c 2⊗b 1⊗b 1⊗c 2⊗e

1 1⊗d 1⊗e 2⊗e 1⊗d 1⊗e

1 2

2 1 1 1 1 1

Table 9. Isomorphism and −H(b1 ⊗ b2 ) on B 1,2 ⊗ B 1,1 b1 11 12 12 11 22 22 21 22 21 11 φ

b2 1 1 ⊗ 11 2 ⊗ 11 2 ⊗ 11 1 ⊗ 11 2 ⊗ 12 2 ⊗ 12 1 ⊗ 12 2 ⊗ 12 1 ⊗ 12 1 ⊗ 11 1 ⊗ 11

2 1 ⊗ 12 1 ⊗ 22 1 ⊗ 22 2 ⊗ 22 2 ⊗ 22 2 ⊗ 22 1 ⊗ 22 2 ⊗ 22 1 ⊗ 22 1 ⊗ 21 1 ⊗ 21

b2 2 1 ⊗ 12 2 ⊗ 12 1 ⊗ 22 2 ⊗ 22 2 ⊗ 11 2 ⊗ 11 2 ⊗ 21 2 ⊗ 22 1 ⊗ 22 1 ⊗ 21 1 ⊗ 21

1 1⊗φ 2⊗φ 2⊗φ 1⊗φ 2 ⊗ 21 2 ⊗ 21 2 ⊗ 11 2 ⊗ 21 2 ⊗ 11 1 ⊗ 11 1 ⊗ 11

1 2 1 1 1 1

2 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

1 1

1 1

Table 10. Isomorphism and −H(b1 ⊗ b2 ) on B 1,2 ⊗ B 2,1 b1 11 12 12 11 22 22 21 22 21 11 φ

a b a ⊗ 11 b ⊗ 11 a ⊗ 12 c ⊗ 11 b ⊗ 12 b ⊗ 12 c ⊗ 12 c ⊗ 12 a ⊗ 22 d ⊗ 11 b ⊗ 22 e ⊗ 11 c ⊗ 22 d ⊗ 12 b ⊗ 22 b ⊗ 22 c ⊗ 22 c ⊗ 22 d ⊗ 22 d ⊗ 22 a ⊗ 22 a ⊗ 22

b2 c d e a ⊗ 12 a ⊗ φ b ⊗ φ a ⊗ 11 a ⊗ 21 c ⊗ φ b ⊗ 11 b ⊗ 21 b ⊗ 21 c ⊗ 11 c ⊗ 21 c ⊗ 21 d ⊗ 12 d ⊗ 22 d ⊗ φ e ⊗ 12 e ⊗ 22 e ⊗ φ d ⊗ 11 d ⊗ 21 c ⊗ 11 e ⊗ 12 e ⊗ 22 e ⊗ 22 e ⊗ 11 e ⊗ 21 e ⊗ 21 d ⊗ 21 d ⊗ 11 e ⊗ 11 a ⊗ 21 a ⊗ 11 b ⊗ 11

a

b

1

1

b2 c 1 1 1 1

d 1 1 1 1

e 1 1 1 1 1 1 1

1

1

1

46

G. HATAYAMA, A. KUNIBA, M. OKADO, T. TAKAGI, AND Y. YAMADA

Appendix C. Explicit form of Q-system Let us write down the Q-system for non-simply laced Xn . Below we assume (a) that Qj = 1 whenever a 6∈ {1, . . . , n}. Xn = Bn : (a)2

(a−1)

(a)

(a)

Qj

= Qj−1 Qj+1 + Qj

(n−1)2 Qj

(n−1) (n−1) Qj−1 Qj+1

(n)

Q2j

(n)

2

=

+

(n−1)2

= Q2j−1 Q2j+1 + Qj

2

(n)

(n)

(1 ≤ a ≤ n − 2),

(n−2) (n) Q2j , Qj

(n)

(n)

(a+1)

Qj

,

(n−1)

(n−1)

Qj+1 .

(a−1)

(a+1)

Q2j+1 = Q2j Q2j+2 + Qj Xn = Cn : (a)2

(a)

(a)

Qj

= Qj−1 Qj+1 + Qj

(n−1)2 Q2j

(n−1) (n−1) Q2j−1 Q2j+1

(n−1)

Q2j+1

2

(n)2

Qj

=

(1 ≤ a ≤ n − 2),

Qj

(n−2) (n)2 Q2j Qj ,

+

(n)

(n)

(n−2)

(n−1)

(n−1)

Q2j+2 + Q2j+1 Qj Qj+1 ,

(n)

(n)

= Q2j

(n−1)

= Qj−1 Qj+1 + Q2j

.

Xn = F4 : (1)2

(1)

(1)

(2)

(2)

(2)

(1)

= Qj−1 Qj+1 + Qj ,

Qj

(2)2

(3)

= Qj−1 Qj+1 + Qj Q2j ,

Qj

(3)2

(2)2

(3)

(3)

= Q2j−1 Q2j+1 + Qj

Q2j

(3)2

(3)

(2)

(3)

(4)

Q2j , (4)

(2)

Q2j+1 = Q2j Q2j+2 + Qj Qj+1 Q2j+1 , (4)2

(4)

(3)

(4)

= Qj−1 Qj+1 + Qj .

Qj Xn = G2 :

(1)2

Qj

(2)2

Q3j

(1)

(2)

(2)

(1)3

(2)

= Q3j−1 Q3j+1 + Qj

2

(2)

(2)2

(2)

(2)

(1)

= Qj−1 Qj+1 + Q3j , (2)

(1)

Q3j+1 = Q3j Q3j+2 + Qj (2)

2

, (1)

Qj+1 ,

(1)

(1)2

Q3j+2 = Q3j+1 Q3j+3 + Qj Qj+1 . References [ABF] G. E. Andrews, R. J. Baxter and P. J. Forrester, Eight vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Stat. Phys. 35, (1984) 193-266. [AK] T. Akasaka and M. Kashiwara, Finite-dimensional representations of quantum affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997) 839-867. [ANOT] T. Arakawa, T. Nakanishi, K. Oshima and A. Tsuchiya, Spectral decomposition of path space in solvable lattice model, Comm. Math. Phys. 181 (1996) 159-182. [B] R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London (1982).

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(1)

[Ya] S. Yamane, Perfect crystals for Uq (G2 ), Thesis, q-alg/9712012. Institute of Physics, University of Tokyo, Tokyo 153-8902, Japan E-mail address: [email protected] Institute of Physics, University of Tokyo, Tokyo 153-8902, Japan E-mail address: [email protected] Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan E-mail address: [email protected] Department of Mathematics and Physics, National Defense Academy, Kanagawa 239-8686, Japan E-mail address: [email protected] Department of Mathematics, Faculty of Science, Kobe University, Hyogo 657-8501, Japan E-mail address: [email protected]