Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 048, 28 pages Combinatorial Formulae for Nested Bethe Vectors?
Vitaly TARASOV †‡ and Alexander VARCHENKO §
† Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA E-mail: [email protected] ‡ St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia E-mail: [email protected] § Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA E-mail: [email protected] Received March 21, 2013, in final form June 27, 2013; Published online July 19, 2013 http://dx.doi.org/10.3842/SIGMA.2013.048
Abstract. We give combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for tensor products of irreducible evaluation modules over the Yangian
Y (glN ) and the quantum affine algebra Uq(glgN ). Key words: weight functions; nested Bethe vectors; algebraic Bethe ansatz
2010 Mathematics Subject Classification: 82B23; 17B80; 17B37; 81R50
Introduction
In this paper we give combinatorial formulae for vector-valued weight functions for tensor products of irreducible evaluation modules over the Yangian Y (glN ) and the quantum affine algebra Uq(glgN ). Those functions are also known as (off-shell) nested Bethe vectors. They play an important role in the theory of quantum integrable models and representation theory of Lie algebras and quantum groups. The nested algebraic Bethe ansatz was developed as a tool to find eigenvectors and eigenvalues of transfer matrices of lattice integrable models associated with higher rank Lie algebras, see [9]. Similar to the regular Bethe ansatz, which is used in the rank one case, eigenvectors are obtained as values of a certain rational function (nested Bethe vector) on solutions of some system of algebraic equations (Bethe ansatz equations). Later, the nested Bethe vectors (also called vector- valued weight functions) were used to construct Jackson integral representations for solutions of the quantized (difference) Knizhnik–Zamolodchikov (qKZ) equations [22]. The results of [9] has been extended to higher transfer matrices in [12]. In the rank one case combinatorial formulae for vector-valued weight function are important in various areas from computation of correlation functions in integrable models, see [8], to eva- luation of some multidimensional generalizations of the Vandermonde determinant [19]. In the glN case considered in this paper, combinatorial formulae, in particular, clarify analytic prop- erties of the vector-valued weight function, which is important for constructing hypergeometric solutions of the qKZ equations associated with glN . Combinatorial formulae for the vector-valued weight functions associated with the differential Knizhnik–Zamolodchikov equations were developed in [10, 17, 18, 15,3].
?This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html 2 V. Tarasov and A. Varchenko
The results of this paper were obtained while the authors were visiting the Max-Planck- Institut f¨urMathematik in Bonn in 1998. The results of the paper were used in [11,7, 20]. The paper has appeared in the arXiv in 2007, but still looks topical. It is published with no intention to give any review of the subject or reflect the state of the art. Let us only mention a few papers making progress in particularly close problems [5,6,4, 14,1,2] and those exploring recently the results of the paper [16, 21, 13]. The paper is organized as follows. In Sections2–5 we consider in detail the Yangian case. In the traditional terminology this case is called rational. In Section6 we formulate the results for the quantum affine algebra case, also called trigonometric. The proofs in that case are very similar to the Yangian case.
1 Basic notation
We will be using the standard superscript notation for embeddings of tensor factors into tensor products. If A1,..., Ak are unital associative algebras, and a ∈ Ai, then
(i) ⊗(i−1) ⊗(k−i) a = 1 ⊗ a ⊗ 1 ∈ A1 ⊗ · · · ⊗ Ak.
(ij) (i) (j) If a ∈ Ai and b ∈ Aj, then (a ⊗ b) = a b , etc.
Example. Let k = 2. Let A1, A2 be two copies of the same algebra A. Then for any a, b ∈ A we have a(1) = a ⊗ 1, b(2) = 1 ⊗ b,(a ⊗ b)(12) = a ⊗ b and (a ⊗ b)(21) = b ⊗ a.
N Fix a positive integer N. All over the paper we identify elements of End C with N × N N matrices using the standard basis of C . We will use the rational and trigonometric R-matrices. The rational R-matrix is
N X R(u) = u + Eab ⊗ Eba, (1.1) a,b=1
N where Eab ∈ End C is a matrix with the only nonzero entry equal to 1 at the intersection of the a-th row and b-th column. The R-matrix satisfies the inversion relation
R(u)R(21)(−u) = 1 − u2 and the Yang–Baxter equation
R(12)(u − v)R(13)(u)R(23)(v) = R(23)(v)R(13)(u)R(12)(u − v). (1.2)
Fix a complex number q not equal to ±1. The trigonometric R-matrix
N −1 X X Rq(u) = uq − q Eaa ⊗ Eaa + (u − 1) (Eaa ⊗ Ebb + Ebb ⊗ Eaa) a=1 16a (21) −1 −1 −1 −1 Rq(u)Rq u = uq − q u q − q and the Yang–Baxter equation (12) (13) (23) (23) (13) (12) Rq (u/v)Rq (u)Rq (v) = Rq (v)Rq (u)Rq (u/v). Combinatorial Formulae for Nested Bethe Vectors 3 Let eab, a, b = 1,...,N, be the standard generators of the Lie algebra glN : [eab, ecd] = δbcead − δadecb. LN ∗ a Let h = a=1 Ceaa be the Cartan subalgebra. For any Λ ∈ h we set Λ = hΛ, eaai, and ∗ N 1 N identify h with C by taking Λ to Λ ,..., Λ . We use the Gauss decomposition glN = L L h ⊕ n+ ⊕ n− where n+ = a 2 Rational weight functions {s} The Yangian Y (glN ) is a unital associative algebra with generators Tab , a, b = 1,...,N, and s = 1, 2,.... Organize them into generating series: ∞ X {s} −s Tab(u) = δab + Tab u , a, b = 1,...,N. (2.1) s=1 The defining relations in Y (glN ) have the form (u − v) Tab(u),Tcd(v) = Tcb(v)Tad(u) − Tcb(u)Tad(v), (2.2) for all a, b, c, d = 1,...,N. N P Combine series (2.1) together into a series T (u) = Eab ⊗ Tab(u) with coefficients in a,b=1 N End C ⊗ Y (glN ). Relations (2.2) amount to the following equality for series with coefficients N N in End C ⊗ End C ⊗ Y (glN ): R(12)(u − v)T (13)(u)T (23)(v) = T (23)(v)T (13)(u)R(12)(u − v). (2.3) The Yangian Y (glN ) is a Hopf algebra. In terms of generating series (2.1), the coproduct ∆ : Y (glN ) → Y (glN ) ⊗ Y (glN ) reads as follows: N X ∆ Tab(u) = Tcb(u) ⊗ Tac(u), a, b = 1,...,N. (2.4) c=1 There is a one-parameter family of automorphisms ρx : Y (glN ) → Y (glN ) defined in terms of −1 the series T (u) by the rule ρx T (u) = T (u − x); in the right side, (u − x) has to be expanded as a power series in u−1. The Yangian Y (glN ) contains the universal enveloping algebra U(glN ) as a Hopf subalgebra. {1} The embedding is given by eab 7→ Tba for all a, b = 1,...,N. We identify U(glN ) with its image in Y (glN ) under this embedding. It is clear from relations (2.2) that for any a, b = 1,...,N, Eab ⊗ 1 + 1 ⊗ eab,T (u) = 0. {1} The evaluation homomorphism : Y (glN ) → U(glN ) is given by the rule : Tab 7→ eba for {s} any a, b = 1,...,N, and : Tab 7→ 0 for any s > 1 and all a, b. Both the automorphisms ρx and the homomorphism restricted to the subalgebra U(glN ) are the identity maps. For a glN -module V denote by V (x) the Y (glN )-module induced from V by the homomor- phism ◦ ρx. The module V (x) is called an evaluation module over Y (glN ). A vector v in a Y (glN )-module is called singular with respect to the action of Y (glN ) if Tba(u)v = 0 for all 1 6 a < b 6 N. A singular vector v that is an eigenvector for the action of T11(u),...,TNN (u) is called a weight singular vector; the respective eigenvalues are denoted by hT11(u)vi,..., hTNN (u)vi. 4 V. Tarasov and A. Varchenko 1 N Example. Let V be a glN -module and let v ∈ V be a singular vector of weight Λ ,..., Λ . Then v is a weight singular vector with respect to the action of Y (glN ) in the evaluation modu- a −1 le V (x) and hTaa(u)vi = 1 + Λ (u − x) , a = 1,...,N. If v1, v2 are weight singular vectors with respect to the action of Y (glN ) in Y (glN )-modules V1, V2, then the vector v1 ⊗ v2 is a weight singular vector with respect to the action of Y (glN ) in the tensor product V1 ⊗ V2, and hTaa(u)v1 ⊗ v2i = hTaa(u)v1ihTaa(u)v2i for all a = 1,...,N. We will use two embeddings of the algebra Y (glN−1) into Y (glN ), called φ and ψ: hN−1i hNi hN−1i hNi φ Tab (u) = Tab (u), ψ Tab (u) = Ta+1,b+1(u), (2.5) hN−1i hNi a, b = 1,...,N − 1. Here Tab (u) and Tab (u) are series (2.1) for the algebras Y (glN−1) and Y (glN ), respectively. Let ξ = (ξ1, . . . , ξN−1) be a collection of nonnegative integers. Set ξ N Here tr : End C → C is the standard trace map, the pairs in the product are ordered lexi- cographically, (a, i) < (b, j) if a < b, or a = b and i < j; the product is taken over all two- element subsets of the set (c, k)|c = 1,...,N − 1, k = 1, . . . , ξc ; in the product, the factor (ξ Example. Let N = 4 and ξ = (1, 1, 1). Then 1 2 3 ⊗3 (14) 1 (24) 2 (34) 3 Bbξ t1, t1, t1 = tr ⊗ id T t1 T t1 T t1 (32) 3 2 (31) 3 1 (21) 2 1 × R t1 − t1 R t1 − t1 R t1 − t1 E21 ⊗ E32 ⊗ E43 ⊗ 1 . 1 N−1 1 N−1 1−1 N−1 −1 Remark. The series Bbξ t1, . . . , tξN−1 belongs to Y (glN ) t1, . . . , tξN−1 t1 ,..., tξN−1 . Remark. Relations (2.3) imply that −→ (1,|ξ|+1) 1 (|ξ|,|ξ|+1) N−1 Y (ξ For instance, (14) 1 (24) 2 (34) 3 (32) 3 2 (31) 3 1 (21) 2 1 T t1 T t1 T t1 R t1 − t1 R t1 − t1 R t1 − t1 (32) 3 2 (31) 3 1 (21) 2 1 (34) 3 (24) 2 (14) 1 = R t1 − t1 R t1 − t1 R t1 − t1 T t1 T t1 T t1 . Combinatorial Formulae for Nested Bethe Vectors 5 Remark. Using the Yang–Baxter equation (1.2) one can rearrange the factors in the product of R-matrices in formulae (2.6), (2.7). For instance, −→ ←− Y (ξ (ξ Example. Let N = 4 and ξ = (2, 1, 1). Then (43) 3 2 (42) 3 1 (41) 3 1 (32) 2 1 (31) 2 1 (21) 1 1 R t1 − t1 R t1 − t2 R t1 − t1 R t1 − t2 R t1 − t1 R t2 − t1 (32) 2 1 (31) 2 1 (21) 1 1 (41) 3 1 (42) 3 1 (43) 3 2 = R t1 − t2 R t1 − t1 R t2 − t1 R t1 − t1 R t1 − t2 R t1 − t1 . 1 N−1 Further on, we will abbreviate, t = t1, . . . , tξN−1 . Set N−1 ξa ξb Y Y 1 Y Y Y 1 Bξ(t) = Bbξ(t) a a b a , (2.8) a tj − ti + 1 tj − ti a=1 16i hNi cf. (2.6). To indicate the dependence on N, if necessary, we will write Bξ (t). 1 h2i 1 1 Example. Let N = 2 and ξ = (ξ ). Then Bξ (t) = T12 t1 ··· T12 tξ1 . Example. Let N = 3 and ξ = (1, 1). Then h3i 1 2 1 1 2 2 1 1 2 1 Bξ (t) = T12 t1 T23 t1 + 2 1 T13 t1 T22 t1 = T23 t1 T12 t1 + 2 1 T13 t1 T22 t1 . t1 − t1 t1 − t1 Example. Let N = 4 and ξ = (1, 1, 1). Then h4i 1 2 3 Bξ (t) = T12 t1 T23 t1 T34 t1 1 1 2 3 1 1 2 3 + 2 1 T13 t1 T22 t1 T34 t1 + 3 2 T12 t1 T24 t1 T33 t1 t1 − t1 t1 − t1 1 1 2 3 1 2 3 + 2 1 3 2 T14 t1 T22 t1 T33 t1 + T13 t1 T24 t1 T32 t1 t1 − t1 t1 − t1 2 1 3 2 t1 − t1 t1 − t1 + 1 1 2 3 + 2 1 3 1 3 2T14 t1 T23 t1 T32 t1 . t1 − t1 t1 − t1 t1 − t1 The direct product of the symmetric groups Sξ1 × · · · × SξN−1 acts on expressions in |ξ| variables, permuting the variables with the same superscript: 1 N−1 1 N−1 1 1 N−1 N−1 σ × · · · × σ : f t1, . . . , t N−1 7→ f t 1 , . . . , t 1 ; ... ; t N−1 , . . . , t N−1 , (2.9) ξ σ1 σ 1 σ σ ξ 1 ξN−1 a where σ ∈ Sξa , a = 1,...,N − 1. 6 V. Tarasov and A. Varchenko Lemma 2.1 ([22, Theorem 3.3.4]). The expression Bξ(t) is invariant under the action of the group Sξ1 × · · · × SξN−1 . P ˇ Proof. Let P = a,b Eab ⊗ Eba be the flip map, and R(u) = PR(u). For any a = 1,...,N − 1, we have Rˇ(u)Ea+1,a ⊗ Ea+1,a = (u + 1)Ea+1,a ⊗ Ea+1,a = Ea+1,a ⊗ Ea+1,aRˇ(u). (2.10) Set −→ (1,|ξ|+1) 1 (|ξ|,|ξ|+1) N−1 Y (ξ ˜ ˜1 ˜N−1 1 N−1 a a Let t = t1,..., tξN−1 be obtained from t = t1, . . . , tξN−1 by the permutation of ti and ti+1. Set j = i + P ξb. The Yang–Baxter equation (1.2) and relations (2.3) yield b ⊗|ξ| ⊗ξ1 ⊗ξN−1 Bbξ(t) = tr ⊗ id T(t)E21 ⊗ · · · ⊗ EN,N−1 ⊗ 1 1 N−1 ⊗|ξ| ˇ(j,j+1) a a ˜ ˇ(j+1,j) a a −1 ⊗ξ ⊗ξ = tr ⊗ id R ti+1 − ti T(t) R ti − ti+1 E21 ⊗ · · · ⊗ EN,N−1 ⊗ 1 a a a a t − t + 1 1 N−1 t − t + 1 i+1 i ⊗|ξ| ˜ ⊗ξ ⊗ξ i+1 i ˜ = a a tr ⊗ id T(t)E21 ⊗ · · · ⊗ EN,N−1 ⊗ 1 = a a Bbξ(t), ti − ti+1 + 1 ti − ti+1 + 1 ˜ by formula (2.10) and the cyclic property of the trace. Therefore, Bξ(t) = Bξ(t), see (2.8). If v is a weight singular vector with respect to the action of Y (glN ), we call the expres- 1 2 1 N−1 N−2 sion Bξ(t)v the (rational) vector-valued weight function of weight ξ , ξ − ξ , . . . , ξ − ξ , −ξN−1 associated with v. Weight functions associated with glN weight singular vectors in evaluation Y (glN )-modules (in particular, highest weight vectors of highest weight glN -modules) can be calculated explicitly by means of the following Theorems 3.1 and 3.3. The theorems express weight functions for Y (glN ) in terms of weight functions for Y (glN−1). Applying the theorems several times one can get 2N−2 combinatorial expressions for the same weight function, the expressions being labeled by subsets of {1,...,N − 2}. The expressions corresponding to the empty set and the whole set are given in Corollaries 3.2 and 3.4. Let v1, . . . , vn be weight singular vectors with respect to the action of Y (glN ). Corol- lary 3.6 expresses the weight function Bξ(t)(v1 ⊗ · · · ⊗ vn) as a sum of the tensor products Bζ1 (t1)v1 ⊗ · · · ⊗ Bζn (tn)vn with ζ1 + ··· + ζn = ξ, and t1, . . . , tn being a partition of the collec- tion t of |ξ| variables into collections of |ζ1|,..., |ζn| variables. This yields combinatorial formulae for weight functions associated with tensor products of highest weight vectors of highest weight evaluation modules. Remark. It is shown in [9] that for a weight singular vector v in a tensor product of evaluation Y (glN )-modules, the values of the weight function Bξ(t)v at solutions of a certain system of algebraic equations (Bethe ansatz equations) are eigenvectors of the transfer matrix of the cor- responding lattice integrable model. This result is extended in [12] to the case of higher transfer matrices. Remark. The weight functions Bξ(t)v are used in [22] to construct Jackson integral represen- tations for solutions of the qKZ equations. Combinatorial Formulae for Nested Bethe Vectors 7 Remark. The expression for a vector-valued weight function used here may differ from the expressions for the corresponding objects used in other papers, see [9, 22]. The discrepancy is not essential and may occur due to the choice of coproduct for the Yangian Y (glN ) as well as the choice of normalization. 3 Combinatorial formulae for rational weight functions For a nonnegative integer k introduce a function Wk(t1, . . . , tk): Y ti − tj − 1 Wk(t1, . . . , tk) = . ti − tj 16i ξ 1 N−1 X 1 1 N−1 N−1 Symt f t1, . . . , tξN−1 = f tσ1 , . . . , tσ1 ; ... ; t N−1 , . . . , t N−1 , (3.1) 1 ξ1 σ1 σ N−1 σ1,...,σN−1 ξ a where σ ∈ Sξa , a = 1,...,N − 1, and N−1 ! ξ ξ Y a a Symt f(t) = Symt f(t) Wξa t1, . . . , tξa . (3.2) a=1 1 N−1 1 1 N−1 Let η 6 ··· 6 η be nonnegative integers. Define a function Xη t1, . . . , tη1 ; ... ; t1 ,... , N−1 tηN−1 , a N−2 η j−1 a+1 a Y Y 1 Y ti − tj + 1 Xη(t) = . (3.3) ta+1 − ta ta+1 − ta a=1 j=1 j j i=1 i j N−1 N−1 The function Xη(t) does not actually depend on the variables tηN−2+1, . . . , tηN−1 . 1 N−1 1 1 N−1 For nonnegative integers η > ··· > η define a function Yη t1, . . . , tη1 ; ... ; t1 ,... , N−1 tηN−1 , N−1 ηa j−1 a a−1 Y Y 1 Y ti − tj+ηa−1−ηa + 1 Yη(t) = . (3.4) ta − ta−1 ta − ta−1 a=2 j=1 j j+ηa−1−ηa i=1 i j+ηa−1−ηa 1 1 The function Yη(t) does not actually depend on the variables t1, . . . , tη1−η2 . For any ξ, η ∈ N−1, define a function Z t1, . . . , tN−1 ; s1, . . . , sN−1 , Z>0 ξ,η 1 ξN−1 1 ηN−1 N−2 ξa+1 ηa a+1 a Y Y Y ti − sj + 1 Zξ,η(t; s) = . (3.5) ta+1 − sa a=1 i=1 j=1 i j 1 1 N−1 N−1 The function Zξ,η(t; s) does not actually depend on the variables t1, . . . , tξ1 and s1 , . . . , sηN−1 . If ξ, η, ζ ∈ N−1 are such that ξ − ζ ∈ N−1 and ζ − η ∈ N−1, and t = (t1, . . . , t1 , . . . , tN−1, Z>0 Z>0 Z>0 1 ξ1 1 N−1 . . . , tξN−1 ), then we set 1 1 N−1 N−1 t[η] = t1, . . . , tη1 ; ... ; t1 , . . . , tηN−1 , 1 1 N−1 N−1 t(η,ζ] = tη1+1, . . . , tζ1 ; ... ; tηN−1+1, . . . , tζN−1 . (3.6) Notice that t[η] = t(0,η]. 8 V. Tarasov and A. Varchenko 1 N−1 ˙ 1 N−2 ¨ 2 N−1 1 1 For any ξ = ξ , . . . , ξ set ξ = ξ , . . . , ξ and ξ = ξ , . . . , ξ . If t = t1, . . . , tξ1 , N−1 N−1 . . . , t1 , . . . , tξN−1 , then we set ˙ 1 1 N−2 N−2 ¨ 2 2 N−1 N−1 t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−2 , t = t1, . . . , tξ2 ; ... ; t1 , . . . , tξN−1 . (3.7) 1 N Theorem 3.1. Let V be a glN -module and v ∈ V a singular vector of weight Λ ,..., Λ . Let 1 N−1 1 1 N−1 N−1 ξ , . . . , ξ be nonnegative integers and t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . In the evaluation Y (glN )-module V (x), one has ξN−1 N−2 Y 1 X 1 Y 1 (t)v = Bξ N−1 η1! (ξa − ηa)!(ηa+1 − ηa)! i=1 ti − x η a=1 " N−2 ηa−1 a a+1 ξ Y Y tξa−i − x + Λ × Sym X (t )Z (t ; t ) t η (ξ−η,ξ] ξ−η,η [ξ−η] (ξ−η,ξ] ta − x a=1 i=0 ξa−i # ηN−1−ηN−2 ηN−2−ηN−3 η1 hN−1i ˙ × eN,N−1 eN,N−2 ··· eN1φ B(ξ−η)· (t[ξ−η]) v , (3.8) the sum being taken over all η = (η1, . . . , ηN−1) ∈ N−1 such that η1 ··· ηN−1 = ξN−1 Z>0 6 6 a a ξ and η 6 ξ for all a = 1,...,N − 2. Other notation is as follows: Symt is defined by (3.2), the functions Xη and Zξ−η,η are respectively given by formulae (3.3) and (3.5), φ is the first of embeddings (2.5), and hN−1i ˙ hN−1i · (t[ξ−η]) = (s) · , B(ξ−η) Bζ ζ=(ξ−η) , s=t˙[ξ−η] hN−1i Bζ (s) coming from (2.8). Remark. For N = 2, the sum in the right side of formula (3.8) contains only one term: η = ξ. h1i Moreover, Xη = Zξ−η,η = 1, and B(ξ−η)· = 1 by convention. 1 N Corollary 3.2. Let V be a glN -module and v ∈ V a singular vector of weight Λ ,..., Λ . Let 1 N−1 1 1 N−1 N−1 ξ , . . . , ξ be nonnegative integers and t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . In the evaluation Y (glN )-module V (x), one has a N−1 ξ ←− Y Y 1 X Y 1 mab−ma,b−1 Bξ(t)v = a ab a,b−1 eab v ti − x (m − m )! a=1 i=1 m 16b N a−2 mab b b+1 b+1 b ξ Y Y Y t ab − x + Λ Y tj − t ab + 1 × Sym i+me i+me . (3.9) t b+1 b b+1 b a=3 b=1 i=1 ti+ma,b+1 − ti+mab a,b+1 tj − ti+mab e e 16j ab Here the sum is taken over all collections of nonnegative integers m , 1 6 b < a 6 N, such a1 a,a−1 a+1,a Na a that m 6 ··· 6 m and m + ··· + m = ξ for all a = 1,...,N − 1; by convention, ma0 = 0 for any a = 2,...,N. Other notation is as follows: in the ordered product the factor ξ ~ ~ eab is to the left of the factor ecd if a > c, or a = c and b > d, Symt is defined by (3.2), and ab b+1,b a−1,b a,a−1 me = m + ··· + m for all 1 6 b < a 6 N, in particular, me = 0. Combinatorial Formulae for Nested Bethe Vectors 9 1 N Theorem 3.3. Let V be a glN -module and v ∈ V a singular vector of weight Λ ,..., Λ . Let 1 N−1 1 1 N−1 N−1 ξ , . . . , ξ be nonnegative integers and t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . In the evaluation Y (glN )-module V (x), one has ξ1 N−1 Y 1 X 1 Y 1 (t)v = Bξ t1 − x ηN−1! (ξa − ηa)!(ηa−1 − ηa)! i=1 i η a=2 a " N−1 η a a ξ Y Y t − x + Λ × Sym Y (t )Z (t ; t ) i t η [η] η,ξ−η [η] (η,ξ] ta − x a=2 i=1 i # η1−η2 η2−η3 ηN−1 hN−1i ¨ × e21 e31 ··· eN1 ψ B(ξ−η)·· (t(η,ξ]) v , (3.10) the sum being taken over all η = (η1, . . . , ηN−1) ∈ N−1 such that ξ1 = η1 ··· ηN−1 and Z>0 > > a a ξ η 6 ξ for all a = 2,...,N − 1. Other notation is as follows: Symt is defined by (3.2), the functions Yη and Zη,ξ−η are respectively given by formulae (3.4) and (3.5), ψ is the second of embeddings (2.5), and hN−1i ¨ hN−1i ·· (t(η,ξ]) = (s) ·· , B(ξ−η) Bζ ζ=(ξ−η) , s=t¨(η,ξ] hN−1i Bζ (s) coming from (2.8). Remark. For N = 2 the sum in the right side of formula (3.10) contains only one term: η = ξ. h1i Moreover, Yη = Zη,ξ−η = 1, and B(ξ−η)·· = 1 by convention. 1 N Corollary 3.4. Let V be a glN -module and v ∈ V a singular vector of weight Λ ,..., Λ . Let 1 N−1 1 1 N−1 N−1 ξ , . . . , ξ be nonnegative integers and t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . In the evaluation Y (glN )-module V (x), one has a N−1 ξ −→ Y Y 1 X Y 1 mab−ma+1,b Bξ(t)v = a ab a+1,b eab v ti − x (m − m )! a=1 i=1 m 16b N−1 a−1 ma+1,b−1 a a a a−1 ξ t a+1,b − x + Λ t a+1,b − tj + 1 Y Y Y mb −i Y mb −i ×Symt . (3.11) ta − ta−1 ta − ta−1 a=2 b=1 i=0 ma+1,b−i mab−i ab a−1 ma+1,b−i j b b mb −i Theorem 3.5 ([22]). Let V1, V2 be Y (glN )-modules and v1 ∈ V1, v2 ∈ V2 weight singular vec- 1 N−1 tors with respect to the action of Y (glN ). Let ξ , . . . , ξ be nonnegative integers and t = 1 1 N−1 N−1 t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . Then a+1 a N−1 N−2 η ξ a+1 a X Y 1 ξ Y Y Y ti − tj + 1 Bξ(t)(v1 ⊗ v2) = Symt (ξa − ηa)!ηa! ta+1 − ta η a=1 a=1 i=1 j=ηa+1 i j N−1 ηa ξa Y Y a Y a × Taa(ti )v2 Ta+1,a+1(tj )v1 Bη(t[η])v1 ⊗ Bξ−η(t(η,ξ])v2 , (3.12) a=1 i=1 j=ηa+1 10 V. Tarasov and A. Varchenko the sum being taken over all η = (η1, . . . , ηN−1) ∈ N−1 such that ξ − η ∈ N−1. In the left side Z>0 Z>0 we assume that Bξ(t) acts in the Y (glN )-module V1 ⊗ V2. To make the paper self-contained we will prove Theorem 3.5 in Section5. Corollary 3.6. Let V1,...,Vn be Y (glN )-modules and vr ∈ Vr, r = 1, . . . , n, weight singular 1 N−1 vectors with respect to the action of Y (glN ). Let ξ , . . . , ξ be nonnegative integers and t = 1 1 N−1 N−1 t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . Then Bξ(t)(v1 ⊗ · · · ⊗ vn) a+1 a N−1 n "N−2 n−1 ηr ξ a+1 a X Y Y 1 ξ Y Y Y Y ti − tj + 1 = Sym a a t a+1 a (ηr − η )! t − t η1,...,ηn−1 a=1 r=1 r−1 a=1 r=1 a+1 j=ηa+1 i j i=ηr−1 +1 r a a N−1 n ηr−1 ξ ! Y Y Y a Y a × Taa(ti )vr Ta+1,a+1(tj )vr a a=1 r=1 i=1 j=ηr +1 # × Bη1 (t[η1])v1 ⊗ Bη2−η1 (t(η1,η2])v2 ⊗ · · · ⊗ Bξ−ηn−1 (t(ηn−1,ξ])vn . (3.13) Here the sum is taken over all η , . . . , η ∈ N−1, η = η1, . . . , ηN−1, such that η − η ∈ 1 n−1 Z>0 r r r r+1 r N−1 for any r = 1, . . . , n − 1, and η = 0, η = ξ, by convention. The sets t , t are Z>0 0 n [η1] (ηr,ηr+1] defined by (3.6). In the left side we assume that Bξ(t) acts in the Y (glN )-module V1 ⊗ · · · ⊗ Vn. Remark. In formulae (3.8)–(3.13), the products of factorials in the denominators of the first factors of summands are equal to the orders of the stationary subgroups of expressions in the square brackets. 4 Proofs of Theorems 3.1 and 3.3 We prove Theorems 3.1 and 3.3 by induction with respect to N, assuming that Theorem 3.5 holds. For the base of induction, N = 2, the claims of Theorems 3.1 and 3.3 coincide with each other and reduce to the identity X Y sσi − sσj − 1 = k!. (4.1) sσi − sσj σ∈Sk 16i N−1 Let w1,..., wN−1 be the standard basis of the space C . The module L(x) is a highest weight evaluation module with highest weight (1, 0,..., 0) and highest weight vector w1. The module L¯(x) is a highest weight evaluation module with highest weight (0,..., 0, −1) and highest weight vector wN−1. N−1 For any X ∈ End C set ν(X) = Xw1 andν ¯(X) = XwN−1. N−1⊗k Consider the maps ψ(x1, . . . , xk): Y (glN−1) → C ⊗ Y (glN ), ⊗k hN−1i(k) ψ(x1, . . . , xk) = ν ⊗ id ◦ π(x1) ⊗ · · · ⊗ π(xk) ⊗ ψ ◦ ∆ , (4.2) N−1⊗k and φ(x1, . . . , xk): Y (glN−1) → Y (glN ) ⊗ C , ⊗k hN−1i(k) φ(x1, . . . , xk) = id ⊗ ν¯ ◦ φ ⊗ $(x1) ⊗ · · · ⊗ $(xk) ◦ ∆ , hN−1i(k) ⊗(k+1) where ψ and φ are embeddings (2.5), and ∆ : Y (glN−1) → Y (glN−1) is the multiple coproduct. ⊗k N−1 a1,...,ak For any element g ∈ C ⊗ Y (glN ) we define its components g by the rule N−1 X a1,...,ak g = wa1 ⊗ · · · ⊗ wak ⊗ g . a1,...,ak=1 N−1⊗k A similar rule defines components of elements of the tensor product Y (glN ) ⊗ C . 1 N−1 1 Proposition 4.1 ([22, Theorem 3.4.2]). Let ξ , . . . , ξ be nonnegative integers and t = t1, 1 N−1 N−1 . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . Then N−1 a ,...,a X 1 1 1 1 hN−1i 1 ξ1 (t) = T1,a +1 t ··· T1,a +1 t 1 ψ t , . . . , t 1 (t¨) , (4.3) Bξ 1 1 ξ1 ξ 1 ξ Bξ¨ a1,...,aξ1 =1 cf. (3.7). Proof. To get formula (4.3) we use formulae (2.6) and (2.8), and compute the trace over the first ξ1 tensor factors, taking into account the properties of the R-matrix (1.1) described below. N Let v1,..., vN be the standard basis of the space C . For any a, b = 1,...,N, the R-matrix R(u) preserves the subspace spanned by the vectors va ⊗ vb and vb ⊗ va. N−1 N−1 N N Let W be the image of C ⊗ C in C ⊗ C under the embedding wa ⊗ wb 7→ va+1 ⊗ vb+1, a, b = 1,...,N − 1. The R-matrix R(u) preserves W and the restriction of R(u) on W hN−1i N−1 N−1 coincides with the image of R (u) in End(C ⊗ C ) induced by the embedding. 1 N−1 1 1 N−1 Proposition 4.2. Let ξ , . . . , ξ be nonnegative integers and t = t1, . . . , tξ1 ; ... ; t1 ,..., N−1 tξN−1 . Then N−1 a1,...,a 1 X N−1 N−1 1 1 hN−1i ˙ ξ ξ(t) = Ta +1,1 t N−1 ··· Ta +1,1 t φ t , . . . , t 1 (t) , (4.4) B ξN−1 ξ 1 1 1 ξ Bξ˙ a1,...,aξ1 =1 cf. (3.7). Proof. To get formula (4.4) we modify formula (2.6) according to relation (2.7), use for- mula (2.8), and compute the trace over the last ξN−1 tensor factors, taking into account the structure of the R-matrix (1.1). 12 V. Tarasov and A. Varchenko b Proof of Theorem 3.3. For a collection a = (a1, . . . , aξ1 ) of positive integers let c (a) = 1 N−1 #{r|ar > b}, and c(a) = c (a), . . . , c (a) . To obtain formula (3.10) we apply both sides of formula (4.3) to the singular vector v in the −1 evaluation module V (x) over Y (glN ). In this case, T1a(u) acts as (u − x) ea1 and we have ξ1 Y 1 (t)v = Bξ t1 − x i=1 i N−1 a ,...,a X 1 2 2 3 N−1 X hN−1i 1 ξ1 × eη −η eη −η ··· eη ψ t1, . . . , t1 (t¨) v, (4.5) 21 31 N1 1 ξ1 Bξ¨ η a1,...,aξ1 =1 c(a)=η the first sum being taken over all η = η1, . . . , ηN−1 ∈ N−1 such that ξ1 = η1 ··· ηN−1. Z>0 > > ψ Let V (x) be the Y (glN−1)-module obtained by pulling V (x) back through the embed- ding ψ. Then ψ t1, . . . , t1 hN−1i(t¨)v is the weight function associated with the vector 1 ξ1 Bξ¨ 1 1 ψ w1 ⊗ · · · ⊗ w1 ⊗ v in the Y (glN−1)-module L t1 ⊗ · · · ⊗ L tξ1 ⊗ V (x). We use Theorem 3.5 to write ψ t1, . . . , t1 hN−1i(t¨)v as a sum of tensor products of weight functions in the tensor 1 ξ1 Bξ¨ factors, that is, as a sum of the following expressions: 1 hN−1i 1 hN−1i hN−1i π t1 (s1) w1 ⊗ · · · ⊗ π t 1 (sξ1 ) w1 ⊗ ψ (s0) v, Bζ1 ξ Bζξ1 Bζ0 where ζ0, . . . , ζξ1 , s0, . . . , sξ1 are suitable parameters, and employ Corollary 3.4, valid by the in- duction assumption, to calculate the weight functions π(t1) hN−1i(s )w in the modules L(t1). j Bζj j 1 j As a result, we get formula (4.7), see Lemma 4.3 below. −1 Observe that in the module L(x) one has hT11(u)w1i = 1+(u−x) and hTaa(u)w1i = 1 for all hN−1i a = 2,...,N. The weight function π(x) Bζ (s) w1 equals zero unless ζ = (1,..., 1, 0,..., 0) (it can be no units or zeros in the sequence). If ζ1 = ··· = ζr = 1 and ζr+1 = ··· = ζN−1 = 0, 1 r then s = (s1, . . . , s1) and e w π(x) hN−1i(s)w = r+1,1 1 . Bζ 1 1 2 1 r r−1 (s1 − x)(s1 − s1) ··· (s1 − s1 ) Fix η = η1, . . . , ηN−1 ∈ N−1 such that η1 ··· ηN−1. Consider a collection l of integers Z>0 > > a a+1 a a a li , a = 1,...,N − 2, i = 1, . . . , η , such that 1 6 l1 < ··· < lηa+1 6 η for all a = 1,...,N − 2. 1 1 N−1 N−1 Introduce a function Fl(s) of the variables s1, . . . , sη1 ; ... ; s1 , . . . , sηN−1 : a+1 N−2 η a+1 a Y Y 1 Y si − sj + 1 F (s) = . (4.6) l a+1 a a+1 a si − sla a a si − sj a=1 i=1 i li There is a bijection between collections l and sequences of integers a = (a1, . . . , aη1 ) such that 1 a 1 6 ai 6 N −1 for all i = 1, . . . , η , and c(a) = η. It is established as follows. Define numbers pi 1 1 2 a a−1 a+1 by the rule: p = l , i = 1, . . . , η , and p = p a , a = 2,...,N − 2, i = 1, . . . , η . Then the i i i li b b sequence a is uniquely determined by the requirement that ai > b iff i ∈ p1, . . . , pηb+1 , for all i = 1, . . . , η1. We will write a(l) for the result of this mapping. Summarizing, we get the following statement. Combinatorial Formulae for Nested Bethe Vectors 13 Lemma 4.3. Let η = (η1, . . . , ηN−1) ∈ N−1 be such that ξ1 = η1 ··· ηN−1. Let l be a Z>0 > > collection of integers as described above, and a(l) = (a1, . . . , aξ1 ). Then a1,...,a 1 ψ t1, . . . , t1 hN−1i(t¨) ξ v (4.7) 1 ξ1 Bξ¨ N−1 N−1 ηb ¨ b b Y 1 ξ Y Y ti − x + Λ hN−1i ¨ = Sym¨ Fl(t[η])Zη,ξ−η(t[η]; t(η,ξ]) ψ B ·· (t(η,ξ]) v , (ξb − ηb)! t tb − x (ξ−η) b=2 b=2 i=1 i cf. (3.5) for Zη,ξ−η(t[η]; t(η,ξ]). Comparing the expressions under Sym in formulae (4.7) and (3.10), and taking into account that the product N−1 ηb Y Y b b hN−1i ¨ Zη,ξ−η(t[η]; t(η,ξ]) ti − x + Λ ψ B(ξ−η)·· (t(η,ξ]) v b=2 i=1 is invariant with respect to the action of the groups Sη1 × · · · × SηN−1 and Sξ1−η1 × · · · × 1 1 N−1 N−1 1 S N−1 N−1 permuting respectively the variables t , . . . , t ; ... ; t , . . . , t and t ,... , ξ −η 1 η1 1 ηN−1 η1+1 t1 ; ... ; tN−1 , . . . , tN−1 , one can see that formula (3.10) follows from formula (4.5) and Lem- ξ1 ηN−1+1 ξN−1 ma 4.4 below. Theorem 3.3 is proved. Lemma 4.4. Let η = (η1, . . . , ηN−1) ∈ N−1 such that η1 ··· ηN−1. Let s = s1, . . . , s1 ; ... ; Z>0 > > 1 η1 N−1 N−1 s1 , . . . , sηN−1 . Then N−2 1 Y 1 η X η¨ Sym Y (s) = Sym F (s), (4.8) ηN−1! (ηa − ηa+1)! s η s¨ l a=1 l a cf. (3.4) for Yη(s). The sum is taken over all collections l of integers li , a = 1,...,N − 2, a+1 a a a i = 1, . . . , η , such that 1 6 l1 < ··· < lηa+1 6 η for all a = 1,...,N − 2. Proof. It is convenient to rewrite formula (3.4) in the form similar to (4.6): N−2 ηa+1 a+1 a Y Y 1 Y si − sj+ηa−ηa+1 + 1 Y (s) = . η a+1 a a+1 a si − s a a+1 a+1 si − s a a+1 a=1 i=1 i+η −η i For positive integers p, r such that p 6 r, consider a function Gp,r(y1, . . . , yp; z1, . . . , zr) p 1 r Y 1 Y yi − zj+r−p + 1 = Symz1,...,zr . (r − p)! yi − zi+r−p yi − zj+r−p i=1 i It is a manifestly symmetric function of z1, . . . , zr, and it is a symmetric function of y1, . . . , yp by Lemma 4.5. To prove Lemma 4.4 we will show that the expressions in both sides of formula (4.8) are equal to N−2 Y a+1 a+1 a a Gηa+1,ηa s1 , . . . , sηa+1 ; s1, . . . , sηa . a=1 14 V. Tarasov and A. Varchenko The proof is by induction with respect to N. The base of induction is N = 3. In this case the claim follows from Lemma 4.5 and identity (4.1). The induction step for the left side of (4.8) is as follows: η η¨ η1 Syms Yη(s) = Syms¨ Syms1,...,s1 Yη(s) 1 η1 " η¨ 2 2 1 1 = Syms¨ Gη2,η1 s1, . . . , sη2 ; s1, . . . , sη1 N−2 ηa+1 a+1 a Y Y 1 Y si − sj+ηa−ηa+1 + 1 × a+1 a a+1 a si − s a a+1 a+1 si − s a a+1 a=2 i=1 i+η −η i 2 2 1 1 In the last two equalities we use the fact that Gη2,η1 s1, . . . , sη2 ; s1, . . . , sη1 is symmetric with 2 2 respect to s1, . . . , sη2 , and the induction assumption. The idea of the induction step for the right side of (4.8) is similar. First, one should sym- N−1 N−1 metrize Fl(s) with respect to the variables s1 , . . . , sηN−1 and sum up over all possible collec- N−2 N−2 tions l1 , . . . , lηN−1 , and then use Lemma 4.5. We leave details to a reader. Lemma 4.5. p X p Y 1 Y yi − zj + 1 Gp,r(y1, . . . , yp; z1, . . . , zr) = Symy1,...,yp , yi − zdi yi − zj d i=1 di Lemma 4.6. Let p, r be positive integers such that p 6 r. Then p 1 r Y 1 Y yi − zj + 1 Symz1,...,zr (r − p)! yi − zi yi − zj i=1 16j Proof. The statement follows from Lemma 4.5 by the change of variables yi → −yp−i, zj → −zr−j, and a suitable change of summation indices. Proof of Theorem 3.1. The proof is similar to the proof of Theorem 3.3, mutatis mutandis. In particular, Lemma 4.5 is replaced by Lemma 4.6. Combinatorial Formulae for Nested Bethe Vectors 15 5 Proof of Theorem 3.5 The theorem is proved by induction with respect to N. The base of induction, the N = 2 case, follows from Proposition 5.3. The induction step is provided by Proposition 5.6. N−1 hN−1i P hN−1i hN−1i hN−1i hN−1i Let P = Eab ⊗ Eba be the flip matrix, and R (u) = u + P be the a,b=1 R-matrix for the Yangian Y (glN−1). −1 In this section we regard T (u) as an N × N matrix over the algebra Y (glN )[u ] with entries Tab(u), a, b = 1,...,N. Let N A(u) = T11(u),B(u) = T12(u),...,T1N (u) ,D(u) = Tij(u) i,j=2, (5.1) be the submatrices of T (u). Set R(u) = u−1RhN−1i(u). Formulae (2.3) and (1.1) imply the following commutation relations for A(u), B(u) and D(u): A(u)A(t) = A(t)A(u), (5.2) u − t (12) B[1](u)B[2](t) = B[2](t)B[1](u)R (u − t), (5.3) u − t + 1 u − t − 1 1 A(u)B(t) = B(t)A(u) + B(u)A(t), (5.4) u − t u − t u − t + 1 (12) 1 D(1)(u)B[2](t) = B[2](t)D(1)(u)R (u − t) − B[1](u)D(2)(t), (5.5) u − t u − t (12) (12) R (u − t)D(1)(u)D(2)(t) = D(2)(t)D(1)(u)R (u − t). (5.6) In this section we use superscripts to deal with tensor products of matrices, writing parentheses for square matrices and brackets for the row matrix B. −1 hN−1i hN−1i Set Rˇ(u) = (u + 1) P R (u). For an expression f(u1, . . . , uk) with matrix coeffi- cients and a simple transposition (i, i + 1), i = 1, . . . , k − 1, set (i,i+1) (i,i+1) f(u1, . . . , uk) = f(u1, . . . , ui−1, ui+1, ui, ui+2, . . . , uk)Rˇ (ui − ui+1), (5.7) if the product in the right side makes sense. The matrix Rˇ(u) has the properties Rˇ(u)Rˇ(−u) = 1 and Rˇ(12)(u − v)Rˇ(23)(u)Rˇ(12)(v) = Rˇ(23)(v)Rˇ(12)(u)Rˇ(23)(u − v), cf. (1.2). This yields the following lemma. Lemma 5.1. Formula (5.7) extends to the action of the symmetric group Sk on expressions σ f(u1, . . . , uk) with appropriate matrix coefficients: f 7→ f, σ ∈ Sk. [1] [k] By formula (5.3) the expression B (u1) ··· B (uk) is invariant under the action (5.7) of the symmetric group Sk. For an expression f(u1, . . . , uk) with suitable matrix coefficients, set X RSym(1,...,k) f(u , . . . , u ) = σf(u , . . . , u ). u1,...,uk 1 k 1 k σ∈Sk 16 V. Tarasov and A. Varchenko Proposition 5.2. k Y u − ui − 1 A(u)B[1](u ) ··· B[k](u ) = B[1](u ) ··· B[k](u )A(u) (5.8) 1 k u − u 1 k i=1 i k ! 1 R (1,...,k) 1 Y u1 − ui − 1 [1] [2] [k] + Sym B (u)B (u2) ··· B (u )A(u1) , (k − 1)! u1,...,uk u − u u − u k 1 i=2 1 i (0) [1] [k] D (u)B (u1) ··· B (uk) k Y u − ui + 1 (0k) (01) = B[1](u ) ··· B[k](u )D(0)(u)R (u − u ) ··· R (u − u ) u − u 1 k k 1 i=1 i k 1 (1,...,k) 1 Y u1 − ui + 1 − RSym (k − 1)! t1,...,tk u − u u − u 1 i=2 1 i ! [0] [2] [k] (1) (1k) (12) × B (u)B (u2) ··· B (uk)D (u1)R (u1 − uk) ··· R (u1 − u2) . (5.9) In the second formula the tensor factors are counted by 0, . . . , k. Proof. The statement follows from relations (5.3)–(5.5) by induction with respect to k. We apply formula (5.4) or (5.5) to the product of the first factors in the left side and then use the induction assumption. Remark. Formulae (5.8) and (5.9) have the following structure. The first term in the right side comes from repeated usage of the first term in the right side of the respective relation (5.4) or (5.5). The second term, involving symmetrization, is effectively determined by the fact that the whole expression in the right side is regular at u = ui for any i = 1, . . . , k, and is invariant with respect to action (5.7) of the symmetric group Sk. The symmetrized expression is obtained by applying once the second term in the right side of the relevant relation (5.4) or (5.5) followed by repeated usage of the first term of the respective relation. Let ∆ be coproduct (2.4) for the Yangian Y (glN ). For a matrix F = (Fij) over Y (glN ), denote by ∆(F ) = ∆(Fij) the corresponding matrix over Y (glN ) ⊗ Y (glN ). We will use subscripts in braces to describe the embeddings Y (glN ) → Y (glN ) ⊗ Y (glN ) as one of the tensor factors: X{1} = X ⊗ 1, X{2} = 1 ⊗ X, X ∈ Y (glN ). For a matrix F over Y (glN ), we apply the embeddings entrywise, writing F{1}, F{2} for the corresponding matrices over Y (glN ) ⊗ Y (glN ). Proposition 5.3. We have k X 1 (1,...,k) Y ti − tj − 1 ∆ B[1](t ) ··· B[k](t ) = RSym 1 k t1,...,tk l!(k − l)! ti − tj l=0 16i Proof. The statement is proved by induction with respect to k. Writing the left side as [1] [2] [k] ∆ B (u1) ∆ B (u2) ··· B (uk) , Combinatorial Formulae for Nested Bethe Vectors 17 we expand the first factor according to (2.4): [1] [1] [1] (1) ∆ B (u1) = B{1}(u1)A{2}(u1) + B{2}(u1)D{1}(u1), and apply the induction assumption to expand the second one. Then we use Proposition 5.2 to transform the obtained expression to the right side of (5.10). N−1 Regard vectors in the space C as (N −1)×1 matrices. Formula (4.3) from Proposition 4.1 can be written as follows: 1 (t) = B[1] t1 ··· B[ξ ] t1 ψ t1, . . . , t1 hN−1i(t¨). (5.11) Bξ 1 ξ1 1 ξ1 Bξ¨ For nonnegative integers k, l such that k > l, define an embedding N−1⊗k ψbl(u1, . . . , uk): Y (glN−1) → C ⊗ Y (glN ) ⊗ Y (glN ), (5.12) ⊗l ⊗(k−l) ψbl(u1, . . . , uk) = ϑl ◦ ν ⊗ id ⊗ ν ⊗ id hN−1i(k+1) ◦ π(u1) ⊗ · · · ⊗ π(ul) ⊗ ψ ⊗ π(ul+1) ⊗ · · · ⊗ π(uk) ⊗ ψ ◦ ∆ , where N−1⊗l N−1⊗(k−l) N−1⊗k ϑl : C ⊗ Y (glN ) ⊗ C ⊗ Y (glN ) → C ⊗ Y (glN ) ⊗ Y (glN ) is given by the rule ϑl(x ⊗ X1 ⊗ y ⊗ X2) = x ⊗ y ⊗ X1 ⊗ X2, N−1⊗l N−1⊗(k−l) for x ∈ C , y ∈ C , X1,X2 ∈ Y (glN ), and hN−1i(k+1) ⊗(k+2) ∆ : Y (glN−1) → Y (glN−1) is the multiple coproduct. Let v1, v2 be weight singular vectors with respect to the action of Y (glN ). Lemma 5.4. For any X ∈ Y (glN−1) we have ∆ ψb(u1, . . . , uk)(X) (v1 ⊗ v2) = ψbk(u1, . . . , uk)(X) (v1 ⊗ v2). hN−1i Proof. Recall that ∆ denotes the coproduct for the Yangian Y (glN−1). Let Y×(glN ) be the left ideal in Y (glN ) generated by the coefficients of the series T21(u),...,TN1(u). It follows from relations (2.4) and (2.2) that hN−1i ∆ ψ(X) − (ψ ⊗ ψ) ∆ (X) ∈ Y (glN ) ⊗ Y×(glN ) for any X ∈ Y (glN−1). Therefore, hN−1i ∆ ψ(X) (v1 ⊗ v2) = (ψ ⊗ ψ) ∆ (X) (v1 ⊗ v2) (5.13) because v2 is a weight singular vector. The lemma follows from formulae (4.2), (5.12) and (5.13). Lemma 5.5. For any X ∈ Y (glN−1) we have (l+1) (k) D{1} (ul+1) ··· D{1}(uk)A{2}(t1) ··· A{2}(tl)ψbk(t1, . . . , tk)(X) (v1 ⊗ v2) l k Y 1 Y 1 (1) (k) = T22 ti v1 T11 tj v2 ψbl(t1, . . . , tk)(X)I ··· I (v1 ⊗ v2). i=1 j=l+1 18 V. Tarasov and A. Varchenko hN−1i hN−1i Proof. Recall that D(u) = (id⊗ψ) T (u) and R(u−ui) = id⊗π(ui) T (u) . Then according to relation (5.6), for any X ∈ Y (glN−1) we have hN−1i hN−1i D(ui) ψ ⊗ π(ui) ∆ (X) = π(ui) ⊗ ψ ∆ (X) D(ui). In addition, remind that D(u)(w1 ⊗ v1) = w1 ⊗ T22(u)v1 = T22(u)v1 (w1 ⊗ v1), because v1 is a weight singular vector. Therefore, (l+1) (k) D{1} (ul+1) ··· D{1}(uk)ψbk(u1, . . . , uk)(X)(v1 ⊗ v2) (l+1) (k) = ψbl(u1, . . . , uk) XD{1} (ul+1) ··· D{1}(uk) (v1 ⊗ v2) k Y = T22(uj)v1 ψbl(u1, . . . , uk)(X)(v1 ⊗ v2). j=l+1 Recall that we regard ψbl(u1, . . . , uk)(X) as a matrix over Y (glN ) ⊗ Y (glN ). All entries of this matrix belong to ψ Y (glN−1)) ⊗ ψ Y (glN−1)) . It follows from relations (2.2) that for any 0 0 0 X ∈ Y (glN−1) the coefficients of the commutator T11(u)ψ(X ) − ψ(X )T11(u) belong to the left ideal in Y (glN ) generated by the coefficients of the series T21(u),...,TN1(u). Therefore, 0 0 0 A(ui)ψ(X )v2 = ψ(X )T11(ui)v2 = T11(ui)v2 ψ(X )v2 because A(ui) = T11(ui), cf. (5.1), and v2 is a weight singular vector. Hence, A{2}(u1) ··· A{2}(ul)ψbl(u1, . . . , uk)(X)(v1 ⊗ v2) l Y = T11(ui)v2 ψbl(u1, . . . , uk)(X)(v1 ⊗ v2), i=1 which proves the lemma. Proposition 5.6. In the notation of Theorem 3.5, we have Bξ(t)(v1 ⊗ v2) (5.14) ξ1 1 1 l ξ1 X 1 (1,...,ξ1) Y ti − tj − 1 Y Y = Sym T t1v T t1v 1 t1,...,t1 1 1 11 i 2 22 j 1 l!(ξ − l)! 1 ξ1 ti − tj l=0 16i Here the space V1 ⊗ V2 is considered as the Y (glN )-module in the left side of the formula, and as the Y (glN ) ⊗ Y (glN )-module in the right side. Proof. Expand Bξ(t) according to formula (5.11). Since Y (glN ) acts in V1 ⊗ V2 via the copro- duct ∆, we have [1] [ξ1] 1 1 hN−1i (t)(v ⊗ v ) = ∆ B (t ) ··· B (t 1 )ψ t , . . . , t (t¨) (v ⊗ v ) Bξ 1 2 1 ξ 1 ξ1 Bξ¨ 1 2 [1] [ξ1] 1 1 hN−1i = ∆ B (t ) ··· B (t 1 ) ∆ ψ t , . . . , t (t¨) (v ⊗ v ). 1 ξ 1 ξ1 Bξ¨ 1 2 Recall that ∆ applies to matrices entrywise. In the last expression, we develop the factor [1] [ξ1] 1 ∆ B (t1) ··· B (tξ1 ) according to Proposition 5.3, and replace the factor ∆ ψ t1,... , 1 hN−1i 1 1 hN−1i t (t¨) by ψ 1 t , . . . , t (t¨) according to Lemma 5.4. After that, we uti- ξ1 Bξ¨ bξ 1 ξ1 Bξ¨ lize Lemma 5.5 to transform the result to the right side of formula (5.14). Combinatorial Formulae for Nested Bethe Vectors 19 6 Trigonometric weight functions Notation in this section may not coincide with the notation in Sections2–5. The quantum loop algebra Uq(glgN ) (the quantum affine algebra without central extension) {±s} is the unital associative algebra with generators Lab , a, b = 1,...,N, and s = 0, 1, 2,.... Organize them into generating series ∞ ± {±0} X {±s} ±s Lab(u) = Lab + Lab u , a, b = 1,...,N, (6.1) s=1 N ± P ± and combine the series into matrices L (u) = Eab ⊗ Lab(u). The defining relations in a,b=1 Uq(glgN ) are {+0} {−0} Lab = Lba = 0, 1 6 a < b 6 N, {−0} {+0} {+0} {−0} Laa Laa = Laa Laa = 1, a = 1,...,N, (12) µ (1) ν (2) ν (2) µ (1) (12) Rq (u/v) L (u) L (v) = L (v) L (u) Rq (u/v), (µ, ν) = (+, +), (+, −), (−, −), where Rq(u) is the trigonometric R-matrix (1.3). The quantum loop algebra Uq(glgN ) is a Hopf algebra. In terms of generating series (6.1), the coproduct ∆ : Uq(glgN ) → Uq(glgN ) ⊗ Uq(glgN ) reads as follows: N ± X ± ± ∆ : Lab(u) 7→ Lcb(u) ⊗ Lac(u). c=1 ± ± The subalgebras Uq (glgN ) ⊂ Uq(glgN ) generated by the coefficients of the respective series Lab(u), a, b = 1,...,N, are Hopf subalgebras. There is a one-parameter family of automorphisms ρx : Uq(glgN ) → Uq(glgN ), defined by the rule ± ± ρx : Lab(u) 7→ Lab(u/x). The quantum loop algebra Uq(glgN ) contains the algebra Uq(glN ) as a Hopf subalgebra. The {+0} {−0} ˆ {−0} subalgebra is generated by the elements Lab , Lab , 1 6 a 6 b 6 N. Set ka = Laa , a = 1,...,N, and L{+0}kˆ kˆ−1L{−0} eˆ = − ba a , eˆ = a ab , 1 a < b N. (6.2) ab q − q−1 ba q − q−1 6 6 ˆ ˆ The elements k1,..., kN ,e ˆ12,..., eˆN−1,N ,e ˆ21,..., eˆN,N−1 are the Chevalley generators of Uq(glN ). We list some of relations for the introduced elements below, subscripts running over all possible values unless the range is specified explicitly: ˆ δab−δac ˆ kaeˆbc = q eˆbcka, eˆa,a+1eˆa+1,b − qeˆa+1,beˆa,a+1 =e ˆa,b−1eˆb−1,b − qeˆb−1,beˆa,b−1 =e ˆab, −1 −1 a + 1 < b, eˆb,a+1eˆa+1,a − q eˆa+1,aeˆb,a+1 =e ˆb,b−1eˆb−1,a − q eˆb−1,aeˆb,b−1 =e ˆba, eˆcaeˆba = qeˆbaeˆca, eˆcbeˆca = qeˆcaeˆcb, a < b < c. 20 V. Tarasov and A. Varchenko The coproduct formulae are ∆(kˆa) = kˆa ⊗ kˆa, ˆ ˆ−1 ˆ ˆ−1 ∆(ˆea,a+1) = 1 ⊗ eˆa,a+1 +e ˆa,a+1 ⊗ kaka+1, ∆(ˆea+1,a) =e ˆa+1,a ⊗ 1 + ka+1ka ⊗ eˆa+1,a. 1 By minor abuse of notation we say that a vector v in a Uq(glN )-module has weight Λ ,... , N ˆ Λa Λ if kav = q v for all a = 1,...,N. A vector v is called a singular vector ife ˆbav = 0 for all 1 6 a < b 6 N. The evaluation homomorphism : Uq(glgN ) → Uq(glN ) is given by the rule + ˆ−1 ˆ − ˆ −1ˆ−1 Laa(u) 7→ ka − uka,Laa(u) 7→ ka − u ka , a = 1,...,N, + −1ˆ − −1ˆ L (u) 7→ −u q − q kaeˆba,L (u) 7→ q − q kaeˆba, ab ab 1 6 a < b 6 N, + −1 ˆ−1 − −1 −1 ˆ−1 Lba(u) 7→ − q − q eˆabka ,Lba(u) 7→ u q − q eˆabka . Both the automorphisms ρx and the homomorphism restricted to the subalgebra Uq(glN ) are the identity maps. For a Uq(glN )-module V denote by V (x) the Uq(glgN )-module induced from V by the homo- morphism ◦ ρx. The module V (x) is called an evaluation module over Uq(glgN ). Remark. In a k-fold tensor product of evaluation modules the series L+(u) and L−(u) act as polynomials in u and u−1, respectively, and the action of L+(u) is proportional to that of ukL−(u). − Let V be a Uq (glgN )-module. A vector v ∈ V is called a weight singular vector with respect to − − the action of Uq (glgN ) if Lba(u)v = 0 for all 1 6 a < b 6 N, and v is an eigenvector for the action − − − − of L11(u),...,LNN (u); the respective eigenvalues are denoted by hL11(u)vi,..., hLNN (u)vi. 1 N Example. Let V be a Uq(glN )-module and let v ∈ V be a singular vector of weight Λ ,..., Λ . − Then v is a weight singular vector with respect to the action of Uq (glgN ) in the evaluation modu- − Λa −Λa −1 le V (x) and hLaa(u)vi = q − q xu , a = 1,...,N. We will use two embeddings of the algebra Uq(gl^N−1) into Uq(glgN ), called φ and ψ: ± hN−1i ± hNi ± hN−1i ± hNi φ Lab(u) = Lab(u) , ψ Lab(u) = La+1,b+1(u) . (6.3) ± hN−1i ± hNi Here Lab(u) and Lab(u) are series (6.1) for the algebras Uq(gl^N−1) and Uq(glgN ), respectively. The constructions and statements in the rest of the section are similar to those of Section2. We will mention only essential points and omit details. Let k be a nonnegative integer. Let ξ = (ξ1, . . . , ξN−1) be a collection of nonnegative integers. Remind that ξ 1 N−1 ⊗|ξ| − 1(1,|ξ|+1) − N−1 (|ξ|,|ξ|+1) Bbξ t1, . . . , tξN−1 = tr ⊗ id L t1 ··· L tξN−1 −→ ! Y (ξ 1 N−1 1 N−1 1−1 N−1 −1 Remark. The series Bbξ t1, . . ., tξN−1 belongs to Uq(glgN ) t1, . . ., tξN−1 t1 , . . ., tξN−1 . Combinatorial Formulae for Nested Bethe Vectors 21 Set a b N−1 a ξ ξ a Y Y ti Y Y Y ti Bξ(t) = Bbξ(t) a −1 a b a , (6.5) a qtj − q ti tj − ti a=1 16i hNi To indicate the dependence on N, if necessary, we will write Bξ (t). 1 h2i − 1 − 1 Example. Let N = 2 and ξ = (ξ ). Then Bξ (t) = L12 t1 ··· L12 tξ1 . Example. Let N = 3 and ξ = (1, 1). Then 2 h3i − 1 − 2 −1 t1 − 1 − 2 Bξ (t) = L12 t1 L23 t1 + q − q 2 1 L13 t1 L22 t1 . t1 − t1 Example. Let N = 4 and ξ = (1, 1, 1). Then h4i − 1 − 2 − 3 Bξ (t) = L12 t1 L23 t1 L34 t1 2 3 −1 t1 − 1 − 2 − 3 t1 − 1 − 2 − 3 + q − q 2 1 L13 t1 L22 t1 L34 t1 + 3 2 L12 t1 L24 t1 L33 t1 t1 − t1 t1 − t1 2 3 −12 t1t1 − 1 − 2 − 3 − 1 − 2 − 3 + q − q 2 1 3 2 L14 t1 L22 t1 L33 t1 + L13 t1 L24 t1 L32 t1 t1 − t1 t1 − t1 2 1 3 2 −12 2 3 −1 3 t1 − t1 t1 − t1 + q − q t1t1 − 1 − 2 − 3 + q − q t1 2 1 3 1 3 2 L14 t1 L23 t1 L32 t1 . t1 − t1 t1 − t1 t1 − t1 Recall that the direct product of the symmetric groups Sξ1 × · · · × SξN−1 acts on expressions in |ξ| variables, permuting the variables with the same superscript, cf. (2.9). Lemma 6.1 ([22, Theorem 3.3.4]). The expression Bξ(t) is invariant under the action of the group Sξ1 × · · · × SξN−1 . − If v is a weight singular vector with respect to the action of Uq (glgN ), we call the expression 1 2 1 N−1 N−2 Bξ(t)v a (trigonometric) vector-valued weight function of weight ξ , ξ − ξ , . . . , ξ − ξ , −ξN−1 associated with v. − Weight functions associated with Uq(glN ) weight singular vectors in evaluation Uq (glgN )-mod- ules (in particular, highest weight vectors of highest weight Uq(glN )-modules) can be calculated explicitly by means of the following Theorems 6.2 and 6.4, which are analogues of Theorems 3.1 and 3.3, respectively. Corollaries 6.3 and 6.5 are the respective counterparts of Corollaries 3.2 and 3.4. Theorem 6.6 and Corollary 6.7 are analogous to Theorem 3.5 and Corollary 3.6 in the Yangian case and yield combinatorial formulae for weight functions associated with tensor products of highest weight vectors of highest weight evaluation modules over the quantum loop algebra. Remark. The expression for a vector-valued weight function used here may differ from the expressions for the corresponding objects used in other papers, see [9, 22]. The discrepancy can occur due to the choice of coproduct for the quantum loop algebra Uq(glgN ) as well as the choice of normalization. For a nonnegative integer k introduce a function Wk(t1, . . . , tk): −1 Y q ti − qtj Wk(t1, . . . , tk) = . ti − tj 16i 1 N−1 For an expression f(t1, . . . , tξN−1 ), set N−1 ! ξ ξ Y a a Symt f(t) = Symt f(t) Wξa t1, . . . , tξa , (6.6) a=1 ξ where Symt is defined by (3.1). 1 N−1 1 1 N−1 Let η 6 ··· 6 η be nonnegative integers. Define a function Xη t1, . . . , tη1 ; ... ; t1 , N−1 . . . , tηN−1 , a N−2 η j−1 a+1 −1 a Y Y 1 Y qti − q tj Xη(t) = . (6.7) ta+1 − ta ta+1 − ta a=1 j=1 j j i=1 i j N−1 N−1 The function Xη(t) does not actually depend on the variables tηN−2+1, . . . , tηN−1 . 1 N−1 1 1 N−1 For nonnegative integers η > ··· > η define a function Yη t1, . . . , tη1 ; ... ; t1 ,... , N−1 tηN−1 , N−1 ηa j−1 a −1 a−1 Y Y 1 Y qti − q tj+ηa−1−ηa Yη(t) = . (6.8) ta − ta−1 ta − ta−1 a=2 j=1 j j+ηa−1−ηa i=1 i j+ηa−1−ηa 1 1 The function Yη(t) does not actually depend on the variables t1, . . . , tη1−η2 . For any ξ, η ∈ N−1, define a function Z t1, . . . , tN−1 ; s1, . . . , sN−1 , Z>0 ξ,η 1 ξN−1 1 ηN−1 N−2 ξa+1 ηa a+1 −1 a Y Y Y qti − q sj Zξ,η(t; s) = . (6.9) ta+1 − sa a=1 i=1 j=1 i j 1 1 N−1 N−1 The function Zξ,η(t; s) does not depend on the variables t1, . . . , tξ1 and s1 , . . . , sηN−1 . n qn−q−n Q We are using the following q-numbers: [n]q = q−q−1 , and q-factorials: [n]q! = [r]q. Recall r=1 that for a collection t of |ξ| variables we introduced the subcollections t[η], t(η,ξ] by (3.7) and t˙, t¨ by (3.6). ˆ For any 1 6 a < b 6 N sete ˇba = kaeˆab, cf. (6.2). 1 N Theorem 6.2. Let V be a Uq(glN )-module and v ∈ V a singular vector of weight Λ ,..., Λ . 1 N−1 1 1 N−1 N−1 Let ξ , . . . , ξ be nonnegative integers and t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . In the eva- luation Uq(glgN )-module V (x), one has N−2 ηa(ηa−ηa+1) X |η| 1 Y q (t)v = q − q−1 Bξ [η1] ! [ξa − ηa] ![ηa+1 − ηa] ! η q a=1 q q " N−2 ηa−1 ξ Y Y Λa+1 a −Λa+1 × Symt Xη(t(ξ−η,ξ])Zξ−η,η(t[ξ−η]; t(ξ−η,ξ]) q tξa−i − q x a=1 i=0 # ηN−1−ηN−2 ηN−2−ηN−3 η1 hN−1i ˙ × eˇN,N−1 eˇN,N−2 ··· eˇN1φ B(ξ−η)· (t[ξ−η]) v , (6.10) the sum being taken over all η = η1, . . . , ηN−1 ∈ N−1 such that η1 ··· ηN−1 = ξN−1 Z>0 6 6 a a ξ and η 6 ξ for all a = 1,...,N − 2. Other notation is as follows: Symt is defined by (6.6), Combinatorial Formulae for Nested Bethe Vectors 23 the functions Xη and Zξ−η,η are respectively given by formulae (6.7) and (6.9), φ is the first of embeddings (6.3), and hN−1i hN−1i B · = B (s) · ˙ , (ξ−η) (t˙[ξ−η]) ζ ζ=(ξ−η) , s=t[ξ−η] hN−1i Bζ (s) coming from (6.5). Remark. For N = 2, the sum in the right side of formula (6.10) contains only one term: η = ξ. h1i: Moreover, Xη = Zξ−η,η = 1, and B(ξ−η)· = 1 by convention. 1 N Corollary 6.3. Let V be a Uq(glN )-module and v ∈ V a singular vector of weight Λ ,..., Λ . 1 N−1 1 1 N−1 N−1 Let ξ , . . . , ξ be nonnegative integers and t = (t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 ). In the eva- luation Uq(glgN )-module V (x), one has ←− ma,b−1(ma,b−1−mab) −1|ξ| X Y q mab−ma,b−1 Bξ(t)v = q − q ab a,b−1 eˇab v (6.11) [m − m ]q! m 16b N−1 ηa(ηa−1−ηa) X |η| 1 Y q (t)v = q − q−1 Bξ [ηN−1] ! [ξa − ηa] ![ηa−1 − ηa] ! η q a=2 q q " N−1 ηa ξ Y Y Λa a −Λa × Symt Yη(t[η])Zη,ξ−η(t[η]; t(η,ξ]) q ti − q x a=2 i=1 # η1−η2 η2−η3 ηN−1 hN−1i ¨ × eˇ21 eˇ31 ··· eˇN1 ψ B(ξ−η)·· (t(η,ξ]) v , (6.12) the sum being taken over all η = η1, . . . , ηN−1 ∈ N−1 such that ξ1 = η1 ··· ηN−1 and Z>0 > > a a ξ η 6 ξ for all a = 2,...,N − 1. Other notation is as follows: Symt is defined by (6.6), the functions Yη and Zη,ξ−η are respectively given by formulae (6.8) and (6.9), ψ is the second of embeddings (6.3), and hN−1i ¨ hN−1i ·· (t(η,ξ]) = (s) ·· , B(ξ−η) Bζ ζ=(ξ−η) , s=t¨(η,ξ] hN−1i Bζ (s) coming from (6.5). Remark. For N = 2, the sum in the right side of formula (6.12) contains only one term: η = ξ. h1i Moreover, Yη = Zη,ξ−η = 1, and B(ξ−η)·· = 1 by convention. 24 V. Tarasov and A. Varchenko 1 N Corollary 6.5. Let V be a Uq(glN )-module and v ∈ V a singular vector of weight Λ ,..., Λ . 1 N−1 1 1 N−1 N−1 Let ξ , . . . , ξ be nonnegative integers and t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . In the eva- luation Uq(glgN )-module V (x), one has −→ ma+1,b(mab−ma+1,b) −1|ξ| X Y q mab−ma+1,b Bξ(t)v = q − q ab a+1,b eˇab v (6.13) [m − m ]q! m 16b ab Here the sum is taken over all collections of nonnegative integers m , 1 6 b < a 6 N, such that a+1,a Na a+1,1 a+1,a a m > ··· > m and m + ··· + m = ξ for all a = 1,...,N − 1; by convention, mN+1,a = 0 for any a = 1,...,N. Other notation is as follows: in the ordered product the ξ ~ ~ factor eˇab is to the left of the factor eˇcd if b < d, or b = d and a < c, Symt is defined by (6.6), ab a1 ab a+1,a a and mb = m + ··· + m for all 1 6 b < a 6 N, in particular, mb = ξ . − Theorem 6.6 ([22]). Let V1, V2 be Uq (glgN )-modules and v1 ∈ V1, v2 ∈ V2 weight singular − 1 N−1 vectors with respect to the action of Uq (glgN ). Let ξ , . . . , ξ be nonnegative integers and 1 1 N−1 N−1 t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . Then N−1 "N−2 ηa+1 ξa a+1 −1 a X Y 1 ξ Y Y Y qti − q tj Bξ(t)(v1 ⊗ v2) = Symt [ξa − ηa] ![ηa] ! ta+1 − ta η a=1 q q a=1 i=1 j=ηa+1 i j N−1 ηa ξa # Y Y − a Y − a × La+1,a+1 tj v1 Laa ti v2 Bη(t[η])v1 ⊗ Bξ−η(t(η,ξ])v2 , (6.14) a=1 i=1 j=ηa+1 the sum being taken over all η = η1, . . . , ηN−1 ∈ N−1 such that ξ − η ∈ N−1. In the left Z>0 Z>0 − side we assume that Bξ(t) acts in the Uq (glgN )-module V1 ⊗ V2. − Corollary 6.7. Let V1,...,Vn be Uq (glgN )-modules and vr ∈ Vr, r = 1, . . . , n, weight singular − 1 N−1 vectors with respect to the action of Uq (glgN ). Let ξ , . . . , ξ be nonnegative integers and 1 1 N−1 N−1 t = t1, . . . , tξ1 ; ... ; t1 , . . . , tξN−1 . Then Bξ(t)(v1 ⊗ · · · ⊗ vn) a+1 a N−1 n "N−2 n−1 ηr ξ a+1 −1 a X Y Y 1 ξ Y Y Y Y qti − q tj = Sym a a t a+1 a [ηr − η ]q! t − t η1,...,ηn−1 a=1 r=1 r−1 a=1 r=1 a+1 j=ηa+1 i j i=ηr−1 +1 r a a N−1 n ηr−1 ξ Y Y Y − a Y − a × Laa ti vr La+1,a+1 tj vr a a=1 r=1 i=1 j=ηr +1 # × Bη1 (t[η1])v1 ⊗ Bη2−η1 (t(η1,η2])v2 ⊗ · · · ⊗ Bξ−ηn−1 (t(ηn−1,ξ])vn . (6.15) Here the sum is taken over all η , . . . , η ∈ N−1, η = η1, . . . , ηN−1, such that η − η ∈ 1 n−1 Z>0 r r r r+1 r N−1 for any r = 1, . . . , n − 1, and η = 0, η = ξ, by convention. The sets t , t are Z>0 0 n [η1] (ηr,ηr+1] − defined by (3.6). In the left side we assume that Bξ(t) acts in the Uq (glgN )-module V1 ⊗ · · · ⊗ Vn. Combinatorial Formulae for Nested Bethe Vectors 25 Remark. Denominators in the right sides of formulae (6.10)–(6.15) contain q-factorials, which can vanish when q is a root of unity. Nevertheless, the right sides remain well-defined at roots of unity. This happens due to the fact that the symmetrized expressions in square brackets have nontrivial stationary subgroups, cf. Remark at the end of Section2, so the result of the ξ symmetrization Symt divided by the product of q-factorials can be replaced by the sum over the cosets. Proofs of Theorems 6.4, 6.2 and 6.6 are similar to those of Theorems 3.3, 3.1 and 3.5, respec- tively. Here we mention only the required modifications of technical facts: identity (4.1) and Lemmas 4.5, 4.6. The analogue of the identity (4.1) is −1 X Y q sσi − qsσj = [k]q!, (6.16) sσi − sσj σ∈Sk 16i Lemma 6.8. Let p, r be positive integers such that p 6 r. Then p −1 −1 Y 1 Y qyi − q zj+r−p Y q zi − qzj Symz1,...,zr yi − zi+r−p yi − zj+r−p zi − zj i=1 i i) f(z1, . . . , zr) is symmetric in z1, . . . , zr. ii) f(z1, . . . , zr) is a rational function of z1 with only simple poles located at z1 = yi, i = 1, . . . , p, and regular as z1 → ∞. 2 iii) Res f z1, q yi, z3, . . . , zr = 0 for any i = 1, . . . , p. z1=yi p−r iv) f(uz1, . . . , uzr) = u 1 + o(1) as u → ∞. Denote by Cr−p(y1, . . . , yp; z1, . . . , zr) the collection of properties i)–iv), the subscript r − p referring to the exponent of u in property iv). Consider the partial fraction decomposition of f(z1, . . . , zr) as a function of z1: p ˜ X fi(z2, . . . , zr) f(z , . . . , z ) = f (z , . . . , z ) + . (6.19) 1 r 0 2 r y − z i=1 i 1 26 V. Tarasov and A. Varchenko Then the function f0(z2, . . . , zr) has the properties Cr−p−1(y1, . . . , yp; z2, . . . , zr), while the func- tion f˜i(z2, . . . , zr), i > 0, has the properties Cr−p(y1, . . . , yp; z2, . . . , zr) and 2 f˜i q yi, z3, . . . , zr = 0, cf. iii). The last claim is equivalent to the fact that the function r Y yi − zj f (z , . . . , z ) = f˜(z , . . . , z ) (6.20) i 2 r i 2 r q−1y − qz j=2 i j has the properties Cr−p(y1,..., ybi, . . . , yp; z2, . . . , zr). We expand the functions f0, . . . , fp similarly to (6.19), (6.20): p r −1 X fij(z3, . . . , zr) Y q yi − qzs f (z , . . . , z ) = f (z , . . . , z ) + , i 2 r i0 3 r y − z y − z j=1 j 2 s=3 i s j6=i and observe that the function f00 has the properties Cr−p−2(y1, . . . , yp; z3, . . . , zr), the functions f0i, fi0, i > 0 have the properties Cr−p−1(y1,..., ybi, . . . , yp; z3, . . . , zr), and the function fij, i, j > 0 has the properties Cr−p(y1,..., ybi,..., ybj, . . . , yp; z3, . . . , zr). Eventually, we obtain the following expansion of the function f(z1, . . . , zr): r −1 X Y Y qyαi − q zj f(z1, . . . , zr) = fα ϕαi (zi) , (6.21) yαi − zj α i=1 i Since the operations Vals,i in the left side of (6.22) can be applied to the function f(z1, . . . , zr) in any order without changing the answer, Valαr,r ··· Valα1,1 f(z1, . . . , zr) equals Val0,τ1 ··· Val0,τp Val1,τp+1 ··· Valp,τr f(z1, . . . , zr) (6.23) for a suitable permutation τ. Since f(z1, . . . , zr) is symmetric in z1, . . . , zr, expression (6.23) does not depend on τ and equals lim ··· lim Res ··· Res f(z1, . . . , zr). (6.24) z1→∞ zr−p→∞ zr−p+1=y1 zr=yp Due to the explicit formula for f(z1, . . . , zr), the terms in Symz1,...,zr which contribute nontrivially to expression (6.24) correspond to permutations that do not move the numbers r − p + 1, . . . , r. Using identity (6.16), we obtain that expression (6.24) equals −1 −1 p −p(r−p) Y q yi − qyj qyi − q yj (−1) q [r − p]q! 2 . 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