Scalar products in models with GL(3) trigonometric R-matrix. Highest coefficient
aa r X i v : . [ m a t h - ph ] N ov LAPTH-065/13
Scalar products in models with GL (3) trigonometric R -matrix. Highest coefficient S. Pakuliak a , E. Ragoucy b , N. A. Slavnov c ∗ a Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow reg., Russia,Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow reg., Russia,Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia b Laboratoire de Physique Th´eorique LAPTH, CNRS and Universit´e de Savoie,BP 110, 74941 Annecy-le-Vieux Cedex, France c Steklov Mathematical Institute, Moscow, Russia
Abstract
We study quantum integrable models with GL (3) trigonometric R-matrix solvable bythe nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can beexpressed in terms of a bilinear combination of the highest coefficients. We show that inthe models with GL (3) trigonometric R-matrix there exist two different highest coefficients.We obtain various representations for them in terms of sums over partitions. We also proveseveral important properties of the highest coefficients, which are necessary for the evaluationof the scalar products. Keywords:
Nested Bethe ansatz, scalar products, highest coefficient.
One of the striking facts about quantum integrable systems is the possibility to find the Hamil-tonian eigenvectors. Then, the knowledge of these eigenvectors allows one to have analyticalinsight on the form and the behavior of correlation functions for these models. The generalframework for such calculation is the Quantum Inverse Scattering Method [1–4], and the use ∗ [email protected], [email protected], [email protected]
1f the Bethe ansatz to construct the eigenvectors of the transfer matrix, which is a generatingfunctional of all commuting Hamiltonians. Unfortunately, if the method works well for the“simplest” cases based on GL (2) or U q ( b gl ) symmetries, it becomes quickly very technical formodels based on algebras of higher rank, and much less is known in these later cases.In the present paper we begin a systematic study of scalar products of the Bethe vectorsin quantum integrable models with GL (3) trigonometric R-matrix. The role of the scalarproducts is extremely important in the study of correlation functions [3, 5–7]. In particular,focusing on the class of quantum integrable models where the inverse scattering problem can besolved [8, 9], one can reduce the problem of calculation of the form factors and the correlationfunctions of local operators to the calculation of the scalar products of the Bethe vectors [8].Furthermore explicit and compact formulas for the scalar products sometimes allow one to studythe correlation functions even in such models, for which the solution of the inverse scatteringproblem is not known [3, 5–7, 10]. This approach was successfully applied for the quantumintegrable models with GL (2)-invariant or GL (2) trigonometric R-matrix [11–22]. In all theseworks a determinant representation for the scalar products of the Bethe vectors obtained in [23]was essentially used.The problem of the scalar products appears to be much more sophisticated in the modelsbased on the higher rank algebras. The first results in this field were obtained by N. Reshetikhinfor the models with GL (3)-invariant R-matrix [24]. There, a formula for the scalar product ofgeneric Bethe vectors and a determinant representation for the norm of the transfer matrixeigenvectors were found. In the Reshetikhin representation for the scalar product, the notion of“highest coefficient” plays the most important role. This function depends on the R-matrix ofthe model and appears to be a rational function of the Bethe parameters. The scalar product is abilinear combination of these highest coefficients. The knowledge of the highest coefficient allowsone, in some important particular cases, to reduce this bilinear combination to a determinantrepresentation [25–27] analogous to the one of [23].It was shown in [24] that the highest coefficient is equal to a partition function of the15-vertex model with special boundary conditions. Using this fact one can obtain explicitrepresentations for the highest coefficient in models with the GL (3)-invariant R-matrix [25, 28].Unfortunately, these results can not be directly extended to the case of models with GL (3)trigonometric R-matrix. The main reason is that the GL (3) trigonometric R-matrix is notsymmetric (see (2.2)). This leads to the fact that in these models actually there are two highestcoefficients, which have essentially different explicit representations. The main purpose of thispaper is to derive these explicit formulas. We also establish a number of important propertiesof the highest coefficients, which are necessary for the calculation of the scalar products of theBethe vectors.In contrast to the Reshitihin’s approach, we do not associate the highest coefficients withsome partition functions. Instead we use a more direct method for their calculation. The firsttool of our approach is an explicit representation for the dual Bethe vectors [29]. It is worthmentioning that in pioneer papers on the nested Bethe ansatz [2, 30, 31] no explicit formulasfor the Bethe vectors and the dual ones were given. More detailed formulas were obtained in[32] in the theory of solutions of the quantum Knizhnik–Zamolodchikov equation. There theBethe vectors were given by certain trace over auxiliary spaces of the products of monodromymatrices and R-matrices. 2xplicit expressions for the Bethe vectors in terms of the monodromy matrix elements forthe models with the GL ( N ) trigonometric R-matrix were obtained in the work [33], where therealization of Bethe vectors in terms of the current generators of the quantum affine algebra U q ( b gl N ) [34] was used (see also [35]).The second tool of our method is based on the formulas of the multiple action of themonodromy matrix entries onto the Bethe vectors [36]. Using these formulas one can calculatenot only the highest coefficients, but the whole scalar product of the Bethe vectors. However,the last problem is much more technical. It requires, in particular, the knowledge of severalnon-obvious properties of the highest coefficients. Therefore we postpone its solution to ourfurther publication. In the present paper we restrict ourselves with the study of the highestcoefficients only.The plan of the paper is as follows. In section 2, we present the model we work with, andintroduce the notations that will be used throughout the paper. We also recall some resultsobtained previously and needed here. In section 3, we exhibit the main result of the paper, asum formulas for the highest coefficients. The proof of the sum formulas is given in section 4.Section 5 gathers different properties of the highest coefficients, as well as some alternativepresentations for them. Appendices collect different formulas or proofs of formulas, needed inthe paper. We consider a quantum integrable model defined by the monodromy matrix T ( u ) with thematrix elements T ij ( u ), i, j = 1 , , u, v ) · ( T ( u ) ⊗ ) · ( ⊗ T ( v )) = ( ⊗ T ( v )) · ( T ( u ) ⊗ ) · R( u, v ) , (2.1)with the GL (3) trigonometric quantum R-matrixR( u, v ) = f ( u, v ) X ≤ i ≤ E ii ⊗ E ii + X ≤ i
3. Namely, whenever such an operator or a scalarfunction depends on a set of variables (for instance, T ij ( ¯ w ), λ i (¯ u ), r k (¯ v )), this means that wedeal with the product of the operators or the scalar functions with respect to the correspondingset: T ij ( ¯ w ) = Y w k ∈ ¯ w T ij ( w k ); λ (¯ u ) = Y u j ∈ ¯ u λ ( u j ); r k (¯ v ℓ ) = Y v j ∈ ¯ vv j = v ℓ r k ( v j ) . (2.8)4ere and below the notation ¯ v ℓ for an arbitrary set ¯ v means the set ¯ v \ v ℓ . A similar conventionwill be used for the products of functions f ( u, v ) f ( w i , ¯ w i ) = Y w j ∈ ¯ ww j = w i f ( w i , w j ); f (¯ u, ¯ v ) = Y u j ∈ ¯ u Y v k ∈ ¯ v f ( u j , v k ) . (2.9)Partitions of sets into two or more subsets will be noted as ¯ u ⇒ { ¯ u I , ¯ u II } . Here the romannumbers are used for the numeration of subsets ¯ u I and ¯ u II . Union of sets is denoted by braces,for example, { ¯ w, ¯ u } = ¯ η .In various formulas the Izergin determinant K k (¯ x | ¯ y ) appears [38]. It is defined for two sets¯ x and ¯ y of the same cardinality x = y = k : K k (¯ x | ¯ y ) = Q ≤ i,j ≤ k ( qx i − q − y j ) Q ≤ i Note that the restrictions on the cardinalities of subsets in the formulas (2.13) and(2.14) are shown explicitly by the subscripts of the Izergin determinants and the superscriptsof the Bethe vectors. However, for convenience we will describe such the restrictions in specialcomments after formulas.If we set n = a in (2.13), then ¯ η III = ∅ , and we obtain T ( ¯ w ) B a,b (¯ u ; ¯ v ) = ( − q ) a λ ( ¯ w ) X r (¯ η I ) f (¯ η II , ¯ η I ) f ( ¯ ξ II , ¯ ξ I ) f ( ¯ ξ II , ¯ η I ) × K ( r ) a ( q − ¯ w | ¯ η II ) K ( l ) a (¯ η I | q ¯ ξ I ) K ( l ) a ( ¯ ξ I | q ¯ w ) B ,b ( ∅ ; ¯ ξ II ) . (2.15)If in addition ¯ v = ∅ and we want to find a coefficient of r ( ¯ w ), then ¯ ξ I = ¯ w , ¯ ξ II = ∅ , and weshould set ¯ η I = ¯ w , ¯ η II = ¯ u . Using (A.3) and (A.4) we obtain T ( ¯ w ) B a, (¯ u ; ∅ ) = λ ( ¯ w ) r ( ¯ w ) K ( l ) a (¯ u | ¯ w ) | i + IT , (2.16)where IT stands for irrelevant terms , i.e. terms that do not contribute to the coefficient weconsider.Similarly, if we set n = b in (2.14), then ¯ ξ III = ∅ , and we obtain T ( ¯ w ) B a,b (¯ u ; ¯ v ) = ( − q ) − b λ ( ¯ w ) X r ( ¯ ξ I ) f ( ¯ ξ I , ¯ ξ II ) f (¯ η I , ¯ η II ) f ( ¯ ξ I , ¯ η II ) × K ( r ) b ( q − ¯ w | ¯ η I ) K ( r ) b ( q − ¯ η I | ¯ ξ I ) K ( l ) b ( ¯ ξ II | q ¯ w ) B a, (¯ η II ; ∅ ) . (2.17)If in addition ¯ u = ∅ and we want to find a coefficient of r ( ¯ w ), then ¯ η I = ¯ w , ¯ η II = ∅ , and weshould set ¯ ξ I = ¯ w , ¯ ξ II = ¯ v . Using (A.3) and (A.4) we obtain T ( ¯ w ) B ,b ( ∅ ; ¯ v ) = λ ( ¯ w ) r ( ¯ w ) K ( r ) b ( ¯ w | ¯ v ) | i + IT . (2.18)Observe that the actions (2.16), (2.18) reproduce the known results for the models with GL (2)trigonometric R-matrix. We have mentioned already that the Bethe vectors are given by certain polynomials in themonodromy matrix elements T ( u ), T ( u ), T ( u ) applied to the vector | i . The explicit6orm of these polynomials is not essential in the formulas for the multiple action (2.13), (2.14).However we need explicit representations for the dual Bethe vectors in terms of the monodromymatrix elements in order to obtain formulas for the highest coefficients. Such representationswere obtained in our work [29]. We give two of them: C a,b (¯ u ; ¯ v ) = X K ( r ) k (¯ v I | ¯ u I ) λ (¯ v II ) λ (¯ u ) f (¯ v II , ¯ v I ) f (¯ u I , ¯ u II ) f (¯ v, ¯ u ) h | T (¯ v II ) T (¯ u II ) T (¯ u I ) , (2.19)and C a,b (¯ u ; ¯ v ) = X K ( l ) k (¯ v I | ¯ u I ) λ (¯ u II ) λ (¯ v ) f (¯ v II , ¯ v I ) f (¯ u I , ¯ u II ) f (¯ v, ¯ u ) h | T (¯ u II ) T (¯ v II ) T (¯ v I ) . (2.20)Here the sum goes over all partitions of the sets ¯ u ⇒ { ¯ u I , ¯ u II } and ¯ v ⇒ { ¯ v I , ¯ v II } such that u I = v I = k , k = 0 , . . . , min( a, b ).Both of these representations are needed for our purpose. They correspond to two differentembeddings of U q ( b gl ) into U q ( b gl ) algebra. It is also easy to check that (2.19) and (2.20) arerelated by the isomorphism ϕ described in [29]. This isomorphism maps the original algebra U q ( b gl ) to the algebra U q − ( b gl ) ϕ (cid:0) T i,j ( u ) (cid:1) = ˜ T − j, − i ( u ) , (2.21)where T ( u ) ∈ U q ( b gl ) and ˜ T ( u ) ∈ U q − ( b gl ) respectively. The map (2.21) is a very powerful toolfor the study of the scalar products. In particular, many properties of the scalar products canbe established via the mapping between U q ( b gl ) and U q − ( b gl ). The scalar products are defined as S a,b (¯ u C ; ¯ v C | ¯ u B ; ¯ v B ) = C a,b (¯ u C ; ¯ v C ) B a,b (¯ u B ; ¯ v B ) , (3.1)where all the Bethe parameters are generic complex numbers. We have added the superscripts C and B to the sets ¯ u , ¯ v in order to stress that the vectors C a,b and B a,b may depend on differentsets of parameters.Knowing the explicit form of the dual Bethe vectors (2.19), (2.20) and the multiple actionof the operators T ij (2.13), (2.14) one can formally calculate the scalar product (3.1). It isclear that the result is given as a sum with respect to partitions of the sets ¯ u C , ¯ u B , ¯ v C , and ¯ v B .The terms of this sum depend on the products of the vacuum eigenvalues r and r , as well ason the functions entering the R-matrix. In complete analogy with the case of GL (3)-invariantR-matrix (see e.g. [24]) one can derive the following representation: S a,b (¯ u C ; ¯ v C | ¯ u B ; ¯ v B ) = X r (¯ u C II ) r (¯ u B I ) r (¯ v C II ) r (¯ v B I ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) W part (cid:18) ¯ u C II , ¯ u B II , ¯ u C I , ¯ u B I ¯ v C I , ¯ v B I , ¯ v C II , ¯ v B II (cid:19) . (3.2) For the complete calculation of the scalar product one should also know the multiple action of the operator T ( u ). This action can be found in [36]. However for the calculation of the highest coefficients this action is notneeded. u C ⇒ { ¯ u C I , ¯ u C II } , ¯ u B ⇒ { ¯ u B I , ¯ u B II } , ¯ v C ⇒ { ¯ v C I , ¯ v C II } and¯ v B ⇒ { ¯ v B I , ¯ v B II } with u C I = u B I and v C I = v B I . The form of the functions W part dependson the partitions, what is shown by the subscript ‘part’. They also depend on the R-matrixentries, but not on the functions r and r . In other words, they depend on the algebra, not onthe representations one chooses.The highest coefficients Z ( l,r ) a,b are defined as particular cases of the functions W part , corre-sponding to special choices of partitions: Z ( l ) a,b (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) = W part (cid:18) ¯ u C , ¯ u B , ∅ , ∅ ¯ v C , ¯ v B , ∅ , ∅ (cid:19) , Z ( r ) a,b (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ) = W part (cid:18) ∅ , ∅ , ¯ u C , ¯ u B , ∅ , ∅ , ¯ v C , ¯ v B (cid:19) . (3.3)Just as in the case of the Izergin determinant, we call these coefficients left and right. Thesubscripts of the highest coefficients shows that u C = u B = a and v C = v B = b .Similarly to the GL (3)-invariant case, all other coefficients W part in (3.2) can be expressedin term of products of the left and the right highest coefficients . Thus, the scalar product is abilinear combination of the highest coefficients, and that is why the role of Z ( l,r ) a,b is so important.In the case of the GL (3)-invariant R-matrix, the two highest coefficients coincide and areequal to a partition function of the 15-vertex model with special boundary conditions [24].However, as we have already mentioned, in the models with GL (3) trigonometric R-matrix theleft and right highest coefficients differ from each other. The main result of the paper is anexplicit expression for them. Proposition 3.1. The left and right highest coefficients have the following representations: Z ( l ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) = ( − q ) − b X K ( r ) b (¯ s | ¯ w I q ) K ( l ) a ( ¯ w II | ¯ t ) K ( l ) b (¯ y | ¯ w I ) f ( ¯ w I , ¯ w II ) , (3.4) Z ( r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) = ( − q ) b X K ( l ) b (¯ s | ¯ w I q ) K ( r ) a ( ¯ w II | ¯ t ) K ( r ) b (¯ y | ¯ w I ) f ( ¯ w I , ¯ w II ) . (3.5) Here ¯ w = { ¯ s, ¯ x } . The sum is taken with respect to partitions of the set ¯ w ⇒ { ¯ w I , ¯ w II } with w I = b and w II = a . We would like to draw the attention of the reader that the difference between Z ( l ) a,b and Z ( r ) a,b is much more essential than the one between K ( l ) n and K ( r ) n . To see this one can consider theexplicit expressions for the highest coefficients in the simplest nontrivial case a = b = 1: Z ( l )1 , ( t ; x | s ; y ) = xy g ( x, t ) g ( y, s ) f ( s, x ) + xys g ( x, s ) g ( s, t ) g ( y, x ) , Z ( r )1 , ( t ; x | s ; y ) = ts g ( x, t ) g ( y, s ) f ( s, x ) + tsx g ( x, s ) g ( s, t ) g ( y, x ) . (3.6)From this we find, for example,( ts ) − Z ( r )1 , ( t ; x | s ; y ) − ( xy ) − Z ( l )1 , ( t ; x | s ; y ) = ( q − q − ) g ( s, t ) g ( y, x ) . (3.7) See (5.29) for the explicit formula. The detailed proof will be given in a forthcoming paper. Z ( l ) a,b and Z ( r ) a,b can not be removed via simple multiplication of them by certain sets ofvariables, like in (2.12).Below, to save space, we will combine the formulas for Z ( l ) and Z ( r ) into one. For instance,the equations (3.4) and (3.5) can be written as follows: Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) = ( − q ) ∓ b X K ( r,l ) b (¯ s | ¯ w I q ) K ( l,r ) a ( ¯ w II | ¯ t ) K ( l,r ) b (¯ y | ¯ w I ) f ( ¯ w I , ¯ w II ) . (3.8)The superscript ( l, r ) on Z a,b means that the equation (3.8) is valid for Z ( l ) a,b and for Z ( r ) a,b sepa-rately. Choosing the first or the second component of ( l, r ) and the corresponding (up or downresp.) exponent of ( − q ) ∓ b in this equation, we obtain either (3.4) or (3.5).Similarly to the case of the GL (3)-invariant R-matrix there exist slightly different represen-tations, so-called twin formula for the highest coefficients: Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) = ( − q ) ∓ a X K ( r,l ) a ( ¯ w II | ¯ xq ) K ( l,r ) a ( ¯ w II | ¯ t ) K ( l,r ) b (¯ y | ¯ w I ) f ( ¯ w I , ¯ w II ) . (3.9)All the notations are the same as in (3.8). This formula follows from the reduction propertiesof the Izergin determinants (A.3). Indeed, we have( − q ) ± b K ( l,r ) b (¯ s | ¯ w I q ) = ( − q ) ± ( a + b ) K ( l,r ) a + b ( { ¯ s, ¯ x }|{ ¯ w I q , ¯ xq } )= ( − q ) ± ( a + b ) K ( l,r ) a + b ( { ¯ w I , ¯ w II }|{ ¯ w I q , ¯ xq } ) = ( − q ) ± a K ( l,r ) a ( ¯ w II | ¯ xq ) , (3.10)where the superscript ( l, r ) has the same meaning as in (3.8). Due to (3.10) the equivalence of(3.8) and (3.9) becomes evident. Other representations for the highest coefficients in terms ofsums over partitions are given in section 5.2. In order to derive (3.4), (3.5) we should calculate the scalar product S a,b (¯ u C ; ¯ v C | ¯ u B ; ¯ v B ) andfind rational coefficients of the products r (¯ u B ) r (¯ v C ) and r (¯ u C ) r (¯ v B ). r (¯ u B ) r (¯ v C ) Here we calculate the coefficient of r (¯ u B ) r (¯ v C ). We start with the dual Bethe vector in theform (2.19). For our goal it is enough to take only one term from the sum over partitionscorresponding to k = 0: C a,b (¯ u C ; ¯ v C ) = h | T (¯ v C ) T (¯ u C ) λ (¯ v C ) λ (¯ u C ) f (¯ v C , ¯ u C ) + IT , (4.1)and we recall that IT means the terms that do not contribute to the coefficient we consider.Indeed, acting with C a,b (¯ u C ; ¯ v C ) on the Bethe vector we want to obtain the product r (¯ v C ) overthe complete set ¯ v C . This is possible if and only if the product T (¯ v C II ) in (2.19) depends onthe complete set ¯ v C , that is ¯ v C II = ¯ v C . Hence, ¯ u C I = ¯ v C I = ∅ , and all other terms in (2.19) are notessential. 9t remains to act successively with T (¯ u C ) and T (¯ v C ) on the Bethe vector. Using (2.15)we obtain C a,b (¯ u C ; ¯ v C ) B a,b (¯ u B ; ¯ v B ) = ( − q ) a λ (¯ v C ) f (¯ v C , ¯ u C ) X r (¯ η I ) f (¯ η II , ¯ η I ) f ( ¯ ξ II , ¯ ξ I ) f ( ¯ ξ II , ¯ η I ) × K ( r ) a ( q − ¯ u C | ¯ η II ) K ( l ) a (¯ η I | q ¯ ξ I ) K ( l ) a ( ¯ ξ I | q ¯ u C ) h | T (¯ v C ) B ,b ( ∅ ; ¯ ξ II ) + IT . (4.2)Here ¯ η = { ¯ u C , ¯ u B } and ¯ ξ = { ¯ u C , ¯ v B } . The sum is taken over partitions ¯ η ⇒ { ¯ η I , ¯ η II } and¯ ξ ⇒ { ¯ ξ I , ¯ ξ II } with η I = ξ I = a .The remaining action of T (¯ v C ) on B ,b ( ∅ ; ¯ ξ II ) should be calculated via (2.18). This gives us C a,b (¯ u C ; ¯ v C ) B a,b (¯ u B ; ¯ v B ) = ( − q ) a r (¯ v C ) f (¯ v C , ¯ u C ) X r (¯ η I ) f (¯ η II , ¯ η I ) f ( ¯ ξ II , ¯ ξ I ) f ( ¯ ξ II , ¯ η I ) × K ( r ) a ( q − ¯ u C | ¯ η II ) K ( l ) a (¯ η I | q ¯ ξ I ) K ( l ) a ( ¯ ξ I | q ¯ u C ) K ( r ) b (¯ v C | ¯ ξ II ) + IT . (4.3)Now we should extract from the sum (4.3) the terms proportional to r (¯ u B ). For this weshould simply set ¯ η I = ¯ u B and ¯ η II = ¯ u C . After elementary algebra based on the use of (A.3) and(A.4) we arrive at C a,b (¯ u C ; ¯ v C ) B a,b (¯ u B ; ¯ v B ) = r (¯ u B ) r (¯ v C ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) Z ( r ) a,b (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ) + IT , (4.4)where Z ( r ) a,b has the following form: Z ( r ) a,b (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ) = ( − q ) a X K ( l ) a ( ¯ ξ I | q ¯ u C ) K ( r ) a ( ¯ ξ I | ¯ u B ) K ( r ) b (¯ v C | ¯ ξ II ) f ( ¯ ξ II , ¯ ξ I ) , (4.5)where ¯ ξ = { ¯ u C , ¯ v B } , and the sum is taken over partitions ¯ ξ ⇒ { ¯ ξ I , ¯ ξ II } with ξ I = a . Thiscoincides with (3.9) up to notations. r (¯ u C ) r (¯ v B ) The calculation is very similar to the previous one. This time we start with the dual Bethevector in the form (2.20). Now we want to obtain the coefficient of r (¯ u C ) r (¯ v B ), therefore wehave C a,b (¯ u C ; ¯ v C ) = h | T (¯ u C ) T (¯ v C ) λ (¯ v C ) λ (¯ u C ) f (¯ v C , ¯ u C ) + IT . (4.6)We should act successively with T (¯ v C ) and T (¯ u C ) on the Bethe vector. Using (2.17) weobtain C a,b (¯ u C ; ¯ v C ) B a,b (¯ u B ; ¯ v B ) = ( − q ) − b λ (¯ u C ) f (¯ v C , ¯ u C ) X r ( ¯ ξ I ) f ( ¯ ξ I , ¯ ξ II ) f (¯ η I , ¯ η II ) f ( ¯ ξ I , ¯ η II ) × K ( r ) b ( q − ¯ v C | ¯ η I ) K ( r ) b ( q − ¯ η I | ¯ ξ I ) K ( l ) b ( ¯ ξ II | q ¯ v C ) h | T (¯ u C ) B a, (¯ η II ; ∅ ) + IT . (4.7)10ere ¯ η = { ¯ v C , ¯ u B } and ¯ ξ = { ¯ v C , ¯ v B } . The sum is taken over partitions ¯ η ⇒ { ¯ η I , ¯ η II } and¯ ξ ⇒ { ¯ ξ I , ¯ ξ II } with η I = ξ I = b .Now we act with T (¯ u C ) on B a, (¯ η II ; ∅ ) via (2.16) C a,b (¯ u C ; ¯ v C ) B a,b (¯ u B ; ¯ v B ) = ( − q ) − b r (¯ u C ) f (¯ v C , ¯ u C ) X r ( ¯ ξ I ) f ( ¯ ξ I , ¯ ξ II ) f (¯ η I , ¯ η II ) f ( ¯ ξ I , ¯ η II ) × K ( r ) b ( q − ¯ v C | ¯ η I ) K ( r ) b ( q − ¯ η I | ¯ ξ I ) K ( l ) b ( ¯ ξ II | q ¯ v C ) K ( l ) a (¯ η II | ¯ u C ) + IT . (4.8)Setting here ¯ ξ I = ¯ v B and ¯ ξ II = ¯ v C we obtain after trivial algebra (and the use of (A.3), (A.4)) C a,b (¯ u C ; ¯ v C ) B a,b (¯ u B ; ¯ v B ) = r (¯ u C ) r (¯ v B ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) Z ( l ) a,b (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) + IT , (4.9)where Z ( l ) a,b has the following form: Z ( l ) a,b (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) = ( − q ) − b X K ( r ) b ( q − ¯ v C | ¯ η I ) K ( l ) b (¯ v B | ¯ η I ) K ( l ) a (¯ η II | ¯ u C ) f (¯ η I , ¯ η II ) , (4.10)and ¯ η = { ¯ u B , ¯ v C } . This coincides with (3.4) up to notations.Thus, we have proved that the coefficient of the product r (¯ u B ) r (¯ v C ) is proportional to thefunction Z ( r ) a,b (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ), while the coefficient of the product r (¯ u C ) r (¯ v B ) is proportional toto the function Z ( l ) a,b (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ). Z ( l ) and Z ( r ) In order to obtain a complete formula for the scalar product one should know some properties ofthe highest coefficients. In particular, different representations for Z ( l,r ) a,b are of great importance.The description of the residues of Z ( l,r ) a,b in their poles, as well as some reduction properties, alsoare useful. In this section we give a list of properties of the highest coefficients. It is easy to see that both highest coefficients are symmetric with respect to all the permutationsof variables in any of the four sets: ¯ t , ¯ x , ¯ s , and ¯ y . It also follows immediately from the definitions(3.8), (3.9) that Z ( l,r ) a, (¯ t ; ¯ x |∅ ; ∅ ) = K ( l,r ) a (¯ x | ¯ t ) , Z ( l,r )0 ,b ( ∅ ; ∅| ¯ s ; ¯ y ) = K ( l,r ) b (¯ y | ¯ s ) . (5.1)Using (A.2) one can easily see that the highest coefficients are invariant under the rescaling ofall arguments Z ( l,r ) a,b ( α ¯ t ; α ¯ x | α ¯ s ; α ¯ y ) = Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) . (5.2)In order to describe more sophisticated properties one should use different representations forthe highest coefficients. 11 .2 Different representations of Z ( l ) and Z ( r ) Just like in the case of the GL (3)-invariant R-matrix there exist several representations for thehighest coefficients in terms of sums over partitions. The original formula (3.8) is given in termsof the sums over partitions of the union of the sets { ¯ x, ¯ s } = ¯ w . There are also representationsin terms of the sums over partitions of the unions of the sets { ¯ tq − , ¯ y } , { ¯ t, ¯ x } , and { ¯ s, ¯ y } . Wegive the complete list of these representations below. • Representations in terms of the partitions of { ¯ tq − , ¯ y } . Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) = ( − q ) ∓ a f (¯ y, ¯ x ) f (¯ s, ¯ t ) X K ( r,l ) a (¯ tq − | ¯ η I q ) K ( l,r ) a (¯ xq − | ¯ η I ) K ( l,r ) b (¯ η II | ¯ s ) f (¯ η I , ¯ η II ) . (5.3)Here ¯ η = { ¯ y, ¯ tq − } . The sum is taken with respect to the partitions ¯ η ⇒ { ¯ η I , ¯ η II } with η I = a and η II = b .These representations also have a twin formula: Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) = ( − q ) ∓ b f (¯ y, ¯ x ) f (¯ s, ¯ t ) X K ( r,l ) b (¯ η II | ¯ yq ) K ( l,r ) a (¯ xq − | ¯ η I ) K ( l,r ) b (¯ η II | ¯ s ) f (¯ η I , ¯ η II ) . (5.4)All the notations are the same as in (5.3). The twin-formula follows from (5.3) due to theidentity ( − q ) ∓ a K ( r,l ) a (¯ tq − | ¯ η I q ) = ( − q ) ∓ b K ( r,l ) b (¯ η II | ¯ yq ). • Representations in terms of the partitions of { ¯ t, ¯ x } . Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) = X ( − q ) ± n f (¯ s, ¯ t I ) f (¯ y, ¯ x II ) f (¯ t I , ¯ t II ) f (¯ x II , ¯ x I ) × K ( l,r ) n (¯ x I | ¯ t I ) K ( l,r ) a − n (¯ x II | ¯ t II q − ) K ( l,r ) b + n ( { ¯ y, ¯ t I q − }|{ ¯ s, ¯ x I } ) . (5.5)The sum is taken with respect to all partitions ¯ t ⇒ { ¯ t I , ¯ t II } and ¯ x ⇒ { ¯ x I , ¯ x II } with t I = x I = n , n = 0 , , . . . , a . • Representations in terms of the partitions of { ¯ s, ¯ y } . Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) = X ( − q ) ± n f (¯ s II , ¯ t ) f (¯ y I , ¯ x ) f (¯ s I , ¯ s II ) f (¯ y II , ¯ y I ) × K ( l,r ) n (¯ y I | ¯ s I ) K ( l,r ) b − n (¯ y II | ¯ s II q − ) K ( l,r ) a + n ( { ¯ s I , ¯ x }|{ ¯ y I q , ¯ t } ) . (5.6)The sum is taken with respect to all partitions ¯ s ⇒ { ¯ s I , ¯ s II } and ¯ y ⇒ { ¯ y I , ¯ y II } with s I = y I = n , n = 0 , , . . . , b .All the representations above follow from the original ones. In complete analogy with thecase of the GL (3)-invariant R-matrix the sums over partitions in (3.8) can be presented asmultiple contour integrals, where the integration contours surround the points ¯ w = { ¯ s, ¯ x } . Thenmoving these contours to the points { ¯ t, ¯ x } or { ¯ s, ¯ y } (depending on the specific representation)one obtains the equations (5.5) or (5.6). We refer the reader to the work [28] for the details ofthis derivation. 12epresentations (5.5) and (5.6) allow us to prove a very important property of Z ( l,r ) : Z ( l,r ) b,a (¯ s ; ¯ y | ¯ tq − ; ¯ xq − ) = f − (¯ y, ¯ x ) f − (¯ s, ¯ t ) Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) . (5.7)This formula can be obtained by substitution of the Z ( l,r ) b,a (¯ s ; ¯ y | ¯ tq − ; ¯ xq − ) into (5.5). This willgive us (5.6) for the Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ).The property (5.7) immediately implies the representations (5.3), (5.4). Indeed, one caneasily check that using (3.8) for Z ( l,r ) b,a (¯ s ; ¯ y | ¯ tq − ; ¯ xq − ) we obtain the representations (5.3) for Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ).One more property of the highest coefficients with respect to re-ordering of their argumentshas the following form: Z ( l,r ) a,b ; q − (¯ t ; ¯ x | ¯ s ; ¯ y ) = Z ( r,l ) b,a ; q (¯ y ; ¯ s | ¯ x ; ¯ t ) . (5.8)Here we have added to the highest coefficients the subscripts q and q − , in order to stress thatin the l.h.s. the function Z ( l,r ) a,b is evaluated with replacement q by q − . On the contrary, in ther.h.s. of (5.8) the highest coefficient is evaluated at the same q , but with replacement left byright, a by b , and re-ordering of the arguments. Usually we omit the additional subscript q inthe formulas. The proof of this property is given in appendix B.1. Using (5.8) and (A.5) onecan easily check that the representation (5.6) follows from (5.5) after the replacement q by q − .We conclude this section by establishing the behavior of the highest coefficients as one oftheir arguments goes to infinity. For this, it is convenient to use the representations (5.5) and(5.6). Due to (A.7) and (A.8) one can easily convince himself that Z ( l ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) → , t i → ∞ or s j → ∞ , i = 1 , . . . , a, Z ( l ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) is bounded , x i → ∞ or y j → ∞ , j = 1 , . . . , b. (5.9)These equations together with the property (5.8) yield Z ( r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) → , x i → ∞ or y j → ∞ , i = 1 , . . . , a, Z ( r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) is bounded , t i → ∞ or s j → ∞ , j = 1 , . . . , b. (5.10) Z ( l,r ) a,b All the sum formulas for Z ( l,r ) a,b involve the products of three Izergin determinants. There exitsmore general formulas with three Izergin determinants, which are also reducible to the highestcoefficients. Such formulas are necessary for the derivation of sum representations for the scalarproduct of the Bethe vectors. Below we give the list of these formulas.Let a ≥ b . Then X K ( r,l ) b (¯ t I | ¯ yq ) K ( l,r ) b (¯ t I | ¯ sq ) K ( l,r ) a − b ( ¯ ξ | ¯ t II ) f (¯ t II , ¯ t I ) = ( − q ) ± b Z ( l,r ) a,b (¯ t ; { ¯ ξ, ¯ y }| ¯ s ; ¯ yq − ) f (¯ y, ¯ t ) f (¯ s, ¯ t ) , (5.11) X K ( l,r ) b (¯ t I | ¯ yq ) K ( r,l ) b (¯ t I | ¯ sq ) K ( l,r ) a − b ( ¯ ξ | ¯ t II ) f (¯ t II , ¯ t I ) = ( − q ) ± b Z ( l,r ) a,b (¯ t ; { ¯ ξ, ¯ s }| ¯ y ; ¯ sq − ) f (¯ y, ¯ t ) f (¯ s, ¯ t ) . (5.12)13ere the sum is taken over partitions ¯ t ⇒ { ¯ t I , ¯ t II } with t I = b and t II = a − b .Let now a ≤ b . Then X K ( r,l ) a ( q − ¯ t | ¯ y I ) K ( l,r ) a (¯ xq − | ¯ y I ) K ( l,r ) b − a (¯ y II | ¯ ξ ) f (¯ y I , ¯ y II ) = ( − q ) ± a Z ( l,r ) a,b (¯ tq ; ¯ x |{ ¯ ξ, ¯ t } ; ¯ y ) f (¯ y, ¯ t ) f (¯ y, ¯ x ) , (5.13) X K ( l,r ) a ( q − ¯ t | ¯ y I ) K ( r,l ) a (¯ xq − | ¯ y I ) K ( l,r ) b − a (¯ y II | ¯ ξ ) f (¯ y I , ¯ y II ) = ( − q ) ± a Z ( l,r ) a,b (¯ xq ; ¯ t |{ ¯ ξ, ¯ x } ; ¯ y ) f (¯ y, ¯ t ) f (¯ y, ¯ x ) . (5.14)Here the sum is taken over partitions ¯ y ⇒ { ¯ y I , ¯ y II } with y I = a and y II = b − a .All these formulas follow from the representations (5.3), (5.4) (see an example of the proofin appendix B.2). Note also that due to (5.8) and (A.5) the equations (5.13) and (5.14) followfrom respectively (5.11) and (5.12) via the replacement q → q − . The highest coefficients Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) have simple poles at t i = x j , t i = s k , x j = y ℓ , and s k = y ℓ . Similarly to the case of the GL (3)-invariant R-matrix the corresponding residues canbe expressed in terms of Z ( l,r ) a − ,b (¯ t ; ¯ x | ¯ s ; ¯ y ) or Z ( l,r ) a,b − (¯ t ; ¯ x | ¯ s ; ¯ y ). In particular, Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) (cid:12)(cid:12)(cid:12) s b → y b = f ( y b , s b ) f ( y b , ¯ s b ) f (¯ y b , y b ) f ( y b , ¯ x ) Z ( l,r ) a,b − (¯ t ; ¯ x | ¯ s b ; ¯ y b ) + reg , (5.15)where reg means regular part. We remind also that ¯ s b = ¯ s \ s b and ¯ y b = ¯ y \ y b .The residue at t a = x a has similar form Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) (cid:12)(cid:12)(cid:12) t a → x a = f ( x a , t a ) f ( x a , ¯ t a ) f (¯ x a , x a ) f (¯ s, x a ) Z ( l,r ) a − ,b (¯ t a ; ¯ x a | ¯ s ; ¯ y ) + reg . (5.16)It is worth mentioning that equations (5.15) and (5.16) are not independent, because they arerelated by the transforms (5.7) and (5.8).The formula for the residue at s b = t a is slightly more sophisticated. Namely, Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) (cid:12)(cid:12)(cid:12) s b → t a = f ( s b , t a ) f (¯ s b , s b ) f ( t a , ¯ t a ) × a X p =1 K ( l,r )1 ( x p | t a ) f (¯ x p , x p ) Z ( l,r ) a − ,b (¯ t a ; ¯ x p |{ ¯ s b , x p } ; ¯ y ) + reg . (5.17)Similarly the residue at y b = x a is given by Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) (cid:12)(cid:12)(cid:12) y b → x a = f ( y b , x a ) f (¯ y b , y b ) f ( x a , ¯ x a ) × b X p =1 K ( l,r )1 ( x a | s p ) f ( s p , ¯ s p ) Z ( l,r ) a,b − (¯ t ; { ¯ x a , s p }| ¯ s p ; ¯ y b ) + reg . (5.18)The derivation of all these formulas is exactly the same as in the case of the GL (3)-invariant R-matrix, therefore we refer the reader to [28] for the corresponding proofs. Note that the formulas(5.17) and (5.18) are related by the properties (5.8), (A.5), and the transform q → q − .14 .5 Multiple poles The residue formulas above imply multiple residue formulas. Namely, if z = n , then it followsfrom (5.15) and (5.16) thatlim ¯ z ′ → ¯ z f − (¯ z ′ , ¯ z ) Z ( l,r ) a,b + n (¯ t ; ¯ x |{ ¯ s, ¯ z } ; { ¯ y, ¯ z ′ } ) = f (¯ z, ¯ x ) f (¯ z, ¯ s ) f (¯ y, ¯ z ) Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) , (5.19)and lim ¯ z ′ → ¯ z f − (¯ z ′ , ¯ z ) Z ( l,r ) a + n,b ( { ¯ t, ¯ z } ; { ¯ x, ¯ z ′ }| ¯ s ; ¯ y ) = f (¯ z, ¯ t ) f (¯ x, ¯ z ) f (¯ s, ¯ z ) Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ y ) . (5.20)For z = 1 the equations (5.19), (5.20) are direct corollaries of (5.15), (5.16) respectively. Thenone can use trivial induction over n = z .The residue formula (5.17) implies the following reduction:lim ¯ z ′ → ¯ z f − (¯ z, ¯ z ′ ) Z ( l,r ) a,b ( { ¯ t, ¯ z ′ } ; ¯ x |{ ¯ s, ¯ z } ; ¯ y ) = f (¯ s, ¯ z ) f (¯ z, ¯ t ) × X K ( l,r ) n (¯ x I | ¯ z ) f (¯ x II , ¯ x I ) Z ( l,r ) a − n,b (¯ t ; ¯ x II |{ ¯ s, ¯ x I } ; ¯ y ) . (5.21)The sum is taken with respect to the partitions ¯ x ⇒ { ¯ x I , ¯ x II } with x I = n .Similarly, starting from (5.18) one can find thatlim ¯ z ′ → ¯ z f − (¯ z, ¯ z ′ ) Z ( l,r ) a,b (¯ t ; { ¯ x, ¯ z ′ }| ¯ s ; { ¯ y, ¯ z } ) = f (¯ y, ¯ z ) f (¯ z, ¯ x ) × X K ( l,r ) n (¯ z | ¯ s I ) f (¯ s I , ¯ s II ) Z ( l,r ) a,b − n (¯ t ; { ¯ x, ¯ s I }| ¯ s II ; ¯ y ) . (5.22)Here the sum is taken with respect to the partitions ¯ s ⇒ { ¯ s I , ¯ s II } with s I = n .For z = 1 the equations (5.21), (5.22) follow immediately from (5.17), (5.18) respectively.Then one can proceed via induction over n = z , using the identities (A.11), (A.12) (see detailsin appendix B.3). The reduction formulas (5.21), (5.22) can be transformed. Namely, one can apply the transform(5.7) to these equations (see appendix B.4). In this way we arrive at the following reductions Z ( l,r ) a,b ( { ¯ t, q ¯ z } ; ¯ x |{ ¯ s, ¯ z } ; ¯ y ) = X K ( l,r ) n (¯ y I | ¯ z ) Z ( l,r ) a,b − n ( { ¯ t, q ¯ y I } ; ¯ x | ¯ s ; ¯ y II ) f (¯ y II , ¯ y I ) f (¯ y I , ¯ x ) f (¯ y I , ¯ s ) . (5.23)The sum is taken with respect to the partitions ¯ y ⇒ { ¯ y I , ¯ y II } with y I = n . One more reductionhas the form Z ( l,r ) a,b (¯ t ; { ¯ x, ¯ z }| ¯ s ; { ¯ y, ¯ zq − } ) = X K ( l,r ) n (¯ z | ¯ t I ) Z ( l,r ) a − n,b (¯ t II ; ¯ x | ¯ s ; { ¯ y, ¯ t I q − } ) f (¯ t I , ¯ t II ) f (¯ x, ¯ t I ) f (¯ s, ¯ t I ) . (5.24)Here the sum is taken with respect to the partitions ¯ t ⇒ { ¯ t I , ¯ t II } with t I = n .15hese formulas have special cases, when the highest coefficients degenerate into the productsof the Izergin determinants. In particular, if b ≤ a and n = b , then in (5.23) ¯ s = ∅ and ¯ y II = ∅ .The equation (5.23) turns into Z ( l,r ) a,b ( { ¯ t, q ¯ z } ; ¯ x | ¯ z ; ¯ y ) = f (¯ y, ¯ x ) K ( l,r ) b (¯ y | ¯ z ) K ( l,r ) a (¯ x |{ ¯ t, q ¯ y } ) . (5.25)Similarly, if a ≤ b and n = a , then in (5.24) ¯ x = ∅ and ¯ t II = ∅ . The equation (5.24) turns into Z ( l,r ) a,b (¯ t ; ¯ z | ¯ s ; { ¯ y, ¯ zq − } ) = f (¯ s, ¯ t ) K ( l,r ) a (¯ z | ¯ t ) K ( l,r ) b ( { ¯ y, ¯ tq − }| ¯ s ) . (5.26)We draw the attention of the reader that (5.26) is the image of (5.25) under the replacementof q by q − and the use of eqs. (5.8) and (A.5). The equations (5.23), (5.24) can be considered as summation identities, which allow one toexpress certain sums involving K ( l,r ) and Z ( l,r ) in terms of the highest coefficient Z ( l,r ) . In theseidentities one takes a sum with respect to partitions of one set of variables. There exist moresophisticated identities of similar type, where one takes a sum with respect to partitions of twosets of variables. In this section we give one of such the identities. It plays very important rolein the calculation of scalar products. Proposition 5.1. Let a , b , n , p be non-negative integers and p ≤ b . Let ¯ t , ¯ x , ¯ s , ¯ y , ¯ w , ¯ z be sixsets of generic complex variables with cardinalities t = a, x = a, z = n, s = b, y = p, w = b − p. (5.27) Then f ( ¯ ξ, ¯ y ) Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; { ¯ y, ¯ w } ) = X ( − q ) ∓ k K ( r,l ) p ( { ¯ s I q − , ¯ ξ I }| ¯ y ) Z ( l,r ) a,b − k (¯ t ; ¯ x | ¯ s II ; { ¯ w, ¯ ξ I } ) × f (¯ s I , ¯ s II ) f ( ¯ ξ II , ¯ ξ I ) f (¯ y, ¯ s I ) f ( ¯ w, ¯ s I ) f − (¯ s I , ¯ z ) . (5.28) Here ¯ ξ is a union of two sets: ¯ ξ = { ¯ xq − , ¯ zq − } . The sum is taken over partitions of the set ¯ s ⇒ { ¯ s I , ¯ s II } with s I = k ∈ [0 , . . . , p ] and the set ¯ ξ ⇒ { ¯ ξ I , ¯ ξ II } with ξ I = p − k . The proof of this identity is given in appendix B.5. Conclusion In this paper we have obtained several explicit representations for the highest coefficients Z ( l,r ) for integrable models based on GL (3) trigonometric R -matrix and found their properties. Ofcourse, this result is only a first step towards the calculation of the scalar products of Bethevectors and then of the correlation functions for local operators. The calculation of the scalarproducts will be done in our forthcoming publication, where we are going to use the presentresults. Indeed, as we have explained in section 3, the scalar product can be presented as a sum16ith respect to partitions of the Bethe parameters (3.2). The rational coefficients W part in thisequation are proportional to the product of the left and the right highest coefficients. Hence theknowledge of highest coefficients is essential in the calculation of the scalar product. To stressthis fact, we announce an explicit expression for W part , that will be proved in our forthcomingpublication. Proposition 5.2. The scalar product of two Bethe vectors (3.1) is given by equation (3.2) .For a fixed partition with u C I = u B I = k and v C I = v B I = n , (where k = 0 , . . . , a and n = 0 , . . . , b ), the rational coefficient W part has the form W part (cid:18) ¯ u C II , ¯ u B II , ¯ u C I , ¯ u B I ¯ v C I , ¯ v B I , ¯ v C II , ¯ v B II (cid:19) = f (¯ u B II , ¯ u B I ) f (¯ u C I , ¯ u C II ) f (¯ v B I , ¯ v B II ) f (¯ v C II , ¯ v C I ) f (¯ v C I , ¯ u C I ) f (¯ v B II , ¯ u B II ) × Z ( l ) a − k,n (¯ u C II ; ¯ u B II | ¯ v C I ; ¯ v B I ) Z ( r ) k,b − n (¯ u B I ; ¯ u C I | ¯ v B II ; ¯ v C II ) . (5.29)In the scaling limit u = e εu ′ , v = e εv ′ , q = e εc/ , ε → 0, the trigonometric R-matrix goesto the GL (3)-invariant R-matrix. Then the functions Z ( l ) and Z ( r ) coincide, and this formulaturns into the representation obtained in [24]. The last one was already found to be useful forthe analysis of form factors of local operators in GL (3)-invariant integrable models. We hopethat the explicit representation (5.29) will be also fruitful for the study of integrable modelsbased on the q -deformed GL (3) symmetry. Acknowledgements We warmly thank S. Belliard for his contribution at the early stage of this work. Work of S.P. wassupported in part by RFBR grant 11-01-00962-a and grant of Scientific Foundation of NRU HSE12-09-0064. E.R. was supported by ANR Project DIADEMS (Programme Blanc ANR SIMI12010-BLAN-0120-02). N.A.S. was supported by the Program of RAS Basic Problems of theNonlinear Dynamics, grants RFBR-11-01-00440-a, RFBR-13-01-12405-ofi-m2, SS-2484.2014.1. A Properties of Izergin determinants Most of the properties of the left and right Izergin determinants easily follow directly fromtheir definitions (2.10), (2.11). We give below a list of these properties. We remind that thesuperscript ( l, r ) on K means that the equality is valid for K ( l ) and for K ( r ) with appropriatechoice of component (first/up or second/down) throughout the equality.Initial condition: K ( l )1 (¯ x | ¯ y ) = x g ( x, y ) , K ( r )1 (¯ x | ¯ y ) = y g ( x, y ) . (A.1)Scaling: K ( l,r ) n ( α ¯ x | α ¯ y ) = K ( l,r ) n (¯ x | ¯ y ) . (A.2)Reduction: K ( l,r ) n +1 ( { ¯ x, q − z }|{ ¯ y, z } ) = K ( l,r ) n +1 ( { ¯ x, z }|{ ¯ y, q z } ) = − q ∓ K ( l,r ) n (¯ x | ¯ y ) . (A.3)17nverse order of arguments: K ( l,r ) n ( q − ¯ x | ¯ y ) = ( − q ) ∓ n f − (¯ y, ¯ x ) K ( r,l ) n (¯ y | ¯ x ) , (A.4) K ( l,r ) n ; q − (¯ x | ¯ y ) = K ( r,l ) n ; q (¯ y | ¯ x ) , (A.5)where K ( l,r ) n ; q − means K ( l,r ) n with q replaced by q − . As for Z ( l,r ) a,b and relation (5.8), we have putan additional index q − or q in (A.5) to stress this replacement.Residues in the poles: K ( l,r ) n +1 ( { ¯ x, z }|{ ¯ y, z ′ } ) (cid:12)(cid:12)(cid:12) z ′ → z = f ( z, z ′ ) f ( z, ¯ y ) f (¯ x, z ) K ( l,r ) n (¯ x | ¯ y ) + reg , (A.6)where reg means the regular part.Behavior at infinity: K ( l ) n (¯ x | ¯ y ) ∼ y − i , y i → ∞ , K ( r ) n (¯ x | ¯ y ) ∼ x − i , x i → ∞ , i = 1 , . . . , n, (A.7)and K ( l ) n (¯ x | ¯ y ) is bounded , x i → ∞ , K ( r ) n (¯ x | ¯ y ) is bounded , y i → ∞ , i = 1 , . . . , n. (A.8) Proposition A.1. Let x = y = n and z = z ′ = m . Then lim ¯ z ′ → ¯ z f − (¯ z, ¯ z ′ ) K ( l,r ) n + m ( { ¯ x, ¯ z }|{ ¯ y, ¯ z ′ } ) = f (¯ x, ¯ z ) f (¯ z, ¯ y ) K ( l,r ) n (¯ x | ¯ y ) . (A.9) Proof . Using (A.4) we have K ( l,r ) n + m ( { ¯ x, ¯ z }|{ ¯ y, ¯ z ′ } ) = ( − q ) ∓ ( m + n ) K ( r,l ) n + m ( { ¯ y, ¯ z ′ }|{ ¯ xq , ¯ zq } ) f (¯ x, ¯ y ) f (¯ x, ¯ z ′ ) f (¯ z, ¯ y ) f (¯ z, ¯ z ′ ) . (A.10)The limit ¯ z ′ → ¯ z becomes trivial, and using successively (A.3), (A.4) we arrive at (A.9).The Izergin determinants satisfy also summation identities. Lemma A.1. Let ¯ γ , ¯ α and ¯ β be three sets of complex variables with α = m , β = m , and γ = m + m . Then X K ( l,r ) m (¯ γ I | ¯ α ) K ( r,l ) m ( ¯ β | ¯ γ II ) f (¯ γ II , ¯ γ I ) = ( − q ) ∓ m f (¯ γ, ¯ α ) K ( r,l ) m + m ( { ¯ αq − , ¯ β }| ¯ γ ) . (A.11) The sum is taken with respect to all partitions of the set ¯ γ ⇒ { ¯ γ I , ¯ γ II } with γ I = m and γ II = m . Due to (A.4) the equation (A.11) can be also written in the form X K ( l,r ) m (¯ γ I | ¯ α ) K ( r,l ) m ( ¯ β | ¯ γ II ) f (¯ γ II , ¯ γ I ) = ( − q ) ± m f ( ¯ β, ¯ γ ) K ( l,r ) m + m (¯ γ |{ ¯ α, ¯ βq } ) . (A.12)This statement is a simple corollary of Lemma 1 of the work [26].18 Some proofs B.1 Proof of (5.8) We take the representation (3.8) and replace there q by q − . Using (A.5) we obtain Z ( l,r ) a,b ; q − (¯ t ; ¯ x | ¯ s ; ¯ y ) = ( − q ) ∓ b X K ( l,r ) b ( ¯ w I q − | ¯ s ) K ( r,l ) a (¯ t | ¯ w II ) K ( r,l ) b ( ¯ w I | ¯ y ) f ( ¯ w II , ¯ w I ) , (B.1)where ¯ w = { ¯ x, ¯ s } . Here we have used the evident property of the function f ( x, y ) under thereplacement q → q − : f q − ( x, y ) = f ( y, x ). Replacing ¯ w I ↔ ¯ w II we find that the equation (B.1)coincides with the representation (3.9) for Z ( r,l ) b,a (¯ y ; ¯ s | ¯ x ; ¯ t ). (cid:3) B.2 Proof of (5.11) Consider the highest coefficient Z ( l,r ) a,b (¯ t ; { ¯ ξ, ¯ y ′ }| ¯ s ; ¯ yq − ) for a ≥ b . Using (5.4) we obtain( − q ) ± b Z ( l,r ) a,b (¯ t ; { ¯ ξ, ¯ y ′ }| ¯ s ; ¯ yq − ) f (¯ y, ¯ t ) f (¯ s, ¯ t ) = f (¯ yq − , ¯ ξ ) f (¯ yq − , ¯ y ′ ) f (¯ y, ¯ t ) × X K ( r,l ) b (¯ η II | ¯ y ) K ( l,r ) b (¯ η II | ¯ s ) K ( l,r ) a ( { ¯ ξ, ¯ y ′ }| ¯ η I q ) f (¯ η I , ¯ η II ) . (B.2)Here ¯ η = { ¯ tq − , ¯ yq − } , t = a , s = y = y ′ = b , ξ = a − b , and η I = a . Considerthe limit ¯ y ′ → ¯ y . Then the product f (¯ yq − , ¯ y ′ ) = f − (¯ y ′ , ¯ y ) vanishes. However the Izergindeterminant K ( l,r ) a ( { ¯ ξ, ¯ y ′ }| ¯ η I q ) may have poles at y ′ i = y i . Evidently, the complete compensationof the vanishing product f − (¯ y ′ , ¯ y ) occurs if and only if ¯ yq − ⊂ ¯ η I . Then we can set ¯ η I = { ¯ yq − , ¯ t II q − } and ¯ η II = ¯ t I q − . Substituting this into (B.2) we obtain( − q ) ± b Z ( l,r ) a,b (¯ t ; { ¯ ξ, ¯ y }| ¯ s ; ¯ yq − ) f (¯ y, ¯ t ) f (¯ s, ¯ t ) = lim ¯ y ′ → ¯ y f − (¯ y ′ , ¯ y ) f − ( ¯ ξ, ¯ y ) f − (¯ y, ¯ t ) × X K ( r,l ) b (¯ t I q − | ¯ y ) K ( l,r ) b (¯ t I q − | ¯ s ) K ( l,r ) a ( { ¯ ξ, ¯ y ′ }|{ ¯ t II , ¯ y } ) f (¯ y, ¯ t I ) f (¯ t II , ¯ t I ) , (B.3)where the sum is taken over partitions ¯ t ⇒ { ¯ t I , ¯ t II } with t I = b . It remains to take the limitvia (A.9), and we arrive at (5.11). (cid:3) B.3 Proof of (5.21) Let (5.21) be valid for z = n − 1. Consider the case z = n . Taking the limit successivelyfirst for ¯ z ′ n → ¯ z n and then for z ′ n → z n we obtainlim ¯ z ′ → ¯ z f − (¯ z, ¯ z ′ ) Z ( l,r ) a,b ( { ¯ t, ¯ z ′ } ; ¯ x |{ ¯ s, ¯ z } ; ¯ y ) = f (¯ s, ¯ z n ) f (¯ z n , ¯ t ) X K ( l,r ) n − (¯ x I | ¯ z n ) f (¯ x II , ¯ x I ) × lim ¯ z ′ n → ¯ z n f − ( z n , z ′ n ) Z ( l,r ) a − n +1 ,b ( { ¯ t, z ′ n } ; ¯ x II |{ ¯ s, ¯ x I , z n } ; ¯ y )= f (¯ s, ¯ z ) f (¯ z, ¯ t ) X Z ( l,r ) a − n,b (¯ t ; ¯ x ii |{ ¯ s, ¯ x I , ¯ x i } ; ¯ y ) × K ( l,r ) n − (¯ x I | ¯ z n ) f (¯ x II , ¯ x I ) K ( l,r )1 (¯ x i | z n ) f (¯ x ii , ¯ x i ) f (¯ x I , z n ) . (B.4)19ere we first divide ¯ x into subsets { ¯ x I , ¯ x II } with x I = n − 1, and then split the subset¯ x II into sub-subsets { ¯ x i , ¯ x ii } with x i = 1. Setting { ¯ x i , ¯ x I } = ¯ x and using K ( l,r )1 (¯ x i | z n ) =( − q ) ∓ f (¯ x i , z n ) K ( r,l )1 ( z n q − | ¯ x i ) we findlim ¯ z ′ → ¯ z f − (¯ z, ¯ z ′ ) Z ( l,r ) a,b ( { ¯ t, ¯ z ′ } ; ¯ x |{ ¯ s, ¯ z } ; ¯ y ) = f (¯ s, ¯ z ) f (¯ z, ¯ t ) × X Z ( l,r ) a − n,b (¯ t ; ¯ x ii |{ ¯ s, ¯ x } ; ¯ y ) f (¯ x ii , ¯ x ) f (¯ x , z n ) × ( − q ) ∓ K ( l,r ) n − (¯ x I | ¯ z n ) K ( r,l )1 ( z n q − | ¯ x i ) f (¯ x i , ¯ x I ) . (B.5)Applying lemma A.1 to the last line of (B.5) we can take the sum with respect to the partitions¯ x ⇒ { ¯ x i , ¯ x I } , that finally giveslim ¯ z ′ → ¯ z f − (¯ z, ¯ z ′ ) Z ( l,r ) a,b ( { ¯ t, ¯ z ′ } ; ¯ x |{ ¯ s, ¯ z } ; ¯ y ) = f (¯ s, ¯ z ) f (¯ z, ¯ t ) × X K ( l,r ) n (¯ x | ¯ z ) f (¯ x ii , ¯ x ) Z ( l,r ) a − n,b (¯ t ; ¯ x ii |{ ¯ s, ¯ x } ; ¯ y ) . (B.6) (cid:3) B.4 Proof of (5.24) Let us simply write (5.22) replacing a by b and setting: ¯ t = q ¯ u , ¯ x = q ¯ v , ¯ s = ¯ α , and ¯ y = ¯ β lim ¯ z ′ → ¯ z f − (¯ z, ¯ z ′ ) Z ( l,r ) b,a ( q ¯ u ; { q ¯ v, ¯ z ′ }| ¯ α ; { ¯ β, ¯ z } ) = f ( ¯ β, ¯ z ) f (¯ z, q ¯ v ) × X K ( l,r ) n (¯ z | ¯ α I ) f ( ¯ α I , ¯ α II ) Z ( l,r ) b,a − n ( q ¯ u ; { q ¯ v, ¯ α I }| ¯ α II ; ¯ β ) , (B.7)where the sum is taken over partitions ¯ α ⇒ { ¯ α I , ¯ α II } with α I = n . Now we can use (5.2) and(5.7) to transform Z ( l,r ) b,a and Z ( l,r ) b,a − n in (B.7). We havelim ¯ z ′ → ¯ z f − (¯ z, ¯ z ′ ) Z ( l,r ) b,a ( q ¯ u ; { q ¯ v, ¯ z ′ }| ¯ α ; { ¯ β, ¯ z } ) = f ( ¯ β, ¯ z ) Z ( l,r ) a,b ( ¯ α ; { ¯ β, ¯ z }| ¯ u ; { ¯ v, q − ¯ z } ) f (¯ u, ¯ α ) f (¯ v, ¯ z ) f (¯ v, ¯ β ) , (B.8)and Z ( l,r ) b,a − n ( q ¯ u ; { q ¯ v, ¯ α I }| ¯ α II ; ¯ β ) = f ( ¯ β, ¯ α I ) Z ( l,r ) a − n,b ( ¯ α II ; ¯ β | ¯ u ; { ¯ v, q − ¯ α I } ) f (¯ v, ¯ β ) f (¯ u, ¯ α II ) . (B.9)Substituting all this into (B.7) after evident cancelations we obtain Z ( l,r ) a,b ( ¯ α ; { ¯ β, ¯ z }| ¯ u ; { ¯ v, q − ¯ z } ) = X K ( l,r ) n (¯ z | ¯ α I ) f ( ¯ α I , ¯ α II ) × f ( ¯ β, ¯ α I ) f (¯ u, ¯ α I ) Z ( l,r ) a − n,b ( ¯ α II ; ¯ β | ¯ u ; { ¯ v, q − ¯ α I } ) . (B.10)It remains to set ¯ α = ¯ t , ¯ β = ¯ x , ¯ u = ¯ s , ¯ v = ¯ y , and we arrive at (5.24). (cid:3) .5 Proof of Proposition 5.1 We use induction over p . Denote the l.h.s. and the r.h.s. of (5.28) by F ( l,r ) a,b,p,n (¯ t ; ¯ x | ¯ s ; ¯ w ; ¯ y | ¯ z ) and˜ F ( l,r ) a,b,p,n (¯ t ; ¯ x | ¯ s ; ¯ w ; ¯ y | ¯ z ) respectively. For p = 0 the equation (5.28) is trivial. Indeed, since k ≤ p we obtain that ¯ s I = ¯ ξ I = ¯ y = ∅ for p = 0. Hence, the sum over partitions in the r.h.s. of (5.28)reduces to the one term, and both sides of this equation give Z ( l,r ) a,b (¯ t ; ¯ x | ¯ s ; ¯ w ).Now let (5.28) be valid for y = p − a , b , and n : F ( l,r ) a,b,p − ,n (¯ t ; ¯ x | ¯ s ; ¯ w ; ¯ y | ¯ z ) = ˜ F ( l,r ) a,b,p − ,n (¯ t ; ¯ x | ¯ s ; ¯ w ; ¯ y | ¯ z ) , ∀ a, b, n . (B.11)The general strategy of the proof is the following. We consider both sides of this equation at y = p as functions of y p , the other variables being fixed. Obviously F and ˜ F are rationalfunctions of y p . We first establish that these functions have their poles in the same points andthen prove that due to the induction assumption (B.11) the residues in these poles coincide.Then it means that the difference F − ˜ F is a polynomial in y p . Finally taking into account thebehavior of this polynomial at y p → ∞ and y p = 0 we conclude that it is identically equal tozero.Obviously, the function F ( l,r ) a,b,p,n has poles at y p = ξ ℓ , ℓ = 1 , . . . , a + n due to the factor f ( ¯ ξ, ¯ y ).The highest coefficient Z ( l,r ) a,b has additional poles at y p = s i , i = 1 , . . . , b . However the poles of Z ( l,r ) a,b at y p = x j , j = 1 , . . . , a are compensated by the zeros of the prefactor: f ( ¯ ξ, ¯ y ) = f (¯ zq − , ¯ y ) f (¯ xq − , ¯ y ) = f − (¯ y, ¯ z ) f − (¯ y, ¯ x ) . (B.12)It is easy to see that the r.h.s. ˜ F ( l,r ) a,b,p,n has poles in the same points. Due to the product f (¯ y, ¯ s I ) it has poles at y p = s i . The function K ( r,l ) p ( { ¯ s I q − , ¯ ξ I }| ¯ y ) has poles at y p = ξ ℓ , howeverthe poles at y p = s i q − are compensated by the product f (¯ y, ¯ s I ).Consider the residues of F ( l,r ) a,b,p,n at y p = s i . Using the reduction property (5.15) we obtain(for shortness here and below we omit the arguments of F ( l,r ) a,b,p,n and ˜ F ( l,r ) a,b,p,n in the l.h.s. ofequations): F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → s i = f ( y p , s i ) f ( s i , ¯ s i ) f (¯ y p , s i ) f ( ¯ w, s i ) (cid:2) f ( s i , ¯ x ) f (¯ xq − , s i ) (cid:3) f (¯ zq − , s i ) × f ( ¯ ξ, ¯ y p ) Z ( l,r ) a,b − (¯ t ; ¯ x | ¯ s i ; { ¯ y p , ¯ w } ) + reg . (B.13)The terms in square brackets cancel each other, the terms in the second line give F ( l,r ) a,b − ,p − ,n : F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → s i = f ( y p , s i ) f ( s i , ¯ s i ) f (¯ y p , s i ) f ( ¯ w, s i ) f − ( s i , ¯ z ) F ( l,r ) a,b − ,p − ,n (¯ t ; ¯ x | ¯ s i ; ¯ w ; ¯ y p | ¯ z )+reg . (B.14)Consider now the residue of ˜ F at y p = s i . The pole occurs if and only if s i ∈ ¯ s I . Setting¯ s I = { s i , ¯ s } and using the property (A.3) of K ( r,l ) we obtain˜ F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → s i = X ( − q ) ∓ ( k − K ( r,l ) p − ( { ¯ s q − , ¯ ξ I }| ¯ y p ) Z ( l,r ) a,b − k (¯ t ; ¯ x | ¯ s II ; { ¯ w, ¯ ξ I } ) f ( s i , ¯ s ) f ( s i , ¯ s II ) × f (¯ s , ¯ s II ) f ( ¯ ξ II , ¯ ξ I ) (cid:2) f ( y p , s i ) f (¯ y p , s i ) f ( ¯ w, s i ) f − ( s i , ¯ z ) (cid:3) f (¯ y p , ¯ s ) f ( ¯ w, ¯ s ) f − (¯ s , ¯ z ) + reg , (B.15)21here the sum is taking over partitions ¯ s i ⇒ { ¯ s , ¯ s II } and ¯ ξ ⇒ { ¯ ξ I , ¯ ξ II } . The terms in squarebrackets can be moved out of the sum. The product f ( s i , ¯ s ) f ( s i , ¯ s II ) combines into f ( s i , ¯ s i ) andalso can be moved out of the sum. We arrive at˜ F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → s i = f ( y p , s i ) f ( s i , ¯ s i ) f (¯ y p , s i ) f ( ¯ w, s i ) f − ( s i , ¯ z ) X ( − q ) ∓ k K ( r,l ) p − ( { ¯ s q − , ¯ ξ I }| ¯ y p ) × Z ( l,r ) a,b − − k (¯ t ; ¯ x | ¯ s II ; { ¯ w, ¯ ξ I } ) f (¯ s , ¯ s II ) f ( ¯ ξ II , ¯ ξ I ) f (¯ y p , ¯ s ) f ( ¯ w, ¯ s ) f − (¯ s , ¯ z ) + reg , (B.16)where k = s = k − 1. Evidently, the sum over partitions in the r.h.s. of (B.16) gives˜ F ( l,r ) a,b − ,p − ,n (¯ t ; ¯ x | ¯ s i ; ¯ w ; ¯ y p | ¯ z ) and we obtain˜ F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → s i = f ( y p , s i ) f ( s i , ¯ s i ) f (¯ y p , s i ) f ( ¯ w, s i ) f − ( s i , ¯ z ) ˜ F ( l,r ) a,b − ,p − ,n (¯ t ; ¯ x | ¯ s i ; ¯ w ; ¯ y p | ¯ z ) + reg . (B.17)Comparing (B.14) and (B.17) and taking into account (B.11) we conclude that the difference F ( l,r ) a,b,p,n − ˜ F ( l,r ) a,b,p,n is a bounded function of y p as y p → s i , i = 1 , . . . , b .Consider now the residues of F ( l,r ) a,b,p,n at y p = ξ ℓ . We have F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → ξ ℓ = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) h f ( ¯ ξ ℓ , ¯ y p ) Z ( l,r ) a,b (cid:0) ¯ t ; ¯ x | ¯ s ; (cid:8) ¯ w, ξ ℓ , ¯ y p (cid:9)(cid:1)i + reg . (B.18)Now one should distinguish between two cases: either ξ ℓ ∈ ¯ zq − or ξ ℓ ∈ ¯ xq − . Let ξ ℓ = z j q − .Then the combination in the square brackets of (B.18) is just F ( l,r ) a,b,p − ,n − (¯ t ; ¯ x | ¯ s ; { ¯ w, ξ ℓ } ; ¯ y p | ¯ z j ).Thus, we obtain F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → z j q − = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) F ( l,r ) a,b,p − ,n − (¯ t ; ¯ x | ¯ s ; { ¯ w, ξ ℓ } ; ¯ y p | ¯ z j ) + reg . (B.19)Let now ξ ℓ = x j q − . In this case the prefactor f ( ¯ ξ ℓ , ¯ y p ) does not compensate the pole of Z ( l,r ) a,b at y p = x j . Therefore the combination in the squared brackets in (B.18) is not directly F ( l,r ) a,b,p − ,n − .In order to overcome this problem we use (5.24) at n = 1. We have Z ( l,r ) a,b (¯ t ; { ¯ x j , x j }| ¯ s ; { ¯ w, ¯ y p , x j q − } ) = a X i =1 K ( l,r )1 ( x j | t i ) f ( t i , ¯ t i ) f (¯ x j , t i ) f (¯ s, t i ) × Z ( l,r ) a − ,b (¯ t i ; ¯ x j | ¯ s ; { ¯ w, t i q − , ¯ y p } ) . (B.20)Substituting (B.20) into (B.18) we obtain F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → x j q − = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) a X i =1 K ( l,r )1 ( x j | t i ) f ( t i , ¯ t i ) f (¯ x j , t i ) f (¯ s, t i ) × f ( ¯ ξ ℓ , ¯ y p ) Z ( l,r ) a − ,b (¯ t i ; ¯ x j | ¯ s ; { ¯ w, t i q − , ¯ y p } ) + reg . (B.21)22ow the combination in the second line of (B.21) gives F ( l,r ) a − ,b,p − ,n (¯ t i ; ¯ x j | ¯ s ; { ¯ w, t i q − } ; ¯ y p | ¯ z ),hence, F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → x j q − = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) a X i =1 K ( l,r )1 ( x j | t i ) f ( t i , ¯ t i ) f (¯ x j , t i ) f (¯ s, t i ) × F ( l,r ) a − ,b,p − ,n (¯ t i ; ¯ x j | ¯ s ; { ¯ w, t i q − } ; ¯ y p | ¯ z ) + reg . (B.22)Thus, we have reduced the residues of F ( l,r ) a,b,p,n at y p = ξ ℓ to the functions F ( l,r ) a,b,p − ,n − or F ( l,r ) a − ,b,p − ,n .Consider now the pole of ˜ F ( l,r ) a,b,p,n at y p = ξ ℓ . It occurs if and only if ξ ℓ ∈ ¯ ξ I . Setting¯ ξ I = { ξ ℓ , ¯ ξ } and using property (A.6) of K ( r,l ) we obtain˜ F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → ξ ℓ = X ( − q ) ∓ k K ( r,l ) p − ( { ¯ s I q − , ¯ ξ }| ¯ y p ) f ( ξ ℓ , y p ) f ( ξ ℓ , ¯ y p ) f (¯ s I q − , ξ ℓ ) f ( ¯ ξ , ξ ℓ ) f (¯ s I , ¯ s II ) × Z ( l,r ) a,b − k (¯ t ; ¯ x | ¯ s II ; { ¯ w, ¯ ξ , ξ ℓ } ) f ( ¯ ξ II , ¯ ξ ) f ( ¯ ξ II , ξ ℓ ) f ( ξ ℓ , ¯ s I ) f (¯ y p , ¯ s I ) f ( ¯ w, ¯ s I ) f − (¯ s I , ¯ z ) + reg , (B.23)where the sum is taking over partitions ¯ s ⇒ { ¯ s I , ¯ s II } and ¯ ξ ℓ ⇒ { ¯ ξ , ¯ ξ II } . The terms f (¯ s I q − , ξ ℓ )and f ( ξ ℓ , ¯ s I ) cancel each other. The terms f ( ξ ℓ , y p ) f ( ξ ℓ , ¯ y p ) can be moved out of the sum. Theproduct f ( ¯ ξ , ξ ℓ ) f ( ¯ ξ II , ξ ℓ ) combines into f ( ¯ ξ ℓ , ξ ℓ ) and also can be moved out of the sum. We arriveat ˜ F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → ξ ℓ = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) X ( − q ) ∓ k K ( r,l ) p − ( { ¯ s I q − , ¯ ξ }| ¯ y p ) × Z ( l,r ) a,b − k (¯ t ; ¯ x | ¯ s II ; { ¯ w, ¯ ξ , ξ ℓ } ) f (¯ s I , ¯ s II ) f ( ¯ ξ II , ¯ ξ ) f (¯ y p , ¯ s I ) f ( ¯ w, ¯ s I ) f − (¯ s I , ¯ z ) + reg . (B.24)Now we set ξ ℓ = z j q − . Then we simply rewrite f ( ¯ w, ¯ s I ) f − (¯ s I , ¯ z ) = h f ( ¯ w, ¯ s I ) f ( ξ ℓ , ¯ s I ) i f − (¯ s I , ¯ z j ) , (B.25)and we see that the sum over partitions in (B.24) gives ˜ F ( l,r ) a,b,p − ,n − (¯ t ; ¯ x | ¯ s ; { ¯ w, ξ ℓ } ; ¯ y p | ¯ z j ):˜ F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → z j q − = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) ˜ F ( l,r ) a,b,p − ,n − (¯ t ; ¯ x | ¯ s ; { ¯ w, ξ ℓ } ; ¯ y p | ¯ z j ) + reg . (B.26)It remains to consider the case ξ ℓ = x j q − . Due to (5.24) we have Z ( l,r ) a,b − k (¯ t ; { ¯ x j , x j }| ¯ s II ; { ¯ w, ¯ ξ , x j q − } ) = a X i =1 K ( l,r )1 ( x j | t i ) f ( t i , ¯ t i ) f (¯ x j , t i ) f (¯ s II , t i ) × Z ( l,r ) a − ,b (¯ t i ; ¯ x j | ¯ s II ; { ¯ w, ¯ ξ , t i q − } ) . (B.27)23ubstituting (B.27) into (B.24) we obtain˜ F ( l,r ) a,b,p,n (cid:12)(cid:12)(cid:12) y p → x j q − = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) a X i =1 K ( l,r )1 ( x j | t i ) f ( t i , ¯ t i ) f (¯ x j , t i ) × X ( − q ) ∓ k K p − ( { ¯ s I q − , ¯ ξ }| ¯ y p ) Z ( l,r ) a − ,b − k (¯ t i ; ¯ x j | ¯ s II ; { ¯ w, ¯ ξ , t i q − } ) × f (¯ s I , ¯ s II ) f ( ¯ ξ II , ¯ ξ ) f (¯ y p , ¯ s I ) f ( ¯ w, ¯ s I ) f (¯ s II , t i ) f − (¯ s I , ¯ z ) . (B.28)Now we observe that f ( ¯ w, ¯ s I ) f (¯ s II , t i ) = h f ( ¯ w, ¯ s I ) f ( t i q − , ¯ s I ) i f (¯ s, t i ) . (B.29)Substituting this into (B.28) we arrive at˜ F ( l,r ) a,b,p,n (¯ t ; ¯ x | ¯ s ; ¯ w ; ¯ y | ¯ z ) (cid:12)(cid:12)(cid:12) y p → ξ ℓ = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) a X i =1 K ( l,r )1 ( x j | t i ) f ( t i , ¯ t i ) f (¯ x j , t i ) f (¯ s, t i ) × X ( − q ) ∓ k K ( r,l ) p − ( { ¯ s I q − , ¯ ξ }| ¯ y p ) Z ( l,r ) a − ,b − k (¯ t i ; ¯ x j | ¯ s II ; { ¯ w, ¯ ξ , t i q − } ) f (¯ s I , ¯ s II ) f ( ¯ ξ II , ¯ ξ ) × f (¯ y p , ¯ s I ) h f ( ¯ w, ¯ s I ) f ( t i q − , ¯ s I ) i f − (¯ s I , ¯ z ) . (B.30)Evidently the sum over partitions in (B.30) gives ˜ F ( l,r ) a − ,b,p − ,n (¯ t i ; ¯ x j | ¯ s ; { ¯ w, t i q − } ; ¯ y p | ¯ z ), therefore˜ F ( l,r ) a,b,p,n (¯ t ; ¯ x | ¯ s ; ¯ w ; ¯ y | ¯ z ) (cid:12)(cid:12)(cid:12) y p → ξ ℓ = f ( ξ ℓ , y p ) f ( ¯ ξ ℓ , ξ ℓ ) f ( ξ ℓ , ¯ y p ) × a X i =1 K ( l,r )1 ( x j | t i ) f ( t i , ¯ t i ) f (¯ x j , t i ) f (¯ s, t i ) F ( l,r ) a − ,b,p − ,n (¯ t i ; ¯ x j | ¯ s ; { ¯ w, t i q − } ; ¯ y p | ¯ z ) . (B.31)Comparing (B.19) with (B.26) and (B.22) with (B.31), and taking into account the inductionassumption (B.11) we come to conclusion that the difference F ( l,r ) a,b,p,n − ˜ F ( l,r ) a,b,p,n is a boundedfunction of y p as y p → ξ ℓ , ℓ = 1 , . . . a + n . Thus, we have proved that the function F ( l,r ) a,b,p,n − ˜ F ( l,r ) a,b,p,n has no poles neither in the points y p = s i nor in y p = ξ ℓ . Hence, this is a polynomial in y p . Due to (A.7) and (5.10) the polynomial F ( r ) a,b,p,n − ˜ F ( r ) a,b,p,n decreases as y p → ∞ . Hence, F ( r ) a,b,p,n − ˜ F ( r ) a,b,p,n = 0. The case of the polynomial F ( l ) a,b,p,n − ˜ F ( l ) a,b,p,n is slightly more sophisticated.Due to (A.8) and (5.9) we conclude that it is bounded as y p → ∞ . Hence, it does not dependon y p . It follows from the representation (5.5) that the highest coefficient Z ( l ) a,b (¯ t ; ¯ x | ¯ s ; { ¯ y, ¯ w } )is proportional to the product of all y i . Hence, the function F ( l ) a,b,p,n vanishes at y p = 0. 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