Scalar products of Bethe vectors in the models with \mathfrak{gl}(m|n) symmetry
A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, N. A. Slavnov
aa r X i v : . [ m a t h - ph ] S e p LAPTH-010/17
Scalar products of Bethe vectorsin the models with gl ( m | n ) symmetry A. Hutsalyuk a,b , A. Liashyk c,d,e , S. Z. Pakuliak a,f ,E. Ragoucy g , N. A. Slavnov h a Moscow Institute of Physics and Technology, Dolgoprudny, Moscow reg., Russia b Fachbereich C Physik, Bergische Universit¨at Wuppertal, 42097 Wuppertal, Germany c Bogoliubov Institute for Theoretical Physics, NAS of Ukraine, Kiev, Ukraine d National Research University Higher School of Economics, Faculty of Mathematics,Moscow, Russia e Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow,Russia f Laboratory of Theoretical Physics, JINR, Dubna, Moscow reg., Russia g Laboratoire de Physique Th´eorique LAPTh, CNRS and USMB,BP 110, 74941 Annecy-le-Vieux Cedex, France h Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Abstract
We study scalar products of Bethe vectors in the models solvable by the nested alge-braic Bethe ansatz and described by gl ( m | n ) superalgebra. Using coproduct propertiesof the Bethe vectors we obtain a sum formula for their scalar products. This formuladescribes the scalar product in terms of a sum over partitions of Bethe parameters. Wealso obtain recursions for the Bethe vectors. This allows us to find recursions for thehighest coefficient of the scalar product. [email protected], [email protected], [email protected], [email protected],[email protected] Introduction
The problem of calculating correlation functions of quantum exactly solvable models is ofgreat importance. The creation of the Quantum Inverse Scattering Method (QISM) in theearly 80s of the last century provided a powerful tool for investigating this problem [1–4].The first works in which QISM was applied to the problem of correlation functions [5, 6]were devoted to the models related to the different deformations of the affine algebra b gl (2).Already in those papers, the key role of Bethe vectors scalar products was established. Inparticular, a sum formula for the scalar product of Bethe vectors was obtained in [5]. Thisformula gives the scalar product as a sum over partitions of Bethe parameters.A generalization of QISM to the models with higher rank symmetry was given in papers[7–9] where the nested algebraic Bethe ansatz was developed. There a recursive procedurewas developed to construct Bethe vectors corresponding to the gl ( N ) algebra from the knownBethe vectors of the gl ( N −
1) algebra. The problem of the scalar products in SU (3)-invariantmodels were studied in [10], where an analog of the sum formula for the scalar product wasobtained and the norm of the transfer matrix eigenstates was computed. Recently in a seriesof papers [11–16] the Bethe vectors scalar products in the models with gl (3) and gl (2 | R -matrix was given in [21, 22].Concerning the scalar products in the models with higher rank (super) symmetries, onlyfew results are known for today. First, it is worth mentioning the papers [23, 24], in which theauthors developed a new approach to the problem based on the quantized Knizhnik–Zamo-lodchikov equation. There the norms of the transfer matrix eigenstates in gl ( N )-based modelswere calculated. Some partial results were also obtained when specializing to fundamentalrepresentations or to particular cases of Bethe vectors [25–28].In this paper we study the Bethe vectors scalar products in the models described by gl ( m | n ) superalgebras. Hence it encompasses the case of gl ( m ) algebras. In spite of wework within the framework of the traditional approach based on the nested algebraic Betheansatz, we essentially use recent results obtained in [29] via the method of projections forconstruction of Bethe vectors. This method was proposed in the paper [30]. It uses therelation between two different realizations of the quantized Hopf algebra U q ( b gl ( N )) associatedwith the affine algebra b gl ( N ), one in terms of the universal monodromy matrix T ( z ) and RT T -commutation relations and second in terms of the total currents, which are defined bythe Gauss decomposition of the monodromy matrix T ( z ) [31]. In [29] we generalized thisapproach to the case of the Yangians of gl ( m | n ) superalgebras. Among the results of [29]that are used in the present paper, we note the formulas for the action of the monodromymatrix entries onto the Bethe vectors, and also the coproduct formula for the Bethe vectors.The main result of this paper is the sum formula for the scalar product of Bethe vec-tors. In our previous publications (see e.g. [15, 21]) we derived it using explicit formulas ofthe monodromy matrix elements multiple actions onto the Bethe vectors. This method isstraightforward, but it becomes rather cumbersome already for gl (3) and gl (2 |
1) based mod-els. Furthermore, the possibility of its application to the models with higher rank symmetriesis under question. Instead, in the present paper we use a method based on the coproductformula for the Bethe vectors. Actually, the structure of the scalar product is encoded in thecoproduct formula. Therefore, this method directly leads to the sum formula, in which thescalar product is given as a sum over partitions of Bethe parameters.The sum formula contains an important object called the highest coefficient (HC) [5]. In2he gl (2) based models and their q -deformation the HC coincides with a partition functionof the six-vertex model with domain wall boundary condition. An explicit representationfor it was found in [32]. In the models with gl (3) symmetry the HC also can be associatedwith a special partition function, however, its explicit form is much more sophisticated (seee.g. [11, 13]). One can expect that in the case of higher rank algebras an analogous explicitformula for the HC becomes too complex. Therefore, in this paper we do not derive suchformulas, but instead, we obtain recursions, which allow one to construct the HC startingwith the ones in the models with lower rank symmetries. These recursions can be derivedfrom recursions on the Bethe vectors that we also obtain in this paper.As we have already mentioned, the Bethe vectors scalar products are of great importancein the problem of correlation functions of quantum integrable models. Certainly, the sumformula is not convenient for its direct applications, as it contains a big number of terms,which grows exponentially in the thermodynamic limit. However, it gives a key for studyingparticular cases of scalar products, in which the sum over partitions can be reduced to asingle determinant. This type of formulas can be used for calculating form factors of variousintegrable models of physical interest, like, for instance, the Hubbard model [33], the t-Jmodel [34–36] or multi-component Bose/Fermi gas [37], not to mention spin chain modelsas they are nowadays tested in condensed matter experiments [38]. We also hope that ourresults will be of some interest in the context of super-Yang-Mills theories, when studied inthe integrable systems framework. Indeed, in these theories, the general approach relies ona spin chain based on the psu (2 , |
4) superalgebra. We believe that the present results willcontribute to a better understanding of the theory.The article is organized as follows. In section 2 we introduce the model under consider-ation. There we also specify our conventions and notation. In section 3 we describe Bethevectors of gl ( m | n )-based models. Section 4 contains the main results of the paper. Here wegive a sum formula for the scalar product of generic Bethe vectors and recursion relationsfor the Bethe vectors and the highest coefficient. The rest of the paper contains the proofsof the results announced in section 4. In section 5 we prove recursion formulas for the Bethevectors. Section 6 contains a proof of the sum formula for the scalar product. In section 7we study highest coefficient and find a recursion for it. Proofs of some auxiliary statementsare gathered in appendices. gl ( m | n ) -based models The R -matrix of gl ( m | n )-based models acts in the tensor product C m | n ⊗ C m | n , where C m | n isthe Z -graded vector space with the grading [ i ] = 0 for 1 ≤ i ≤ m , [ i ] = 1 for m < i ≤ m + n .Here, we assume that m ≥ n ≥
1, but we want to stress that our considerations areapplicable to the case m = 0 or n = 0 as well, i.e. to the non-graded algebras. Matricesacting in this space are also graded. We define this grading on the basis of elementary units E ij as [ E ij ] = [ i ] + [ j ] ∈ Z (recall that ( E ij ) ab = δ ia δ jb ). The tensor products of C m | n spacesare graded as follows:( ⊗ E ij ) · ( E kl ⊗ ) = ( − ([ i ]+[ j ])([ k ]+[ l ]) E kl ⊗ E ij . (2.1)The R -matrix of gl ( m | n )-invariant models has the form R ( u, v ) = I + g ( u, v ) P, g ( u, v ) = cu − v . (2.2)3ere c is a constant, I and P respectively are the identity matrix and the graded permutationoperator [39]: I = ⊗ = n + m X i,j =1 E ii ⊗ E jj , P = n + m X i,j =1 ( − [ j ] E ij ⊗ E ji . (2.3)The key object of QISM is a quantum monodromy matrix T ( u ). Its matrix elements T i,j ( u ) are graded in the same way as the matrices [ E ij ]: [ T i,j ( u )] = [ i ] + [ j ]. The grading is amorphism, i.e. [ T i,j ( u ) · T k,l ( v )] = [ T i,j ( u )] + [ T k,l ( v )]. Their commutation relations are givenby the RT T -relation R ( u, v ) (cid:0) T ( u ) ⊗ (cid:1)(cid:0) ⊗ T ( v ) (cid:1) = (cid:0) ⊗ T ( v ) (cid:1)(cid:0) T ( u ) ⊗ (cid:1) R ( u, v ) . (2.4)Equation (2.4) holds in the tensor product C m | n ⊗ C m | n ⊗ H , where H is a Hilbert space ofthe Hamiltonian under consideration. Here all the tensor products are graded.The RT T -relation (2.4) yields a set of commutation relations for the monodromy matrixelements[ T i,j ( u ) , T k,l ( v ) } = ( − [ i ]([ k ]+[ l ])+[ k ][ l ] g ( u, v ) (cid:16) T k,j ( v ) T i,l ( u ) − T k,j ( u ) T i,l ( v ) (cid:17) = ( − [ l ]([ i ]+[ j ])+[ i ][ j ] g ( u, v ) (cid:16) T i,l ( u ) T k,j ( v ) − T i,l ( v ) T k,j ( u ) (cid:17) , (2.5)where we introduced the graded commutator[ T i,j ( u ) , T k,l ( v ) } = T i,j ( u ) T k,l ( v ) − ( − ([ i ]+[ j ])([ k ]+[ l ]) T k,l ( v ) T i,j ( u ) . (2.6)The graded transfer matrix is defined as the supertrace of the monodromy matrix T ( u ) = str T ( u ) = m + n X j =1 ( − [ j ] T j,j ( u ) . (2.7)One can easily check [39] that [ T ( u ) , T ( v )] = 0. Thus the transfer matrix can be used as agenerating function of integrals of motion of an integrable system. In this paper we use notation and conventions of the work [29]. Besides the function g ( u, v )we introduce two rational functions f ( u, v ) = 1 + g ( u, v ) = u − v + cu − v ,h ( u, v ) = f ( u, v ) g ( u, v ) = u − v + cc . (2.8)In order to make formulas uniform we also use ‘graded’ functions g [ i ] ( u, v ) = ( − [ i ] g ( u, v ) = ( − [ i ] cu − v ,f [ i ] ( u, v ) = 1 + g [ i ] ( u, v ) = u − v + ( − [ i ] cu − v ,h [ i ] ( u, v ) = f [ i ] ( u, v ) g [ i ] ( u, v ) = ( u − v ) + ( − [ i ] c ( − [ i ] c , (2.9)4nd γ i ( u, v ) = f [ i ] ( u, v ) h ( u, v ) δ i,m , ˆ γ i ( u, v ) = f [ i +1] ( u, v ) h ( v, u ) δ i,m . (2.10)Observe that we use the subscript i for the functions γ and ˆ γ instead of the subscript [ i ].This is because these functions actually take three values. For example, γ i ( u, v ) = f ( u, v ) for i < m , γ i ( u, v ) = g ( u, v ) for i = m , and γ i ( u, v ) = f ( v, u ) for i > m . It is also easy to seethat ˆ γ i ( u, v ) = ( − δ i,m γ i ( u, v ).Let us formulate now a convention on the notation. We denote sets of variables by bar,for example, ¯ u . When dealing with several of them, we may equip these sets or subsets withadditional superscript: ¯ s i , ¯ t ν , etc. Individual elements of the sets or subsets are denoted byLatin subscripts, for instance, u j is an element of ¯ u , t ik is an element of ¯ t i etc. As a rule, thenumber of elements in the sets is not shown explicitly in the equations, however we give thesecardinalities in special comments to the formulas. We assume that the elements in everysubset of variables are ordered in such a way that the sequence of their subscripts is strictlyincreasing: ¯ t i = { t i , t i , . . . , t ir i } . We call this ordering the natural order.We use a shorthand notation for products of the rational functions (2.8)–(2.10). Namely,if some of these functions depends on a set of variables (or two sets of variables), this meansthat one should take the product over the corresponding set (or double product over twosets). For example, g (¯ u, v ) = Y u j ∈ ¯ u g ( u j , v ) ,f [ i ] ( t i − k , ¯ t i ) = Y t iℓ ∈ ¯ t i f [ i ] ( t i − k , t iℓ ) ,γ ℓ (¯ s i , ¯ t ℓ ) = Y s ij ∈ ¯ s i Y t ℓk ∈ ¯ t ℓ γ ℓ ( s ij , t ℓk ) . (2.11)By definition, any product over the empty set is equal to 1. A double product is equal to 1if at least one of the sets is empty.Below we will extend this convention to the products of monodromy matrix entries andtheir eigenvalues (see (3.3) and (3.4)). Bethe vectors belong to the space H in which the monodromy matrix entries act. We do notspecify this space, however, we assume that it contains a pseudovacuum vector | i , such that T i,i ( u ) | i = λ i ( u ) | i , i = 1 , . . . , m + n,T i,j ( u ) | i = 0 , i > j , (3.1)where λ i ( u ) are some scalar functions. In the framework of the generalized model [5] con-sidered in this paper, they remain free functional parameters. Below it will be convenient todeal with ratios of these functions α i ( u ) = λ i ( u ) λ i +1 ( u ) , i = 1 , . . . , m + n − . (3.2)We extend the convention on the shorthand notation (2.11) to the products of the func-tions introduced above, for example, λ k (¯ u ) = Y u j ∈ ¯ u λ k ( u j ) , α i (¯ t i ) = Y t iℓ ∈ ¯ t i α i ( t iℓ ) . (3.3)5e use the same convention for the products of commuting operators T i,j (¯ u ) = Y u j ∈ ¯ u T i,j ( u j ) , for [ i ] + [ j ] = 0 , mod 2 . (3.4)Finally, for the product of odd operators T i,j with [ i ] + [ j ] = 1 we introduce a special notation T i,j (¯ u ) = T i,j ( u ) . . . T i,j ( u p ) Q ≤ k<ℓ ≤ p h ( u ℓ , u k ) , [ i ] + [ j ] = 1 , i < j, T i,j (¯ u ) = T i,j ( u ) . . . T i,j ( u p ) Q ≤ k<ℓ ≤ p h ( u k , u ℓ ) , [ i ] + [ j ] = 1 , i > j. (3.5)Due to the commutation relations (2.5) the operator products (3.5) are symmetric over per-mutations of the parameters ¯ u . In physical models, vectors of the space H describe states with quasiparticles of differenttypes (colors). In gl ( m | n )-based models quasiparticles may have N = m + n − { r , . . . , r N } be a set of non-negative integers. We say that a state has coloring { r , . . . , r N } ,if it contains r i quasiparticles of the color i . A state with a fixed coloring can be obtained bysuccessive application of the creation operators T i,j with i < j to the vector | i , which has zerocoloring. Acting on this state, an operator T i,j adds quasiparticles with the colors i, . . . , j − T i,i +1 creates one quasiparticle of thecolor i , the operator T ,n + m creates N quasiparticles of N different colors. The diagonaloperators T i,i are neutral, the matrix elements T i,j with i > j play the role of annihilationoperators. Acting on the state of a fixed coloring, the annihilation operator T i,j removesfrom this state the quasiparticles with the colors j, . . . , i −
1, one particle of each color. Inparticular, if j − < k < i , and the annihilation operator T i,j acts on a state in which thereis no particles of the color k , then this action vanishes.This definition can be formalized at the level of the Yangian through the Cartan generatorsof the Lie superalgebra gl ( m | n ). Indeed, the zero modes T ij [0] = lim u →∞ uc (cid:0) T ij ( u ) − δ ij (cid:1) form a gl ( m | n ) superalgebra, with commutation relations[ T ij [0] , T kl [0] } = ( − [ i ]([ k ]+[ l ])+[ k ][ l ] (cid:16) δ il T kj [0] − δ jk T il [0] (cid:17) , i, j, k, l = 1 , ..., m + n. (3.6)This superalgebra is a symmetry of the generalized model, since it commutes with the transfermatrix, [ T ij [0] , T ( z )] = 0, i, j = 1 , ..., m + n . In fact the monodromy matrix entries form arepresentation of this superalgebra:[ T ij [0] , T kl ( z ) } = ( − [ i ]([ k ]+[ l ])+[ k ][ l ] (cid:16) δ il T kj ( z ) − δ jk T il ( z ) (cid:17) , i, j, k, l = 1 , ..., m + n. (3.7)In particular, for the Cartan generators T jj [0] we obtain[ T jj [0] , T kl ( z )] = ( − [ j ] (cid:0) δ jl − δ jk (cid:1) T kl ( z ) , j, k, l = 1 , ..., m + n. (3.8)Then, the colors correspond to the eigenvalues under the Cartan generators h j = j X k =1 ( − [ k ] T kk [0] , j = 1 , ..., m + n − . (3.9) The last generator h m + n is central, see (3.10). h j , T kl ( z )] = ε j ( k, l ) T kl ( z ) with ε j ( k, l ) = − k ≤ j < lε j ( k, l ) = +1 if l ≤ j < kε j ( k, l ) = 0 otherwise (3.10)These eigenvalues just correspond to creation/annihilation operators as described above.Bethe vectors are certain polynomials in the creation operators T i,j applied to the vector | i . Since Bethe vectors are eigenvectors under the Cartan generators T kk [0], they are alsoeigenvectors of the color generators h j , and hence contain only terms with the same coloring. Remark.
In various models of physical interest the coloring of the Bethe vectors obeyscertain constraints, for instance, r ≥ r ≥ · · · ≥ r N . In particular, this case occurs ifthe monodromy matrix of the model is given by the product of the R -matrices (2.2) in thefundamental representation. We do not restrict ourselves with this particular case and donot impose any restriction for the coloring of the Bethe vectors. Thus, in what follows r i arearbitrary non-negative integers.In this paper we do not use an explicit form of the Bethe vectors, however, the reader canfind it in [29]. A generic Bethe vector of gl ( m | n )-based model depends on N = m + n − t , ¯ t , . . . , ¯ t N called Bethe parameters. We denote Bethe vectors by B (¯ t ), where¯ t = { t , . . . , t r ; t , . . . , t r ; . . . ; t N , . . . , t Nr N } , (3.11)and the cardinalities r i of the sets ¯ t i coincide with the coloring. Thus, each Bethe parameter t ik can be associated with a quasiparticle of the color i .Bethe vectors are symmetric over permutations of the parameters t ik within the set ¯ t i ,however, they are not symmetric over permutations over parameters belonging to differentsets ¯ t i and ¯ t j . For generic Bethe vectors the Bethe parameters t ik are generic complex num-bers. If these parameters satisfy a special system of equations (Bethe equations), then thecorresponding vector becomes an eigenvector of the transfer matrix (2.7). In this case it iscalled on-shell Bethe vector . In this paper we consider generic Bethe vectors, however, someformulas (for instance, the sum formula for the scalar product (4.11), (4.15)) can be specifiedto the case of on-shell Bethe vectors as well.Though we do not use the explicit form of the Bethe vectors, we should fix their normal-ization. We have already mentioned that a generic Bethe vector has the form of a polynomialin T i,j with i < j applied to the pseudovacuum | i . Among all the terms of this polynomialthere is one monomial that contains the operators T i,j with j − i = 1 only. Let us call thisterm the main term and denote it by e B (¯ t ). Then B (¯ t ) = e B (¯ t ) + . . . . (3.12)where ellipsis means all the terms containing at least one operator T i,j with j − i >
1. We willfix the normalization of the Bethe vectors by fixing a numeric coefficient of the main term e B (¯ t ) = T , (¯ t ) . . . T N,N +1 (¯ t N ) | i Q Ni =1 λ i +1 (¯ t i ) Q N − i =1 f [ i +1] (¯ t i +1 , ¯ t i ) , (3.13)where T i,i +1 (¯ t i ) = T i,i +1 ( t i ) . . . T i,i +1 ( t ir i ) (cid:16)Q ≤ j 1. Here we also have equipped the operators T ij withadditional superscripts showing the corresponding Yangians. This mapping also acts on thevacuum eigenvalues λ i ( u ) (3.1) and their ratios α i ( u ) (3.2) ϕ : ( λ i ( u ) → − λ N +2 − i ( u ) , i = 1 , . . . , N + 1 ,α i ( u ) → α N +1 − i ( u ) , i = 1 , . . . , N . (3.16)Morphism ϕ induces a mapping of Bethe vectors B m | n of Y ( gl ( m | n )) to Bethe vectors B n | m of Y ( gl ( n | m )). To describe this mapping we introduce special orderings of the sets ofBethe parameters. Namely, let −→ t = { ¯ t , ¯ t , . . . , ¯ t N } and ←− t = { ¯ t N , . . . , ¯ t , ¯ t } . (3.17)The ordering of the Bethe parameters within every set ¯ t k is not essential. Then ϕ (cid:16) B m | n ( −→ t ) (cid:17) = ( − r m B n | m ( ←− t ) Q Nk =1 α N +1 − k (¯ t k ) . (3.18)Applying the mapping (3.18) to B m | n and then replacing m ↔ n we obtain an alternativedescription of the Bethe vectors corresponding to the embedding of gl ( m | n − 1) to gl ( m | n ).The use of ϕ (3.18) allows one to establish important properties of the Bethe vectors scalarproducts (see section 7.2). Dual Bethe vectors belong to the dual space H ∗ , and they are polynomials in T i,j with i > j applied from the right to the dual pseudovacuum vector h | . This vector possesses propertiessimilar to (3.1) h | T i,i ( u ) = λ i ( u ) h | , i = 1 , . . . , m + n, h | T i,j ( u ) = 0 , i < j , (3.19)where the functions λ i ( u ) are the same as in (3.1).We denote dual Bethe vectors by C (¯ t ), where the set of Bethe parameters ¯ t consists ofseveral sets ¯ t i as in (3.11). Similarly to how it was done for Bethe vectors, we can introduce8he coloring of the dual Bethe vectors. At the same time the role of creation and annihilationoperators are reversed.One can obtain dual Bethe vectors via a special antimorphism of the algebra (2.4) [40]Ψ : T i,j ( u ) → ( − [ i ]([ j ]+1) T j,i ( u ) . (3.20)This antimorphism is nothing but a super (or equivalently, graded) transposition compatiblewith the notion of supertrace. It satisfies a propertyΨ( A · B ) = ( − [ A ][ B ] Ψ( B ) · Ψ( A ) , (3.21)where A and B are arbitrary elements of the monodromy matrix. If we extend the action ofthis antimorphism to the pseudovacuum vectors byΨ (cid:0) | i (cid:1) = h | , Ψ (cid:0) A | i (cid:1) = h | Ψ (cid:0) A (cid:1) , Ψ (cid:0) h | (cid:1) = | i , Ψ (cid:0) h | A (cid:1) = Ψ (cid:0) A (cid:1) | i , (3.22)then it turns out that [29]Ψ (cid:0) B (¯ t ) (cid:1) = C (¯ t ) , Ψ (cid:0) C (¯ t ) (cid:1) = ( − r m B (¯ t ) , (3.23)where r m = t m . Remark. It should not be surprising that Ψ (cid:0) B (¯ t ) (cid:1) = B (¯ t ). The point is that the an-timorphism Ψ is idempotent of order 4 and its square is the parity operator (counting thenumber of odd monodromy matrix elements modulo 2).Thus, dual Bethe vectors are polynomials in T i,j with i > j acting from the right onto h | .They also contain the main term e C (¯ t ), which now consists of the operators T i,j with i − j = 1.The main term of the dual Bethe vector can be obtained from (3.13) via the mapping Ψ: e C (¯ t ) = ( − r m ( r m − / h | T N +1 ,N (¯ t N ) . . . T , (¯ t ) Q Ni =1 λ i +1 (¯ t i ) Q N − i =1 f [ i +1] (¯ t i +1 , ¯ t i ) , (3.24)where T i +1 ,i (¯ t i ) = T i +1 ,i ( t i ) . . . T i +1 ,i ( t ir i ) (cid:16)Q ≤ j Proposition 4.1. Bethe vectors of gl ( m | n ) -based models satisfy a recursion B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ) = N +1 X j =2 T ,j ( z ) λ ( z ) X part(¯ t ,..., ¯ t j − ) B ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj ) × Q j − ν =2 α ν (¯ t ν I ) g [ ν ] (¯ t ν I , ¯ t ν − I ) γ ν (¯ t ν II , ¯ t ν I ) h (¯ t , z ) δ m, Q j − ν =1 f [ ν +1] (¯ t ν +1 , ¯ t ν I ) . (4.1) Here for j > the sets of Bethe parameters ¯ t , . . . , ¯ t j − are divided into disjoint subsets ¯ t ν I and ¯ t ν II ( ν = 2 , . . . , j − ) such that the subset ¯ t ν I consists of one element only: t ν I = 1 . Thesum is taken over all partitions of this type. We set by definition ¯ t I ≡ z and ¯ t N +1 = ∅ . We used the following notation in proposition 4.1 B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ) = B ( (cid:8) z, ¯ t (cid:9) ; ¯ t ; . . . ; ¯ t N ) , B ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj ) = B (¯ t ; ¯ t II ; . . . ; ¯ t j − II ; ¯ t j ; . . . ; ¯ t N ) . (4.2)This and similar notation will be used throughout of the paper. Remark. We stress that each of the subsets ¯ t I , . . . , ¯ t N I in (4.1) must consist of exactly oneelement. However, this condition is not feasible, if the original Bethe vector B ( t ) contains anempty set ¯ t k = ∅ for some k ∈ [2 , . . . , N ]. In this case, the sum over j in (4.1) breaks off at j = k . Indeed, the action of the operators T ,j ( z ) with j > k on a Bethe vector necessarilycreates a quasiparticle of the color k . Since this quasiparticle is absent in the lhs of (4.1), wecannot have the operators T ,j ( z ) with j > k in the rhs. Similar consideration shows that if B ( t ) contains several empty sets ¯ t k , . . . , ¯ t k ℓ , then the sum ends at j = min( k , . . . , k ℓ ). Remark. One can notice that for m = 1 an additional factor h (¯ t , z ) − appears in therecursion. The point is that with this recursion we add a quasiparticle of the color 1 tothe original set of quasiparticles via the actions of the operators T ,j . For m = 1 all theseoperators are odd, which explains the appearance of the factor h (¯ t , z ) − . This difference canalso be seen explicitly in the example of recursion for the main term (3.13) e B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ) = T , ( z ) e B (¯ t ) h (¯ t , z ) δ m, λ ( z ) f [2] (¯ t , z ) . (4.3)Using the mappings (3.15) and (3.20) one can obtain one more recursion for the Bethevectors and two recursions for the dual ones. Proposition 4.2. Bethe vectors of gl ( m | n ) -based models satisfy a recursion B ( (cid:8) ¯ t k (cid:9) N − ; (cid:8) z, ¯ t N (cid:9) ) = N X j =1 T j,N +1 ( z ) λ N +1 ( z ) X part(¯ t j ,..., ¯ t N − ) B ( (cid:8) ¯ t k (cid:9) j − ; (cid:8) ¯ t k II (cid:9) N − j ; ¯ t N ) × Q N − ν = j g [ ν +1] (¯ t ν +1 I , ¯ t ν I )ˆ γ ν (¯ t ν I , ¯ t ν II ) h (¯ t N , z ) δ m,N Q Nν = j f [ ν ] (¯ t ν I , ¯ t ν − ) . (4.4)10 ere for j < N the sets of Bethe parameters ¯ t j , . . . , ¯ t N − are divided into disjoint subsets ¯ t ν I and ¯ t ν II ( ν = j, . . . , N − ) such that the subset ¯ t ν I consists of one element: t ν I = 1 . The sumis taken over all partitions of this type. We set by definition ¯ t N I ≡ z and ¯ t = ∅ . Remark. If the Bethe vector B ( t ) contains several empty sets ¯ t k , . . . , ¯ t k ℓ , then the sumover j in (4.4) begins with j = max( k , . . . , k ℓ ) + 1.Acting with antimorphism (3.20) onto equations (4.1) and (4.4) we immediately arrive atrecursions for the dual Bethe vectors. Corollary 4.1. Dual Bethe vectors of gl ( m | n ) -based models satisfy recursions C ( (cid:8) z, ¯ s (cid:9) ; (cid:8) ¯ s k (cid:9) N ) = N +1 X j =2 X part(¯ s ,..., ¯ s j − ) C ( (cid:8) ¯ s (cid:9) ; (cid:8) ¯ s k II (cid:9) j − ; (cid:8) ¯ s k (cid:9) Nj ) T j, ( z ) λ ( z ) ( − r δ m, × Q j − ν =2 α ν (¯ s ν I ) g [ ν ] (¯ s ν I , ¯ s ν − I )ˆ γ ν (¯ s ν II , ¯ s ν I ) h (¯ s , z ) δ m, Q j − ν =1 f [ ν +1] (¯ s ν +1 , ¯ s ν I ) , (4.5) and C ( (cid:8) ¯ s k (cid:9) N − ; (cid:8) z, ¯ s N (cid:9) ) = N X j =1 X part(¯ s j ,..., ¯ s N − ) C ( (cid:8) ¯ s k (cid:9) j − ; (cid:8) ¯ s k II (cid:9) N − j ; ¯ s N ) T N +1 ,j ( z ) λ N +1 ( z ) ( − r N δ m,N × Q N − ν = j g [ ν ] (¯ s ν +1 I , ¯ s ν I ) γ ν (¯ s ν I , ¯ s ν II ) h (¯ s N , z ) δ m,N Q Nν = j f [ ν ] (¯ s ν I , ¯ s ν − ) . (4.6) Here the summation over the partitions occurs as in the formulas (4.1) and (4.4) . Thenumbers r (resp. r N ) are the cardinalities of the sets ¯ s (resp. ¯ s N ). The subsets ¯ s ν I consistof one element: s ν I = 1 . If C (¯ s ) contains empty sets of the Bethe parameters, then the sumscut similarly to the case of the Bethe vectors B (¯ t ) . By definition ¯ s I ≡ z in (4.5) , ¯ s N I ≡ z in (4.6) , and ¯ s = ¯ s N +1 = ∅ . The proof of corollary 4.1 is given in section 5.2.Using recursion (4.1) one can express a Bethe vector with t = r in terms of Bethevectors with t = r − 1. Applying this recursion successively we eventually express theoriginal Bethe vector in terms of a linear combination of terms that are products of the mon-odromy matrix elements T ,j acting onto Bethe vectors with t = 0. The latter effectivelycorresponds to the Yangian Y ( gl ( m − | n )) (see [29]): B m | n ( ∅ ; { ¯ t k } N ) = B m − | n (¯ t ) (cid:12)(cid:12)(cid:12) ¯ t k → ¯ t k +1 . (4.7)Thus, continuing this process we formally can reduce Bethe vectors of Y ( gl ( m | n )) to the onesof Y ( gl (1 | n )).Similarly, using recursion (4.4) and B m | n ( { ¯ t k } N − ; ∅ ) = B m | n − (¯ t ) , (4.8)we eventually reduce Bethe vectors of Y ( gl ( m | n )) to the ones of Y ( gl ( m | Y ( gl (1 | B | (¯ t ) = T , (¯ t ) | i /λ (¯ t ). Similarly, one can builtdual Bethe vectors via (4.5), (4.6). These procedures, of course, are of little use for practicalpurposes, however, they can be used to prove various assertions by induction.11 .2 Sum formula for the scalar product Let B (¯ t ) be a generic Bethe vector and C (¯ s ) be a generic dual Bethe vector such that t k = s k = r k , k = 1 , . . . , N . Then their scalar product is defined by S (¯ s | ¯ t ) = C (¯ s ) B (¯ t ) . (4.9)Note that if t k = s k for some k ∈ { , . . . , N } , then the scalar product vanishes. Indeed,in this case the numbers of creation and annihilation operators of the color k do not coincide.Applying (3.22) to the scalar product and using (cid:2) B (¯ t ) (cid:3) = (cid:2) C (¯ t ) (cid:3) = r m [29] we find that S (¯ s | ¯ t ) = C (¯ t ) B (¯ s ) = S (¯ t | ¯ s ) . (4.10)Computing the scalar product one should use commutation relations (2.5) and move alloperators T i,j with i > j from the dual vector C (¯ s ) to the right through the operators T i,j with i < j , which are in the vector B (¯ t ). In the process of commutation, new operators willappear, which should be moved to the right or left, depending on the relation between theirsubscripts. Once an operator T i,j with i ≥ j reaches the vector | i , it either annihilates it for i > j , or gives a function λ i for i = j . The argument of the function λ i can a priori be anyBethe parameter t kℓ or s kℓ . Similarly, if an operator T i,j with i ≤ j reaches the vector h | , iteither annihilates it for i < j , or gives a function λ i for i = j , which depends on one of theBethe parameters.Due to the normalization of the Bethe vectors the functions λ i then turn into the ratios α i .Thus, the scalar product eventually depends on the functions α i and some rational functionswhich appear in the process of commutating the monodromy matrix entries.The following proposition specifies how the scalar product depends on the functions α i . Proposition 4.3. Let B (¯ t ) be a generic Bethe vector and C (¯ s ) be a generic dual Bethe vectorsuch that t k = s k = r k , k = 1 , . . . , N . Then their scalar product is given by S (¯ s | ¯ t ) = X W m | n part (¯ s I , ¯ s II | ¯ t I , ¯ t II ) N Y k =1 α k (¯ s k I ) α k (¯ t k II ) . (4.11) Here all the sets of the Bethe parameters ¯ t k and ¯ s k are divided into two subsets ¯ t k ⇒ { ¯ t k I , ¯ t k II } and ¯ s k ⇒ { ¯ s k I , ¯ s k II } , such that t k I = s k I . The sum is taken over all possible partitions of thistype. The rational coefficients W m | n part depend on the partition. They are completely determinedby the R -matrix of the model and do not depend on the ratios of the vacuum eigenvalues α k . Proposition 4.3 states that after calculating the scalar product the Bethe parameters of thetype k ( t kj or s kj ) can be arguments of functions λ k +1 or λ k only. Due to the normalization ofthe Bethe vectors these functions respectively cancel in the first case or produce the functions α k in the second case. We prove proposition 4.3 in section 6.1.We would like to stress that the rational functions W m | n part are model independent. Indeed,within the QISM framework the Hamiltonian of a quantum model is encoded in the supertraceof the monodromy matrix T ( u ). Thus, one can say that the quantum model is defined by T ( u ). Looking at presentation (4.11) one can notice that the model dependent part of thescalar product entirely lies in the α k functions, because only these functional parametersdepend on the monodromy matrix. On the other hand, the coefficients W m | n part are completelydetermined by the R -matrix, that is, they depend only on the underlying algebra. Thus,if two different quantum integrable models have the same R -matrix (2.2), then the scalar12roducts of Bethe vectors in these models are given by (4.11) with the same coefficients W m | n part .The Highest Coefficient (HC) of the scalar product is defined as a rational coefficientcorresponding to the partition ¯ s I = ¯ s , ¯ t I = ¯ t , and ¯ s II = ¯ t II = ∅ . We denote the HC by Z m | n (¯ s | ¯ t ). Then, the HC is a particular case of the rational coefficient W m | n part : W m | n part (¯ s, ∅| ¯ t, ∅ ) = Z m | n (¯ s | ¯ t ) . (4.12)Similarly one can define a conjugated HC Z m | n (¯ s | ¯ t ) as a coefficient corresponding to thepartition ¯ s II = ¯ s , ¯ t II = ¯ t , and ¯ s I = ¯ t I = ∅ . W m | n part ( ∅ , ¯ s |∅ , ¯ t ) = Z m | n (¯ s | ¯ t ) . (4.13)Due to (4.10) one can easily show that Z m | n (¯ s | ¯ t ) = Z m | n (¯ t | ¯ s ) . (4.14)The following proposition determines the general coefficient W m | n part in terms of the HC. Proposition 4.4. For a fixed partition ¯ t k ⇒ { ¯ t k I , ¯ t k II } and ¯ s k ⇒ { ¯ s k I , ¯ s k II } in (4.11) the rationalcoefficient W m | n part has the following presentation in terms of the HC: W m | n part (¯ s I , ¯ s II | ¯ t I , ¯ t II ) = Z m | n (¯ s I | ¯ t I ) Z m | n (¯ t II | ¯ s II ) Q Nk =1 γ k (¯ s k II , ¯ s k I ) γ k (¯ t k I , ¯ t k II ) Q N − j =1 f [ j +1] (¯ s j +1 II , ¯ s j I ) f [ j +1] (¯ t j +1 I , ¯ t j II ) . (4.15)The proof of proposition 4.4 is given in section 6.2.Explicit expressions for the HC are known for small m and n [15]. In particular, Z | (¯ s | ¯ t ) = g (¯ s, ¯ t ) . (4.16)Determinant representations for Z | or Z | were obtained in [32]. Relatively compact for-mulas for Z m | n at m + n = 3 were found in [11, 14, 15], however, representations for theHC in the general gl ( m | n ) case are very cumbersome. Instead, one can use relatively simplerecursions established by the following propositions. Proposition 4.5. The HC Z m | n (¯ s | ¯ t ) possesses the following recursion over the set ¯ s : Z m | n (¯ s | ¯ t ) = N +1 X p =2 X part(¯ s ,..., ¯ s p − )part(¯ t ,..., ¯ t p − ) g [2] (¯ t I , ¯ s I ) γ (¯ t I , ¯ t II ) f (¯ t II , ¯ s I ) f [ p ] (¯ s p , ¯ s p − I ) h (¯ s , ¯ s I ) δ m, × p − Y ν =2 g [ ν ] (¯ s ν I , ¯ s ν − I ) g [ ν +1] (¯ t ν I , ¯ t ν − I ) γ ν (¯ s ν II , ¯ s ν I ) γ ν (¯ t ν I , ¯ t ν II ) f [ ν ] (¯ s ν , ¯ s ν − I ) f [ ν ] (¯ t ν I , ¯ t ν − ) × Z m | n ( (cid:8) ¯ s k II (cid:9) p − , (cid:8) ¯ s k (cid:9) Np | (cid:8) ¯ t k II (cid:9) p − ; (cid:8) ¯ t k (cid:9) Np ) . (4.17) Here for every fixed p ∈ { , . . . , m + n } the sums are taken over partitions ¯ t k ⇒ { ¯ t k I , ¯ t k II } with k = 1 , . . . , p − and ¯ s k ⇒ { ¯ s k I , ¯ s k II } with k = 2 , . . . , p − , such that t k I = s k I = 1 for k = 2 , ..., p − . The subset ¯ s I is a fixed Bethe parameter from the set ¯ s . There is no sumover partitions of the set ¯ s in (4.17) . Note that we have changed the definition of the HC with respect to the one that we used in our previouspublications. Now it involves a normalization factor Q N − j =1 f [ j +1] (¯ s j +1 , ¯ s j ) f [ j +1] (¯ t j +1 , ¯ t j ). Corollary 4.2. The HC Z m | n (¯ s | ¯ t ) satisfies the following recursion over the set ¯ t N : Z m | n (¯ s | ¯ t ) = N X p =1 X part(¯ s p ,..., ¯ s N )part(¯ t p ,..., ¯ t N − ) g (¯ s N I , ¯ t N I )ˆ γ N (¯ s N II , ¯ s N I ) f (¯ s N II , ¯ t N I ) f [ p ] (¯ t p I , ¯ t p − ) h (¯ t N , ¯ t N I ) δ m,N × N − Y ν = p g [ ν +1] (¯ s ν +1 I , ¯ s ν I ) g [ ν +1] (¯ t ν +1 I , ¯ t ν I )ˆ γ ν (¯ s ν II , ¯ s ν I )ˆ γ ν (¯ t ν I , ¯ t ν II ) f [ ν +1] (¯ s ν +1 , ¯ s ν I ) f [ ν +1] (¯ t ν +1 I , ¯ t ν ) × Z m | n ( (cid:8) ¯ s k (cid:9) p − , (cid:8) ¯ s k II (cid:9) Np | (cid:8) ¯ t k (cid:9) p − ; (cid:8) ¯ t k II (cid:9) Np ) . (4.18) Here for every fixed p ∈ { , . . . , m + n − } the sums are taken over partitions ¯ t k ⇒ { ¯ t k I , ¯ t k II } with k = p, . . . , N − and ¯ s k ⇒ { ¯ s k I , ¯ s k II } with k = p, . . . , N , such that t k I = s k I = 1 for k = p, . . . , N − . The subset ¯ t N I is a fixed Bethe parameter from the set ¯ t N . There is no sumover partitions of the set ¯ t N in (4.18) . This recursion follows from (4.17) and a symmetry property of the HC (7.14) proved insection 7.2. Remark. Similarly to the recursions for the Bethe vectors the sums over p in (4.17), (4.18)break off, if HC Z m | n (¯ s | ¯ t ) contains empty sets of the Bethe parameters. If the colors of theempty sets are { k , . . . , k ℓ } , then the sum over p ends at p = min( k , . . . , k ℓ ) in the recursion(4.17), while in the recursion (4.18) it begins at p = max( k , . . . , k ℓ ) + 1 . These restrictionsfollow from the corresponding restrictions in the recursions for the Bethe vectors.Using proposition 4.5 one can built the HC with s = t = r in terms of the HC with s = t = r − 1. In particular, Z m | n with s = t = 1 can be expressed in terms of Z m | n with s = t = 0. It is obvious, however, that Z m | n ( ∅ , { ¯ s k } N |∅ , { ¯ t k } N ) = Z m − | n ( { ¯ s k } N |{ ¯ t k } N ) . (4.19)due to (4.7). Thus, equation (4.17) allows one to perform recursion over m as well.Similarly, corollary 4.2 allows one to find the HC with s N = t N = r N in terms of theHC with s N = t N = r N − n .Thus, using recursions (4.17) and (4.18) one can eventually express Z m | n (¯ s | ¯ t ) in terms ofknown HC, say, for m + n = 2. However, the corresponding explicit expressions hardly canbe used in practice, because they are too bulky. At the same time, these recursions appearbe very useful for proofs of some important properties of HC. gl ( m ) symmetry As already mentioned, the results stated above are also valid for the case of gl ( m ) Lie algebraswith m > 1, simply by setting n = 0. This implies N = m − 1. In that case, most ofexpressions simplify, due to the absence of grading. We present here the simplified resultsoccurring for gl ( m ). • Bethe vectors of gl ( m )-based models satisfy the recursions B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) m − ) = m X j =2 T ,j ( z ) λ ( z ) X part(¯ t ,..., ¯ t j − ) B ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) m − j ) × Q j − ν =2 α ν (¯ t ν I ) g (¯ t ν I , ¯ t ν − I ) f (¯ t ν II , ¯ t ν I ) Q j − ν =1 f (¯ t ν +1 , ¯ t ν I ) , (4.20)14here the conditions on sets of Bethe parameters are the same as in Proposition 4.1, B ( (cid:8) ¯ t k (cid:9) m − ; (cid:8) z, ¯ t m − (cid:9) ) = m − X j =1 T j,m ( z ) λ m ( z ) X part(¯ t j ,..., ¯ t m − ) B ( (cid:8) ¯ t k (cid:9) j − ; (cid:8) ¯ t k II (cid:9) m − j ; ¯ t m − ) × Q m − ν = j g (¯ t ν +1 I , ¯ t ν I ) f (¯ t ν I , ¯ t ν II ) Q m − ν = j f (¯ t ν I , ¯ t ν − ) , (4.21)where the conditions on sets of Bethe parameters are the same as in Proposition 4.2.The starting point for these recursions is the gl (2) Bethe vector B (¯ t ) = T (¯ t ) | i /λ (¯ t ). • Dual Bethe vectors of gl ( m )-based models satisfy the recursions C ( (cid:8) z, ¯ s (cid:9) ; (cid:8) ¯ s k (cid:9) m − ) = m X j =2 X part(¯ s ,..., ¯ s j − ) C ( (cid:8) ¯ s (cid:9) ; (cid:8) ¯ s k II (cid:9) j − ; (cid:8) ¯ s k (cid:9) m − j ) T j, ( z ) λ ( z ) × Q j − ν =2 α ν (¯ s ν I ) g (¯ s ν I , ¯ s ν − I ) f (¯ s ν II , ¯ s ν I ) Q j − ν =1 f (¯ s ν +1 , ¯ s ν I ) , (4.22)and C ( (cid:8) ¯ s k (cid:9) m − ; (cid:8) z, ¯ s m − (cid:9) ) = m − X j =1 X part(¯ s j ,..., ¯ s m − ) C ( (cid:8) ¯ s k (cid:9) j − ; (cid:8) ¯ s k II (cid:9) m − j ; ¯ s m − ) T m,j ( z ) λ m ( z ) × Q m − ν = j g (¯ s ν +1 I , ¯ s ν I ) f (¯ s ν I , ¯ s ν II ) Q m − ν = j f (¯ s ν I , ¯ s ν − ) . (4.23)The conditions on the sets of parameters and partitions are given in Corollary 4.1. Thestarting point for these recursions is the gl (2) dual Bethe vector C (¯ t ) = h | T (¯ t ) /λ (¯ t ). • For a fixed partition ¯ t k ⇒ { ¯ t k I , ¯ t k II } and ¯ s k ⇒ { ¯ s k I , ¯ s k II } in (4.11) the rational coefficient W m part has the following presentation in terms of the HC: W m part (¯ s I , ¯ s II | ¯ t I , ¯ t II ) = Z m (¯ s I | ¯ t I ) Z m (¯ t II | ¯ s II ) Q m − k =1 f (¯ s k II , ¯ s k I ) f (¯ t k I , ¯ t k II ) Q m − j =1 f (¯ s j +1 II , ¯ s j I ) f (¯ t j +1 I , ¯ t j II ) . (4.24)In the gl (2) and gl (3) cases this expression reduces to the formulas respectively obtainedin [5] and [10]. • The HC Z m (¯ s | ¯ t ) possesses the following recursions: Z m (¯ s | ¯ t ) = m X p =2 X part(¯ s ,..., ¯ s p − )part(¯ t ,..., ¯ t p − ) g (¯ t I , ¯ s I ) f (¯ t I , ¯ t II ) f (¯ t II , ¯ s I ) f (¯ s p , ¯ s p − I ) × p − Y ν =2 g (¯ s ν I , ¯ s ν − I ) g (¯ t ν I , ¯ t ν − I ) f (¯ s ν II , ¯ s ν I ) f (¯ t ν I , ¯ t ν II ) f (¯ s ν , ¯ s ν − I ) f (¯ t ν I , ¯ t ν − ) × Z m ( (cid:8) ¯ s k II (cid:9) p − , (cid:8) ¯ s k (cid:9) m − p | (cid:8) ¯ t k II (cid:9) p − ; (cid:8) ¯ t k (cid:9) m − p ) , (4.25)15nd Z m (¯ s | ¯ t ) = m − X p =1 X part(¯ s p ,..., ¯ s m − )part(¯ t p ,..., ¯ t m − ) g (¯ t m − I , ¯ s m − I ) f (¯ s m − II , ¯ s m − I ) f (¯ t m − I , ¯ s m − II ) f (¯ t p I , ¯ t p − ) × m − Y ν = p g (¯ s ν +1 I , ¯ s ν I ) g (¯ t ν +1 I , ¯ t ν I ) f (¯ s ν II , ¯ s ν I ) f (¯ t ν I , ¯ t ν II ) f (¯ s ν +1 , ¯ s ν I ) f (¯ t ν +1 I , ¯ t ν ) × Z m ( (cid:8) ¯ s k (cid:9) p − , (cid:8) ¯ s k II (cid:9) m − p | (cid:8) ¯ t k (cid:9) p − ; (cid:8) ¯ t k II (cid:9) m − p ) . (4.26)The conditions on the sets of parameters and partitions are given in Proposition 4.5 andCorollary 4.2. Here, the starting point corresponds to the gl (2) case, in which Z (¯ s | ¯ t )is equal to the partition function of the six-vertex model with domain wall boundaryconditions [5, 32]. One can prove proposition 4.1 via the formulas of the operators T ,j ( z ) action onto the Bethevector. These formulas were derived in [29] T ,j ( z ) B (¯ t ) = η j B ( { z, ¯ t k } j − ; { ¯ t k } Nj )+ N +1 X q = j +1 X part(¯ t j ,..., ¯ t q − ) H q,j (part) B ( { z, ¯ t k } j − ; { z, ¯ t k II } q − j ; { ¯ t k } Nq ) . (5.1)Here in the second line for every q we have a sum over partitions of the sets ¯ t j , . . . , ¯ t q − . Thecoefficient η j in (5.1) is η j = λ j ( z ) f [ j ] (¯ t j , z ) h (¯ t m , z ) [ j ] . (5.2)The coefficient H q,j depends on the partitions and has the form H q,j (part) = f [ q ] (¯ t q , z ) h (¯ t m , z ) [ j ] h (¯ t m II , z ) [ q ] − [ j ] λ q ( z ) g [ j ] ( z, ¯ t q − I ) q − Y ν = j +1 g [ ν ] (¯ t ν I , ¯ t ν − I ) q − Y ν = j Ω ν , (5.3)where Ω ν = α ν (¯ t ν I ) γ ν (¯ t ν II , ¯ t ν I ) f [ ν +1] (¯ t ν +1 , ¯ t ν I ) . (5.4)Note that in (5.1) the operators T ,j ( z ) act onto B (¯ t ), while in (4.1) these operators actonto B ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj ). Therefore, we can directly use the action formula (5.1) for j = 2 only. For j > t , . . . , ¯ t j − with thesubsets ¯ t II , . . . , ¯ t j − II before substituting (5.1) into recursion (4.1).We look for the terms in the formulas (5.2) and (5.3) where we should do the replacement { ¯ t , . . . , ¯ t j − } → { ¯ t II , . . . , ¯ t j − II } . The sets { ¯ t , . . . , ¯ t j − } appear only in the factors h (¯ t m , z ) [ j ] and h (¯ t m II , z ) [ q ] − [ j ] , and provided that m ∈ { , . . . , j − } . This implies that for m = 1 there isno replacement to do. For m > 1, we have [ j ] = 1, because j > m , and [ q ] = [ j ], because q > j .Then, the factor h (¯ t m II , z ) [ q ] − [ j ] drops out, and we should only replace h (¯ t m , z ) [ j ] → h (¯ t m II , z ) [ j ] .16hus, we arrive at the following action formula: T ,j ( z ) B ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj ) = ˜ η j B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) z, ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj )+ N +1 X q = j +1 X part(¯ t j ,..., ¯ t q − ) ˜ H q,j (part) B ( (cid:8) z, ¯ t (cid:9) ; { z, ¯ t k II } q − ; { ¯ t k } Nq ) , (5.5)where ˜ η j = λ j ( z ) f [ j ] (¯ t j , z ) h (¯ t m II , z ) [ j ] h (¯ t m I , z ) δ m, , (5.6)and˜ H q,j (part) = f [ q ] (¯ t q , z ) h (¯ t m II , z ) [ q ] h (¯ t m I , z ) δ m, λ q ( z ) g [ j ] ( z, ¯ t q − I ) q − Y ν = j +1 g [ ν ] (¯ t ν I , ¯ t ν − I ) q − Y ν = j Ω ν . (5.7)Now everything is ready for substituting the action formula (5.5) into recursion (4.1). Let X = N +1 X j =2 T ,j ( z ) X part(¯ t ,..., ¯ t j − ) Q j − ν =2 g [ ν ] (¯ t ν I , ¯ t ν − I )Ω ν λ ( z ) h (¯ t , z ) δ m, f [2] (¯ t , z ) B ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj ) . (5.8)It is easy to see that X is nothing else but the r.h.s. of recursion (4.1). Thus, our goal is toshow that X = B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ). Substituting (5.5) into (5.8) we obtain X = N +1 X j =2 X part(¯ t ,..., ¯ t j − ) ˜ η j Q j − ν =2 g [ ν ] (¯ t ν I , ¯ t ν − I )Ω ν λ ( z ) h (¯ t , z ) δ m, f [2] (¯ t , z ) B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) z, ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj )+ N +1 X j =2 N +1 X q = j +1 X part(¯ t ,..., ¯ t q − ) ˜ H q,j (part) Q j − ν =2 g [ ν ] (¯ t ν I , ¯ t ν − I )Ω ν λ ( z ) h (¯ t , z ) δ m, f [2] (¯ t , z ) B ( (cid:8) z, ¯ t (cid:9) ; { z, ¯ t k II } q − ; { ¯ t k } Nq ) . (5.9)It is convenient to divide X into three contributions X = X (1) + X (2) + X (3) . (5.10)The first term X (1) corresponds to j = 2 in the first line of (5.9): X (1) = ˜ η B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ) λ ( z ) h (¯ t , z ) δ m, f [2] (¯ t , z ) . (5.11)Substituting here ˜ η we see that X (1) = B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ) . (5.12)The contribution X (2) includes the terms with j > X (3) comes from the second line of (5.9). Consider X (3) changing the order ofsummation and substituting there (5.7). We have X (3) = N +1 X q =3 q − X j =2 X part(¯ t ,..., ¯ t q − ) λ q ( z ) f [ q ] (¯ t q , z ) h (¯ t m II , z ) [ q ] h (¯ t m I , z ) δ m, λ ( z ) h (¯ t , z ) δ m, f [2] (¯ t , z ) × g ( z, ¯ t q − I ) g (¯ t j I , ¯ t j − I ) q − Y ν =2 g [ ν ] (¯ t ν I , ¯ t ν − I )Ω ν ! B ( (cid:8) z, ¯ t (cid:9) ; { z, ¯ t k II } q − ; { ¯ t k } Nq ) . (5.13)17he sum over j can be easily computed q − X j =2 g (¯ t j I , ¯ t j − I ) = 1 c q − X j =2 (¯ t j I − ¯ t j − I ) = 1 c (¯ t q − I − ¯ t I ) = − /g ( z, ¯ t q − I ) , (5.14)and we recall that by definition ¯ t I = z . Thus, X (3) = − N +1 X q =3 X part(¯ t ,..., ¯ t q − ) λ q ( z ) f [ q ] (¯ t q , z ) h (¯ t m II , z ) [ q ] λ ( z ) h (¯ t II , z ) δ m, f [2] (¯ t , z ) q − Y ν =2 g [ ν ] (¯ t ν I , ¯ t ν − I )Ω ν × B ( (cid:8) z, ¯ t (cid:9) ; { z, ¯ t k II } q − ; { ¯ t k } Nq ) . (5.15)On the other hand, the contribution X (2) is X (2) = N +1 X j =3 X part(¯ t ,..., ¯ t j − ) λ j ( z ) f [ j ] (¯ t j , z ) h (¯ t m II , z ) [ j ] λ ( z ) h (¯ t II , z ) δ m, f [2] (¯ t , z ) j − Y ν =2 g [ ν ] (¯ t ν I , ¯ t ν − I )Ω ν × B ( (cid:8) z, ¯ t (cid:9) ; { z, ¯ t k II } j − ; { ¯ t k } Nj ) . (5.16)Comparing (5.16) and (5.15) we see that they cancel each other. Thus, X = B ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ). (cid:3) Let us derive now recursion (4.4) starting with (4.1) and using morphism (3.15). Sincethe mapping (3.15) relates two different Yangians Y ( gl ( m | n )) and Y ( gl ( n | m )), we use hereadditional superscripts for the functions g ( u, v ), f ( u, v ), γ ( u, v ), and ˆ γ ( u, v ). For example,notation f m | n [ ν ] ( u, v ) means that the function f [ ν ] ( u, v ) is defined with respect to Y ( gl ( m | n )): f m | n [ ν ] ( u, v ) = (cid:20) f ( u, v ) , ν ≤ m,f ( v, u ) , ν > m. (5.17)At the same time the notation f n | m [ ν ] ( u, v ) means that the function f [ ν ] ( u, v ) is defined withrespect to Y ( gl ( n | m )): f n | m [ ν ] ( u, v ) = (cid:20) f ( u, v ) , ν ≤ n,f ( v, u ) , ν > n. (5.18)The other rational functions should be understood similarly. It is easy to see that g m | n [ ν ] ( u, v ) = g n | m [ N +2 − ν ] ( v, u ) ,f m | n [ ν ] ( u, v ) = f n | m [ N +2 − ν ] ( v, u ) ,γ m | nν ( u, v ) = ˆ γ n | mN +1 − ν ( v, u ) . (5.19)Let us act with ϕ onto (4.1). Due to (3.15)–(3.18) we have ϕ T m | n ,j ( z ) λ ( z ) ! = ( − [ j ] T n | mN +2 − j,N +1 ( z ) λ N ( z ) , (5.20)18 (cid:16) B m | n ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ) (cid:17) = ( − r m + δ m, B n | m ( (cid:8) ¯ t k (cid:9) N ; (cid:8) z, ¯ t (cid:9) ) α N ( z ) Q Nk =1 α N +1 − k (¯ t k ) , (5.21)and ϕ B m | n ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj ) j − Y ν =2 α ν (¯ t ν I ) ! = ( − r m + δ m, +[ j ] B n | m ( (cid:8) ¯ t k (cid:9) jN ; (cid:8) ¯ t k II (cid:9) j − ; ¯ t ) Q Nk =1 α N +1 − k (¯ t k ) (5.22)Thus, the action of the morphism ϕ onto (4.1) gives B n | m ( (cid:8) ¯ t k (cid:9) N ; (cid:8) z, ¯ t (cid:9) ) = N +1 X j =2 T N +2 − j,N +1 ( z ) λ N +1 ( z ) X part(¯ t ,..., ¯ t j − ) B n | m ( (cid:8) ¯ t k (cid:9) jN ; (cid:8) ¯ t k II (cid:9) j − ; ¯ t ) × Q j − ν =2 g m | n [ ν ] (¯ t ν I , ¯ t ν − I ) γ m | nν (¯ t ν II , ¯ t ν I ) h (¯ t , z ) δ m, Q j − ν =1 f m | n [ ν +1] (¯ t ν +1 , ¯ t ν I ) . (5.23)Using the relations (5.19) and the trivial identity δ m, = δ n,N we recast (5.23) as B n | m ( (cid:8) ¯ t k (cid:9) N ; (cid:8) z, ¯ t (cid:9) ) = N +1 X j =2 T N +2 − j,N +1 ( z ) λ N +1 ( z ) X part(¯ t ,..., ¯ t j − ) B n | m ( (cid:8) ¯ t k (cid:9) jN ; (cid:8) ¯ t k II (cid:9) j − ; ¯ t ) × Q j − ν =2 g n | m [ N +2 − ν ] (¯ t ν − I , ¯ t ν I )ˆ γ n | mN +1 − ν (¯ t ν I , ¯ t ν II ) h (¯ t , z ) δ n,N Q j − ν =1 f n | m [ N +1 − ν ] (¯ t ν I , ¯ t ν +1 ) . (5.24)Finally, relabeling the sets of the Bethe parameters ¯ t k → ¯ t N +1 − k and changing ν → N + 1 − ν we obtain B n | m ( (cid:8) ¯ t k (cid:9) N − ; { z, ¯ t N } ) = N X j =1 T j,N +1 ( z ) λ N +1 ( z ) X part(¯ t j ,..., ¯ t N − ) B n | m ( (cid:8) ¯ t k (cid:9) j − ; (cid:8) ¯ t k II (cid:9) N − j ; ¯ t N ) × Q N − ν = j g n | m [ ν +1] (¯ t ν +1 I , ¯ t ν I )ˆ γ n | mν (¯ t ν I , ¯ t ν II ) h (¯ t N , z ) δ n,N Q Nν = j f n | m [ ν ] (¯ t ν I , ¯ t ν − ) . (5.25)It remains to replace m ↔ n , and we arrive at (4.4). (cid:3) To obtain recursion for dual Bethe vectors it is enough to act with antimorphism (3.20) ontorecursions (4.1) and (4.4). Consider in details the action of Ψ onto (4.1).Acting with Ψ on the lhs of (4.1) we obtain a dual vector C ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ) due to (3.23).In the rhs we have Ψ( T ,j B ) = ( − [ j ][ B ] C T j, . (5.26)The parity of the Bethe vector can be determined via the coloring arguments. Recall thatBethe vectors are polynomials in the operators T i,j acting on the vector | i , and all the termsof these polynomials have the same coloring. Due to the general rule, a quasiparticle of thecolor m can be created by the operators T i,j with i ≤ m and j > m . Hence, all these operatorsare odd, because [ i ] = 0 for i ≤ m and [ j ] = 1 for j > m . On the other hand, the action of19n even operator T i,j cannot create a quasiparticle of the color m due to similar arguments.Thus, if a Bethe vector has a coloring { r , . . . , r N } , then all the terms of the polynomial in T i,j contain exactly r m odd operators, where r m = t m . Thus, (cid:2) B (¯ t ) (cid:3) = r m , mod 2.In the case under consideration we should find the number r ′ m of the odd operators in theBethe vector B ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj ). Let r m = t m in the original vector B (¯ t ). If m = 1,then r ′ m = r m . If 1 < m < j , then r ′ m = r m − 1. Finally, if m ≥ j , then r ′ m = r m . All thesecases can be described by the formula r ′ m = r m − [ j ] + δ m, . Thus, we obtain C ( (cid:8) z, ¯ t (cid:9) ; (cid:8) ¯ t k (cid:9) N ) = N +1 X j =2 X part(¯ t ,..., ¯ t j − ) C ( (cid:8) ¯ t (cid:9) ; (cid:8) ¯ t k II (cid:9) j − ; (cid:8) ¯ t k (cid:9) Nj ) T j, ( z ) λ ( z ) ( − [ j ] r ′ m × Q j − ν =2 α ν (¯ t ν I ) g [ ν ] (¯ t ν I , ¯ t ν − I ) γ ν (¯ t ν II , ¯ t ν I ) h (¯ t , z ) δ m, Q j − ν =1 f [ ν +1] (¯ t ν +1 , ¯ t ν I ) , (5.27)where r ′ m = r m − [ j ] + δ m, .This expression can be slightly simplified. Recall that ˆ γ i ( x, y ) = ( − δ m,i γ i ( x, y ). Thus,changing γ ν (¯ t ν II , ¯ t ν I ) → ˆ γ ν (¯ t ν II , ¯ t ν I ) in (5.27) we obtain j − Y ν =2 γ ν (¯ t ν II , ¯ t ν I ) = ( − ([ j ] − [2]) r ′ m j − Y ν =2 ˆ γ ν (¯ t ν II , ¯ t ν I ) . (5.28)It remains to observe that [2] = δ m, . Thus, substituting (5.28) into (5.27) and replacing thesets ¯ t k with ¯ s k we arrive at (4.5). Recursion (4.6) can be obtained exactly in the same way. λ i ( z ) In this section, we investigate the functional dependence of the scalar product on the functions α i . Proposition 4.3 states that the Bethe parameters from the sets ¯ s i and ¯ t i can be thearguments of the functions α i only. In other words, the scalar product does not depend on α i ( s ℓk ) or α i ( t ℓk ) with ℓ = i .We prove this statement via induction over N = m + n − 1. For N = 1 it becomes obvious.Assume that it is valid for some N − C m | n (¯ s )and B m | n (¯ t ) with m + n − N . Observe that we added superscripts to the Bethe vectorsin order to distinguish them from the vectors corresponding to gl ( m − | n ) algebra. Wefirst prove that the scalar product does not depend on the functions α i ( s ℓk ) with ℓ = i for i = 2 , . . . , N .Successive application of the recursion (4.5) allows one to express a dual Bethe vector C m | n (¯ s ) in terms of dual Bethe vectors C m − | n (¯ σ ). Schematically this expression can bewritten in the following form C m | n (¯ s ) = m + n X j ,...,j r =2 X { ¯ σ ,..., ¯ σ N } Θ (¯ s ) j ,...,j r (¯ σ ) C m − | n ( (cid:8) ¯ σ (cid:9) N ) T j , ( s ) . . . T j r , ( s r ) λ (¯ s ) . (6.1)Here r = s and ¯ σ i ⊂ ¯ s i for i = 2 , . . . , N . The sum is taken over multi-index { j , . . . , j r } .Every term of this sum contains also a sum over partitions of the sets ¯ s , . . . , ¯ s N into subsets¯ σ , . . . , ¯ σ N and their complementary subsets. The factors Θ (¯ s ) j ,...,j r (¯ σ ) are some numerical20oefficients whose explicit form is not essential. It is important, however, to note that in(4.5) they depend on α i ( s ik ) with i = 2 , . . . , N and do not depend on the functions α i withother arguments.Let us multiply (6.1) from the right by a Bethe vector B m | n (¯ t ) and act with the operators T j p , ( s p ) onto this vector. Due to the results of [29] the action of any operator T ij ( z ) ontothe Bethe vector B m | n (¯ t ) gives a linear combination of new Bethe vectors B m | n (¯ τ ), such that¯ τ = { ¯ τ , . . . , ¯ τ N } and ¯ τ i ⊂ { ¯ t i ∪ z } . In the case under consideration each of the operators T j p , ( s p ) annihilates a particle of color 1. Hence, the total action of T j , ( s ) . . . T j r , ( s r )annihilates all the particles of color 1 in the vector B m | n (¯ t ). Thus, after this action the Bethevector B m | n (¯ t ) turns into B m − | n (¯ τ ), where ¯ τ = { ¯ τ , . . . , ¯ τ N } and ¯ τ i ⊂ { ¯ t i ∪ ¯ s } T j , ( s ) . . . T j r , ( s r ) λ (¯ s ) B m | n (¯ t ) = X { ¯ τ ,..., ¯ τ N } Θ (¯ t ) (¯ τ ) B m − | n ( (cid:8) ¯ τ k (cid:9) N ) . (6.2)Here the coefficients Θ (¯ t ) (¯ τ ) of the linear combination depend on the original sets ¯ t k andsubsets ¯ τ k . They involve the functions α i whose arguments belong to the set { ¯ s ∪ ¯ t } .Therefore, the factors Θ (¯ t ) (¯ τ ) do not depend on α j ( s ik ) with i, j = 2 , . . . , N .Thus, we obtain a recursion for the scalar product C m | n (¯ s ) B m | n (¯ t ) = X { ¯ σ ,..., ¯ σ N }{ ¯ τ ,..., ¯ τ N } Θ (¯ s ) j ,...,j r (¯ σ )Θ (¯ t ) (¯ τ ) C m − | n ( (cid:8) ¯ σ k (cid:9) N ) B m − | n ( (cid:8) ¯ τ k (cid:9) N ) , (6.3)where ¯ σ k ⊂ ¯ s k and ¯ τ k ⊂ { ¯ s ∪ ¯ t k } . The sum is taken over subsets ¯ σ k and ¯ τ k .Due to the induction assumption, the scalar product C m − | n ( (cid:8) ¯ σ k (cid:9) N ) B m − | n ( (cid:8) ¯ τ k (cid:9) N ) de-pends on the functions α i with arguments σ ik and τ ik . Since σ ik ∈ ¯ s i , we conclude that theBethe parameters s ik for i = 2 , . . . , N can become the arguments of the functions α i only. Thenumerical coefficients Θ (¯ s ) j ,...,j r (¯ σ ) and Θ (¯ t ) (¯ τ ) do not break this type of dependence. Thus, weprove that in the scalar product C m | n (¯ s ) B m | n (¯ t ) the Bethe parameters s ik with i = 2 , . . . , N can become the arguments of the functions α i only.Due to the symmetry (4.10), an analogous property holds for the Bethe parameters ¯ t i with i = 2 , . . . , N . Namely, these parameters can be the arguments of the functions α i only.It remains to prove that the Bethe parameters from the sets ¯ s and ¯ t can be the argumentsof the function α . For this we use the second recursion for the dual Bethe vector (4.6)and repeat all the considerations above. Then we find that the Bethe parameters s ik with i = 1 , . . . , N − α i only. Then, the use of (4.10)completes the proof of proposition 4.3. (cid:3) Consider a composite model, in which the monodromy matrix T ( u ) is presented as a productof two partial monodromy matrices [6, 20, 29, 41]: T ( u ) = T (2) ( u ) T (1) ( u ) . (6.4)Within the framework of the composite model, it is assumed that the matrix elements ofevery T ( l ) ( u ) ( l = 1 , 2) act in some Hilbert space H ( l ) , such that H = H (1) ⊗ H (2) . Each of T ( l ) ( u ) satisfies the RT T -relation (2.4) and has its own pseudovacuum vector | i ( l ) and dual21ector h | ( l ) , such that | i = | i (1) ⊗ | i (2) and h | = h | (1) ⊗ h | (2) . Since the operators T (2) i,j ( u )and T (1) k,l ( v ) act in different spaces, they supercommute with each other. We assume that T ( l ) i,i ( u ) | i ( l ) = λ ( l ) i ( u ) | i ( l ) , h | ( l ) T ( l ) i,i ( u ) = λ ( l ) i ( u ) h | ( l ) , i = 1 , . . . , m + n, l = 1 , , (6.5)where λ ( l ) i ( u ) are new free functional parameters. We also introduce α ( l ) k ( u ) = λ ( l ) k ( u ) λ ( l ) k +1 ( u ) , l = 1 , , k = 1 , . . . , N. (6.6)Obviously λ i ( u ) = λ (1) i ( u ) λ (2) i ( u ) , α k ( u ) = α (1) k ( u ) α (2) k ( u ) . (6.7)The partial monodromy matrices T ( l ) ( u ) have the corresponding Bethe vectors B ( l ) (¯ t )and dual Bethe vectors C ( l ) (¯ s ). A Bethe vector of the total monodromy matrix T ( u ) can beexpressed in terms partial Bethe vectors B ( l ) (¯ t ) via coproduct formula [29, 41] B (¯ t ) = X Q Nν =1 α (2) ν (¯ t ν i ) γ ν (¯ t ν ii , ¯ t ν i ) Q N − ν =1 f [ ν +1] (¯ t ν +1ii , ¯ t ν i ) B (1) (¯ t i ) ⊗ B (2) (¯ t ii ) . (6.8)Here all the sets of the Bethe parameters ¯ t ν are divided into two subsets ¯ t ν ⇒ { ¯ t ν i , ¯ t ν ii } , andthe sum is taken over all possible partitions.Similar formula exists for the dual Bethe vectors C (¯ s ) (see appendix A) C (¯ s ) = X Q Nν =1 α (1) ν (¯ s ν ii ) γ ν (¯ s ν i , ¯ s ν ii ) Q N − ν =1 f [ ν +1] (¯ s ν +1i , ¯ s ν ii ) C (2) (¯ s ii ) ⊗ C (1) (¯ s i ) , (6.9)where the sum is organized in the same way as in (6.8).Then the scalar product of the total Bethe vectors C (¯ s ) and B (¯ t ) takes the form S (¯ s | ¯ t ) = X Q Nν =1 α (1) ν (¯ s ν ii ) α (2) ν (¯ t ν i ) γ ν (¯ s ν i , ¯ s ν ii ) γ ν (¯ t ν ii , ¯ t ν i ) Q N − ν =1 f [ ν +1] (¯ s ν +1i , ¯ s ν ii ) f [ ν +1] (¯ t ν +1ii , ¯ t ν i ) S (1) (¯ s i | ¯ t i ) S (2) (¯ s ii | ¯ t ii ) , (6.10)where S (1) (¯ s i | ¯ t i ) = C (1) (¯ s i ) B (1) (¯ t i ) , S (2) (¯ s ii | ¯ t ii ) = C (2) (¯ s ii ) B (2) (¯ t ii ) . (6.11)Note that in this formula s ν i = t ν i , (and hence, s ν ii = t ν ii ), otherwise the scalar products S (1) and S (2) vanish. Let s ν i = t ν i = k ′ ν , where k ′ ν = 0 , , . . . , r ν . Then s ν ii = t ν ii = r ν − k ′ ν .Now let us turn to equation (4.11). Our goal is to express the rational coefficients W m | n part in terms of the HC. For this we use the fact that W m | n part are model independent. Therefore, wecan find them in some special model whose monodromy matrix satisfies the RT T -relation. The terminology coproduct formula is used for historical reason, because (6.8) was derived for the firsttime in [29] (see also [30] for the non-graded case) as a property of the Bethe vectors induced by the Yangiancoproduct. s ν ⇒ { ¯ s ν I , ¯ s ν II } and ¯ t ν ⇒{ ¯ t ν I , ¯ t ν II } such that s ν I = t ν I = k ν , where k ν = 0 , , . . . , r ν . Hence, s ν II = t ν II = r ν − k ν .Consider a concrete model, in which α (1) ν ( z ) = 0 , if z ∈ ¯ s ν II ; α (2) ν ( z ) = 0 , if z ∈ ¯ t ν I . (6.12)Due to (6.7) these conditions imply α ν ( z ) = 0 , if z ∈ ¯ s ν II ∪ ¯ t ν I . (6.13)Then the scalar product is proportional to the coefficient W m | n part (¯ s I , ¯ s II | ¯ t I , ¯ t II ), because all otherterms in the sum over partitions (4.11) vanish due to the condition (6.13). Thus, S (¯ s | ¯ t ) = W m | n part (¯ s I , ¯ s II | ¯ t I , ¯ t II ) N Y k =1 α k (¯ s k I ) α k (¯ t k II ) . (6.14)On the other hand, (6.12) implies that a non-zero contribution in (6.10) occurs if and only if¯ s ν ii ⊂ ¯ s ν I and ¯ t ν i ⊂ ¯ t ν II . Hence, r ν − k ′ ν ≤ k ν and k ′ ν ≤ r ν − k ν . But this is possible if and onlyif k ′ ν + k ν = r ν . Thus, ¯ s ν ii = ¯ s ν I and ¯ t ν i = ¯ t ν II . Then, for the complementary subsets we obtain¯ s ν i = ¯ s ν II and ¯ t ν ii = ¯ t ν I . Thus, we arrive at S (¯ s | ¯ t ) = Q Nν =1 α (1) ν (¯ s ν I ) α (2) ν (¯ t ν II ) γ ν (¯ s ν II , ¯ s ν I ) γ ν (¯ t ν I , ¯ t ν II ) Q N − ν =1 f [ ν +1] (¯ s ν +1 II , ¯ s ν I ) f [ ν +1] (¯ t ν +1 I , ¯ t ν II ) S (1) (¯ s II | ¯ t II ) S (2) (¯ s I | ¯ t I ) . (6.15)It is easy to see that calculating the scalar product S (1) (¯ s II | ¯ t II ) we should take only theterm corresponding to the conjugated HC. Indeed, all other terms are proportional to α (1) ν ( z )with z ∈ ¯ s ν II , therefore, they vanish. Hence S (1) (¯ s II | ¯ t II ) = N Y ν =1 α (1) ν (¯ t ν II ) · Z m | n (¯ s II | ¯ t II ) . (6.16)Similarly, calculating the scalar product S (2) (¯ s I | ¯ t I ) we should take only the term correspondingto the HC: S (2) (¯ s I | ¯ t I ) = N Y ν =1 α (2) ν (¯ s ν I ) · Z m | n (¯ s I | ¯ t I ) . (6.17)Substituting this into (6.15) and using (6.7), (6.14) we arrive at W m | n part (¯ s I , ¯ s II | ¯ t I , ¯ t II ) = Z m | n (¯ s I | ¯ t I ) Z m | n (¯ s II | ¯ t II ) Q Nk =1 γ k (¯ s k II , ¯ s k I ) γ k (¯ t k I , ¯ t k II ) Q N − j =1 f [ j +1] (¯ s j +1 II , ¯ s j I ) f [ j +1] (¯ t j +1 I , ¯ t j II ) . (6.18)This expression obviously coincides with (4.15) due to (4.14). This choice of the functions α k is always possible, for example, within the framework of inhomogeneousmodel with spins in higher dimensional representations, in which inhomogeneities coincide with some of theBethe parameters. Highest coefficient It follows from proposition 4.3 that the scalar product is a sum, in which every term is pro-portional to a product of the functions α k . Let us call a term unwanted , if the correspondingproduct of the functions α k contains at least one α k ( t kj ), where t kj ∈ ¯ t . Respectively, a termis wanted , if all functions α k depend on the Bethe parameters s kj from the set ¯ s .Below we consider some equations modulus unwanted terms. In this case we use a symbol ∼ =. Thus, an equation of the type lhs ∼ = rhs means that the lhs is equal to the rhs modulusunwanted terms.Using the notion of unwanted terms one can redefine the HC (4.12) as follows: S (¯ s | ¯ t ) ∼ = N Y k =1 α k (¯ s k ) · Z m | n (¯ s | ¯ t ) . (7.1)On the other hand, it follows from the explicit form of Bethe vectors [29] that B (¯ t ) ∼ = e B (¯ t ) = T , (¯ t ) . . . T N,N +1 (¯ t N ) | i Q Nj =1 λ j +1 (¯ t j ) Q N − j =1 f [ j +1] (¯ t j +1 , ¯ t j ) , (7.2)because all other terms in the Bethe vector contain factors α k ( t kj ), and thus, they are un-wanted. Hence, in order to find the HC it is enough to consider a reduced scalar product˜ S (¯ s | ¯ t ) S (¯ s | ¯ t ) ∼ = ˜ S (¯ s | ¯ t ) = C (¯ s ) e B (¯ t ) . (7.3)In order to calculate the reduced scalar product (7.3) we can use the recursion (4.5) forthe dual Bethe vector C (¯ s ). We write it in the form C (¯ s ) = N +1 X p =2 X part(¯ s ,..., ¯ s p − ) C ( (cid:8) ¯ s k II (cid:9) p − ; (cid:8) ¯ s k (cid:9) Np ) T p, (¯ s I ) λ (¯ s I ) ( − ( r − δ m, × Q p − ν =2 α ν (¯ s ν I ) g [ ν ] (¯ s ν I , ¯ s ν − I )ˆ γ ν (¯ s ν II , ¯ s ν I ) h (¯ s , ¯ s I ) δ m, Q p − ν =1 f [ ν +1] (¯ s ν +1 , ¯ s ν I ) . (7.4)Here the sum is taken over partitions of the sets ¯ s k ⇒ { ¯ s k I , ¯ s k II } for k = 2 , . . . , p , such that s k I = 1. The Bethe parameter ¯ s I is fixed, and hence, the subset ¯ s II also is fixed. There isno the sum over partitions of the set ¯ s in (7.4).Thus, we obtain˜ S (¯ s | ¯ t ) = N +1 X p =2 X part(¯ s ,..., ¯ s p − ) ( − ( r − δ m, C ( (cid:8) ¯ s k II (cid:9) p − , (cid:8) ¯ s k (cid:9) Np ) T p, (¯ s I ) e B (¯ t ) × Q p − ν =2 α ν (¯ s ν I ) g [ ν ] (¯ s ν I , ¯ s ν − I )ˆ γ ν (¯ s ν II , ¯ s ν I ) λ (¯ s I ) h (¯ s , ¯ s I ) δ m, Q p − ν =1 f [ ν +1] (¯ s ν +1 , ¯ s ν I ) . (7.5)The action of T p, (¯ s I ) onto the vector e B (¯ t ) modulus unwanted terms is given by proposi-24ion B.1. Thus, we obtain˜ S (¯ s | ¯ t ) ∼ = α (¯ s I ) N +1 X p =2 X part(¯ s ,..., ¯ s p − )part(¯ t ,..., ¯ t p − ) ( − ( r − δ m, g [2] (¯ t I , ¯ s I )ˆ γ (¯ t I , ¯ t II ) f [1] (¯ t II , ¯ s I ) f [ p ] (¯ s p , ¯ s p − I ) h (¯ s , ¯ s I ) δ m, × p − Y ν =2 α ν (¯ s ν I ) g [ ν ] (¯ s ν I , ¯ s ν − I ) g [ ν +1] (¯ t ν I , ¯ t ν − I )ˆ γ ν (¯ s ν II , ¯ s ν I )ˆ γ ν (¯ t ν I , ¯ t ν II ) f [ ν ] (¯ s ν , ¯ s ν − I ) f [ ν ] (¯ t ν I , ¯ t ν − ) × C ( (cid:8) ¯ s k II (cid:9) p − , (cid:8) ¯ s k (cid:9) Np ) e B ( (cid:8) ¯ t k II (cid:9) p − ; (cid:8) ¯ t k (cid:9) Np ) . (7.6)Here ¯ t m + n = ¯ s m + n = ∅ . Calculating the reduced scalar products in (7.6) modulus unwantedterms C ( (cid:8) ¯ s k II (cid:9) p − , (cid:8) ¯ s k (cid:9) Np ) e B ( (cid:8) ¯ t k II (cid:9) p − ; (cid:8) ¯ t k (cid:9) Np ) ∼ = p − Y k =1 α k (¯ s k II ) N Y ℓ = p α ℓ (¯ s ℓ ) × Z m | n ( (cid:8) ¯ s k II (cid:9) p − , (cid:8) ¯ s k (cid:9) Np | (cid:8) ¯ t k II (cid:9) p − ; (cid:8) ¯ t k (cid:9) Np ) , (7.7)and substituting this into (7.6) we immediately arrive at the recursion (4.17).We have also used( − ( r − δ m, ˆ γ (¯ t I , ¯ t II ) = γ (¯ t I , ¯ t II ) , ˆ γ ν (¯ s ν II , ¯ s ν I )ˆ γ ν (¯ t ν I , ¯ t ν II ) = γ ν (¯ s ν II , ¯ s ν I ) γ ν (¯ t ν I , ¯ t ν II ) . Due to isomorphism (3.15) between Yangians Y ( gl ( m | n )) and Y ( gl ( n | m )) one can find asimple relation between the HC corresponding to these algebras. In this section we obtainthis relation.Consider the sum formula (4.11) for the scalar product of gl ( m | n ) Bethe vectors S m | n ( −→ s | −→ t ) = X W m | n part ( −→ s I , −→ s II | −→ t I , −→ t II ) N Y k =1 α k (¯ s k I ) α k (¯ t k II ) , (7.8)where we have stressed the ordering (3.17) of the Bethe parameters. Let us act with themorphism ϕ (3.15) on the scalar product S m | n ( −→ s | −→ t ). This can be done in two ways. First,using (3.18) and (3.26) we obtain ϕ (cid:16) S m | n ( −→ s | −→ t ) (cid:17) = ϕ (cid:16) C m | n ( −→ s ) B m | n ( −→ t ) (cid:17) = ( − r m C n | m ( ←− s ) B n | m ( ←− t ) Q Nk =1 α N +1 − k (¯ s k ) α N +1 − k (¯ t k )= ( − r m S n | m ( ←− s | ←− t ) Q Nk =1 α N +1 − k (¯ s k ) α N +1 − k (¯ t k ) . (7.9)The scalar product S n | m ( ←− s | ←− t ) has the standard representation (4.11). Thus, we find ϕ (cid:16) S m | n ( −→ s | −→ t ) (cid:17) = X part ( − r m W n | m part ( ←− s I , ←− s II | ←− t I , ←− t II ) Q Nk =1 α N +1 − k (¯ s k ) α N +1 − k (¯ t k ) N Y k =1 α k (¯ s N − k +1 I ) α k (¯ t N − k +1 II ) . (7.10)On the other hand, acting with ϕ directly on the sum formula (7.8) we have ϕ (cid:16) S m | n ( −→ s | −→ t ) (cid:17) = X part W m | n part ( −→ s I , −→ s II | −→ t I , −→ t II ) N Y k =1 (cid:16) α N +1 − k (¯ s k I ) α N +1 − k (¯ t k II ) (cid:17) − . (7.11)25omparing (7.10) and (7.11) we arrive at( − r m X part W n | m part ( ←− s I , ←− s II | ←− t I , ←− t II ) N Y k =1 α N +1 − k (¯ s k I ) α N +1 − k (¯ t k II )= X part W m | n part ( −→ s I , −→ s II | −→ t I , −→ t II ) N Y k =1 α N +1 − k (¯ s k II ) α N +1 − k (¯ t k I ) (7.12)Since α i are free functional parameters, the coefficients of the same products of α i must beequal. Hence, W m | n part ( −→ s I , −→ s II | −→ t I , −→ t II ) = ( − r m W n | m part ( ←− s II , ←− s I | ←− t II , ←− t I ) , (7.13)for arbitrary partitions of the sets ¯ s and ¯ t . In particular, setting ¯ s II = ¯ t II = ∅ we obtain Z m | n ( −→ s | −→ t ) = ( − r m Z n | m ( ←− s | ←− t ) = ( − r m Z n | m ( ←− t | ←− s ) . (7.14)Using this property one can obtain recursion (4.18) for the highest coefficient. Indeed, onecan easily see that applying (4.17) to the rhs of (7.14) we obtain (4.18) for the lhs of thisequation. Conclusion In the present paper we have considered the Bethe vectors scalar products in the integrablemodels solvable by the nested algebraic Bethe ansatz and possessing gl ( m | n ) supersymmetry.The main result of the paper is the sum formula given by equations (4.11) and (4.15). Weobtained it using the coproduct formula for the Bethe vectors. This way certainly is moredirect and simple than the methods used before for the derivation of the sum formulas.The sum formula is obtained for the Bethe vectors with arbitrary coloring. However, aswe have mentioned in section 3.1, in various models of physical interest the coloring of theBethe vectors is restricted by the condition r ≥ r ≥ · · · ≥ r N . A peculiarity of these modelsis that only the ratio α ( u ) is a non-trivial function of u , while all other α ’s are identicallyconstants: α k ( u ) = α k , k > α k ( u ) = 1, k > gl (3)-based models one hardly can expect to obtain a relativelysimple closed formula for it in the general gl ( m | n ) case. On the other hand, the recursionsobtained in this paper allow one to study analytical properties of the HC, in particular to findthe residues in the poles of this rational function. Using these results it is possible to derivean analog of Gaudin formula for on-shell Bethe vectors in the gl ( m | n ) based models exactlyin the same way as it was done in [5, 10]. We will consider this question in our forthcomingpublication.As we have already mentioned in Introduction, the sum formula itself is not very con-venient for use. One should remember, however, that the sum formula describes the scalarproduct of generic Bethe vectors, where we have no restriction for the Bethe parameters.26t the same time, in most cases of physical interest one deals with Bethe vectors, in whichmost of the Bethe parameters satisfy Bethe equations. In particular, this situation occurs incalculating form factors. Then one can hope to obtain a significant simplification of the sumformula, as it was shown for the models with gl (3) and gl (2 | 1) symmetries. We are planningto study this problem in our further publications.In conclusion we would like to discuss one more possible direction of generalization ofour results. In this paper we considered the so-called distinguished gradation, that is to saythe special grading [ i ] = 0 for 1 ≤ i ≤ m , [ i ] = 1 for m < i ≤ m + n . However, this isnot the only possible choice of grading. Other gradings induce different inequivalent pre-sentations of the superalgebra, where the number of fermionic simple roots can vary from apresentation to another. These different presentations are labelled by the different Dynkindiagrams associated to the superalgebra. Obviously, since the different presentations dealwith the same superalgebra, they are isomorphic. However, the mapping between two pre-sentations is based on a generalized Weyl transformation acting on their Dynkin diagrams,lifted at the level of the superalgebra. These generalized Weyl transformations, in particular,affect the bosonic/fermionic nature of the generators, and thus can change commutators toanti-commutators (and vice-versa). Then, the precise expression of the mapping is heavy toformulate for all the generators of the Yangian. This is also true for Bethe vectors and Betheparameters, a precise correspondence can be quite intricate to formulate. However, from theLie superalgebra theory one knows that such a correspondence must exist. These consid-erations have been developed in [45] for the construction of the mapping on the particularcase of the gl (1 | 2) algebra. The general case of generic gl ( m | n ) superalgebra is presented in[46] for the form of the Bethe equations, but open spin chains (see also [47] where the peri-odic case is reviewed). In conclusion, if a qualitative generalization of the present results tothe superalgebras with different gradings is rather straightforward, a precise correspondenceremains open. Acknowledgements The work of A.L. has been funded by Russian Academic Excellence Project 5-100, by YoungRussian Mathematics award and by joint NASU-CNRS project F14-2017. The work of S.P.was supported in part by the RFBR grant 16-01-00562-a. A Coproduct formula for the Bethe vectors The presentation (6.8) for the Bethe vector of the composite model can be treated as a co-product formula for the Bethe vector. Indeed, equation (6.4) formally determines a coproduct∆ of the monodromy matrix entries∆( T i,j ( u )) = m + n X k =1 ( − ([ j ]+[ k ])([ i ]+[ k ]) T k,j ( u ) ⊗ T i,k ( u ) . (A.1)Then (6.8) is nothing but the action of ∆ onto the Bethe vector [29].The action of the coproduct onto the dual Bethe vectors can be obtained via antimorphism(3.20). It was proved in [42] (see also similar consideration in prop. 1.5.4 of [43]) that∆ ◦ Ψ = (Ψ ⊗ Ψ) ◦ ∆ ′ , (A.2)where ∆ ′ ( T i,j ( u )) = X T i,k ( u ) ⊗ T k,j ( u ) . (A.3)27hen∆( C (¯ t )) = ∆(Ψ( B (¯ t ))) = (Ψ ⊗ Ψ) ◦ ∆ ′ ( B (¯ t ))= (Ψ ⊗ Ψ) X Q Nν =1 α (1) ν (¯ t ν I ) γ ν (¯ t ν II , ¯ t ν I ) Q N − ν =1 f [ ν +1] (¯ t ν +1 II , ¯ t ν I ) B (2) (¯ t I ) ⊗ B (1) (¯ t II ) ! = X Q Nν =1 α (1) ν (¯ t ν I ) γ ν (¯ t ν II , ¯ t ν I ) Q N − ν =1 f [ ν +1] (¯ t ν +1 II , ¯ t ν I ) C (2) (¯ t I ) ⊗ C (1) (¯ t II ) . (A.4)Relabeling here the subsets ¯ t ν I ↔ ¯ t ν II we arrive at (6.9). (cid:3) B Action formulas In this section we derive the action of the operators T p, on the main term (3.13). For thiswe first consider some multiple commutation relations in the RT T -algebra (2.4). B.1 Multiple commutation relations Multiple commutation relations of the monodromy matrix entries in superalgebras were stu-died in [44]. Here we consider several particular cases of commutation relations with theoperators T i,i +1 (¯ v ) (3.14).It follows from (2.5) that T i,i ( u ) T i,i +1 ( v ) = f [ i ] ( v, u ) T i,i +1 ( v ) T i,i ( u ) + g [ i ] ( u, v ) T i,i +1 ( u ) T i,i ( v ) ,T i,i ( u ) T i − ,i ( v ) = f [ i ] ( u, v ) T i − ,i ( v ) T i,i ( u ) + g [ i ] ( v, u ) T i − ,i ( u ) T i,i ( v ) . (B.1)We see that these commutation relations look exactly the same as in the case of algebra gl ( n ).The only difference is that the functions f and g acquire an additional subscript indicatingparity. Therefore, for commutation relations, we can apply the standard arguments of thealgebraic Bethe ansatz [1, 3, 4]. In particular, let us consider commutation of the operator T i,i ( t i − α ) with the product T i,i +1 (¯ t i ), where t i − α is a fixed parameter of the set ¯ t i − . Let uscall a term wanted, if it contains the operator T i,i ( t i − α ) in the extreme right position. Thenmoving T i,i ( t i − α ) through the product T i,i +1 (¯ t i ) we should keep the original argument of T i,i leading to T i,i ( t i − α ) T i,i +1 (¯ t i ) ∼ = f [ i ] (¯ t i , t i − α ) T i,i +1 (¯ t i ) T i,i ( t i − α ) . (B.2)Consider now commutation of the operator T i +1 ,i ( t i − α ) with the product T i,i +1 (¯ t i ) using T i +1 ,i ( u ) T i,i +1 ( v ) − ( − δ i,m T i,i +1 ( v ) T i +1 ,i ( u )= g [ i +1] ( u, v ) (cid:0) T i +1 ,i +1 ( u ) T i,i ( v ) − T i +1 ,i +1 ( v ) T i,i ( u ) (cid:1) . (B.3)Let, as before, a term be wanted, if it contains the operator T i,i ( t i − α ) in the extreme rightposition. Moving T i +1 ,i ( t i − α ) through the product T i,i +1 (¯ t i ) we can obtain the terms of thefollowing type: (i) T i +1 ,i ( t i − α );(ii) T i +1 ,i +1 ( t ij ) T i,i ( t i − α ) , j = 1 , . . . , r i ;(iii) T i +1 ,i +1 ( t i − α ) T i,i ( t ij ) , j = 1 , . . . , r i ;(iv) T i +1 ,i +1 ( t ij ) T i,i ( t ij ) , j , j = 1 , . . . , r i . (B.4)28mong all these contributions only the terms (ii) are wanted. Thus, we have T i +1 ,i ( t i − α ) T i,i +1 (¯ t i ) ∼ = r i X j =1 Λ j T i,i +1 (¯ t i \ t ij ) T i +1 ,i +1 ( t ij ) T i,i ( t i − α ) , (B.5)where Λ j are rational coefficients to be determined. Due to the symmetry of T i,i +1 (¯ t i ) over¯ t i it is sufficient to find Λ only. Then a wanted term must contain T i +1 ,i +1 ( t i ) T i,i ( t i − α ) inthe extreme right position. We have T i +1 ,i ( t i − α ) T i,i +1 (¯ t i ) = T i +1 ,i ( t i − α ) T i,i +1 ( t i ) T i,i +1 (¯ t i \ t i ) h (¯ t i , t i ) δ m,i ∼ = g [ i +1] ( t i − α , t i ) (cid:0) T i +1 ,i +1 ( t i − α ) T i,i ( t i ) − T i +1 ,i +1 ( t i ) T i,i ( t i − α ) (cid:1) T i,i +1 (¯ t i \ t i ) h (¯ t i , t i ) δ m,i . (B.6)The term T i +1 ,i +1 ( t i − α ) T i,i ( t i ) obviously gives unwanted contribution. The remaining oper-ators T i +1 ,i +1 ( t i ) T i,i ( t i − α ) should move through the product T i,i +1 (¯ t i \ t i ) via (B.1) keepingtheir arguments. This leads to T i +1 ,i ( t i − α ) T i,i +1 (¯ t i ) ∼ = g [ i +1] ( t i , t i − α ) r i Y k =2 f [ i ] ( t ik , t i − α ) f [ i +1] ( t i , t ik ) × T i,i +1 (¯ t i \ t i ) h (¯ t i , t i ) δ m,i T i +1 ,i +1 ( t i ) T i,i ( t i − α ) . (B.7)Thus, using (2.10) we arrive atΛ = g [ i +1] ( t i , t i − α ) r i Y k =2 f [ i ] ( t ik , t i − α )ˆ γ i ( t i , t ik ) . (B.8)The final result can be written as a sum over partitions of the set ¯ t i : T i +1 ,i ( t i − α ) T i,i +1 (¯ t i ) ∼ = X g [ i +1] (¯ t i I , t i − α ) f [ i ] (¯ t i II , t i − α )ˆ γ i (¯ t i I , ¯ t i II ) × T i,i +1 (¯ t i II ) T i +1 ,i +1 (¯ t i I ) T i,i ( t i − α ) . (B.9)Here the set ¯ t i is divided into subsets ¯ t i I and ¯ t i II such that t i I = 1 . B.2 Action formulas In this section we consider the action of the operators T p, ( s ) onto the main term of theBethe vector (3.13). Here p > s is a generic complex number. The result of this actioncontains various terms, among which we will distinguish wanted and unwanted terms. Let aterm be wanted, if it is proportional to λ ( s ) and does not contain any α i ( t kℓ ). Otherwise aterm is unwanted. Proposition B.1. Let e B (¯ t ) be the main term of a Bethe vector (3.13) . Then the wantedterm of the action of T p, onto e B (¯ t ) reads T p, ( s ) e B (¯ t ) ∼ = λ ( s ) X part(¯ t ) p − Y ℓ =2 g [ ℓ +1] (¯ t ℓ I , ¯ t ℓ − I )ˆ γ ℓ (¯ t ℓ I , ¯ t ℓ II ) f [ ℓ ] (¯ t ℓ I , ¯ t ℓ − ) × g [2] (¯ t I , s )ˆ γ (¯ t I , ¯ t II ) f [1] (¯ t II , s ) e B ( (cid:8) ¯ t k II (cid:9) p − ; (cid:8) ¯ t k (cid:9) Np ) . (B.10) Here the sum is taken over partitions of the sets ¯ t k with k = 1 , . . . , p − into subsets ¯ t k I and ¯ t k II such that t k I = 1 . 29o prove proposition B.1 we introduce for 1 ≤ i < k ≤ m + n e B ik ( { ¯ t ν } k − i ) = T i,i +1 (¯ t i ) . . . T k − ,k (¯ t k − ) | i Q k − j = i λ j +1 (¯ t j ) Q k − j = i f [ j +1] (¯ t j +1 , ¯ t j ) , (B.11)where T j,j +1 is defined by (3.14). Obviously, e B ,n + m ( { ¯ t ν } N ) = e B (¯ t ). We first prove severalauxiliary lemmas. Lemma B.1. Let j < ℓ and j < i . Then T ℓ,j ( s ) e B ik ( { ¯ t ν } k − i ) = 0 . (B.12) Proof. The proof is based on the arguments of the coloring. The operator T ℓ,j annihilatesthe particles of the colors j, . . . , ℓ − 1. On the other hand, for i > j the state e B ik ( { ¯ t ν } k − i )does not contain the particles of the color j . Hence, the action of T ℓ,j onto e B ik ( { ¯ t ν } k − i )vanishes. (cid:3) Lemma B.2. Let j < i . Then T j,j ( s ) e B ik ( { ¯ t ν } k − i ) = λ j ( s ) e B ik ( { ¯ t ν } k − i ) . (B.13) Proof. Obviously, e B ik ( { ¯ t ν } k − i ) = T i,i +1 (¯ t i ) λ i +1 (¯ t i ) f [ i +1] (¯ t i +1 , ¯ t i ) e B i +1 ,k ( { ¯ t ν } k − i +1 ) . (B.14)When one commutes T j,j with one of the operators in the product T i,i +1 (¯ t i ), then from (2.5),we obtain the operators T i,j or T i +1 ,j acting on e B i +1 ,k (¯ t ). Due to lemma B.1 this actionvanishes, because i > j . Thus, T j,j ( s ) e B ik ( { ¯ t ν } k − i ) = T i,i +1 (¯ t i ) λ i +1 (¯ t i ) f [ i +1] (¯ t i +1 , ¯ t i ) T j,j ( s ) e B i +1 ,k ( { ¯ t ν } k − i +1 ) . (B.15)Continuing this process we eventually move T j,j to the vacuum vector, where it gives λ j ( s ). (cid:3) In the following lemmas the actions are considered modulus unwanted terms. Let t i − α bea fixed parameter of the set ¯ t i − . We say that a term is wanted , if a Bethe parameter t jℓ for j = i, . . . , k − λ j +1 . Otherwise, a term is unwanted . Lemma B.3. The wanted term of the action of T i,i ( t i − α ) onto e B ik ( { ¯ t ν } k − i ) is given by T i,i ( t i − α ) e B ik ( { ¯ t ν } k − i ) ∼ = λ i ( t i − α ) f [ i ] (¯ t i , t i − α ) e B ik ( { ¯ t ν } k − i ) . (B.16) Proof. We present e B ik ( { ¯ t ν } k − i ) in the form (B.14). Then, moving T i,i ( t i − α ) through theproduct T i,i +1 (¯ t i ) we should use (B.2), otherwise we obtain unwanted terms. Therefore, atthe first step we obtain T i,i ( t i − α ) e B ik ( { ¯ t ν } k − i ) ∼ = f [ i ] (¯ t i , t i − α ) T i,i +1 (¯ t i ) λ i +1 (¯ t i ) f [ i +1] (¯ t i +1 , ¯ t i ) T i,i ( t i − α ) e B i +1 ,k ( { ¯ t ν } k − i +1 ) . (B.17)Then application of lemma B.2 completes the proof. (cid:3) emma B.4. The wanted term of the action of T i +1 ,i ( t i − α ) onto e B ik ( { ¯ t ν } k − i ) is given by T i +1 ,i ( t i − α ) e B ik ( { ¯ t ν } k − i ) ∼ = X λ i ( t i − α ) g [ i +1] (¯ t i I , t i − α ) f [ i ] (¯ t i II , t i − α )ˆ γ i (¯ t i I , ¯ t i II ) e B ik (¯ t i II ; { ¯ t ν } k − i +1 ) . (B.18) Here the sum is taken over partitions ¯ t i ⇒ { ¯ t i I , ¯ t i II } such that t i I = 1 . Proof. We again present e B ik ( { ¯ t ν } k − i ) in the form (B.14). Then, moving T i +1 ,i ( t i − α )through the product T i,i +1 (¯ t i ) we should use (B.9), otherwise we obtain unwanted terms.Thus, we obtain T i +1 ,i ( t i − α ) e B i +1 ,k (¯ t ) ∼ = X g [ i +1] (¯ t i I , t i − α ) f [ i ] (¯ t i II , t i − α )ˆ γ i (¯ t i I , ¯ t i II ) × T i,i +1 (¯ t i II ) T i +1 ,i +1 (¯ t i I ) T i,i ( t i − α ) λ i +1 (¯ t i ) f [ i +1] (¯ t i +1 , ¯ t i ) e B i +1 ,k ( { ¯ t ν } k − i +1 ) . (B.19)Then application of lemmas B.2 and B.3 completes the proof. (cid:3) Lemma B.5. Let i < p < k . Then T p,i ( t i − α ) e B ik ( { ¯ t ν } k − i ) ∼ = λ i ( t i − α ) X part(¯ t ) e B ik ( { ¯ t ν II } p − i ; { ¯ t ν } k − p ) × g [ i +1] (¯ t i I , t i − α )ˆ γ i (¯ t i I , ¯ t i II ) f [ i ] (¯ t i II , t i − α ) p − Y ν = i +1 g [ ν +1] (¯ t ν I , ¯ t ν − I )ˆ γ ν (¯ t ν I , ¯ t ν II ) f [ ν ] (¯ t ν I , ¯ t ν − ) . (B.20) Here the sum is taken over partitions of the sets ¯ t ν ⇒ { ¯ t ν I , ¯ t ν II } for ν = i, . . . , p − , such that t ν I = 1 . Proof. The proof uses induction over p − i . If p − i = 1, then the statement coincideswith the one of lemma B.4. Assume that (B.20) is valid for i replaced with i + 1. Then weuse presentation (B.14) T p,i ( t i − α ) e B ik ( { ¯ t ν } k − i ) = T p,i ( t i − α ) T i,i +1 (¯ t i ) λ i +1 (¯ t i ) f [ i +1] (¯ t i +1 , ¯ t i ) e B i +1 ,k ( { ¯ t ν } k − i +1 ) . (B.21)Moving T p,i ( t i − α ) through the product T i,i +1 (¯ t i ) we can obtain the terms of the followingtype: (i) T p,i ( t i − α );(ii) T p,i +1 ( t ij ) T i,i ( t i − α );(iii) T p,i +1 ( t i − α ) T i,i ( t ij );(iv) T p,i +1 ( t ij ) T i,i ( t ij ) . (B.22)The term (i) vanishes due to lemma B.1. The terms (iii) and (iv) give unwanted terms dueto lemma B.2. Hence, only the term (ii) survives. Using the arguments similar to the onesthat we used for obtaining equation (B.9) we arrive at T p,i ( t i − α ) e B ik ( { ¯ t ν } k − i ) ∼ = X g [ i +1] (¯ t i I , t i − α ) f [ i ] (¯ t i II , t i − α )ˆ γ i (¯ t i I , ¯ t i II ) × T i,i +1 (¯ t i II ) T p,i +1 (¯ t i I ) T i,i ( t i − α ) λ i +1 (¯ t i ) f [ i +1] (¯ t i +1 , ¯ t i ) e B i +1 ,k ( { ¯ t ν } k − i +1 ) . (B.23)31ere the sum is taken over partitions ¯ t i ⇒ { ¯ t i I , ¯ t i II } such that t i I = 1. Applying lemma B.2we find T p,i ( t i − α ) e B ik ( { ¯ t ν } k − i ) ∼ = X λ i ( t i − α ) g [ i +1] (¯ t i I , t i − α ) f [ i ] (¯ t i II , t i − α )ˆ γ i (¯ t i I , ¯ t i II ) × T i,i +1 (¯ t i II ) T p,i +1 (¯ t i I ) λ i +1 (¯ t i ) f [ i +1] (¯ t i +1 , ¯ t i ) e B i +1 ,k ( { ¯ t ν } k − i +1 ) . (B.24)The action of T p,i +1 (¯ t i I ) onto e B i +1 ,k ( { ¯ t ν } k − i +1 ) is known due to the induction assumption.Substituting this known action into (B.23) we prove lemma B.5. (cid:3) In fact, lemma B.5 gives the proof of proposition B.1. Indeed, it is enough to set i = 1and k = m + n in (B.20). We also set by definition t α = s and introduce an auxiliary emptyset ¯ t m + n ≡ ∅ . Then lemma B.5 describes the action of T p, ( s ) onto the main term e B (¯ t ). References [1] L. D. Faddeev, E. K. Sklyanin and L. A. Takhtajan, Quantum Inverse Problem. I , Theor.Math. Phys. (1979) 688–706.[2] L. D. Faddeev and L. A. 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