We derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic.

The Gaudin Hamiltonians are an important example of a family of commuting operators. We study the case when the Gaudin Hamiltonians possess a symmetry given by the diagonal action of 𝔀. In this case the Gaudin Hamiltonians depend on a choice of a simple Lie algebra 𝔀, 𝔀-modules V1, …, Vn, and distinct complex numbers z1, …, zn, see (2.1).

The problem of studying the spectrum of the Gaudin Hamiltonians has received a lot of attention. However, the majority of the work has been done in type A. In this paper we study the cases of types B, C, and D.

The main approach is the Bethe ansatz method. Our goal is to establish the method when all but one module Vi are isomorphic to the first fundamental representation VΟ‰1. Namely, we show that the Bethe ansatz equations have sufficiently many solutions and that the Bethe vectors constructed from those solutions form a basis in the space of singular vectors of V1 βŠ— β‹― βŠ— Vn.

The solution of a similar problem in type A in Ref. 7 led to several important results, such as a proof of the strong form of the Shapiro-Shapiro conjecture for Grassmannians, simplicity of the spectrum of higher Gauding Hamiltonians, the bijection between Fuchsian differential operators without monodromy with the Bethe vectors, etc, see Ref. 4 and references therein. We hope that this paper will give a start to similar studies in type B. In addition, the explicit formulas for simplest examples outside type A are important as experimental data for testing various conjectures.

By the standard methods, the problem is reduced to the case of n = 2, with V1 being an arbitrary finite-dimensional module, V2 = Vω1 and z1 = 0, z2 = 1. The reduction involves taking appropriate limits, when all points zi go to the same number with different rates. Then the n = 2 problems are observed in the leading order and the generic situation is recovered from the limiting case by the usual argument of deformations of isolated solutions of algebraic systems, see Ref. 7 and Section IV for details.

For the 2-point case when one of the modules is the defining representation VΟ‰1, the spaces of singular vectors of a given weight are either trivial or one-dimensional. Then, according to the general philosophy, see Ref. 5, one would expect to solve the Bethe ansatz equations explicitly. In type A it was done in Ref. 5. In the supersymmetric case of 𝔀𝔩(p|q) the corresponding Bethe ansatz equations are solved in Ref. 10. The other known cases with one dimensional spaces include tensor products of two arbitrary irreducible 𝔰𝔩2 modules, see Ref. 13 and tensor products of an arbitrary module with a symmetric power VkΟ‰1 of the vector representation in the case of 𝔰𝔩r+1, see Ref. 8. Interestingly, in the latter case the solutions of the Bethe ansatz equations are related to zeros of Jacobi-Pineiro polynomials which are multiple orthogonal polynomials.

In all previously known cases when the dimension of the space of singular vectors of a given weight is one, the elementary symmetric functions of solutions of Bethe ansatz equations completely factorize into products of linear functions of the parameters. This was one of the main reasons the formulas were found essentially by brute force. However, unexpectedly, the computer experiments showed that in types B, C, and D, the formulas do not factorize, see also Theorem 5.5 in Ref. 7, and therefore, the problem remained unsolved. In this paper we present a method to compute the answer systematically.

Our idea comes from the reproduction procedure studied in Ref. 9. Let V1 = VΞ» be the irreducible module of highest weight Ξ», let V2, …, Vn be finite-dimensional irreducible modules, and let l1, …, lr be non-negative integers, where r is the rank of 𝔀. Fix distinct complex numbers z1 = 0, z1, …, zn. Consider the Bethe ansatz equation, see (2.2), associated to these data. Set V = V2 βŠ— β‹― βŠ— Vn, denote the highest weight vector of V by v+, the weight of v+ by ΞΌ+, and set ΞΌ = ΞΌ + βˆ’ βˆ‘ i = 1 r l i Ξ± i ⁠. Here Ξ±i are simple roots of 𝔀.

Given an isolated solution of the Bethe ansatz equations we can produce two Bethe vectors: one in the space of singular vectors in VΞ» βŠ— V of weight ΞΌ + Ξ» and another one in the space of vectors in V of weight ΞΌ. The first Bethe vector, see (2.3), is an eigenvector of the standard Gaudin Hamiltonians, see (2.1), acting in VΞ» βŠ— V and the second Bethe vector is an eigenvector of trigonometric Gaudin Hamiltonians, see Ref. 9. The second vector is a projection of the first vector to the space v+ βŠ— V ≃ V.

Then the reproduction procedure of Ref. 9 in the jth direction allows us to construct a new solution of the Bethe ansatz equation associated to new data: representations V1 = Vsjβ‹…Ξ», V2, …, Vn and integers l 1 , … , l Μƒ j , … , l r so that the new weight ΞΌ Μƒ = ΞΌ + βˆ’ βˆ‘ i β‰  j l i Ξ± i βˆ’ l Μƒ j Ξ± j is given by ΞΌ Μƒ = s j ΞΌ ⁠. This construction is quite general, it works for all symmetrizable Kac-Moody algebras provided that the weight Ξ» is generic, see Theorem 2.6 below. It gives a bijection between solutions corresponding to weights ΞΌ of V in the same Weyl orbit.

Note that in the case μ = μ+, the Bethe ansatz equations are trivial. Therefore, using the trivial solution and the reproduction procedure, we, in principle, can obtain solutions for all weights of the form μ = wμ+. Note also that in the case of the vector representation, V = Vω1, all weights in V are in the Weyl orbit of μ+ = ω1 (with the exception of weight μ = 0 in type B). Therefore, we get all the solutions we need that way (the exceptional weight is easy to treat separately).

In contrast to Ref. 9, we do not have the luxury of generic weight Ξ», and we have to check some technical conditions on each reproduction step. It turns out, such checks are easy when going to the trivial solution, but not the other way, see Section III C. We manage to solve the recursion and obtain explicit formulas, see Corollary 3.10 for type B, Theorem 5.1 for type C, and Theorem 5.4 for type D. We complete the check using these formulas, see Section III E 4.

To each solution of Bethe ansatz, one can associate an oper. For types A, B, C the oper becomes a scalar differential operator with rational coefficients, see Ref. 6, and Sections III F and V A. In fact, the coefficients of this operator are eigenvalues of higher Gaudin Hamiltonians, see Ref. 3 for type A and Ref. 2 for types B and C. The differential operators for the solutions obtained via the reproduction procedure are closely related. It allows us to give simple formulas for the differential operators related to our solutions, see Propositions 3.11 and 5.3. According to Ref. 6, the kernel of the differential operator is a space of polynomials with a symmetry, called a self-dual space. We intend to discuss the self-dual spaces related to our situation in detail elsewhere.

The paper is constructed as follows. In Section II we describe the problem and set our notation. We study in detail the case of type B in Sections III and IV. In Section III we solve the Bethe ansatz equation for n = 2 when one of the modules is Vω1. In Section IV, we use the results of Section III to show the completeness and simplicity of the spectrum of Gaudin Hamiltonians acting in tensor products where all but one factors are Vω1, for generic values of zi. In Section V we give the corresponding formulas and statements in types C and D.

Let 𝔀 be a simple Lie algebra over β„‚ with Cartan matrix A = ( a i , j ) i , j = 1 r ⁠. Denote the universal enveloping algebra of 𝔀 by 𝒰(𝔀). Let D = diag{d1, …, dr} be the diagonal matrix with positive relatively prime integers di such that B = DA is symmetric.

Let π”₯ βŠ‚ 𝔀 be the Cartan subalgebra. Fix simple roots Ξ±1, …, Ξ±r in π”₯βˆ—. Let Ξ± 1 ∨ , … , Ξ± r ∨ ∈ h be the corresponding coroots. Fix a nondegenerate invariant bilinear form (, ) in 𝔀 such that ( Ξ± i ∨ , Ξ± j ∨ ) = a i , j / d j ⁠. Define the corresponding invariant bilinear forms in π”₯βˆ— such that (Ξ±i, Ξ±j) = diai,j. We have γ€ˆ Ξ» , Ξ± i ∨ 〉 = 2 ( Ξ» , Ξ± i ) / ( Ξ± i , Ξ± i ) for Ξ» ∈ π”₯βˆ—. In particular, γ€ˆ Ξ± j , Ξ± i ∨ 〉 = a i , j ⁠. Let Ο‰1, …, Ο‰r ∈ π”₯βˆ— be the fundamental weights, γ€ˆ Ο‰ j , Ξ± i ∨ 〉 = Ξ΄ i , j ⁠.

Let P = { Ξ» ∈ h βˆ— | γ€ˆ Ξ» , Ξ± i ∨ 〉 ∈ Z } and P + = { Ξ» ∈ h βˆ— | γ€ˆ Ξ» , Ξ± i ∨ 〉 ∈ Z β©Ύ 0 } be the weight lattice and the set of dominant integral weights. The dominance order > on π”₯βˆ— is defined by ΞΌ > Ξ½ if and only if ΞΌ βˆ’ Ξ½ = βˆ‘ i = 1 r a i Ξ± i ⁠, ai ∈ β„€β©Ύ0 for i = 1, …, r.

Let ρ ∈ π”₯βˆ— be such that γ€ˆ ρ , Ξ± i ∨ 〉 = 1 ⁠, i = 1, …, r. We have (ρ, Ξ±i) = (Ξ±i, Ξ±i)/2.

For Ξ» ∈ π”₯βˆ—, let VΞ» be the irreducible 𝔀-module with highest weight Ξ». We denote γ€ˆ Ξ» , Ξ± i ∨ 〉 by Ξ»i and sometimes write V(Ξ»1,Ξ»2,…,Ξ»r) for VΞ».

The Weyl group 𝒲 βŠ‚ Aut(π”₯βˆ—) is generated by reflections si, i = 1, …, r,

s i ( Ξ» ) = Ξ» βˆ’ γ€ˆ Ξ» , Ξ± i ∨ 〉 Ξ± i , Ξ» ∈ h βˆ— .

We use the notation

w β‹… Ξ» = w ( Ξ» + ρ ) βˆ’ ρ , w ∈ W , Ξ» ∈ h βˆ— ,

for the shifted action of the Weyl group.

Let E1, …, Er ∈ 𝔫+, H1, …, Hr ∈ π”₯, F1, …, Fr ∈ π”«βˆ’ be the Chevalley generators of 𝔀.

The coproduct Ξ” : 𝒰(𝔀) β†’ 𝒰(𝔀) βŠ— 𝒰(𝔀) is defined to be the homomorphism of algebras such that Ξ”x = 1 βŠ— x + x βŠ— 1, for all x ∈ 𝔀.

Let (xi)i∈O be an orthonormal basis with respect to the bilinear form (, ) in 𝔀.

Let Ξ© 0 = βˆ‘ i ∈ O x i 2 ∈ U ( g ) be the Casimir element. For any u ∈ 𝒰(𝔀), we have uΞ©0 = Ξ©0u. Let Ξ© = βˆ‘ i ∈ O x i βŠ— x i ∈ g βŠ— g βŠ‚ U ( g ) βŠ— U ( g ) ⁠. For any u ∈ 𝒰(𝔀), we have Ξ”(u)Ξ© = ΩΔ(u).

The following lemma is well-known, see, for example, Ref. 1, Ex. 23.4.

Lemma 2.1.

Letβ€ˆVΞ»β€ˆbe an irreducible module of highest weightβ€ˆΞ». Then Ξ©0β€ˆacts onβ€ˆVΞ»β€ˆby the constant (Ξ» + ρ, Ξ» + ρ) βˆ’ (ρ, ρ).β–‘

Let V be a 𝔀-module. Let Sing V = {v ∈ V|𝔫+v = 0} be the subspace of singular vectors in V. For ΞΌ ∈ π”₯βˆ— let V[ΞΌ] = {v ∈ V|hv = γ€ˆΞΌ, h〉v} be the subspace of V of vectors of weight ΞΌ. Let Sing V[ΞΌ] = (Sing V)∩(V[ΞΌ]) be the subspace of singular vectors in V of weight ΞΌ.

Let n be a positive integer and Ξ› = (Ξ›1, …, Ξ›n), Ξ›i ∈ π”₯βˆ—, a sequence of weights. Denote by VΞ› the 𝔀-module VΞ›1 βŠ— β‹― βŠ— VΞ›n.

If X ∈ End(VΞ›i), then we denote by X(i) ∈ End(VΞ›) the operator idβŠ—iβˆ’1 βŠ— X βŠ— idβŠ—nβˆ’i acting non-trivially on the ith factor of the tensor product. If X = βˆ‘ k X k βŠ— Y k ∈ End ( V Ξ› i βŠ— V Ξ› j ) ⁠, then we set X ( i , j ) = βˆ‘ k X k ( i ) βŠ— Y k ( j ) ∈ End ( V Ξ› ) ⁠.

Let z = (z1, …, zn) be a point in β„‚n with distinct coordinates. Introduce linear operators H 1 ( z ) , … , H n ( z ) on VΞ› by the formula

H i ( z ) = βˆ‘ j , j β‰  i Ξ© ( i , j ) z i βˆ’ z j , i = 1 , … , n .
(2.1)

The operators H 1 ( z ) , … , H n ( z ) are called the Gaudin Hamiltonians of the Gaudin model associated with VΞ›. One can check that the Hamiltonians commute, [ H i ( z ) , H j ( z ) ] = 0 for all i, j. Moreover, the Gaudin Hamiltonians commute with the action of 𝔀, [ H i ( z ) , x ] = 0 for all i and x ∈ 𝔀. Hence for any ΞΌ ∈ π”₯βˆ—, the Gaudin Hamiltonians preserve the subspace Sing VΞ›[ΞΌ] βŠ‚ VΞ›.

Fix a sequence of weights Ξ› = ( Ξ› i ) i = 1 n ⁠, Ξ›i ∈ π”₯βˆ—, and a sequence of non-negative integers l = (l1, …, lr). Denote l = l1 + β‹― + lr, Ξ› = Ξ›1 + β‹― + Ξ›n and Ξ±(l) = l1Ξ±1 + β‹― + lrΞ±r.

Let c be the unique non-decreasing function from {1, …, l} to {1, …, r}, such that #cβˆ’1(i) = li for i = 1, …, r. The master function Ξ¦(t, z, Ξ›, l) is defined by

Ξ¦ ( t , z , Ξ› , l ) = ∏ 1 β©½ i < j β©½ n ( z i βˆ’ z j ) ( Ξ› i , Ξ› j ) ∏ i = 1 l ∏ s = 1 n ( t i βˆ’ z s ) βˆ’ ( Ξ± c ( i ) , Ξ› s ) ∏ 1 β©½ i < j β©½ l ( t i βˆ’ t j ) ( Ξ± c ( i ) , Ξ± c ( j ) ) .

The function Ξ¦ is a function of complex variables t = (t1, …, tl), z = (z1, …, zn), weights Ξ›, and discrete parameters l. The main variables are t, the other variables will be considered as parameters.

We call Ξ›iβ€ˆthe weight at a point zi, and we also call c(i) the color of variable ti.

A point t ∈ β„‚l is called a critical point associated to z, Ξ›, l, if the following system of algebraic equations is satisfied:

βˆ’ βˆ‘ s = 1 n ( Ξ± c ( i ) , Ξ› s ) t i βˆ’ z s + βˆ‘ j , j β‰  i ( Ξ± c ( i ) , Ξ± c ( j ) ) t i βˆ’ t j = 0 i = 1 , … , l .
(2.2)

In other words, a point t is a critical point if

Ξ¦ βˆ’ 1 βˆ‚ Ξ¦ βˆ‚ t i ( t ) = 0 for i = 1 , … , l .

Equation (2.2) is called the Bethe ansatz equation associated to Ξ›, z, l.

By definition, if t = (t1, …, tl) is a critical point and (Ξ±c(i), Ξ±c(j))β‰ 0 for some i, j, then tiβ‰ tj. Also if (Ξ±c(i), Ξ›s)β‰ 0 for some i, s, then tiβ‰ zs.

Let Ξ£l be the permutation group of the set {1, …, l}. Denote by Ξ£l βŠ‚ Ξ£l the subgroup of all permutations preserving the level sets of the function c. The subgroup Ξ£l is isomorphic to Ξ£l1 Γ— β‹― Γ— Ξ£lr. The action of the subgroup Ξ£l preserves the set of critical points of the master function. All orbits of Ξ£l on the critical set have the same cardinality l1!…lr! In what follows we do not distinguish between critical points in the same Ξ£l-orbit.

The following lemma is known.

Lemma 2.2
Ref. 6Β 

If weight Ξ› βˆ’ Ξ±(l) is a dominant integral, then the set of critical points is finite. β–‘

Consider highest weight irreducible 𝔀-modules VΞ›1, …, VΞ›n, the tensor product VΞ›, and its weight subspace VΞ›[Ξ› βˆ’ Ξ±(l)]. Fix a highest weight vector vΞ›i in VΞ›i for i = 1, …, n.

Following Ref. 12, we consider a rational map

Ο‰ : C n Γ— C l β†’ V Ξ› [ Ξ› βˆ’ Ξ± ( l ) ]

called the canonical weight function.

Let P(l, n) be the set of sequences I = ( i 1 1 , … , i j 1 1 ; … ; i 1 n , … , i j n n ) of integers in {1, …, r} such that for all i = 1, …, r, the integer i appears in I precisely li times. For I ∈ P(l, n), and a permutation Οƒ ∈ Ξ£l, set Οƒ1(i) = Οƒ(i) for i = 1, …, j1 and Οƒs(i) = Οƒ(j1 + β‹― + jsβˆ’1 + i) for s = 2, …, n and i = 1, …, js. Define

Ξ£ ( I ) = { Οƒ ∈ Ξ£ l | c ( Οƒ s ( j ) ) = i s j for s = 1 , … , n and j = 1 , … , j s } .

To every I ∈ P(l, n) we associate a vector

F I v = F i 1 1 … F i j 1 1 v Ξ› 1 βŠ— β‹― βŠ— F i 1 n … F i j n n v Ξ› n

in VΞ›[Ξ› βˆ’ Ξ±(l)], and rational functions

Ο‰ I , Οƒ = Ο‰ Οƒ 1 ( 1 ) , … , Οƒ 1 ( j 1 ) ( z 1 ) … Ο‰ Οƒ n ( 1 ) , … , Οƒ n ( j n ) ( z n ) ,

labeled by Οƒ ∈ Ξ£(I), where

Ο‰ i 1 , … , i j ( z ) = 1 ( t i 1 βˆ’ t i 2 ) … ( t i j βˆ’ 1 βˆ’ t i j ) ( t i j βˆ’ z ) .

We set

Ο‰ ( z , t ) = βˆ‘ I ∈ P ( l , n ) βˆ‘ Οƒ ∈ Ξ£ ( I ) Ο‰ I , Οƒ F I v .
(2.3)

Let t ∈ β„‚l be a critical point of the master function Ξ¦(β‹…, z, Ξ›, l). Then the value of the weight function Ο‰(z, t) ∈ VΞ›[Ξ› βˆ’ Ξ±(l)] is called the Bethe vector. Note that the Bethe vector does not depend on a choice of the representative in the Ξ£l-orbit of critical points.

The following facts about Bethe vectors are known. Assume that z ∈ β„‚n has distinct coordinates. Assume that t ∈ β„‚l is an isolated critical point of the master function Ξ¦(β‹…, z, Ξ›, l).

Lemma 2.3
Ref. 7Β 

The Bethe vectorβ€ˆΟ‰(z, t) is well defined.β–‘

Theorem 2.4
Ref. 14Β 

The Bethe vectorβ€ˆΟ‰(z, t) is non-zero.β–‘

Theorem 2.5
Ref. 11Β 
The Bethe vectorβ€ˆΟ‰(z, t) is singular,β€ˆΟ‰(z, t) ∈ Sing VΞ›[Ξ› βˆ’ Ξ±(l)]. Moreover,β€ˆΟ‰(z, t) is a common eigenvector of the Gaudin Hamiltonians,β€ˆ
H i ( z ) Ο‰ ( z , t ) = Ξ¦ βˆ’ 1 βˆ‚ Ξ¦ βˆ‚ z i ( t , z ) Ο‰ ( z , t ) i = 1 , … , n .
β–‘

Let t = (t1, …, tl) be a critical point of a master function Ξ¦(t, z, Ξ›, l). Introduce a sequence of polynomials y = (y1(x), …, yr(x)) in a variable x by the formula

y i ( x ) = ∏ j , c ( j ) = i ( x βˆ’ t j ) .

We say that the r-tuple of polynomials yrepresents a critical point t of the master function Ξ¦(t, z, Ξ›, l). Note that the r-tuple y does not depend on a choice of the representative in the Ξ£l-orbit of the critical point t.

We have l = βˆ‘ i = 1 r deg y i = βˆ‘ i = 1 r l i ⁠. We call lβ€ˆthe length of y. We use notation y(l) to indicate the length of y.

Introduce functions

T i ( x ) = ∏ s = 1 n ( x βˆ’ z s ) γ€ˆ Ξ› s , Ξ± i ∨ 〉 , i = 1 , … , r .
(2.4)

We say that a given r-tuple of polynomials y ∈ P(β„‚[x])r is generic with respect to Ξ›, z if

  • polynomials yi(x) have no multiple roots;

  • roots of yi(x) are different from roots and singularities of the function Ti;

  • if aij < 0 then polynomials yi(x), yj(x) have no common roots.

If y represents a critical point of Ξ¦, then y is generic.

Following Ref. 9, we reformulate the property of y to represent a critical point for the case when all but one weights are dominant integral.

We denote by W(f, g) the Wronskian of functions f and g, W(f, g) = fβ€²g βˆ’ fgβ€².

Theorem 2.6
Ref. 9Β 
Assume thatβ€ˆz ∈ β„‚nβ€ˆhas distinct coordinates andβ€ˆz1 = 0. Assume thatβ€ˆ Ξ› i ∈ P + ,β€ˆi = 2, …, n. A genericβ€ˆr-tupleβ€ˆyβ€ˆrepresents a critical point associated toβ€ˆΞ›, z, lβ€ˆif and only if for everyβ€ˆi = 1, …, rβ€ˆthere exists a polynomialβ€ˆ y Μƒ i β€ˆsatisfyingβ€ˆ
W ( y i , x γ€ˆ Ξ› 1 + ρ , Ξ± i ∨ 〉 y Μƒ i ) = T i ∏ j β‰  i y j βˆ’ γ€ˆ Ξ± j , Ξ± i ∨ 〉 .
(2.5)
Moreover, if theβ€ˆr-tupleβ€ˆ y Μƒ i = ( y 1 , … , y Μƒ i , … , y r ) β€ˆis generic, then it represents a critical point associated to data (si β‹… Ξ›1, Ξ›2, …, Ξ›n), z, li, whereβ€ˆliβ€ˆis determined by equationβ€ˆ
Ξ› βˆ’ Ξ› 1 βˆ’ Ξ± ( l i ) = s i ( Ξ› βˆ’ Ξ› 1 βˆ’ Ξ± ( l ) ) .
β–‘
We say that the r-tuple y Μƒ i (and the critical point it represents) is obtained from the r-tuple y (and the critical point it represents) by the reproduction procedure in the ith direction.

Note that the reproduction procedure can be iterated. The reproduction procedure in the ith direction applied to r-tuple y Μƒ i returns back the r-tuple y. More generally, it is shown in Ref. 9 that the r-tuples obtained from y by iterating a reproduction procedure are in a bijective correspondence with the elements of the Weyl group.

We call a function f(x) a quasi-polynomial if it has the form xap(x), where a ∈ β„‚ and p(x) ∈ β„‚[x]. Under the assumptions of Theorem 2.6, all Ti are quasi-polynomials.

In Sections III and IV we work with Lie algebra of type Br.

Let 𝔀 = 𝔰𝔬(2r + 1). We have (Ξ±i, Ξ±i) = 4, i = 1, …, r βˆ’ 1, and (Ξ±r, Ξ±r) = 2.

In this section we work with data Ξ› = (Ξ», Ο‰1), z = (0, 1). The main result of the section is the explicit formulas for the solutions of the Bethe ansatz equations, see Corollary 3.10.

One of our goals is to diagonalize the Gaudin Hamiltonians associated to Ξ› = (Ξ», Ο‰1), z = (0, 1). It is sufficient to do that in the spaces of singular vectors of a given weight.

Let Ξ» ∈ P + ⁠. We write the decomposition of finite-dimensional 𝔀-module VΞ» βŠ— VΟ‰1. We have

V Ξ» βŠ— V Ο‰ 1 = V Ξ» + Ο‰ 1 βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βŠ• β‹― βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βˆ’ β‹― βˆ’ Ξ± r βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βˆ’ β‹― βˆ’ Ξ± r βˆ’ 1 βˆ’ 2 Ξ± r βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βˆ’ β‹― βˆ’ Ξ± r βˆ’ 2 βˆ’ 2 Ξ± r βˆ’ 1 βˆ’ 2 Ξ± r βŠ• β‹― βŠ• V Ξ» + Ο‰ 1 βˆ’ 2 Ξ± 1 βˆ’ β‹― βˆ’ 2 Ξ± r βˆ’ 1 βˆ’ 2 Ξ± r = V ( Ξ» 1 + 1 , Ξ» 2 , … , Ξ» r ) βŠ• V ( Ξ» 1 βˆ’ 1 , Ξ» 2 + 1 , Ξ» 3 , … , Ξ» r ) βŠ• V ( Ξ» 1 , … , Ξ» k βˆ’ 1 , Ξ» k βˆ’ 1 , Ξ» k + 1 + 1 , … , Ξ» r ) βŠ• β‹― βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 2 , Ξ» r βˆ’ 1 βˆ’ 1 , Ξ» r + 2 ) βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 1 , Ξ» r ) βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 2 , Ξ» r βˆ’ 1 + 1 , Ξ» r βˆ’ 2 ) βŠ• V ( Ξ» 1 , … , Ξ» r βˆ’ 2 + 1 , Ξ» r βˆ’ 1 βˆ’ 1 , Ξ» r ) βŠ• β‹― βŠ• V ( Ξ» 1 βˆ’ 1 , Ξ» 2 , … , Ξ» r ) ,
(3.1)

with the convention that the summands with non-dominant highest weights are omitted. In addition, if Ξ»r = 0, then the summand VΞ»βˆ’Ξ±1βˆ’β‹―βˆ’Ξ±r = V(Ξ»1,Ξ»2,…,Ξ»rβˆ’1,Ξ»r) is absent.

Note, in particular, that all multiplicities are 1.

By Theorem 2.5, to diagonalize the Gaudin Hamiltonians, it is sufficient to find a solution of the Bethe ansatz equation (2.2) associated to Ξ›, z, and l corresponding to the summands in the decomposition (3.1). We call an r-tuple of integers lβ€ˆadmissible if VΞ»+Ο‰1βˆ’Ξ±(l) βŠ‚ VΞ» βŠ— VΟ‰1.

The admissible r-tuples l have the form

l = ( 1 , … , 1 οΈΈ k ones , 0 , … , 0 ) or l = ( 1 , … , 1 οΈΈ k ones , 2 , … , 2 ) ,
(3.2)

where k = 0, …, r. In the first case the length l = l1 + β‹― + lr is k and in the second case 2r βˆ’ k. It follows that different admissible r-tuples have different length and, therefore, admissible tuples l are parameterized by length l ∈ {0, 1, …, 2r}. We call a non-negative integer ladmissible if it is the length of an admissible r-tuple l. More precisely, a non-negative integer l is admissible if l = 0 or if l β©½ r, Ξ»l > 0 or if l = r + 1, Ξ»r > 1 or if r + 1 < l β©½ 2r, Ξ»2rβˆ’l+1 > 0.

In terms of y = (y1, …, yr), we have the following cases, corresponding to (3.2).

For l β©½ r, the polynomials y1, …, yl are linear and yl+1, …, yr are all equal to one.

For l > r, the polynomials y1, …, y2rβˆ’l are linear and y2rβˆ’l+1, …, yr are quadratic.

Remark 3.1.

Forβ€ˆl β©½ rβ€ˆthe Bethe ansatz equations for type Brβ€ˆcoincide with the Bethe ansatz equations for type Arβ€ˆwhich were solved directly in Ref.β€ˆ5,. In what follows, we recover the result forβ€ˆl < r, and we refer to Ref.β€ˆ5Β  for the case ofβ€ˆl = r.

We illustrate our approach in the case of B2, l = 4. We have n = 2, Ξ› 1 = Ξ» ∈ P + ⁠, Ξ›2 = Ο‰1, z1 = 0, z2 = 1. We write Ξ» = (Ξ»1, Ξ»2), where Ξ» i = γ€ˆ Ξ» , Ξ± i ∨ 〉 ∈ Z β©Ύ 0 ⁠.

Suppose the Bethe ansatz equation has a solution with l = 4. Then it is represented by quadratic polynomials y 1 ( 4 ) and y 2 ( 4 ) ⁠. By Theorem 2.6, it means that there exist polynomials y Μƒ 1 , y Μƒ 2 such that

W ( y 1 ( 4 ) , y Μƒ 1 ) = x Ξ» 1 ( x βˆ’ 1 ) y 2 ( 4 ) , W ( y 2 ( 4 ) , y Μƒ 2 ) = x Ξ» 2 y 1 ( 4 ) 2 .

Note we have Ξ»1, Ξ»2 ∈ β„€β©Ύ0, but for Ξ»1 = 0 the first equation is impossible for degree reasons. Therefore, there are no solutions with l = 4 for Ξ»1 = 0 which is exactly when the corresponding summand is absent in (3.1) and when l = 4 is not admissible.

Step 1: There exists a unique monic linear polynomial u1 such that βˆ’ Ξ» 1 y Μƒ 1 = x Ξ» 1 + 1 u 1 ⁠. Clearly, the only root of u1 cannot coincide with the roots of x Ξ» 1 ( x βˆ’ 1 ) y 2 ( 4 ) ⁠, therefore the pair ( u 1 , y 2 ( 4 ) ) is generic. It follows from Theorem 2.6 that the pair ( u 1 , y 2 ( 4 ) ) solves the Bethe ansatz equation with l = 3 and Ξ» replaced by s1 β‹… Ξ» = (βˆ’Ξ»1 βˆ’ 2, 2Ξ»1 + Ξ»2 + 2).

In terms of Wronskians, it means that there exist quasi-polynomials y Λ† 1 and y Λ† 2 such that

W ( u 1 , y Λ† 1 ) = x βˆ’ Ξ» 1 βˆ’ 2 ( x βˆ’ 1 ) y 2 ( 4 ) , W ( y 2 ( 4 ) , y Λ† 2 ) = x 2 Ξ» 1 + Ξ» 2 + 2 u 1 2 .

The procedure we just described corresponds to the reproduction in the first direction, we have s1(Ο‰1 βˆ’ 2Ξ±1 βˆ’ 2Ξ±2) = Ο‰1 βˆ’ Ξ±1 βˆ’ 2Ξ±2.

Note that s2(Ο‰1 βˆ’ 2Ξ±1 βˆ’ 2Ξ±2) = Ο‰1 βˆ’ 2Ξ±1 βˆ’ 2Ξ±2 and the reproduction in the second direction applied to ( y 1 ( 4 ) , y 2 ( 4 ) ) does not change l = 4. We do not use it.

Step 2: We apply the reproduction in the second direction to the l = 3 solution ( u 1 , y 2 ( 4 ) ) ⁠.

For degree reasons, we have βˆ’ ( Ξ» 2 + 2 Ξ» 1 + 1 ) y Λ† 2 = x Ξ» 2 + 2 Ξ» 1 + 3 β‹… 1 ⁠. Set u2 = 1. Clearly, the pair (u1, u2) is generic. By Theorem 2.6, the pair (u1, u2) solves Bethe ansatz equation with l = 1 and Ξ›1 = (s2s1) β‹… Ξ» = (Ξ»1 + Ξ»2 + 1, βˆ’ 2Ξ»1 βˆ’ Ξ»2 βˆ’ 4).

It means we have s2(Ο‰1 βˆ’ Ξ±1 βˆ’ 2Ξ±2) = Ο‰1 βˆ’ Ξ±1 and there exist quasi-polynomials y Μ„ 1 , y Μ„ 2 such that

W ( u 1 , y Μ„ 1 ) = x Ξ» 1 + Ξ» 2 + 1 ( x βˆ’ 1 ) u 2 = x Ξ» 1 + Ξ» 2 + 1 ( x βˆ’ 1 ) , W ( u 2 , y Μ„ 2 ) = x βˆ’ 2 Ξ» 1 βˆ’ Ξ» 2 βˆ’ 4 u 1 2 .

Note that we also have Ξ» 1 y Λ† 1 = x βˆ’ Ξ» 1 βˆ’ 1 y 1 ( 4 ) ⁠. Therefore, we can recover the initial solution ( y 1 ( 4 ) , y 2 ( 4 ) ) from ( u 1 , y 2 ( 4 ) ) ⁠. In general, if we start with an arbitrary l = 3 solution and use the reproduction in the first direction, we obtain a pair of quadratic polynomials. If this pair is generic, then it represents an l = 4 solution associated to the data Ξ› = (Ξ», Ο‰1), z, l = (2, 2). However, we have no easy argument to show that it is generic. Thus, our procedure gives an inclusion of all l = 4 solutions to the l = 3 solutions and we need an extra argument to show this inclusion is a bijection.

Step 3: Finally, we apply the reproduction in the first direction to the l = 1 solution (u1, u2).

We have βˆ’ ( Ξ» 1 + Ξ» 2 + 1 ) y Μ„ 1 = x Ξ» 1 + Ξ» 2 + 2 β‹… 1 ⁠. Set v1 = 1. Clearly, the pair (v1, u2) = (1, 1) is generic and represents the only solution of the Bethe ansatz equation with l = 0 and Ξ›1 = (s1s2s1) β‹… Ξ» = (βˆ’Ξ»1 βˆ’ Ξ»2 βˆ’ 3, Ξ»2). We denote the final weight (s1s2s1) β‹… Ξ» by ΞΈ = (ΞΈ1, ΞΈ2).

It means we have s1(Ο‰1 βˆ’ Ξ±1) = Ο‰1, and there exist quasi-polynomials y ∘ 1 , y ∘ 2 such that

W ( v 1 , y ∘ 1 ) = x βˆ’ Ξ» 1 βˆ’ Ξ» 2 βˆ’ 3 ( x βˆ’ 1 ) u 2 2 , W ( u 2 , y 2 ∘ ) = x Ξ» 2 v 1 = x Ξ» 2 .

As before, we have ( Ξ» 1 + Ξ» 2 + 1 ) y ∘ 1 = x βˆ’ Ξ» 1 βˆ’ Ξ» 2 βˆ’ 2 u 1 ⁠, and therefore using reproduction in the first direction to pair (v1, u2) we recover the pair (u1, u2).

To sum up, we have the inclusions of solutions for l = 4 to l = 3 to l = 1 to l = 0 with the Ξ›1 varying by the shifted action of the Weyl group. Since for l = 0 the solution is unique, it follows that for l = 1, 3, 4 the solutions are at most unique. Moreover, if it exists, it can be computed recursively.

We proceed with the direct computation of y 1 ( 4 ) , y 2 ( 4 ) ⁠. From Step 3, we have v1 = u2 = 1. Then we compute

u 1 = x βˆ’ Ξ» 1 + Ξ» 2 + 1 Ξ» 1 + Ξ» 2 + 2 .

From Step 2, we get

y 2 ( 4 ) = x 2 βˆ’ 2 ( 2 Ξ» 1 + Ξ» 2 + 1 ) ( Ξ» 1 + Ξ» 2 + 1 ) ( 2 Ξ» 1 + Ξ» 2 + 2 ) ( Ξ» 1 + Ξ» 2 + 2 ) x + ( 2 Ξ» 1 + Ξ» 2 + 1 ) ( Ξ» 1 + Ξ» 2 + 1 ) 2 ( 2 Ξ» 1 + Ξ» 2 + 3 ) ( Ξ» 1 + Ξ» 2 + 2 ) 2 .

Finally, from Step 1,

y 1 ( 4 ) = x 2 βˆ’ ( 2 Ξ» 1 + Ξ» 2 + 1 ) ( 2 Ξ» 1 2 + 2 Ξ» 1 Ξ» 2 + 4 Ξ» 1 + Ξ» 2 + 2 ) ( Ξ» 1 + 1 ) ( Ξ» 1 + Ξ» 2 + 2 ) ( 2 Ξ» 1 + Ξ» 2 + 2 ) x + Ξ» 1 ( Ξ» 1 + Ξ» 2 + 1 ) ( 2 Ξ» 1 + Ξ» 2 + 1 ) ( Ξ» 1 + 1 ) ( Ξ» 1 + Ξ» 2 + 2 ) ( 2 Ξ» 1 + Ξ» 2 + 3 ) .

From the formula it is easy to check that the pair ( y 1 ( 4 ) , y 2 ( 4 ) ) is generic if Ξ»1 > 0 and therefore represents a solution of the Bethe ansatz equation associated to Ξ›, z, and l = 4.

Thus the Bethe ansatz equation associated to Ξ›, z, l = (2, 2) has a unique solution given by the formulas above.

Let l ∈ {0, …, r βˆ’ 1, r + 2, …, 2r}, we establish a reproduction procedure which produces solutions of length l βˆ’ 1 from the ones of length l. For l = r + 1, the reproduction procedure goes from l = r + 1 to l = r βˆ’ 1. We recover the special case l = r directly from Ref. 5, see Remark 3.1. By Theorem 2.6 it is sufficient to check that the new r-tuple of polynomial is generic with respect to new data. It is done with the help of following series of lemmas.

For brevity, we denote x βˆ’ 1 by y0 for this section.

The first lemma describes the reproduction in the kth direction from l = 2r βˆ’ k + 1 to l = 2r βˆ’ k, where k = 1, …, r βˆ’ 1.

Lemma 3.2.

Letβ€ˆk ∈ {1, …, r βˆ’ 1}. Letβ€ˆΞ½ = (Ξ½1, …, Ξ½r) be an integral weight such thatβ€ˆΞ½k β©Ύ 0. Letβ€ˆy1, …, ykβˆ’1β€ˆbe linear polynomials andβ€ˆyk, …, yrβ€ˆbe quadratic polynomials. Suppose theβ€ˆr-tuple of polynomialsβ€ˆy(2rβˆ’k+1) = (y1, …, yr) represents a critical point associated to (Ξ½, Ο‰1), zβ€ˆandβ€ˆl = 2r βˆ’ k + 1. Then there exists a unique monic linear polynomialβ€ˆukβ€ˆsuch thatβ€ˆW(yk, xΞ½k+1uk) = βˆ’ Ξ½kxΞ½kykβˆ’1yk+1. Moreover,β€ˆΞ½k > 0 and theβ€ˆr-tuple of polynomialsβ€ˆy(2rβˆ’k) = (y1, …, ykβˆ’1, uk, yk+1, …, yr) represents a critical point associated to (sk β‹… Ξ½, Ο‰1), zβ€ˆandβ€ˆl = 2r βˆ’ k.

Proof.

The existence of polynomial y Μƒ k such that W ( y k , y Μƒ k ) = x Ξ½ k y k βˆ’ 1 y k + 1 implies Ξ½k > 0. Indeed, if deg y Μƒ k β©Ύ 3 ⁠, then deg W ( y k , y Μƒ k ) β©Ύ 4 ; if deg y Μƒ k β©½ 2 ⁠, then deg W ( y k , y Μƒ k ) β©½ 2 ⁠. Hence degxΞ½kykβˆ’1yk+1β‰ 3 , it follows that Ξ½kβ‰ 0.

By Theorem 2.6, it is enough to show y(2rβˆ’k) is generic. If ykβˆ’1yk+1 is divisible by uk, then yk has a common root with ykβˆ’1yk+1 which is impossible since (y1, …, yr) is generic. Since uk is linear, it cannot have a multiple root. β–‘

Note that we do not have such a lemma for the reproduction in the kth direction which goes from l βˆ’ 1 to l since unlike uk the new polynomial is quadratic and we cannot immediately conclude that it has distinct roots. We overcome this problem using the explicit formulas in Section III E 4.

The next lemma describes the reproduction in the rth direction from l = r + 1 to l = r βˆ’ 1.

Lemma 3.3.

Letβ€ˆΞ½ = (Ξ½1, …, Ξ½r) be an integral weight such thatβ€ˆΞ½r β©Ύ 0. Letβ€ˆy1, …, yrβˆ’1β€ˆbe linear polynomials andβ€ˆyrβ€ˆbe a quadratic polynomial. Suppose theβ€ˆr-tuple of polynomialsβ€ˆy(r+1) = (y1, …, yr) represents a critical point associated to (Ξ½, Ο‰1), zβ€ˆandβ€ˆl = r + 1. Thenβ€ˆ W ( y r , x Ξ½ r + 1 ) = βˆ’ ( Ξ½ r βˆ’ 1 ) x Ξ½ r y r βˆ’ 1 2 . Moreover,β€ˆΞ½r β©Ύ 2 and theβ€ˆr-tuple of polynomialsβ€ˆy(rβˆ’1) = (y1, …, yrβˆ’2, yrβˆ’1, 1) represents a critical point associated to (sr β‹… Ξ½, Ο‰1), zβ€ˆandβ€ˆl = r βˆ’ 1.β–‘

Finally, we discuss the reproduction in the kth direction from l = k to l = k βˆ’ 1, where k = 1, …, r βˆ’ 1.

Lemma 3.4.

Letβ€ˆk ∈ {1, …, r βˆ’ 1}. Letβ€ˆΞ½ = (Ξ½1, …, Ξ½r) be an integral weight such thatβ€ˆΞ½k β©Ύ 0. Letβ€ˆy1, …, ykβ€ˆbe linear polynomials andβ€ˆyk+1 = β‹― = yr = 1. Suppose theβ€ˆr-tuple of polynomialsβ€ˆy(k) = (y1, …, yr) represents a critical point associated to (Ξ½, Ο‰1), zβ€ˆandβ€ˆl = k. Thenβ€ˆW(yk, xΞ½k+1) = βˆ’ Ξ½kxΞ½kykβˆ’1yk+1. Moreover,β€ˆΞ½k > 0 and theβ€ˆr-tuple of polynomialsβ€ˆy(kβˆ’1) = (y1, …, ykβˆ’1, 1, 1, …, 1) represents a critical point associated to (sk β‹… Ξ½, Ο‰1), zβ€ˆandβ€ˆl = k βˆ’ 1. β–‘

In this section, we show that there exists at most one solution of the Bethe ansatz equation (2.2).

We start with the explicit formulas for the shifted action of the Weyl group.

Lemma 3.5.

Letβ€ˆΞ» = (Ξ»1, …, Ξ»r) ∈ π”₯βˆ—.

We haveβ€ˆ
( s 1 … s k ) β‹… Ξ» = ( βˆ’ Ξ» 1 βˆ’ β‹― βˆ’ Ξ» k βˆ’ k βˆ’ 1 , Ξ» 1 , … , Ξ» k βˆ’ 1 , Ξ» k + Ξ» k + 1 + 1 , Ξ» k + 2 , … , Ξ» r ) ,
whereβ€ˆk = 1, …, r βˆ’ 2,β€ˆ
( s 1 … s r βˆ’ 1 ) β‹… Ξ» = ( βˆ’ Ξ» 1 βˆ’ β‹― βˆ’ Ξ» r βˆ’ 1 βˆ’ r , Ξ» 1 , … , Ξ» r βˆ’ 2 , 2 Ξ» r βˆ’ 1 + Ξ» r + 2 ) ,
β€ˆ
( s 1 … s r ) β‹… Ξ» = ( βˆ’ Ξ» 1 βˆ’ β‹― βˆ’ Ξ» r βˆ’ r βˆ’ 1 , Ξ» 1 , … , Ξ» r βˆ’ 2 , 2 Ξ» r βˆ’ 1 + Ξ» r + 2 ) ,
andβ€ˆ
( s 1 … s r s r βˆ’ 1 … s 2 r βˆ’ k ) β‹… Ξ» = ( βˆ’ Ξ» 1 βˆ’ β‹― βˆ’ Ξ» 2 r βˆ’ k βˆ’ 1 βˆ’ 2 Ξ» 2 r βˆ’ k βˆ’ β‹― βˆ’ 2 Ξ» r βˆ’ 1 βˆ’ Ξ» r βˆ’ k βˆ’ 1 , Ξ» 1 , … , Ξ» 2 r βˆ’ k βˆ’ 2 , Ξ» 2 r βˆ’ k βˆ’ 1 + Ξ» 2 r βˆ’ k + 1 , Ξ» 2 r βˆ’ k + 1 , … , Ξ» r ) ,
whereβ€ˆk = r + 1, …, 2r βˆ’ 1.

Proof.
If k = 1, …, r βˆ’ 2, r, the action of a simple reflection is given by
s k β‹… Ξ» = ( Ξ» 1 , … , Ξ» k βˆ’ 2 , Ξ» k βˆ’ 1 + Ξ» k + 1 , βˆ’ Ξ» k βˆ’ 2 , Ξ» k + Ξ» k + 1 + 1 , Ξ» k + 2 , … , Ξ» r ) .
In addition,
s r βˆ’ 1 β‹… Ξ» = ( Ξ» 1 , … , Ξ» r βˆ’ 3 , Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + 1 , βˆ’ Ξ» r βˆ’ 1 βˆ’ 2 , 2 Ξ» r βˆ’ 1 + Ξ» r + 2 ) .
The lemma follows.β–‘

We also prepare the inverse formulas.

Lemma 3.6.
Letβ€ˆΞΈ = (ΞΈ1, …, ΞΈr) ∈ π”₯βˆ—. We haveβ€ˆ
( s k … s 1 ) β‹… ΞΈ = ( ΞΈ 2 , … , ΞΈ k , βˆ’ ΞΈ 1 βˆ’ β‹― βˆ’ ΞΈ k βˆ’ k βˆ’ 1 , ΞΈ 1 + β‹― + ΞΈ k + 1 + k , ΞΈ k + 2 , … , ΞΈ r ) ,
whereβ€ˆk = 1, …, r βˆ’ 2,β€ˆ
( s r βˆ’ 1 … s 1 ) β‹… ΞΈ = ( ΞΈ 2 , … , ΞΈ r βˆ’ 1 , βˆ’ ΞΈ 1 βˆ’ β‹― βˆ’ ΞΈ r βˆ’ 1 βˆ’ r , 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ 2 ) ,
β€ˆ
( s r … s 1 ) β‹… ΞΈ = ( ΞΈ 2 , … , ΞΈ r βˆ’ 1 , ΞΈ 1 + β‹― + ΞΈ r + r βˆ’ 1 , βˆ’ 2 ΞΈ 1 βˆ’ β‹― βˆ’ 2 ΞΈ r βˆ’ 1 βˆ’ ΞΈ r βˆ’ 2 r ) ,
andβ€ˆ
( s 2 r βˆ’ k s 2 r βˆ’ k + 1 … s r s r βˆ’ 1 … s 1 ) β‹… ΞΈ = ( ΞΈ 2 , … , ΞΈ 2 r βˆ’ k βˆ’ 1 , ΞΈ 1 + β‹― + ΞΈ 2 r βˆ’ k βˆ’ 1 + 2 ΞΈ 2 r βˆ’ k + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + k βˆ’ 1 , βˆ’ ΞΈ 1 βˆ’ β‹― βˆ’ ΞΈ 2 r βˆ’ k βˆ’ 2 ΞΈ 2 r βˆ’ k + 1 βˆ’ β‹― βˆ’ 2 ΞΈ r βˆ’ 1 βˆ’ ΞΈ r βˆ’ k , ΞΈ 2 r βˆ’ k + 1 , … , ΞΈ r ) ,
whereβ€ˆk = r + 1, …, 2r βˆ’ 1. In particular,β€ˆ
( s 1 s 2 … s r s r βˆ’ 1 … s 1 ) β‹… ΞΈ = ( βˆ’ ΞΈ 1 βˆ’ 2 ΞΈ 2 βˆ’ β‹― βˆ’ 2 ΞΈ r βˆ’ 1 βˆ’ ΞΈ r βˆ’ 2 r + 1 , ΞΈ 2 , … , ΞΈ r ) .
β–‘

Lemma 3.7.

Letβ€ˆ Ξ» ∈ P + β€ˆand letβ€ˆlβ€ˆbe as in (3.2). Suppose the Bethe ansatz equation associated toβ€ˆΞ› = (Ξ», Ο‰1), z = (0, 1), l, whereβ€ˆ Ξ» ∈ P + , has solutions. Thenβ€ˆlβ€ˆis admissible. Moreover, ifβ€ˆl β©Ύ r + 1, then we can perform the reproduction procedure in the (2r βˆ’ l + 1) th, (2r βˆ’ l + 2) th, …, (r βˆ’ 1) th,β€ˆrβ€ˆth, (r βˆ’ 1) th, …, 1-st directions successively. Ifβ€ˆl < r, we can perform the reproduction procedure in theβ€ˆlβ€ˆth, (l βˆ’ 1) th, …, 1-st directions successively.

Proof.

We use Lemmas 3.2-3.4. The condition of the lemmas of the form Ξ½k β©Ύ 0 follows from Lemmas 3.5 and 3.6. β–‘

Corollary 3.8.

Letβ€ˆ Ξ» ∈ P + β€ˆandβ€ˆlβ€ˆas in (3.2). The Bethe ansatz equation (2.2) associated toβ€ˆΞ›, z, l, has at most one solution. Ifβ€ˆlβ€ˆis not admissible it has no solutions.

Proof.

If l β‰  r, then by Lemma 3.7, every solution of the Bethe ansatz equations by a series of reproduction procedures produces a solution for l = 0. These reproduction procedures are invertible, and for l = 0 we clearly have only one solution (1, …, 1). Therefore the conclusion.

For l = r the corollary follows from Theorem 2 in Ref. 5, see also Remark 3.1.β–‘

In this section, we give explicit formulas for the solution of the Bethe ansatz equation corresponding to data Ξ› = (Ξ», Ο‰1), z = (0, 1) and l, Ξ» ∈ P + ⁠, l ∈ {0, …, 2r}.

We denote by ΞΈ the weight obtained from Ξ» after the successive reproduction procedures as in Lemma 3.7. Explicitly, if l β©½ r βˆ’ 1, then ΞΈ = (s1…slβˆ’1sl) β‹… Ξ»; if l β©Ύ r + 1, then ΞΈ = (s1…srβˆ’1srsrβˆ’1…s2rβˆ’l+1) β‹… Ξ». We recover the solution starting from data (ΞΈ, Ο‰1), z = (0, 1) and l = 0, where the solution is (1, …, 1) by applying the reproduction procedures in the opposite direction explicitly. In the process we obtain monic polynomials ( y 1 ( l ) , … , y r ( l ) ) representing a critical point.

Recall that for l β©½ r, y1, …, yl are linear polynomials and yl+1, …, yr are all equal to one. We use the notation y i ( l ) = x βˆ’ c i ( l ) ⁠, i = 1, …, l.

Recall further that for l > r, the polynomials y1, …, y2rβˆ’l are linear and y2rβˆ’l+1, …, yr are quadratic. We use the notation y i ( l ) = x βˆ’ c i ( l ) ⁠, i = 1, …, 2r βˆ’ l and y i ( l ) = ( x βˆ’ a i ( l ) ) ( x βˆ’ b i ( l ) ) ⁠, i = 2r βˆ’ l + 1, …, r.

Formulas for c i ( l ) ⁠, a i ( l ) ⁠, and b i ( l ) in terms of θi, clearly, do not depend on l, in such cases we simply write ci, ai, and bi.

Denote y 0 ( k ) = x βˆ’ 1 ⁠, c0 = 1, and T1(x) = xΞ»1. Also let

A ( k ) ( ΞΈ ) = { ( s k … s 1 ) β‹… ΞΈ if k β©½ r , ( s 2 r βˆ’ k … s r βˆ’ 1 s r s r βˆ’ 1 … s 1 ) β‹… ΞΈ if k β©Ύ r + 1 .

Explicitly, A(k)(ΞΈ) are given in Lemma 3.6.

1. Constant term of yi in terms of ΞΈ

For brevity, we write simply A(k) for A(k)(ΞΈ). We also use A i ( k ) for components of the weight A(k): A ( k ) = ( A 1 ( k ) , … , A r ( k ) ) ⁠.

For l β©½ r βˆ’ 1, we have y(lβˆ’1) = (x βˆ’ c1, …, x βˆ’ clβˆ’1, 1, …, 1). It is easy to check that if l is admissible and Ξ» is dominant then A l ( l βˆ’ 1 ) = ΞΈ 1 + β‹― + ΞΈ l + ( l βˆ’ 1 ) is a negative integer.

We solve for y Μƒ l ( l βˆ’ 1 ) ⁠,

W ( y l ( l βˆ’ 1 ) , y Μƒ l ( l βˆ’ 1 ) ) = T l ( l βˆ’ 1 ) y l βˆ’ 1 ( l βˆ’ 1 ) y l + 1 ( l βˆ’ 1 ) = x A l ( l βˆ’ 1 ) ( x βˆ’ c l βˆ’ 1 ) .

In other words

βˆ’ ( y Μƒ l ( l βˆ’ 1 ) ) β€² = x A l ( l βˆ’ 1 ) + 1 βˆ’ c l βˆ’ 1 x A l ( l βˆ’ 1 ) .

Choosing the solution which is a quasi-polynomial, we obtain

y Μƒ l ( l βˆ’ 1 ) = βˆ’ x A l ( l βˆ’ 1 ) + 1 A l ( l βˆ’ 1 ) + 2 x βˆ’ A l ( l βˆ’ 1 ) + 2 A l ( l βˆ’ 1 ) + 1 c l βˆ’ 1 .

Therefore, the reproduction procedure in the lth direction gives y(l) = (x βˆ’ c1, …, x βˆ’ cl, 1, …, 1), where c l = A l ( l βˆ’ 1 ) + 2 A l ( l βˆ’ 1 ) + 1 c l βˆ’ 1 ⁠. Substituting the value for A l ( l βˆ’ 1 ) and using induction, we have

c k = ∏ j = 1 k ΞΈ 1 + β‹― + ΞΈ j + j + 1 ΞΈ 1 + β‹― + ΞΈ j + j ,

for k = 1, …, r βˆ’ 1.

For l = r + 1 we have y(rβˆ’1) = (x βˆ’ c1, …, x βˆ’ crβˆ’1, 1) and A r ( r βˆ’ 1 ) = 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ 2 ∈ Z < 0 ⁠. We solve for y Μƒ r ( r βˆ’ 1 ) ⁠,

W ( y r ( r βˆ’ 1 ) , y Μƒ r ( r βˆ’ 1 ) ) = T r ( r βˆ’ 1 ) ( y r βˆ’ 1 ( r βˆ’ 1 ) ) 2 = x A r ( r βˆ’ 1 ) ( x βˆ’ c r βˆ’ 1 ) 2 .

This implies

y Μƒ r ( r βˆ’ 1 ) = βˆ’ x A r ( r βˆ’ 1 ) + 1 A r ( r βˆ’ 1 ) + 3 x 2 βˆ’ 2 ( A r ( r βˆ’ 1 ) + 3 ) A r ( r βˆ’ 1 ) + 2 c r βˆ’ 1 x + A r ( r βˆ’ 1 ) + 3 A r ( r βˆ’ 1 ) + 1 c r βˆ’ 1 2 .

Therefore, after performing the reproduction procedure in the rth direction to y(rβˆ’1), we obtain the r-tuple y(r+1) = (x βˆ’ c1, …, x βˆ’ crβˆ’1, (x βˆ’ ar)(x βˆ’ br)), where

a r b r = A r ( r βˆ’ 1 ) + 3 A r ( r βˆ’ 1 ) + 1 c r βˆ’ 1 2 = ∏ j = 1 r βˆ’ 1 ΞΈ 1 + β‹― + ΞΈ j + j + 1 ΞΈ 1 + β‹― + ΞΈ j + j 2 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r + 1 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ 1 .

For l such that r + 2 β©½ l β©½ 2r, let k = 2r βˆ’ l, then y(2rβˆ’k) = (x βˆ’ c1, …, x βˆ’ ck, (x βˆ’ ak+1)(x βˆ’ bk+1), …, (x βˆ’ ar)(x βˆ’ br)) and A k ( 2 r βˆ’ k βˆ’ 1 ) = ΞΈ 1 + β‹― + ΞΈ k + 2 ΞΈ k + 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ k βˆ’ 2 ∈ Z < 0 ⁠.

We have

W ( y k ( 2 r βˆ’ k ) , y Μƒ k ( 2 r βˆ’ k ) ) = x A k ( 2 r βˆ’ k βˆ’ 1 ) y k βˆ’ 1 ( 2 r βˆ’ k ) y k + 1 ( 2 r βˆ’ k ) ,

substituting βˆ’ ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) y Μƒ k ( 2 r βˆ’ k ) = x A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ( x βˆ’ a k ) ( x βˆ’ b k ) ⁠, we get

( A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ) ( x βˆ’ c k ) ( x βˆ’ a k ) ( x βˆ’ b k ) + x ( x βˆ’ c k ) ( x βˆ’ a k )
+ x ( x βˆ’ c k ) ( x βˆ’ b k ) βˆ’ x ( x βˆ’ a k ) ( x βˆ’ b k )
(3.3)
= ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) ( x βˆ’ a k + 1 ) ( x βˆ’ b k + 1 ) ( x βˆ’ c k βˆ’ 1 ) .

Substituting x = 0 into (3.3), we obtain

( A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ) c k a k b k = ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) c k βˆ’ 1 a k + 1 b k + 1 .
(3.4)

It results in

a k b k = c r βˆ’ 1 c k βˆ’ 1 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r + 1 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ 1 Γ— ∏ i = 1 r βˆ’ k ( ΞΈ 1 + β‹― + ΞΈ r βˆ’ 1 ) + ( ΞΈ r + ΞΈ r βˆ’ 1 + β‹― + ΞΈ r + 1 βˆ’ i ) + r + i ( ΞΈ 1 + β‹― + ΞΈ r βˆ’ 1 ) + ( ΞΈ r + ΞΈ r βˆ’ 1 + β‹― + ΞΈ r + 1 βˆ’ i ) + r + i βˆ’ 1 = 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r + 1 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ 1 ∏ j = 1 r βˆ’ 1 ΞΈ 1 + β‹― + ΞΈ j + j + 1 ΞΈ 1 + β‹― + ΞΈ j + j ∏ j = 1 k βˆ’ 1 ΞΈ 1 + β‹― + ΞΈ j + j + 1 ΞΈ 1 + β‹― + ΞΈ j + j Γ— ∏ i = 1 r βˆ’ k ( ΞΈ 1 + β‹― + ΞΈ r βˆ’ 1 ) + ( ΞΈ r + ΞΈ r βˆ’ 1 + β‹― + ΞΈ r + 1 βˆ’ i ) + r + i ( ΞΈ 1 + β‹― + ΞΈ r βˆ’ 1 ) + ( ΞΈ r + ΞΈ r βˆ’ 1 + β‹― + ΞΈ r + 1 βˆ’ i ) + r + i βˆ’ 1 .
(3.5)

2. The formula for ak + bk in terms of ΞΈ

Comparing the coefficient of x2 in (3.3), we obtain

( A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ) ( a k + b k + c k ) + 2 c k = ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) ( a k + 1 + b k + 1 + c k βˆ’ 1 ) .
(3.6)

Comparing the coefficient of x in (3.3), we obtain

( A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ) ( c k ( a k + b k ) + a k b k ) + c k ( a k + b k ) βˆ’ a k b k
= ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) ( c k βˆ’ 1 ( a k + 1 + b k + 1 ) + a k + 1 b k + 1 ) .
(3.7)

Solving (3.6) and (3.7) for ak + bk, one has

a k + b k = ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) ( a k + 1 b k + 1 βˆ’ c k βˆ’ 1 2 ) + ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 3 ) c k βˆ’ 1 c k βˆ’ A k ( 2 r βˆ’ k βˆ’ 1 ) a k b k ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) c k βˆ’ ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ) c k βˆ’ 1 .

This gives the explicit formulas

a k + b k = 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + 2 r + 1 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + 2 r ∏ j = 1 k βˆ’ 1 ΞΈ 1 + β‹― + ΞΈ j + j + 1 ΞΈ 1 + β‹― + ΞΈ j + j Γ— 1 + ∏ j = k r βˆ’ 1 ΞΈ 1 + β‹― + ΞΈ j + j + 1 ΞΈ 1 + β‹― + ΞΈ j + j Γ— ∏ j = k r βˆ’ 1 ΞΈ 1 + β‹― + ΞΈ j + 2 ΞΈ j + 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ k ΞΈ 1 + β‹― + ΞΈ j + 2 ΞΈ j + 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ k βˆ’ 1 .

These solutions indeed satisfy (3.3) for each k. This can be checked by a direct computation.

3. Final formulas

We use Lemma 3.5 to express ΞΈi by Ξ»j. Here are the final formulas.

If l < r, then

c j ( l ) = ∏ i = 1 j Ξ» i + β‹― + Ξ» l + l βˆ’ i Ξ» i + β‹― + Ξ» l + l βˆ’ i + 1 j = 1 , … , l .
(3.8)

We also borrow from Ref. 5 the l = r result.

c j ( r ) = ∏ i = 1 j Ξ» i + β‹― + Ξ» r βˆ’ 1 + Ξ» r / 2 + r βˆ’ i Ξ» i + β‹― + Ξ» r βˆ’ 1 + Ξ» r / 2 + r βˆ’ i + 1 j = 1 , … , r .
(3.9)

If l β©Ύ r + 1, then

c k ( l ) = ∏ j = 1 k Ξ» j + β‹― + Ξ» 2 r βˆ’ l + 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 1 Ξ» j + β‹― + Ξ» 2 r βˆ’ l + 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j ,
(3.10)

for k = 1, …, 2r βˆ’ l. Finally, for 2r βˆ’ l + 1 β©½ k β©½ r, we have

a k ( l ) b k ( l ) = ∏ j = 1 2 r βˆ’ l Ξ» j + β‹― + Ξ» 2 r βˆ’ l + 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 1 Ξ» j + β‹― + Ξ» 2 r βˆ’ l + 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j 2
Γ— ∏ j = 2 r βˆ’ l + 1 r βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 2 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 1
Γ— ∏ j = 2 r βˆ’ l + 1 k βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 2 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 1
Γ— ∏ i = 1 r βˆ’ k Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» r βˆ’ i + l βˆ’ r βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» r βˆ’ i + l βˆ’ r βˆ’ i
Γ— 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r βˆ’ 3 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r βˆ’ 1
(3.11)

and

a k ( l ) + b k ( l ) = 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r βˆ’ 3 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r βˆ’ 2
Γ— ∏ j = 1 2 r βˆ’ l Ξ» j + β‹― + Ξ» 2 r βˆ’ l + 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 1 Ξ» j + β‹― + Ξ» 2 r βˆ’ l + 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j
Γ— ∏ j = 2 r βˆ’ l + 1 k βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 2 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 1
+ ∏ j = 2 r βˆ’ l + 1 r βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 2 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 1
Γ— ∏ i = 1 r βˆ’ k Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» r βˆ’ i + l βˆ’ r βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» r βˆ’ i + l βˆ’ r βˆ’ i .
(3.12)

4. The solutions are generic

In this section we show the solutions are generic.

Theorem 3.9.

Supposeβ€ˆ Ξ» ∈ P + β€ˆandβ€ˆlβ€ˆis admissible, thenβ€ˆy(l)β€ˆin Section III E 3 represents a critical point associated toβ€ˆΞ› = (Ξ», Ο‰1),β€ˆz = (0, 1), andβ€ˆl.

Proof.

It is sufficient to show that y is generic with respect to Ξ›, z.

Let us first consider G2. For Ξ» ∈ P + ⁠, G2 is equivalent to y 1 ( l ) ( 1 ) β‰  0 and y i ( l ) ( 0 ) β‰  0 if Ξ»iβ‰ 0.

If l β©½ r βˆ’ 1, then the admissibility of l implies Ξ»l > 0. To prove G2, it suffices to show c j ( l ) β‰  0 if Ξ»lβ‰ 0 and c 1 ( l ) β‰  1 ⁠, see (3.8). Note that if Ξ»l > 0, then
0 < Ξ» i + β‹― + Ξ» l + l βˆ’ i Ξ» i + β‹― + Ξ» l + l βˆ’ i + 1 < 1
for all i ∈ {1, …, l}, therefore all c j ( l ) ∈ ( 0 , 1 ) ⁠.

If l = r, this is similar to the previous situation.

If l = r + 1, the admissibility of l implies Ξ»r β©Ύ 2. G2 is obviously true.

If l β©Ύ r + 2, the admissibility of l implies Ξ»2rβˆ’l+1 > 0. One has y k ( l ) ( 0 ) β‰  0 since we have a k ( l ) b k ( l ) β‰  0 ⁠. As for y 1 ( l ) ( 1 ) β‰  0 in the case l = 2r, we delay the proof until after the case G1.

Now, we consider G1. Suppose a k ( l ) = b k ( l ) for some 2r βˆ’ l + 1 β©½ k β©½ r. Observe that
W ( y k ( l ) , y Μƒ k ( l ) ) = T k ( l ) y k βˆ’ 1 ( l ) y k + 1 ( l ) .
By G2, y k ( l ) and T k ( l ) have no common roots. In addition if l = 2r and k = 1, we have y 1 ( l ) ( 1 ) = 0 ⁠, then a 1 ( l ) b 1 ( l ) = 1 ⁠, while as above we have a 1 ( l ) b 1 ( l ) ∈ ( 0 , 1 ) ⁠. It follows that we must have a k ( l ) = a k + 1 ( l ) or a k ( l ) = a k βˆ’ 1 ( l ) (⁠ a k ( l ) = c k βˆ’ 1 ( l ) ⁠, if k = 2r βˆ’ l + 1).
We work in terms of ΞΈ. We have ak = bk = ak+1 or ak = bk = akβˆ’1 or a2rβˆ’l+1 = c2rβˆ’l. If ak = bk = ak+1, then substituting x = ck into (3.3), we get
βˆ’ c k ( c k βˆ’ a k + 1 ) = ( c k βˆ’ b k + 1 ) ( c k βˆ’ c k βˆ’ 1 ) ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) .
(3.13)
Solving (3.4) and (3.13) for ak+1 = ak = bk and bk+1 in terms of ck, ckβˆ’1, and A k ( 2 r βˆ’ k βˆ’ 1 ) ⁠, we obtain
a k + 1 b k + 1 = ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ) c k βˆ’ 1 c k ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 3 ) c k βˆ’ ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) c k βˆ’ 1 ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ) c k βˆ’ A k ( 2 r βˆ’ k βˆ’ 1 ) c k βˆ’ 1 2 .
Comparing it with (3.5) and canceling common factors, we obtain
( A k ( 2 r βˆ’ k βˆ’ 1 ) + 2 ) ( A k ( 2 r βˆ’ k βˆ’ 1 ) + 1 ) 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r + 1 2 ΞΈ 1 + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ 1 = ∏ i = k r ΞΈ 1 + β‹― + ΞΈ i + i + 1 ΞΈ 1 + β‹― + ΞΈ i + i ∏ i = k + 2 r βˆ’ 1 ΞΈ 1 + β‹― + ΞΈ i βˆ’ 1 + 2 ΞΈ i + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ i + 1 ΞΈ 1 + β‹― + ΞΈ i βˆ’ 1 + 2 ΞΈ i + β‹― + 2 ΞΈ r βˆ’ 1 + ΞΈ r + 2 r βˆ’ i .
Substituting ΞΈi in terms of Ξ»j, we have
( Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» k + k + l βˆ’ 2 r βˆ’ 1 ) ( Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» k + k + l βˆ’ 2 r )
β€ˆ
Γ— 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r βˆ’ 3 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r βˆ’ 1
β€ˆ
= ∏ j = k r βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 2 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» j + 2 Ξ» j + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ j βˆ’ 1
β€ˆ
Γ— ∏ i = 1 r βˆ’ k βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» r βˆ’ i + l βˆ’ r βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» r βˆ’ i + l βˆ’ r βˆ’ i .
(3.14)
By our assumption, we have Ξ»2rβˆ’l+1 β©Ύ 1, k β©Ύ 2r βˆ’ l + 1, and l β©Ύ r + 2. It is easily seen that
( Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» k + k + l βˆ’ 2 r βˆ’ 1 ) ( Ξ» 2 r βˆ’ l + 1 + β‹― + Ξ» k + k + l βˆ’ 2 r ) Γ— 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r βˆ’ 3 2 Ξ» 2 r βˆ’ l + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r βˆ’ 1 β©Ύ 1 Γ— 2 Γ— 3 5 > 1 .
Therefore (3.14) is impossible. Similarly, we can exclude a k ( l ) = a k βˆ’ 1 ( l ) ⁠. As for a 2 r βˆ’ l + 1 ( l ) = c 2 r βˆ’ l ( l ) ⁠, by (3.4), it is impossible since each fractional factor is strictly less than 1.

Finally, we prove G3. The nontrivial cases are a k ( l ) = a k + 1 ( l ) and a 2 r βˆ’ l + 1 ( l ) = c 2 r βˆ’ l ( l ) for l β©Ύ r + 1, where k β©Ύ 2r βˆ’ l + 1.

If ak = ak+1, then by (3.3) we have that x βˆ’ ak divides x(x βˆ’ ck)(x βˆ’ bk). As we already proved akβ‰ bk and akβ‰ 0, it follows that ak = ck. This again implies that (x βˆ’ ak)2 divides (x βˆ’ ak+1)(x βˆ’ bk+1)(x βˆ’ ckβˆ’1) as A k ( 2 r βˆ’ k + 1 ) + 2 β‰  0 ⁠. If bk+1 = ak = ak+1, then we are done. If ak = ckβˆ’1, then ckβˆ’1 = ck. It is impossible by the argument used in G2.

If ak = ckβˆ’1, then by (3.3) one has x βˆ’ ak divides x(x βˆ’ ck)(x βˆ’ bk). Since l is admissible, a k ( l ) β‰  0 ⁠. Then akβ‰ bk implies ckβˆ’1 = ck. It is also a contradiction.

In particular, this shows that y 1 ( l ) and y 0 ( l ) have no common roots, i.e., y 1 ( l ) ( 1 ) β‰  0 ⁠. β–‘

Corollary 3.10.

Supposeβ€ˆ Ξ» ∈ P + . Then the Bethe ansatz equation (2.2) associated toβ€ˆΞ›, z, l, whereβ€ˆlβ€ˆis admissible, has exactly one solution. Explicitly, forβ€ˆl β©½ r βˆ’ 1, the correspondingβ€ˆr-tupleβ€ˆy(l)β€ˆwhich represents the solution is described by (3.8), forβ€ˆl = rβ€ˆby (3.9), for 2r β©Ύ l β©Ύ r + 1 by (3.10)–(3.12).β–‘

Let y be an r-tuple of quasi-polynomials. Following Ref. 6, we introduce a linear differential operator D(y) of order 2r by the formula

D Ξ» ( y ) = βˆ‚ βˆ’ ln β€² T 1 2 … T r βˆ’ 1 2 T r y 1 βˆ‚ βˆ’ ln β€² y 1 T 1 2 … T r βˆ’ 1 2 T r y 2 T 1 Γ— βˆ‚ βˆ’ ln β€² y 2 T 1 2 … T r βˆ’ 1 2 T r y 3 T 1 T 2 … βˆ‚ βˆ’ ln β€² y r βˆ’ 1 T 1 … T r βˆ’ 1 T r y r Γ— βˆ‚ βˆ’ ln β€² y r T 1 … T r βˆ’ 1 y r βˆ’ 1 βˆ‚ βˆ’ ln β€² y r βˆ’ 1 T 1 … T r βˆ’ 2 y r βˆ’ 2 … Γ— ( βˆ‚ βˆ’ ln β€² ( y 1 ) ) ,

where Ti, i = 1, …, r, are given by (2.4).

If y is an r-tuple of polynomials representing a critical point associated to integral dominant weights Ξ›1, …, Ξ›n and points z1, …, zn of type Br, then by Ref. 6, the kernel of DΞ»(y) is a self-dual space of polynomials. By Ref. 2 the coefficients of DΞ»(y) are eigenvalues of higher Gaudin Hamiltonians acting on the Bethe vector related to y.

For admissible l and Ξ» ∈ π”₯βˆ—, define a Ξ» l ( 1 ) , … , a Ξ» l ( r ) as the following.

For l = 0, …, r βˆ’ 1, i = 1, …, l, set a Ξ» l ( i ) = Ξ» i + β‹― + Ξ» l + l + 1 βˆ’ i ⁠. For l = 0, …, r βˆ’ 1, i = l + 1, …, r, set a Ξ» l ( i ) = 0 ⁠.

For l = r + 1, …, 2r, set k = 2r βˆ’ l. Then for i = 1, …, k, set

a Ξ» l ( i ) = Ξ» i + β‹― + Ξ» k + 2 Ξ» k + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 r βˆ’ k βˆ’ i

and for i = k + 1, …, r, set a Ξ» l ( i ) = 2 Ξ» k + 1 + β‹― + 2 Ξ» r βˆ’ 1 + Ξ» r + 2 r βˆ’ 2 k βˆ’ 1 ⁠.

Proposition 3.11.

Let the r-tupleβ€ˆyβ€ˆrepresent the solution of the Bethe ansatz equation (2.2) associated toβ€ˆΞ›,β€ˆz, and admissibleβ€ˆl, whereβ€ˆ Ξ» ∈ P + β€ˆandβ€ˆlβ‰ r. Thenβ€ˆ D Ξ» ( y ) = D Ξ» ( x a Ξ» l ( 1 ) , … , x a Ξ» l ( r ) ) .

Proof.

The (2r βˆ’ 1)-tuple (y1, ..., yrβˆ’1, yr, yrβˆ’1, ..., y1) represents a critical point of type A2rβˆ’1. Then the reproduction procedure in direction i of type Br corresponds to a composition of reproduction procedures of type A2rβˆ’1 in directions i and 2r βˆ’ i for i = 1, …, r βˆ’ 1, and to reproduction procedure of type A2rβˆ’1 in direction r for i = r, see Refs. 6 and 9. Proposition follows from Lemma 4.2 in Ref. 9. β–‘

In this section we continue to study the case of 𝔀 = 𝔰𝔬(2r + 1). The main result of the section is Theorem 4.5.

Let Ξ» ∈ P + ⁠. Consider the tensor product of a finite-dimensional irreducible module with highest weight Ξ», VΞ», and the vector representation VΟ‰1.

Recall that the value of the weight function Ο‰(z1, z2, t) at a solution of the Bethe ansatz equations (2.2) is called the Bethe vector. We have the following result, which is usually referred to as completeness of the Bethe ansatz.

Theorem 4.1.

The set of Bethe vectorsβ€ˆΟ‰(z1, z2, t), whereβ€ˆtβ€ˆruns over the solutions to the Bethe ansatz equations (2.2) with admissible lengthβ€ˆl, forms a basis of Sing (VΞ» βŠ— VΟ‰1).

Proof.

All multiplicities in the decomposition of VΞ» βŠ— VΟ‰1 are 1. By Corollary 3.10 for each admissible length l we have a solution of the Bethe ansatz equation. The theorem follows from Theorems 2.4 and 2.5.β–‘

We have the following standard fact.

Lemma 4.2.

Letβ€ˆ ΞΌ , Ξ½ ∈ P + . Ifβ€ˆΞΌ > Ξ½β€ˆthen (ΞΌ + ρ, ΞΌ + ρ) > (Ξ½ + ρ, Ξ½ + ρ).

Proof.

The lemma follows from the proof of Lemma 13.2B in Ref. 1.β–‘

Proposition 4.3.

Letβ€ˆΟ‰, Ο‰β€² ∈ VΞ» βŠ— VΟ‰1β€ˆbe Bethe vectors corresponding to solutions to the Bethe ansatz equations of two different lengths. Thenβ€ˆΟ‰, Ο‰β€² are eigenvectors of the Gaudin Hamiltonianβ€ˆ H ≔ H 1 = βˆ’ H 2 β€ˆwith distinct eigenvalues.

Proof.
Recall the relation
Ξ© ( 1 , 2 ) = 1 2 Ξ” Ξ© 0 βˆ’ 1 βŠ— Ξ© 0 βˆ’ Ξ© 0 βŠ— 1 .
Since Ξ©0 acts as a constant in any irreducible module, 1 βŠ— Ξ©0 + Ξ©0 βŠ— 1 acts as a constant on VΞ» βŠ— VΟ‰1. It remains to consider the spectrum of the diagonal action of ΔΩ0. By Theorem 2.5, Ο‰ and Ο‰β€² are highest weight vectors of two non-isomorphic irreducible submodules of VΞ» βŠ— VΟ‰1. By Lemmas 2.1 and 4.2 the values of ΔΩ0 on Ο‰ and Ο‰β€² are different.β–‘

We use the following well-known lemma from algebraic geometry.

Lemma 4.4.
Letβ€ˆn ∈ β„€β©Ύ1β€ˆand supposeβ€ˆ f k ( Ο΅ ) ( x 1 , … , x l ) = 0 ,β€ˆk = 1, …, n, is a system ofβ€ˆnβ€ˆalgebraic equations forβ€ˆlβ€ˆcomplex variablesβ€ˆx1, …, xl, depending on a complex parameterβ€ˆΟ΅β€ˆalgebraically. Letβ€ˆ ( x 1 ( 0 ) , … , x l ( 0 ) ) β€ˆbe an isolated solution withβ€ˆΟ΅ = 0. Then for sufficiently smallβ€ˆΟ΅, there exists an isolated solutionβ€ˆ ( x 1 ( Ο΅ ) , … , x l ( Ο΅ ) ) , depending algebraically onβ€ˆΟ΅, such thatβ€ˆ
x k ( Ο΅ ) = x k ( 0 ) + o ( 1 ) .
β–‘
Our main result is the following theorem.

Theorem 4.5.

Let 𝔀 = 𝔰𝔬(2r + 1),β€ˆ Ξ» ∈ P + , andβ€ˆN ∈ β„€β©Ύ0. For a generic (N + 1)-tuple of distinct complex numbersβ€ˆz = (z0, z1, …, zN), the Gaudin Hamiltoniansβ€ˆ ( H 0 , H 1 , … , H N ) β€ˆacting inβ€ˆ Sing V Ξ» βŠ— V Ο‰ 1 βŠ— N β€ˆare diagonalizable and have a simple joint spectrum. Moreover, for genericβ€ˆzβ€ˆthere exists a set of solutions {ti, i ∈ I} of the Bethe ansatz equation (2.2) such that the corresponding Bethe vectors {Ο‰(z, ti), i ∈ I} form a basis ofβ€ˆ Sing V Ξ» βŠ— V Ο‰ 1 βŠ— N .

Proof.

Our proof follows that of Theorem 5.2 of Ref. 10, see also of Section 4 in Ref. 7.

Pick distinct non-zero complex numbers z Μƒ 1 , … , z Μƒ N ⁠. We use Theorem 4.1 to define a basis in the space of singular vectors Sing ( V Ξ» βŠ— V Ο‰ 1 βŠ— N ) as follows.

We call a (k + 1)-tuple of weights ΞΌ 0 , ΞΌ 1 , … , ΞΌ k ∈ P + β€ˆadmissible if ΞΌ0 = Ξ» and for i = 1, …, k, we have a submodule VΞΌi βŠ‚ VΞΌiβˆ’1 βŠ— VΟ‰1, see (3.1).

For an admissible tuple of weights, we define a singular vector v ΞΌ 0 , … , ΞΌ k ∈ V Ξ» βŠ— V Ο‰ 1 βŠ— k of weight ΞΌk using induction on k as follows. Let vΞΌ0 = vΞ» be the highest weight vector for module VΞ». Let k be such that 1 β©½ k β©½ N. Suppose we have the singular vector v ΞΌ 0 , … , ΞΌ k βˆ’ 1 ∈ V Ξ» βŠ— V Ο‰ 1 βŠ— k βˆ’ 1 ⁠. It generates a submodule V ΞΌ 0 , … , ΞΌ k βˆ’ 1 βŠ‚ V Ξ» βŠ— V Ο‰ 1 βŠ— k βˆ’ 1 of highest weight ΞΌkβˆ’1.

Let t Μ„ k = ( t Μ„ k , j ( b ) ) ⁠, where b = 1, …, r and j = 1, …, lk,b, be the solution of the Bethe ansatz equation associated to VΞΌkβˆ’1 βŠ— VΟ‰1, z = ( 0 , z Μƒ k ) and lk = (lk,1, …, lk,r) such that ΞΌkβˆ’1 + Ο‰1 βˆ’ Ξ±(lk) = ΞΌk. Note that t Μ„ k depends on ΞΌkβˆ’1 and ΞΌk, even though we do not indicate this dependence explicitly. Note also that in all cases lk,b ∈ {0, 1, 2}.

Then, define vΞΌ0,…,ΞΌk to be the Bethe vector
v ΞΌ 0 , … , ΞΌ k = Ο‰ ( 0 , z Μƒ k , t Μ„ k ) ∈ V ΞΌ 0 , … , ΞΌ k βˆ’ 1 βŠ— V Ο‰ 1 βŠ‚ V Ξ» βŠ— V Ο‰ 1 βŠ— k .
We denote by VΞΌ0,…,ΞΌk the submodule of V Ξ» βŠ— V Ο‰ 1 βŠ— k generated by vΞΌ0,…,ΞΌk.

The vectors v ΞΌ 0 , … , ΞΌ N ∈ V Ξ» βŠ— V Ο‰ 1 βŠ— N are called the iterated singular vectors. To each iterated singular vector vΞΌ0,…,ΞΌN we have an associated collection t Μ„ = ( t Μ„ 1 , … , t Μ„ N ) consisting of all the Bethe roots used in its construction.

Clearly, the iterated singular vectors corresponding to all admissible (N + 1)-tuples of weights form a basis in Sing ( V Ξ» βŠ— V Ο‰ 1 βŠ— N ) ⁠, so we have
V Ξ» βŠ— V Ο‰ 1 βŠ— N = ⨁ ΞΌ 0 , … , ΞΌ N V ΞΌ 0 , ΞΌ 1 , … , ΞΌ N ,
where the sum is over all admissible (N + 1)-tuples of weights.

To prove the theorem, we show that in some region of parameters z for any admissible (N + 1)-tuple of weights ΞΌ0, …, ΞΌN, there exists a Bethe vector ωμ1,…,ΞΌN which tends to vΞΌ1,…,ΞΌN when approaching a certain point (independent on ΞΌi) on the boundary of the region.

To construct the Bethe vector ωμ1,…,ΞΌN associated to vΞΌ1,…,ΞΌN, we need to find a solution to the Bethe equations associated to V Ξ» βŠ— V Ο‰ 1 βŠ— N with Bethe roots, t = ( t j ( b ) ) ⁠, where b = 1, …, r and j = 1 , … , βˆ‘ k = 1 N l k , b ⁠.

We do it for z of the form
z 0 = z and z k = z + Ξ΅ N + 1 βˆ’ k z Μƒ k k = 1 , … , N
(4.1)
for sufficiently small Ξ΅ ∈ β„‚Γ—. Here z ∈ β„‚ is an arbitrary fixed number and z Μƒ k are as above.
Then, similarly to t Μ„ we write t = (t1, …, tN) where t k = ( t k , j ( b ) ) ⁠, b = 1, …, r, and j = 1, …, lk,b is constructed in the form
t k , j ( b ) = z + Ξ΅ N + 1 βˆ’ k t Μƒ k , j ( b ) , k = 1 , … , N , j = 1 , … , l k , b , b = 1 , … , r .
(4.2)
The variables t k , j ( b ) satisfy the system of Bethe ansatz equations
βˆ’ ( Ξ» , Ξ± b ) t k , j ( b ) βˆ’ z 0 + βˆ‘ s = 1 N βˆ’ 2 Ξ΄ b , 1 t k , j ( b ) βˆ’ z s + βˆ‘ q = 1 ( s , q ) β‰  ( k , j ) l s , b ( Ξ± b , Ξ± b ) t k , j ( b ) βˆ’ t s , q ( b ) + βˆ‘ q = 1 l s , b + 1 ( Ξ± b , Ξ± b + 1 ) t k , j ( b ) βˆ’ t s , q ( b + 1 ) + βˆ‘ q = 1 l s , b βˆ’ 1 ( Ξ± b , Ξ± b βˆ’ 1 ) t k , j ( b ) βˆ’ t s , q ( b βˆ’ 1 ) = 0
(4.3)
for b = 1, …, r, k = 1, …, N, j = 1, …, lk,b. Here we agree that ls,0 = ls,N+1 = 0 for all s.

Consider the leading asymptotic behavior of the Bethe ansatz equations as Ξ΅ β†’ 0. We claim that in the leading order, the Bethe ansatz equations for t reduce to the Bethe ansatz equations obeyed by the variables t Μ„ ⁠.

Consider, for example, the leading order of the Bethe equation for t k , j ( 1 ) ⁠. Note that
( Ξ» , Ξ± 1 ) t k , j ( 1 ) βˆ’ z 0 + βˆ‘ s = 1 N 2 t k , j ( 1 ) βˆ’ z s = ( Ξ» , Ξ± 1 ) t Μƒ k , j ( 1 ) + 2 ( k βˆ’ 1 ) t Μƒ k , j ( 1 ) + 2 t Μƒ k , j ( 1 ) βˆ’ z Μƒ k + O ( Ξ΅ ) Ξ΅ βˆ’ N βˆ’ 1 + k ,
β€ˆ
βˆ‘ s = 1 N βˆ‘ q = 1 ( s , q ) β‰  ( k , j ) l s , 1 ( Ξ± 1 , Ξ± 1 ) t k , j ( 1 ) βˆ’ t s , q ( 1 ) = βˆ‘ q = 1 q β‰  j l k , 1 ( Ξ± 1 , Ξ± 1 ) t Μƒ k , j ( 1 ) βˆ’ t Μƒ k , q ( 1 ) + βˆ‘ s = 1 k βˆ’ 1 βˆ‘ q = 1 l s , 1 ( Ξ± 1 , Ξ± 1 ) t Μƒ k , j ( 1 ) + O ( Ξ΅ ) Ξ΅ βˆ’ N βˆ’ 1 + k ,
and similarly
βˆ‘ s = 1 N βˆ‘ q = 1 l s , 2 ( Ξ± 1 , Ξ± 2 ) t k , j ( 1 ) βˆ’ t s , q ( 2 ) = βˆ‘ q = 1 l k , 2 ( Ξ± 1 , Ξ± 2 ) t Μƒ k , j ( 1 ) βˆ’ t Μƒ k , q ( 2 ) + βˆ‘ s = 1 k βˆ’ 1 βˆ‘ q = 1 l s , 2 ( Ξ± 1 , Ξ± 2 ) t Μƒ k , j ( 1 ) + O ( Ξ΅ ) Ξ΅ βˆ’ N βˆ’ 1 + k .
Then by definition of the numbers ls,b, we have
ΞΌ k βˆ’ 1 = Ξ» + ( k βˆ’ 1 ) Ο‰ 1 βˆ’ βˆ‘ b = 1 r βˆ‘ s = 1 k βˆ’ 1 βˆ‘ q = 1 l s , b Ξ± b
and, in particular,
( ΞΌ k βˆ’ 1 , Ξ± 1 ) = ( Ξ» , Ξ± 1 ) + 2 ( k βˆ’ 1 ) βˆ’ βˆ‘ s = 1 k βˆ’ 1 βˆ‘ q = 1 l s , 1 ( Ξ± 1 , Ξ± 1 ) βˆ’ βˆ‘ q = 1 l s , 2 ( Ξ± 1 , Ξ± 2 ) .
Therefore
βˆ’ ( ΞΌ k βˆ’ 1 , Ξ± 1 ) t Μƒ k , j ( 1 ) βˆ’ 2 t Μƒ k , j ( 1 ) βˆ’ z Μƒ k + βˆ‘ q = 1 q β‰  j l k , 1 ( Ξ± 1 , Ξ± 1 ) t Μƒ k , j ( 1 ) βˆ’ t Μƒ k , q ( 1 ) + βˆ‘ q = 1 l k , 2 ( Ξ± 1 , Ξ± 2 ) t Μƒ k , j ( 1 ) βˆ’ t Μƒ k , q ( 2 ) = O ( Ξ΅ ) .
At leading order this is indeed the Bethe equation for t Μ„ k , j ( 1 ) from the set of Bethe equations for the tensor product VΞΌkβˆ’1 βŠ— VΟ‰1, with the tensor factors assigned to the points 0 and z Μƒ k ⁠, respectively. The other equations work similarly.

By Lemma 4.4 it follows that for sufficiently small Ξ΅ there exists a solution to the Bethe equations (4.3) of the form t Μƒ j , k ( a ) = t Μ„ j , k ( a ) + o ( 1 ) ⁠.

Now we claim that the Bethe vector ωμ1,…,ΞΌN = Ο‰(z, t) associated to t has leading asymptotic behavior
Ο‰ ΞΌ 1 , … , ΞΌ N = Ξ΅ K ( v ΞΌ 1 , … , ΞΌ N + o ( 1 ) ) ,
(4.4)
as Ξ΅ β†’ 0, for some K. Consider the definition (2.3) of Ο‰(z, t). We write ωμ1,…,ΞΌN = w1 + w2 where w1 contains only those summands in which every factor in the denominator is of the form
t k , j ( a ) βˆ’ t k , q ( b ) or t k , j ( a ) βˆ’ z k .
The term w2 contains terms where at least one factor is of the form t k , j ( a ) βˆ’ t s , q ( b ) or t k , j ( a ) βˆ’ z s ⁠, sβ‰ k. After substitution using (4.1) and (4.2), one finds that
w 1 = ∏ k = 1 N ∏ j = 1 r Ξ΅ βˆ’ N βˆ’ 1 + k l k , j v ΞΌ 1 , … , ΞΌ N ,
and that w2 is subleading to w1, which establishes our claim.
Consider two distinct Bethe vectors ωμ1,…,ΞΌN and Ο‰ ΞΌ 1 β€² , … , ΞΌ N β€² constructed as above. By Theorem 2.5 both are simultaneous eigenvectors of the quadratic Gaudin Hamiltonians H 0 , H 1 , … , H N ⁠. Let k be the largest possible number in {1, …, N} such that ΞΌ i = ΞΌ i β€² for all i = 1, …, k βˆ’ 1. Consider the Hamiltonian H k ⁠. When the zi are chosen as in (4.1) then one finds
H k = Ξ΅ βˆ’ N βˆ’ 1 + k βˆ‘ j = 0 k βˆ’ 1 Ξ© ( k , j ) z Μƒ k + o ( 1 ) .
(4.5)
The sum βˆ‘ j = 0 k βˆ’ 1 Ξ© ( k , j ) z Μƒ k coincides with the action of the quadratic Gaudin Hamiltonian H of the spin chain VΞΌkβˆ’1 βŠ— VΟ‰1 with sites at 0 and z Μƒ k ⁠, embedded in VΞ» βŠ— (VΟ‰1)βŠ—k via
V ΞΌ k βˆ’ 1 βŠ— V Ο‰ 1 ≃ V ΞΌ 1 , … , ΞΌ k βˆ’ 1 βŠ— V Ο‰ 1 βŠ‚ V Ξ» βŠ— ( V Ο‰ 1 ) βŠ— k .
Since ΞΌ k β‰  ΞΌ k β€² ⁠, vΞΌ1,…,ΞΌk and v ΞΌ 1 β€² , … , ΞΌ k β€² are eigenvectors of βˆ‘ j = 0 k βˆ’ 1 Ξ© ( k , j ) z Μƒ k with distinct eigenvalues by Proposition 4.3. By (4.4) and (4.5), we have that the eigenvalues of H k on ωμ1,…,ΞΌN and Ο‰ ΞΌ 1 β€² , … , ΞΌ N β€² are distinct.

The argument above establishes that the set of points z = (z0, z1, …, zN) for which the Gaudin Hamiltonians are diagonalizable with joint simple spectrum is non-empty. It is a Zariski-open set, therefore the theorem follows. β–‘

Let 𝔀 = 𝔰𝔭(2r) be the simple Lie algebra of type Cr, r β©Ύ 3. We have (Ξ±i, Ξ±i) = 2, i = 1, …, r βˆ’ 1 and (Ξ±r, Ξ±r) = 4. We work with data Ξ› = (Ξ», Ο‰1), z = (0, 1), where Ξ» ∈ P + ⁠.

We have

V Ξ» βŠ— V Ο‰ 1 = V Ξ» + Ο‰ 1 βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βŠ• β‹― βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βˆ’ β‹― βˆ’ Ξ± r βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βˆ’ β‹― βˆ’ Ξ± r βˆ’ 2 βˆ’ 2 Ξ± r βˆ’ 1 βˆ’ Ξ± r βŠ• β‹― βŠ• V Ξ» + Ο‰ 1 βˆ’ 2 Ξ± 1 βˆ’ β‹― βˆ’ 2 Ξ± r βˆ’ 1 βˆ’ Ξ± r = V ( Ξ» 1 + 1 , Ξ» 2 , … , Ξ» r ) βŠ• V ( Ξ» 1 βˆ’ 1 , Ξ» 2 + 1 , Ξ» 3 , … , Ξ» r ) βŠ• V ( Ξ» 1 , … , Ξ» k βˆ’ 1 , Ξ» k βˆ’ 1 , Ξ» k + 1 + 1 , … , Ξ» r ) βŠ• β‹― βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 1 βˆ’ 1 , Ξ» r + 1 ) βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 2 , Ξ» r βˆ’ 1 + 1 , Ξ» r βˆ’ 1 ) βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 3 , Ξ» r βˆ’ 2 + 1 , Ξ» r βˆ’ 1 βˆ’ 1 , Ξ» r ) βŠ• β‹― βŠ• V ( Ξ» 1 + 1 , Ξ» 2 βˆ’ 1 , Ξ» 3 , … , Ξ» r ) βŠ• V ( Ξ» 1 βˆ’ 1 , Ξ» 2 , … , Ξ» r ) ,
(5.1)

with the convention that the summands with non-dominant highest weights are omitted. Note, in particular, all multiplicities are 1.

We call an r-tuple of integers l = (l1, …, lr) admissible if the VΞ»+Ο‰1βˆ’Ξ±(l) appears in (5.1).

The admissible r-tuples l have the form

( 1 , … , 1 οΈΈ k 1 ones , 0 , … , 0 ) or ( 1 , … , 1 οΈΈ k 2 ones , 2 , … , 2 , 1 ) ,
(5.2)

where k1 = 0, 1, …, r and k2 = 0, 1, …, r βˆ’ 2. In the first case the length l = l1 + β‹― + lr is k1 and in the second case 2r βˆ’ k2 βˆ’ 1. It follows that different admissible r-tuples have different lengths and, therefore, admissible tuples l are parametrized by length l ∈ {0, 1, …, 2r βˆ’ 1}. We call a non-negative integer ladmissible if it is the length of an admissible r-tuple l. More precisely, a non-negative integer l is admissible if l = 0 or if l β©½ r, Ξ»l > 0 or if r < l β©½ 2r βˆ’ 1, Ξ»2rβˆ’l > 0.

Similarly to the case of type Br, see Theorem 3.9 and Corollary 3.10, we obtain the solutions to the Bethe ansatz equations for VΞ» βŠ— VΟ‰1.

Theorem 5.1.

Let 𝔀 = 𝔰𝔭(2r). Letβ€ˆlβ€ˆbe as in (5.2). Ifβ€ˆlβ€ˆis not admissible then the Bethe ansatz equation (2.2) associated toβ€ˆΞ›, z, lβ€ˆhas no solutions. Ifβ€ˆlβ€ˆis admissible then the Bethe ansatz equation (2.2) associated toβ€ˆΞ›, z, lβ€ˆhas exactly one solution represented by the followingβ€ˆr-tuple of polynomialsβ€ˆy(l).

Forβ€ˆl = 0, 1, …, r βˆ’ 1, we haveβ€ˆ y ( l ) = ( x βˆ’ c 1 ( l ) , … , x βˆ’ c l ( l ) , 1 , … , 1 ) , whereβ€ˆ c j ( l ) β€ˆare given by (3.8).

Forβ€ˆl = r, we haveβ€ˆ y ( l ) = ( x βˆ’ c 1 ( r ) , … , x βˆ’ c r ( r ) ) , whereβ€ˆ
c j ( r ) = ∏ i = 1 j Ξ» i + β‹― + Ξ» r βˆ’ 1 + 2 Ξ» r + r + 1 βˆ’ i Ξ» i + β‹― + Ξ» r βˆ’ 1 + 2 Ξ» r + r + 2 βˆ’ i j = 1 , … , r βˆ’ 1 ,
β€ˆ
c r ( r ) = Ξ» r Ξ» r + 1 ∏ i = 1 r βˆ’ 1 Ξ» i + β‹― + Ξ» r βˆ’ 1 + 2 Ξ» r + r + 1 βˆ’ i Ξ» i + β‹― + Ξ» r βˆ’ 1 + 2 Ξ» r + r + 2 βˆ’ i .
Forβ€ˆl = r + 1, …, 2r βˆ’ 1, we haveβ€ˆ y ( l ) = ( x βˆ’ c 1 ( l ) , … , x βˆ’ c 2 r βˆ’ l βˆ’ 1 ( l ) , ( x βˆ’ a 2 r βˆ’ l ( l ) ) ( x βˆ’ b 2 r βˆ’ l ( l ) ) , … , ( x βˆ’ a r βˆ’ 1 ( l ) ) ( x βˆ’ b r βˆ’ 1 ( l ) ) , x βˆ’ c r ( l ) ) , whereβ€ˆ
c j ( l ) = ∏ i = 1 j Ξ» i + β‹― + Ξ» 2 r βˆ’ 1 βˆ’ l + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + l + 1 βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 1 βˆ’ l + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + l + 2 βˆ’ i , j = 1 , … , 2 r βˆ’ l βˆ’ 1 ,
β€ˆ
c r ( l ) = ∏ i = 1 2 r βˆ’ l βˆ’ 1 Ξ» i + β‹― + Ξ» 2 r βˆ’ 1 βˆ’ l + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + l + 1 βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 1 βˆ’ l + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + l + 2 βˆ’ i Γ— ∏ i = 2 r βˆ’ l r Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i + 1 , a k ( l ) b k ( l ) = ∏ i = 1 2 r βˆ’ l βˆ’ 1 Ξ» i + β‹― + Ξ» 2 r βˆ’ 1 βˆ’ l + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + l + 1 βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 1 βˆ’ l + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + l + 2 βˆ’ i 2 Γ— ∏ i = 2 r βˆ’ l k βˆ’ 1 Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i + 1 Γ— ∏ i = 2 r βˆ’ l r Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i + 1 Γ— ∏ i = 1 r + 1 βˆ’ k Ξ» 2 r βˆ’ l + β‹― + Ξ» r + 1 βˆ’ i + l + 1 βˆ’ i βˆ’ r Ξ» 2 r βˆ’ l + β‹― + Ξ» r + 1 βˆ’ i + l + 2 βˆ’ i βˆ’ r
andβ€ˆ
a k ( l ) + b k ( l ) = ∏ i = 1 2 r βˆ’ l βˆ’ 1 Ξ» i + β‹― + Ξ» 2 r βˆ’ 1 βˆ’ l + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + l + 1 βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 1 βˆ’ l + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + l + 2 βˆ’ i Γ— ∏ i = 2 r βˆ’ l k βˆ’ 1 Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i + 1 Γ— 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + 2 l βˆ’ 2 r 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + 2 l + 1 βˆ’ 2 r + 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + 2 l + 2 βˆ’ 2 r 2 Ξ» 2 r βˆ’ l + β‹― + 2 Ξ» r + 2 l + 1 βˆ’ 2 r Γ— ∏ i = k r Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i Ξ» 2 r βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r + l βˆ’ i + 1 Γ— ∏ i = k r Ξ» 2 r βˆ’ l + β‹― + Ξ» i + l + i βˆ’ 2 r Ξ» 2 r βˆ’ l + β‹― + Ξ» i + l + i + 1 βˆ’ 2 r ,
forβ€ˆk = 2r βˆ’ l, …, r βˆ’ 1. β–‘
Therefore, in parallel to Theorem 4.5, we have the completeness of Bethe ansatz.

Theorem 5.2.

Let 𝔀 = 𝔰𝔭(2r) andβ€ˆ Ξ» ∈ P + . For a generic (N + 1)-tuple of distinct complex numbersβ€ˆz = (z0, z1, ..., zN), the Gaudin Hamiltoniansβ€ˆ ( H 0 , H 1 , . . . , H N ) β€ˆacting inβ€ˆ Sing V Ξ» βŠ— V Ο‰ 1 βŠ— N β€ˆare diagonalizable and have a simple joint spectrum. Moreover, for genericβ€ˆzβ€ˆthere exists a set of solutions {ti, i ∈ I} of the Bethe ansatz equation (2.2) such that the corresponding Bethe vectors {Ο‰(z, ti), i ∈ I} form a basis ofβ€ˆ Sing V Ξ» βŠ— V Ο‰ 1 βŠ— N .β–‘

Similarly to Section III F, following Ref. 6, we introduce a linear differential operator D(y) of order 2r + 1 by the formula

D Ξ» ( y ) = βˆ‚ βˆ’ ln β€² T 1 2 … T r βˆ’ 1 2 T r 2 y 1 βˆ‚ βˆ’ ln β€² y 1 T 1 2 … T r βˆ’ 1 2 T r 2 y 2 T 1 … Γ— βˆ‚ βˆ’ ln β€² y r βˆ’ 2 T 1 2 … T r βˆ’ 1 2 T r 2 y r βˆ’ 1 T 1 … T r βˆ’ 2 … βˆ‚ βˆ’ ln β€² y r βˆ’ 1 T 1 2 … T r βˆ’ 1 2 T r 2 y r 2 T 1 … T r βˆ’ 1 Γ— βˆ‚ βˆ’ ln β€² T 1 … T r βˆ‚ βˆ’ ln β€² y r 2 T 1 … T r βˆ’ 1 y r βˆ’ 1 Γ— βˆ‚ βˆ’ ln β€² y r βˆ’ 1 T 1 … T r βˆ’ 2 y r βˆ’ 2 … βˆ‚ βˆ’ ln β€² y 2 T 1 y 1 ( βˆ‚ βˆ’ ln β€² ( y 1 ) ) ,

where Ti, i = 1, …, r, are given by (2.4).

If y is an r-tuple of polynomials representing a critical point associated with integral dominant weights Ξ›1, …, Ξ›n and points z1, …, zn of type Cr, then by Ref. 6, the kernel of DΞ»(y) is a self-dual space of polynomials. By Ref. 2 the coefficients of DΞ»(y) are eigenvalues of higher Gaudin Hamiltonians acting on the Bethe vector related to y.

For admissible l and Ξ» ∈ π”₯βˆ—, define a Ξ» l ( 1 ) , … , a Ξ» l ( r ) as follows.

For l = 0, …, r βˆ’ 1, i = 1, …, l, set a Ξ» l ( i ) = Ξ» i + β‹― + Ξ» l + l + 1 βˆ’ i ⁠. For l = 0, …, r, i = l + 1, …, r, set a Ξ» l ( i ) = 0 ⁠.

For l = r, i = 1, …, r βˆ’ 1, set a Ξ» l ( i ) = Ξ» i + β‹― + Ξ» r βˆ’ 1 + 2 Ξ» r + r + 2 βˆ’ i and a Ξ» r ( r ) = Ξ» r + 1 ⁠.

For l = r + 1, …, 2r βˆ’ 1, set k = 2r βˆ’ l βˆ’ 1. Then for i = 1, …, k, set

a Ξ» l ( i ) = Ξ» i + β‹― + Ξ» k + 2 Ξ» k + 1 + β‹― + 2 Ξ» r βˆ’ 1 + 2 Ξ» r + 2 r + 1 βˆ’ k βˆ’ i

and for i = k + 1, …, r βˆ’ 1, set a Ξ» l ( i ) = 2 Ξ» k + 1 + β‹― + 2 Ξ» r βˆ’ 1 + 2 Ξ» r + 2 r βˆ’ 2 k and a Ξ» l ( r ) = Ξ» k + 1 + β‹― + Ξ» r βˆ’ 1 + Ξ» r + r βˆ’ k ⁠.

Proposition 5.3.

Let the r-tupleβ€ˆyβ€ˆrepresent the solution of the Bethe ansatz equation (2.2) associated toβ€ˆΞ›,β€ˆz, and admissibleβ€ˆl, whereβ€ˆ Ξ» ∈ P + . Thenβ€ˆ D Ξ» ( y ) = D Ξ» ( x a Ξ» l ( 1 ) , … , x a Ξ» l ( r ) ) . β–‘

Let 𝔀 = 𝔰𝔬(2r) be the simple Lie algebra of type Dr, where r β©Ύ 4. We have (Ξ±i, Ξ±i) = 2, i = 1, …, r, (Ξ±i, Ξ±iβˆ’1) = 1, i = 1, …, r βˆ’ 1, and (Ξ±r, Ξ±rβˆ’2) = 1, (Ξ±r, Ξ±rβˆ’1) = 0. We work with data Ξ› = (Ξ», Ο‰1), z = (0, 1), where Ξ» ∈ P + ⁠.

We have

V Ξ» βŠ— V Ο‰ 1 = V Ξ» + Ο‰ 1 βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βŠ• β‹― βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βˆ’ β‹― βˆ’ Ξ± r βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βˆ’ β‹― βˆ’ Ξ± r βˆ’ 2 βˆ’ Ξ± r βŠ• V Ξ» + Ο‰ 1 βˆ’ Ξ± 1 βˆ’ β‹― βˆ’ Ξ± r βˆ’ 3 βˆ’ 2 Ξ± r βˆ’ 2 βˆ’ Ξ± r βˆ’ 1 βˆ’ Ξ± r βŠ• β‹― βŠ• V Ξ» + Ο‰ 1 βˆ’ 2 Ξ± 1 βˆ’ β‹― βˆ’ 2 Ξ± r βˆ’ 2 βˆ’ Ξ± r βˆ’ 1 βˆ’ Ξ± r = V ( Ξ» 1 + 1 , Ξ» 2 , … , Ξ» r ) βŠ• V ( Ξ» 1 βˆ’ 1 , Ξ» 2 + 1 , Ξ» 3 , … , Ξ» r ) βŠ• β‹― βŠ• V ( Ξ» 1 , … , Ξ» k βˆ’ 1 , Ξ» k βˆ’ 1 , Ξ» k + 1 + 1 , Ξ» k + 2 , … , Ξ» r ) βŠ• β‹― βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 2 βˆ’ 1 , Ξ» r βˆ’ 1 + 1 , Ξ» r + 1 ) βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 2 , Ξ» r βˆ’ 1 βˆ’ 1 , Ξ» r + 1 ) βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 2 + 1 , Ξ» r βˆ’ 1 βˆ’ 1 , Ξ» r βˆ’ 1 ) βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 2 , Ξ» r βˆ’ 1 + 1 , Ξ» r βˆ’ 1 ) βŠ• V ( Ξ» 1 , Ξ» 2 , … , Ξ» r βˆ’ 4 , Ξ» r βˆ’ 3 + 1 , Ξ» r βˆ’ 2 βˆ’ 1 , Ξ» r βˆ’ 1 , Ξ» r ) βŠ• β‹― βŠ• V ( Ξ» 1 , … , Ξ» k βˆ’ 2 , Ξ» k βˆ’ 1 + 1 , Ξ» k βˆ’ 1 , Ξ» k + 1 , … , Ξ» r ) βŠ• β‹― βŠ• V ( Ξ» 1 + 1 , Ξ» 2 βˆ’ 1 , Ξ» 3 , … , Ξ» r ) βŠ• V ( Ξ» 1 βˆ’ 1 , Ξ» 2 , … , Ξ» r ) ,
(5.3)

with the convention that the summands with non-dominant highest weights are omitted. Note, in particular, all multiplicities are 1.

We call an r-tuple of integers l = (l1, …, lr) admissible if the VΞ»+Ο‰1βˆ’Ξ±(l) appears in (5.3).

The admissible r-tuples l have the form

( 1 , … , 1 οΈΈ k 1 ones , 0 , … , 0 ) or ( 1 , … , 1 οΈΈ r βˆ’ 2 ones , 1 , 0 ) or ( 1 , … , 1 οΈΈ r βˆ’ 2 ones , 0 , 1 ) or ( 1 , … , 1 οΈΈ k 2 ones , 2 , … , 2 , 1 , 1 ) ,
(5.4)

where k1 = 0, …, r βˆ’ 2, r and k2 = 0, …, r βˆ’ 2. In the first case the length l = l1 + β‹― + lr is k1, in the second and third cases r βˆ’ 1 and in the fourth case 2r βˆ’ k2 βˆ’ 2. It follows that different admissible r-tuples in the first and fourth cases have different length and, therefore, admissible tuples l of these types are parametrized by length l ∈ {0, 1, …, r βˆ’ 2, r, …, 2r βˆ’ 2}. We denote the lengths in the second and third cases by r βˆ’ 1 and r βˆ’ 1 Β― ⁠, respectively. More precisely, for l ∈ { 0 , 1 , … , r βˆ’ 1 , r βˆ’ 1 Β― , r , … , 2 r βˆ’ 2 } ⁠, l is a length of an admissible r-tuple l if l = 0 or l β©½ r βˆ’ 1, Ξ»l > 0 or if l = r βˆ’ 1 Β― ⁠, Ξ»r > 0 or if l = r, Ξ»rβˆ’1 > 0 and Ξ»r > 0 or if l β©Ύ r + 1, Ξ»2rβˆ’lβˆ’1 > 0. We call such ladmissible.

Similarly to the case of type Br, see Theorem 3.9 and Corollary 3.10, we obtain the solutions to Bethe ansatz equations for VΞ» βŠ— VΟ‰1.

Theorem 5.4.

Let 𝔀 = 𝔰𝔬(2r). Letβ€ˆlβ€ˆbe as in (5.4). Ifβ€ˆlβ€ˆis not admissible then the Bethe ansatz equation (2.2) associated toβ€ˆΞ›, z, lβ€ˆhas no solutions. Ifβ€ˆlβ€ˆis admissible then the Bethe ansatz equation (2.2) associated toβ€ˆΞ›, z, lβ€ˆhas exactly one solution represented by the followingβ€ˆr-tuple of polynomialsβ€ˆy(l).

Forβ€ˆl = 0, 1, …, r βˆ’ 1, we haveβ€ˆ y ( l ) = ( x βˆ’ c 1 ( l ) , … , x βˆ’ c l ( l ) , 1 , … , 1 ) , whereβ€ˆ c j ( l ) β€ˆare given by (3.8).

Forβ€ˆ l = r βˆ’ 1 Β― , we haveβ€ˆ y ( r βˆ’ 1 Β― ) = ( x βˆ’ c 1 ( r βˆ’ 1 Β― ) , … , x βˆ’ c r βˆ’ 2 ( r βˆ’ 1 Β― ) , 1 , x βˆ’ c r ( r βˆ’ 1 Β― ) ) , whereβ€ˆ
c j ( r βˆ’ 1 Β― ) = ∏ i = 1 j Ξ» i + β‹― + Ξ» r βˆ’ 2 + Ξ» r + r βˆ’ 1 βˆ’ i Ξ» i + β‹― + Ξ» r βˆ’ 2 + Ξ» r + r βˆ’ i j = 1 , … , r βˆ’ 2
andβ€ˆ
c r ( r βˆ’ 1 Β― ) = Ξ» r Ξ» r + 1 ∏ i = 1 r βˆ’ 2 Ξ» i + β‹― + Ξ» r βˆ’ 2 + Ξ» r + r βˆ’ 1 βˆ’ i Ξ» i + β‹― + Ξ» r βˆ’ 2 + Ξ» r + r βˆ’ i .
Forβ€ˆl = r, we haveβ€ˆ y ( r ) = ( x βˆ’ c 1 ( r ) , … , x βˆ’ c r ( r ) ) , whereβ€ˆ
c j ( r ) = ∏ i = 1 j Ξ» i + β‹― + Ξ» r + r βˆ’ i Ξ» i + β‹― + Ξ» r + r + 1 βˆ’ i , j = 1 , … , r βˆ’ 2 ,
β€ˆ
c r βˆ’ 1 ( r ) = Ξ» r βˆ’ 1 Ξ» r βˆ’ 1 + 1 ∏ i = 1 r βˆ’ 2 Ξ» i + β‹― + Ξ» r + r βˆ’ i Ξ» i + β‹― + Ξ» r + r + 1 βˆ’ i ,
β€ˆ
andβ€ˆ
c r ( r ) = Ξ» r Ξ» r + 1 ∏ i = 1 r βˆ’ 2 Ξ» i + β‹― + Ξ» r + r βˆ’ i Ξ» i + β‹― + Ξ» r + r + 1 βˆ’ i .
Forβ€ˆl = r + 1, …, 2r βˆ’ 2, we haveβ€ˆ
y ( l ) = ( x βˆ’ c 1 ( l ) , . . . , x βˆ’ c 2 r βˆ’ l βˆ’ 2 ( l ) , ( x βˆ’ a 2 r βˆ’ l βˆ’ 1 ( l ) ) ( x βˆ’ b 2 r βˆ’ l βˆ’ 1 ( l ) ) , . . . , ( x βˆ’ a r βˆ’ 2 ( l ) ) ( x βˆ’ b r βˆ’ 2 ( l ) ) , x βˆ’ c r βˆ’ 1 ( l ) , x βˆ’ c r ( l ) ) ,
whereβ€ˆ
c j ( l ) = ∏ i = 1 j Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l + 1 βˆ’ i ,
β€ˆj = 1, …, 2r βˆ’ l βˆ’ 2,β€ˆ
c r βˆ’ 1 ( l ) = ∏ i = 1 2 r βˆ’ 2 βˆ’ l Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l + 1 βˆ’ i Γ— ∏ i = 2 r βˆ’ 1 βˆ’ l r βˆ’ 2 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Γ— Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + l βˆ’ r Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + l βˆ’ r + 1 , c r ( l ) = ∏ i = 1 2 r βˆ’ 2 βˆ’ l Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l + 1 βˆ’ i Γ— ∏ i = 2 r βˆ’ 1 βˆ’ l r βˆ’ 2 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Γ— Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r + l βˆ’ r Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r + l βˆ’ r + 1 , a k ( l ) b k ( l ) = ∏ i = 1 2 r βˆ’ l βˆ’ 2 Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i + 1 2 Γ— ∏ i = 2 r βˆ’ 1 βˆ’ l r βˆ’ 2 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Γ— ∏ i = 2 r βˆ’ 1 βˆ’ l k βˆ’ 1 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Γ— ∏ i = k r βˆ’ 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + Ξ» i + l + i + 1 βˆ’ 2 r Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + Ξ» i + l + i + 2 βˆ’ 2 r Γ— Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + l βˆ’ r Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + l βˆ’ r + 1 β‹… Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r + l βˆ’ r Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r + l βˆ’ r + 1
andβ€ˆ
a k ( l ) + b k ( l ) = ∏ i = 1 2 r βˆ’ l βˆ’ 2 Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Ξ» i + β‹― + Ξ» 2 r βˆ’ 2 βˆ’ l + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i + 1 Γ— ∏ i = 2 r βˆ’ 1 βˆ’ l k βˆ’ 1 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Γ— 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r + 1 + 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r + 2 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + 2 l βˆ’ 2 r + 1 Γ— ∏ i = k r βˆ’ 2 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i βˆ’ 1 Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» i + 2 Ξ» i + 1 + β‹― + 2 Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + Ξ» r + l βˆ’ i Γ— Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + l βˆ’ r Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r βˆ’ 1 + l βˆ’ r + 1 β‹… Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r + l βˆ’ r Ξ» 2 r βˆ’ 1 βˆ’ l + β‹― + Ξ» r βˆ’ 2 + Ξ» r + l βˆ’ r + 1 Γ— ∏ i = k r βˆ’ 2 Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + Ξ» i + l + i + 1 βˆ’ 2 r Ξ» 2 r βˆ’ l βˆ’ 1 + β‹― + Ξ» i + l + i + 2 βˆ’ 2 r
β€ˆk = 2r βˆ’ 1 βˆ’ l, …, r βˆ’ 2.β–‘
Note that the formulas above with r = 3 correspond to solutions of the Bethe ansatz equations of type A3 and Ξ› = (Ξ», Ο‰2). These formulas were given in Theorem 5.5, Ref. 7.

Then we deduce the analog of Theorem 4.5.

Theorem 5.5.

Let 𝔀 = 𝔰𝔬(2r) andβ€ˆ Ξ» ∈ P + . For a generic (N + 1)-tuple of distinct complex numbersβ€ˆz = (z0, z1, …, zN), the Gaudin Hamiltoniansβ€ˆ ( H 0 , H 1 , … , H N ) β€ˆacting inβ€ˆ Sing V Ξ» βŠ— V Ο‰ 1 βŠ— N β€ˆare diagonalizable. Moreover, for genericβ€ˆzβ€ˆthere exists a set of solutions {ti, i ∈ I} of the Bethe ansatz equation (2.2) such that the corresponding Bethe vectors {Ο‰(z, ti), i ∈ I} form a basis ofβ€ˆ Sing V Ξ» βŠ— V Ο‰ 1 βŠ— N .β–‘

For type D, the algebra has a non-trivial diagram automorphism which leads to degeneracy of the spectrum. For example, if Ξ»rβˆ’1 = Ξ»r, then the Bethe vectors corresponding to the critical points y(rβˆ’1) and y ( r βˆ’ 1 Β― ) are eigenvectors of the Gaudin Hamiltonian H ≔ H 1 = βˆ’ H 2 with the same eigenvalue. In particular Proposition 4.3 is not applicable since the two corresponding summands in (5.3) have non-comparable highest weights.

This work was partially supported by a grant from the Simons Foundation (Nos. 336826 to Alexander Varchenko and 353831 to Evgeny Mukhin).” The research of A.V. is supported by NSF Grant No. DMS-1362924. A.V. thanks the MPI in Bonn for hospitality during his visit.

1.
Humphreys
,
J. E.
,
Introduction to Lie Algebras and Representation Theory
, 2nd ed. (
Springer-Verlag
,
New York
,
1978
).
2.
Molev
,
A.
and
Mukhin
,
E.
, β€œ
Eigenvalues of Bethe vectors in the Gaudin model
,” e-print arXiv:math.RT/1506.01884.
3.
Mukhin
,
E.
,
Tarasov
,
V.
, and
Varchenko
,
A.
, β€œ
Bethe eigenvectors of higher transfer matrices
,”
J. Stat. Mech.: Theory Exp.
β€ˆ
2006
,
P08002
; e-print arXiv:math.QA/0605015.
4.
Mukhin
,
E.
,
Tarasov
,
V.
, and
Varchenko
,
A.
, β€œ
Schubert calculus and representations of the general linear group
,”
J. Am. Math. Soc.
β€ˆ
22
(
4
),
909
–
940
(
2009
).
5.
Mukhin
,
E.
and
Varchenko
,
A.
, β€œ
Remarks on critical points of phase functions and norms of Bethe vectors
,” in
Arrangements–Tokyo 1998
,
Advanced Studies in Pure Mathematics
Vol.
27
(
Kinokuniya
,
Tokyo
,
2000
), pp.
239
–
246
; e-print arXiv:math.RT/9810087.
6.
Mukhin
,
E.
and
Varchenko
,
A.
, β€œ
Critical points of master functions and flag varieties
,”
Commun. Contemp. Math.
β€ˆ
6
(
1
),
111
–
163
(
2004
); e-print arXiv:math.QA/0209017.
7.
Mukhin
,
E.
and
Varchenko
,
A.
, β€œ
Norm of a Bethe vector and the Hessian of the master function
,”
Compos. Math.
β€ˆ
141
(
4
),
1012
–
1028
(
2005
); e-print arXiv:math.QA/0402349.
8.
Mukhin
,
E.
and
Varchenko
,
A.
, β€œ
Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture
,”
Trans. Am. Math. Soc.
β€ˆ
359
(
11
),
5383
–
5418
(
2007
); e-print arXiv:math.QA/0501144.
9.
Mukhin
,
E.
and
Varchenko
,
A.
, β€œ
Quasi-polynomials and the Bethe ansatz
,”
Geom. Topol. Monogr.
β€ˆ
13
,
385
–
420
(
2008
); e-print arXiv:math.QA/0604048.
10.
Mukhin
,
E.
,
Vicedo
,
B.
, and
Young
,
C.
, β€œ
Gaudin model for 𝔀𝔩(m|n)
,”
J. Math. Phys.
β€ˆ
56
(
5
),
051704
(
2015
); e-print arXiv:math.QA/1404.3526.
11.
Reshetikhin
,
N.
and
Varchenko
,
A.
, β€œ
Quasiclassical asymptotics of solutions to the KZ equations
,” in
Geometry, Topology and Physics for R. Bott
(
International Press
,
1995
), pp.
293
–
322
; e-print arXiv:hep-th/9402126.
12.
Schechtman
,
V.
and
Varchenko
,
A.
, β€œ
Arrangements of hyperplanes and Lie algebra homology
,”
Invent. Math.
β€ˆ
106
,
139
–
194
(
1991
).
13.
Varchenko
,
A.
, β€œ
Critical points of the product of powers of linear functions and families of bases of singular vectors
,”
Compos. Math.
β€ˆ
97
(
3
),
385
–
401
(
1995
); e-print arXiv:hep-th/9312119.
14.
Varchenko
,
A.
, β€œ
Quantum integrable model of an arrangement of hyperplanes
,”
Symmetry, Integrability Geom.: Methods Appl.
β€ˆ
7
,
1
–
55
(
2011
); e-print arXiv:math.QA/1001.4553.