On the supersymmetric XXX spin chains associated to \mathfrak{gl}_{1|1}
aa r X i v : . [ m a t h . QA ] O c t ON THE SUPERSYMMETRIC XXX SPIN CHAINS ASSOCIATED TO gl | KANG LU AND EVGENY MUKHINA
BSTRACT . We study the gl | supersymmetric XXX spin chains. We give an explicit description of thealgebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation gl | Yangian modules.It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra ofHamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. Inparticular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one.We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible,then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated tosymmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the centerof the Yangian.
Keywords: supersymmetric spin chains, Bethe ansatz, rational difference operators.
1. I
NTRODUCTION
Spin chains have been at the center of the study of the integrable models since the introduction of Heisen-berg magnet by H. Bethe back in 1931. The literature on the subject is immense and keeps growing. Whilethe even case is by far the most popular, it is now clear that the spin chains associated to Lie superalgebrasare the integral part of the picture.The supersymmetric spin chains were introduced back to 1980s, see [KS82, Kul85]. They have enjoyeda surge of interest in the recent years, see e.g. [BR09, HLM19, HLPRS16, HLPRS18, MR14, TZZ15].However, we know much less about these models compared to the even case. The presence of fermionicroots creates a number of new features which are not yet well understood.This paper is devoted to the study of the supersymmetric spin chains associated to the super Yangian Y( gl | ) . This case is remarkable as it is sufficiently simple on one hand and it is complex enough to haveall the phenomena of supersymmetry on the other hand. So it provides a perfect testing ground for methodsand conjectures.The simplicity of the Y( gl | ) spin chain is apparent as the model can be written in terms of free fermionsand the corresponding Bethe ansatz equations decouple. Unsurprisingly, for generic values of parameters, theBethe ansatz method gives the complete information about the spectrum of the model. However, when theBethe equation has roots of non-trivial multiplicity, the situation is more subtle as Hamiltonians develop Jor-dan blocks. Moreover, the algebra of Hamiltonians apriori is not finitely generated since all anti-symmetricpowers of the superspace C | are non-trivial and each power gives a non-zero transfer matrix. Finally, thestandard geometric Langlands philosophy asks for a description of the eigenvectors of the Hamiltonians interms of “opers”. The Y( gl | ) “oper” is expected to be a ratio of two order-one difference operators andhave a universal nature.We are able to clarify all these points. Let us discuss our findings in more detail.We consider tensor products L ( λ , b ) = N ks =1 L λ ( s ) ( b s ) of polynomial evaluation modules of Y( gl | ) ,where λ = ( λ (1) , . . . , λ ( k ) ) is a sequence of polynomial gl | -weights and b = ( b , . . . , b k ) is a sequence ofcomplex numbers. KANG LU AND EVGENY MUKHIN
We have the action of the commutative algebra of transfer matrices corresponding to polynomial moduleswhich we call the Bethe algebra, see Section 4.3. The Bethe algebra commutes with the diagonal action of gl | . It turns out that the image of the Bethe algebra in End( L ( λ , b )) is generated by the first transfer matrix str T ( x ) = T ( x ) − T ( x ) , where T ( x ) is the matrix generating Y( gl | ) . More precisely, we show if oneadds to Y( gl | ) the inverse of the central element T (1)11 − T (1)22 , then all transfer matrices correspondingto polynomial modules can be explicitly expressed in terms of the first transfer matrix and the quantumBerezinian (which is central in Y( gl | ) ), see Theorem 6.10.This result is understood as a relation in the Grothendieck ring of the category of finite-dimensionalrepresentations of Y( gl | ) . Namely, the composition factors of a certain tensor product of evaluation vectorrepresentations are all isomorphic to the evaluation of a given symmetric power of the vector representation,up to one-dimensional modules, see Example 2.2. However, instead of discussing the universal R-matrix,we give a Bethe ansatz method proof.Then we need to find the spectrum of the first transfer matrix acting on the subspace ( L ( λ , b )) sing( n − l,l ) ofsingular vectors in L ( λ , b ) of a weight P ki =1 λ ( i ) − lα = ( n − l, l ) , l = 1 , . . . , k − . We manage morethan that: we give an explicit description of the image of the Bethe algebra in the endomorphism ring of thesubspace ( L ( λ , b )) sing( n − l,l ) . Namely, we show that if L ( λ , b ) is cyclic, then the image of the Bethe algebra in End(( L ( λ , b )) sing( n − l,l ) ) has dimension (cid:0) k − l (cid:1) and is isomorphic to the algebra C [ w , . . . , w k − ] S l × S k − l − / h n k − Y i =1 ( x − w i ) − γ λ , b ( x ) i , (1.1)where polynomial γ λ , b ( x ) is constructed from λ , b : γ λ , b ( x ) = k Y s =1 ( x − b s + λ ( s )1 ) − k Y s =1 ( x − b s − λ ( s )2 ) . Moreover, the space ( L ( λ , b )) sing( n − l,l ) is identified with the regular representation over this algebra as a moduleover the Bethe algebra, see Theorem 4.9.It follows that the eigenvectors (up to proportionality) of the Bethe algebra in ( L ( λ , b )) sing( n − l,l ) are ina bijective correspondence with the monic polynomials of degree l which divide γ λ , b ( x ) . In the genericsituation, the roots of polynomial γ λ , b are simple. Therefore, it has exactly (cid:0) k − l (cid:1) distinct monic divisors ofdegree l and the corresponding Bethe vectors form a basis of the space ( L ( λ , b )) sing( n − l,l ) . This is exactly thestandard Bethe ansatz. However if γ λ , b ( x ) has multiple roots, the number of monic divisors is smaller. Andto each monic divisor y we have exactly one eigenvector of the Bethe algebra and a generalized eigenspaceof dimension given by the product of binomial coefficients Y a ∈ C (cid:18) Mult a ( γ λ , b ( x )) Mult a ( y ( x )) (cid:19) , where Mult a ( f ) denotes the order of zero of f ( x ) at x = a . Moreover, when the tensor product is irreducible,then the eigenvalues and eigenvectors of the Bethe algebra can be found by the usual formulas, see (2.21),(2.22) and Theorem 2.11.We prove this result by adopting a method of [MTV09].Finally, let us discuss the Y( gl | ) “opers”. Given a monic divisor y of γ λ , b ( x ) , or, in other words, aneigenvector v y of the Bethe algebra, following [HLM19], we have a rational difference operator D y , see(6.5). From the explicit formula for the eigenvalue, one sees that the coefficients of the rational differenceoperator in this case are essentially eigenvalues of the first transfer matrix acting on v y . Again, we improve UPERSYMMETRIC XXX SPIN CHAINS 3 on that as follows. By [MR14], the Berezinian Ber (1 − T t ( x ) e − ∂ x ) is a generating function for the transfermatrices, see (6.2). We show that Ber (1 − T t ( x ) e − ∂ x ) v y = D y v y by a brute force (and somewhat tedious)computation. It follows that there is a universal formula for the rational difference operator in terms of thefirst transfer matrix and the quantum Berezinian, which produces D y when applied to the vector v y for all y ,see Corollary 6.13.We also describe and prove similar results for the quasi-periodic case, where the monodromy matrix T ( x ) is twisted by a diagonal invertible two-by-two matrix Q , so that the first transfer matrix has the form q T ( x ) − q T ( x ) , see Section 4.5 and Theorem 4.11.This paper deals with tensor products of polynomial modules which provide a natural supersymmetricanalog of the finite-dimensional modules in the even case. We expect that for gl | spin chain in the case ofarbitrary tensor products of finite-dimensional modules, the image of Bethe algebra still has the form (1.1),except for the case q = q = 1 , n = P ks =1 ( λ ( s )1 + λ ( s )2 ) = 0 . The exceptional case is even more interesting,since the tensor product is not semisimple as a gl | -module and some singular vectors are in the image of thecreation operator e . Then the Bethe ansatz is expected to describe the eigenvectors of the transfer matrixwhich are not in that image, see [HMVY19, Section 8.3]. The methods of this paper are not applicable forarbitrary tensor products and one needs a different approach.The paper is organized as follows. In Section 2, we fix notations and discuss basic facts of the algebraicBethe ansatz. Then we study the space V S and its properties in Section 3. Section 4 contains the maintheorems. Section 5 is dedicated to the proofs of main theorems. We discuss the higher transfer matrices andthe relations between higher transfer matrices and the Bethe algebra in Section 6. Acknowledgments.
We thank V. Tarasov for interesting discussions. This work was partially supportedby a grant from the Simons Foundation
OTATION
Lie superalgebra gl | and its representations. A vector superspace V = V ¯0 ⊕ V ¯1 is a Z -gradedvector space. Elements of V ¯0 are called even ; elements of V ¯1 are called odd . We write | v | ∈ { ¯0 , ¯1 } for theparity of a homogeneous element v ∈ V . Set ( − ¯0 = 1 and ( − ¯1 = − .Consider the vector superspace C | , where dim( C | ) = 1 and dim( C | ) = 1 . We choose a homo-geneous basis v , v of C | such that | v | = ¯0 and | v | = ¯1 . For brevity we shall write their parities as | v i | = | i | . Denote by E ij ∈ End( C | ) the linear operator of parity | i | + | j | such that E ij v r = δ jr v i for i, j, r = 1 , .The Lie superalgebra gl | is spanned by elements e ij , i, j = 1 , , with parities e ij = | i | + | j | and thesupercommutator relations are given by [ e ij , e rs ] = δ jr e is − ( − ( | i | + | j | )( | r | + | s | ) δ is e rj . Let h be the commutative Lie subalgebra of gl | spanned by e , e . Denote the universal envelopingalgebras of gl | and h by U( gl | ) and U( h ) , respectively.We call a pair λ = ( λ , λ ) of complex numbers a gl | - weight . A gl | -weight λ is non-degenerate if λ + λ = 0 .Let M be a gl | -module. A non-zero vector v ∈ M is called singular if e v = 0 . Denote the subspaceof all singular vectors of M by ( M ) sing . A non-zero vector v ∈ M is called of weight λ = ( λ , λ ) if e v = λ v and e v = λ v . Denote by ( M ) λ the subspace of M spanned by vectors of weight λ . Set ( M ) sing λ = ( M ) sing ∩ ( M ) λ . KANG LU AND EVGENY MUKHIN
Let λ = ( λ (1) , . . . , λ ( k ) ) be a sequence of gl | -weights, we denote by | λ | the sum | λ | = k X s =1 ( λ ( s )1 + λ ( s )2 ) . Denote by L λ the irreducible gl | -module generated by an even singular vector v λ of weight λ . Then L λ is two-dimensional if λ is non-degenerate and one-dimensional otherwise. Clearly, C | ∼ = L ω , where ω = (1 , .A gl | -module M is called a polynomial module if M is a submodule of ( C | ) ⊗ n for some n ∈ Z > .We say that λ is a polynomial weight if L λ is a polynomial module. Weight λ = ( λ , λ ) is a polynomialweight if and only if λ , λ ∈ Z > and either λ > or λ = λ = 0 . We also write L ( λ ,λ ) for L λ .For non-degenerate polynomial weights λ = ( λ , λ ) and µ = ( µ , µ ) , we have L ( λ ,λ ) ⊗ L ( µ ,µ ) = L ( λ + µ ,λ + µ ) ⊕ L ( λ + µ − ,λ + µ +1) . Define a supertrace str : End( C | ) → C , which is supercyclic, str( E ij ) = ( − | j | δ ij , str([ E ij , E rs ]) = 0 , where [ · , · ] is the supercommutator.Define the supertranspose t , t : End( C | ) → End( C | ) , E ij ( − | i || j | + | i | E ji . The supertranspose is an anti-homomorphism and respects the supertrace, ( AB ) t = ( − | A || B | B t A t , str( A ) = str( A t ) , (2.1)for all × matrices A , B .2.2. Current superalgebra gl | [ t ] . Denote by gl | [ t ] the Lie superalgebra gl | ⊗ C [ t ] of gl | -valued poly-nomials with the point-wise supercommutator. Call gl | [ t ] the current superalgebra of gl | . We identify gl | with the subalgebra gl | ⊗ of constant polynomials in gl | [ t ] .We write e ij [ r ] for e ij ⊗ t r , r ∈ Z > . A basis of gl | [ t ] is given by e ij [ r ] , i, j = 1 , and r ∈ Z > . Theysatisfy the supercommutator relations [ e ij [ r ] , e kl [ s ]] = δ jk e il [ r + s ] − ( − ( | i | + | j | )( | k | + | l | ) δ il e kj [ r + s ] . In particular, one has ( e [ r ]) = ( e [ r ]) = 0 and e [ r ] e [ s ] = − e [ s ] e [ r ] in the universal envelopingsuperalgebra U( gl | [ t ]) .The universal enveloping superalgebra U( gl | [ t ]) is a Hopf superalgebra with the coproduct given by ∆( X ) = X ⊗ ⊗ X, for X ∈ gl | [ t ] . There is a natural Z > -gradation on U( gl | [ t ]) such that deg( e ij [ r ]) = r .Let M be a Z > -graded space with finite-dimensional homogeneous components. Let M j ⊂ M be thehomogeneous component of degree j . We call the formal power series in variable q , ch( M ) = ∞ X j =0 dim( M j ) q j (2.2)the graded character of M . UPERSYMMETRIC XXX SPIN CHAINS 5
Super Yangian Y( gl | ) . We recall the standard facts about super Yangian Y( gl | ) and fix notation,see e.g. [Naz91]. Let P ∈ End( C | ) ⊗ End( C | ) be the graded flip operator given by P = X i,j =1 ( − | j | E ij ⊗ E ji . For two homogeneous vectors v , v in C | , we have P ( v ⊗ v ) = ( − | v || v | v ⊗ v . Define the rational R-matrix R ( x ) ∈ End( C | ) ⊗ End( C | ) by R ( x ) = 1 + P/x . The R-matrix R ( x ) satisfies the Yang-Baxterequation R (1 , ( x − x ) R (1 , ( x ) R (2 , ( x ) = R (2 , ( x ) R (1 , ( x ) R (1 , ( x − x ) . (2.3)The super Yangian Y( gl | ) is a unital associative superalgebra generated by the generators T ( r ) ij of parity | i | + | j | , i, j = 1 , and r > , with the defining relation R (1 , ( x − x ) T (1 , ( x ) T (2 , ( x ) = T (2 , ( x ) T (1 , ( x ) R (1 , ( x − x ) , (2.4)where T ( x ) ∈ End( C | ) ⊗ Y( gl | )[[ x − ]] is the monodromy matrix T ( x ) = X i,j =1 E ij ⊗ T ij ( x ) , T ij ( x ) = ∞ X r =0 T ( r ) ij x − r , T (0) ij = δ ij . Note that the monodromy matrix is even.The defining relation (2.4) gives ( x − x )[ T ij ( x ) , T rs ( x )] = ( − | i || r | + | s || i | + | s || r | (cid:0) T rj ( x ) T is ( x ) − T rj ( x ) T is ( x ) (cid:1) = ( − | i || j | + | s || i | + | s || j | (cid:0) T is ( x ) T rj ( x ) − T is ( x ) T rj ( x ) (cid:1) . (2.5)Equivalently, [ T ( a ) ij , T ( b ) rs ] = ( − | i || j | + | s || i | + | s || j | min( a,b ) X ℓ =1 (cid:16) T ( ℓ − is T ( a + b − ℓ ) rj − T ( a + b − ℓ ) is T ( ℓ − rj (cid:17) . (2.6)The super Yangian Y( gl | ) is a Hopf superalgebra with a coproduct and an opposite coproduct given by ∆ : T ij ( x ) X r =1 T rj ( x ) ⊗ T ir ( x ) , ∆( T ( s ) ij ) = X r =1 s X a =0 T ( a ) rj ⊗ T ( s − a ) ir , i, j = 1 , , (2.7) ∆ op : T ij ( x ) X r =1 ( − ( | i | + | r | )( | r | + | j | ) T ir ( x ) ⊗ T rj ( x ) , i, j = 1 , . The super Yangian Y( gl | ) contains U( gl | ) as a Hopf subalgebra with the embedding given by e ij ( − | i | T (1) ji . By (2.5), one has [ T (1) ij , T rs ( x )] = ( − | i || r | + | s || i | + | s || r | (cid:0) δ is T rj ( x ) − δ rj T is ( x ) (cid:1) . (2.8)The relation (2.8) implies that for any r, s = 1 , , [ E rs ⊗ ⊗ e rs , T ( x )] = 0 . (2.9)Define the degree function on Y( gl | ) by deg( T ( r ) ij ) = r − , then the super Yangian Y( gl | ) is a filteredalgebra. Let F s Y( gl | ) be the subspace spanned by elements of degree s , one has the increasing filtration F Y( gl | ) ⊂ F Y( gl | ) ⊂ · · · ⊂ Y( gl | ) . KANG LU AND EVGENY MUKHIN
The corresponding graded algebra grY( gl | ) inherits from Y( gl | ) the Hopf structure. It is known grY( gl | ) is isomorphic to the universal enveloping superalgebra U( gl | [ t ]) . For instance, the graded prod-uct comes from (2.6) while the graded coproduct comes from (2.7).For any complex number z ∈ C , there is an automorphism ζ z : Y( gl | ) → Y( gl | ) , T ij ( x ) → T ij ( x − z ) , where ( x − z ) − is expanded as a power series in x − . The evaluation homomorphism ev : Y( gl | ) → U( gl | ) is defined by the rule: T ( r ) ji ( − | i | δ r e ij , for r ∈ Z > .For any gl | -module M denote by M ( z ) the Y( gl | ) -module obtained by pulling back of M through thehomomorphism ev ◦ ζ z . The module M ( z ) is called an evaluation module with the evaluation point z .The map ̟ ξ : T ( x ) → ξ ( x ) T ( x ) , where ξ ( x ) is any formal power series in x − with the leading term 1, ξ ( x ) = 1 + ξ x − + ξ x − + · · · ∈ C [[ x − ]] , (2.10)defines an automorphism of Y( gl | ) .There is a one-dimensional module C v given by T ( x ) v = T ( x ) v = ξ ( x ) v , T ( x ) v = T ( x ) v = 0 ,where v is a homogeneous vector of parity ¯ i . We denote this module by C ¯ i,ξ .Let a = ( a , . . . , a n ) be a sequence of complex numbers. Let b = ( b , . . . , b k ) be another sequenceof complex numbers, λ = ( λ (1) , . . . , λ ( k ) ) a sequence of gl | -weights. Set L ( λ , b ) = N ks =1 L λ ( s ) ( b s ) , V ( a ) = N ni =1 C | ( a i ) , and ϕ λ , b ( x ) = k Y s =1 ( x − b s + λ ( s )1 ) , ψ λ , b ( x ) = k Y s =1 ( x − b s − λ ( s )2 ) . (2.11)When we write L ( λ , b ) , we shall always assume the participating gl | -weights λ ( s ) are non-degenerate.We also denote by | i the vector v λ (1) ⊗ · · · ⊗ v λ ( k ) ∈ L ( λ , b ) . We call | i a vacuum vector . We call the Y( gl | ) -module L ( λ , b ) cyclic if it is generated by | i .It is known from [Zhrb95] that up to twisting by an automorphism ̟ ξ with proper ξ ( x ) ∈ C [[ x − ]] as in(2.10), every finite-dimensional irreducible representation of Y( gl | ) is isomorphic to a tensor product ofevaluation Y( gl | ) -modules. In other words, every finite-dimensional irreducible representation of Y( gl | ) is of form L ( λ , b ) ⊗ C ¯ i,ξ .The following explicit description of irreducible and cyclic Y( gl | ) -modules in term of the highestweights is similar to the one in the quantum affine case, see [Zhhf16]. Lemma 2.1.
The Y( gl | ) -module L ( λ , b ) is irreducible if and only if ϕ λ , b ( x ) and ψ λ , b ( x ) are relativelyprime.The Y( gl | ) -module L ( λ , b ) is cyclic if and only if b j = b i + λ ( i )2 + λ ( j )1 for i < j k .Proof. The first statement follows from [Zhrb95, Theorems 4 and 5]. Proof of the second statement is similarto the proof of [Zhhf16, Theorem 4.2]. (cid:3)
We give decomposition of some tensor products of evaluation vector representations which are importantfor Section 6.
Example 2.2.
We have the following equality in the Grothendieck ring, L nω ( z ) ⊗ L ω ( z − n ) = L ( n +1) ω ( z ) + (cid:0) C ¯1 ,ξ n ⊗ L ( n +1) ω ( z ) (cid:1) , (2.12) UPERSYMMETRIC XXX SPIN CHAINS 7 where ξ n ( x ) = ( x − z + n − / ( x − z + n ) . Moreover, (cid:0) C ¯1 ,ξ n ⊗ L ( n +1) ω ( z ) (cid:1) is the unique irreducible Y( gl | ) -submodule of L nω ( z ) ⊗ L ω ( z − n ) while L ( n +1) ω ( z ) is the simple quotient module.Inductively using (2.12), we have the equality in the Grothendieck ring, n O i =1 L ω ( z − i + 1) = n − X ℓ =0 X i < ···
We write v ( s )1 for v λ ( s ) . Let v ( s )2 = e v ( s )1 . Then v ( s )1 , v ( s )2 is a basis of L λ ( s ) .Let E ij , i, j = 1 , , be the linear operator in End( L λ ( s ) ) of parity | i | + | j | such that E ij v ( s ) r = δ jr v ( s ) i for r = 1 , .The R-matrix R ( x ) ∈ End( L λ ( i ) ( b + x )) ⊗ End( L λ ( j ) ( b )) is given by R ( x ) = E ⊗ E − λ ( i )1 + λ ( j )2 − xλ ( j )1 + λ ( i )2 + x E ⊗ E + λ ( j )1 − λ ( i )1 + xλ ( j )1 + λ ( i )2 + x E ⊗ E + λ ( i )2 − λ ( j )2 + xλ ( j )1 + λ ( i )2 + x E ⊗ E − λ ( i )1 + λ ( i )2 λ ( j )1 + λ ( i )2 + x E ⊗ E + λ ( j )1 + λ ( j )2 λ ( j )1 + λ ( i )2 + x E ⊗ E . (2.14)For ( λ ( i )1 , λ ( i )2 ) = ( λ ( j )1 , λ ( j )2 ) = (1 , , the R -matrix is a scalar multiple of the one used in the definitionof the super Yangian Y( gl | ) in Section 2.3: R ( x ) = R ( x ) x/ (1 + x ) = ( x + P ) / (1 + x ) .The R-matrix satisfies ∆ op ( X ) R ( b i − b j ) = R ( b i − b j )∆( X ) , (2.15)for all X ∈ Y( gl | ) . The module L λ ( i ) ( b i ) ⊗ L λ ( j ) ( b j ) is irreducible if and only if R ( b i − b j ) is well-definedand invertible.Define an anti-automorphism ι : Y( gl | ) → Y( gl | ) by the rule, ι ( T ij ( x )) = ( − | i || j | + | i | T ji ( x ) , i, j = 1 , . (2.16)One has ι ( X X ) = ( − | X || X | ι ( X ) ι ( X ) for X , X ∈ Y( gl | ) . Note that for all X ∈ Y( gl | ) wealso have ∆ ◦ ι ( X ) = ( ι ⊗ ι ) ◦ ∆ op ( X ) . (2.17)The Shapovalov form B λ ( s ) on L λ ( s ) is a unique symmetric bilinear form such that B λ ( s ) ( e ij w , w ) = ( − ( | i | + | j | ) | w | B λ ( s ) ( w , ( − | i || j | + | i | e ji w ) , for all i, j and w , w ∈ L λ ( i ) , and B λ ( s ) ( v ( s )1 , v ( s )1 ) = 1 . Explicitly, it is given by B λ ( s ) ( v ( s )1 , v ( s )1 ) = 1 , B λ ( s ) ( v ( s )1 , v ( s )2 ) = B λ ( s ) ( v ( s )2 , v ( s )1 ) = 0 , B λ ( s ) ( v ( s )2 , v ( s )2 ) = − ( λ ( i )1 + λ ( i )2 ) . The Shapovalov forms B λ ( s ) on L λ ( s ) induce a bilinear form B λ = N ks =1 B λ ( s ) (respecting the usual signconvention) on L ( λ ) = N ks =1 L λ ( s ) .Let R λ , b ∈ End( L ( λ )) be the product of R-matrices, R λ , b = −→ Y i k −→ Y i The bilinear form B λ , b satisfies B λ , b ( | i , | i ) = 1 , B λ , b ( Xw , w ) = ( − | X || w | B λ , b ( w , ι ( X ) w ) , for all X ∈ Y( gl | ) , w , w ∈ L ( λ , b ) , see (2.15), (2.17). Note that if L ( λ , b ) is irreducible, then B λ , b iswell-defined and non-degenerate.2.5. Bethe ansatz. Let Q = diag( q , q ) be an invertible diagonal matrix, where q , q ∈ C × . Define the( twisted ) transfer matrix T Q ( x ) to be the supertrace of QT ( x ) , T Q ( x ) = str( QT ( x )) = q T ( x ) − q T ( x ) = q − q + ∞ X r =1 ( q T ( r )11 − q T ( r )22 ) x − r ∈ Y( gl | )[[ x − ]] . Note that the transfer matrix T Q ( x ) essentially depends on the ratio q /q .We write simply T ( x ) for T I ( x ) , where I is the identity matrix.The twisted transfer matrices commute, [ T Q ( x ) , T Q ( x )] = 0 . Moreover, T Q ( x ) commutes with U( h ) if q = q , and U( gl | ) if q = q .Let b = ( b , . . . , b k ) be a sequence of complex numbers, λ = ( λ (1) , . . . , λ ( k ) ) a sequence of gl | -weights. We are interested in finding eigenvalues and eigenvectors of T Q ( x ) in L ( λ , b ) . To be more precise,we call f ( x ) = q − q + ∞ X r =1 f r x − r , f r ∈ C , (2.18)an eigenvalue of T Q ( x ) if there exists a non-zero vector v ∈ L ( λ , b ) such that ( q T ( r )11 − q T ( r )22 ) v = f r v forall r ∈ Z > . If f ( x ) is a rational function, we consider it as a power series in x − as (2.18). The vector v iscalled an eigenvector of T Q ( x ) corresponding to eigenvalue f ( x ) . We also define the eigenspace of T Q ( x ) in L ( λ , b ) corresponding to eigenvalue f ( x ) as T ∞ r =1 ker(( q T ( r )11 − q T ( r )22 ) | L ( λ , b ) − f r ) .It is sufficient to consider L ( λ , b ) with λ ( i )2 = 0 for all i . Indeed, if L ( λ , b ) is an arbitrary tensor productand ξ ( x ) = k Y s =1 x − b s x − b s − λ ( s )2 , then L ( λ , b ) ⊗ C ¯0 ,ξ = L (˜ λ , ˜ b ) , where ˜ λ ( s ) = ( λ ( s )1 + λ ( s )2 , , ˜ b s = b s + λ ( s )2 , s = 1 , . . . , k. Identify L ( λ , b ) ⊗ C ¯0 ,ξ with L ( λ , b ) as vector spaces. Then T Q ( x ) acting on L ( λ , b ) ⊗ C ¯0 ,ξ coincides with ξ ( x ) T Q ( x ) acting on L ( λ , b ) and therefore the problem of diagonalization of the transfer matrix in L ( λ , b ) is reduced to diagonalization of the transfer matrix in L (˜ λ , ˜ b ) .The main method to find eigenvalues and eigenvectors of T Q ( x ) in L ( λ , b ) is the algebraic Bethe ansatz.Here we recall it from [Kul85, BR09].Fix a non-negative integer l . Let t = ( t , . . . , t l ) be a sequence of complex numbers. Define the polyno-mial y t = Q li =1 ( x − t i ) . We say that polynomial y t represents t .A sequence of complex numbers t is called a solution to the Bethe ansatz equation associated to λ , b , l if y t divides the polynomial q ϕ λ , b ( x ) − q ψ λ , b ( x ) , (2.19)see (2.11). We do not distinguish solutions which differ by a permutation of coordinates (that is representedby the same polynomial). UPERSYMMETRIC XXX SPIN CHAINS 9 Remark . In the literature, see e.g. [BR09], one often simply calls t a solution of the Bethe ansatz equationif its coordinates satisfy the following system of algebraic equations: q q k Y s =1 t j − b s + λ ( s )1 t j − b s − λ ( s )2 = 1 , j = 1 , . . . , l. (2.20)Note that (2.20) involves a single t j only, therefore t is a solution of the Bethe ansatz equation if and only ifall t j are roots of the polynomial q ϕ λ , b ( x ) − q ψ λ , b ( x ) . In a generic situation, q ϕ λ , b ( x ) − q ψ λ , b ( x ) hasno multiple roots, and it is sufficient to consider t with distinct coordinates. However, in general we needto allow the same number appear in t several times. According to our definition, a root t of q ϕ λ , b ( x ) − q ψ λ , b ( x ) , occurs in t at most as many times as the order of x − t in q ϕ λ , b ( x ) − q ψ λ , b ( x ) , see [HLM19,Section 3.2]. (cid:3) Let λ ( ∞ ) be the gl | -weight given by λ ( ∞ )1 = k X s =1 λ ( s )1 − l, λ ( ∞ )2 = k X s =1 λ ( s )2 + l. Define the off-shell Bethe vector B l ( t ) ∈ ( L ( λ , b )) λ ( ∞ ) by B l ( t ) = l Y i =1 k Y s =1 t i − b s t i − b s + λ ( s )1 Y i If the on-shell Bethe vector b B l ( t ) is non-zero, then b B l ( t ) is an eigenvector of the transfermatrix T Q ( x ) with the corresponding eigenvalue E Qy t , λ , b ( x ) = y t ( x − y t ( x ) ( q ϕ λ , b ( x ) − q ψ λ , b ( x )) k Y s =1 ( x − b s ) − , (2.22) where ϕ λ , b ( x ) and ψ λ , b ( x ) are given by (2.11) . (cid:3) Different on-shell Bethe vectors are orthogonal with respect to the bilinear form B λ , b ( · , · ) . Corollary 2.5. Let t and t be two different solutions of the Bethe ansatz equation, then B λ , b ( b B l ( t ) , b B l ( t )) = 0 . Proof. It follows from the equality B λ , b ( T Q ( x ) b B l ( t ) , b B l ( t )) = B λ , b ( b B l ( t ) , ι ( T Q ( x )) b B l ( t )) = B λ , b ( b B l ( t ) , T Q ( x ) b B l ( t )) and Theorem 2.4 since T Q ( x ) has different eigenvalues corresponding to b B l ( t ) and b B l ( t ) . (cid:3) Proposition 2.6 ([HLM19]) . If q = q , then the on-shell Bethe vector b B l ( t ) is singular. It is important to know if the on-shell Bethe vectors are non-zero. The following theorem is a particularcase of [HLPRS18, Theorem 4.1] which asserts that the square of the norm of the on-shell Bethe vector isessentially given by the Jacobian of the Bethe ansatz equation.Let θ , θ be differentiable functions in x . Denote by Wr( θ , θ ) the Wronskian of θ and θ , Wr( θ , θ ) = θ θ ′ − θ ′ θ . Theorem 2.7 ([HLPRS18]) . The square of the norm of the on-shell Bethe vector b B l ( t ) is given by B λ , b ( b B l ( t ) , b B l ( t )) = (cid:16) q q (cid:17) l l Y i =1 Wr( ϕ λ , b , ψ λ , b )( t i ) y ′ t ( t i ) . Proof. The statement for the case of q = q is proved in [HLPRS18]. For the case of q = q , the proof issimilar. One only needs to change the Bethe ansatz equation used in the proof of [HLPRS18, Lemma 7.1] tothe twisted case and alter the Korepin criteria (iii) and (iv) with appropriate multiple. (cid:3) Remark . Theorems 2.4 and 2.7 were proved for the case that t i = t j for i = j . However, they still holdwith the definition (2.19) which can be seen by analytic continuation, c.f. [Tar18, Theorem 3.2].2.6. Completeness of Bethe ansatz. We say that the Bethe ansatz is complete if the following conditionsare sastified,(1) on-shell Bethe vectors b B l ( t ) = 0 for all t such that y t divides q ϕ λ , b − q ψ λ , b are non-zero;(2) all eigenvectors of the transfer matrix T Q ( x ) in L ( λ , b ) if q = q and in ( L ( λ , b )) sing if q = q are of the form c b B l ( t ) where c ∈ C × and y t divides q ϕ λ , b − q ψ λ , b .Bethe vectors are obtained from the action of the Yangian on the vacuum vector | i . Therefore we restrictour interest to the case when L ( λ , b ) is cyclic. The cyclic modules are described in Lemma 2.1. Note that dim L ( λ , b ) = 2 k and dim( L ( λ , b )) sing = 2 k − . For generic b , the polynomial q ϕ λ , b ( x ) − q ψ λ , b ( x ) has no multiple roots and hence has the desired number of distinct monic divisors. Thus the algebraicBethe ansatz works well for generic situation. The following theorem is a minor generalization of [HLM19,Theorem A.6]. Theorem 2.9. Suppose L ( λ , b ) is an irreducible Y( gl | ) -module. In the case of q = q we assumein addition that | λ | 6 = 0 . If q ϕ λ , b − q ψ λ , b has no multiple roots, then the transfer matrix T Q ( x ) isdiagonalizable and the Bethe ansatz is complete. In particular, for any given λ and generic b , the transfermatrix T Q ( x ) is diagonalizable and the Bethe ansatz is complete.Proof. The proof is similar to that of [HLM19, Theorem A.6]. Here we only show that the on-shell Bethevectors b B l ( t ) are non-zero by using Theorem 2.7. Note that B λ , z ( b B l ( t ) , b B l ( t )) = q l q l l Y i =1 Wr( q ϕ λ , b − q ψ λ , b , ψ λ , b )( t i ) y ′ ( t i ) . It suffices to show Wr( q ϕ λ , b − q ψ λ , b , ψ λ , b )( t i ) = 0 . Since L ( λ , b ) is an irreducible Y( gl | ) -module, ϕ λ , b and ψ λ , b are relatively prime. So are q ϕ λ , b − q ψ λ , b and ψ λ , b . Note that (cid:0) q ϕ λ , b − q ψ λ , b (cid:1) ( t i ) = 0 and q ϕ λ , b − q ψ λ , b has no multiple roots, we have (cid:0) q ϕ λ , b − q ψ λ , b (cid:1) ′ ( t i ) = 0 and ψ λ , b ( t i ) = 0 . Hence Wr( q ϕ λ , b − q ψ λ , b , ψ λ , b )( t i ) = 0 .The second statement follows from the fact that the discriminant of q ϕ λ , b − q ψ λ , b is a non-zero poly-nomial in b = ( b , . . . , b k ) . It is not hard to prove this fact by considering the leading coefficient of b andusing induction on k . (cid:3) Now we study what happens if polynomial q ϕ λ , b − q ψ λ , b has multiple roots. UPERSYMMETRIC XXX SPIN CHAINS 11 Lemma 2.10. If L ( λ , b ) is an irreducible Y( gl | ) -module, then all on-shell Bethe vectors are non-zero.Proof. Let t be the solution of the Bethe ansatz equation represented by the polynomial q ϕ λ , b ( x ) − q ψ λ , b ( x ) itself (the largest monic divisor). By Theorem 2.7 and comparing the order of zeros and poles,one shows that the norm of the on-shell Bethe vector b B k ( t ) ( b B k − ( t ) if q = q ) is non-zero, c.f. the proofof Theorem 2.9. Since b B k ( t ) is obtained from all other on-shell Bethe vectors by applying a sequence of T ( x ) for proper x ’s with some scalar, all on-shell Bethe vectors are also non-zero. (cid:3) Since different on-shell Bethe vectors correspond to different eigenvalues of T Q ( x ) , see Theorem 2.4, theyare linearly independent. To show the completeness of Bethe ansatz, it suffices to show that all eigenvaluesof T Q ( x ) have the form (2.22) with a monic divisor y t of q ϕ λ , b ( x ) − q ψ λ , b ( x ) and that the correspondingeigenspaces have dimension one.Our first main theorem asserts that the Bethe ansatz is complete for irreducible tensor products of polyno-mial evaluation modules. Theorem 2.11. Let λ be a sequence of polynomial gl | -weights. If L ( λ , b ) is an irreducible Y( gl | ) -module, then the Bethe ansatz is complete. We will prove Theorem 2.11 in Section 5.3. 3. S PACE V S In this section, we discuss the super-analog of V + in [GRTV12, Section 2], c.f. [MTV14, Section 2]. Fix n ∈ Z > .The symmetric group S n acts naturally on C [ z , . . . , z n ] by permuting variables. We call it the standard action of S n on C [ z , . . . , z n ] . Denote by σ i ( z ) the i -th elementary symmetric polynomial in z , . . . , z n .The algebra of symmetric polynomials C [ z , . . . , z n ] S is freely generated by σ ( z ) , . . . , σ n ( z ) .Fix ℓ ∈ { , , . . . , n } . We have a subgroup S ℓ × S n − ℓ ⊂ S n . Then S ℓ permutes the first ℓ vari-ables while S n − ℓ permutes the last n − ℓ variables. Denote by C [ z , . . . , z n ] S ℓ × S n − ℓ the subalgebra of C [ z , . . . , z n ] consisting of S ℓ × S n − ℓ -invariant polynomials. It is known that C [ z , . . . , z n ] S ℓ × S n − ℓ is afree C [ z , . . . , z n ] S -module of rank (cid:0) nℓ (cid:1) .3.1. Definition of V S . Let V = ( C | ) ⊗ n be the tensor power of the vector representation of gl | . The gl | -module V has weight decomposition V = n M ℓ =0 ( V ) ( n − ℓ,ℓ ) . The space V has a basis v i ⊗ · · · ⊗ v i n , where i j ∈ { , } . Define I = { j | i j = 1 } and I = { j | i j =2 } . Then I = ( I , I ) gives a two-partition of the set { , , . . . , n } . We simply write v I for the vector v i ⊗ · · · ⊗ v i n . Denote by I ℓ the set of all two-partitions I of { , , . . . , n } such that | I | = ℓ . Then the setof vectors { v I | I ∈ I ℓ } forms a basis of ( V ) ( n − ℓ,ℓ ) .Let V be the space of polynomials in variables z = ( z , z , . . . , z n ) with coefficients in V , V = V ⊗ C [ z , z , . . . , z n ] . The space V is identified with the subspace V ⊗ of constant polynomials in V . The space V has a naturalgrading induced from the grading on C [ z , . . . , z n ] with deg( z i ) = 1 . Namely, the degree of an element v ⊗ p in V is given by the degree of the polynomial p , deg( v ⊗ p ) = deg p . Denote by F s V the subspacespanned by all elements of degree s . One has the increasing filtration F V ⊂ F V ⊂ · · · ⊂ V . Clearly,the space End( V ) has a filtration structure induced from that on V . Let P ( i,j ) be the graded flip operator which acts on the i -th and j -th factors of V . Let s , s , . . . , s n − bethe simple permutations of the symmetric group S n . Define the modified S n -action on V by the rule: ˆ s i : f ( z , . . . , z n ) P ( i,i +1) f ( z , . . . , z i +1 , z i , . . . , z n )+ f ( z , . . . , z n ) − f ( z , . . . , z i +1 , z i , . . . , z n ) z i − z i +1 , (3.1)for f ( z , . . . , z n ) ∈ V . Note that the modified S n -action respects the filtration on V . Denote the subspaceof all vectors in V invariant with respect to the modified S n -action by V S .Clearly, the gl | -action on V commutes with the modified S n -action on V and preserves the grading.Therefore, V S is a filtered gl | -module. Hence we have the weight decomposition for both V S and ( V S ) sing , V S = n M ℓ =0 ( V S ) ( n − ℓ,ℓ ) , ( V S ) sing = n M ℓ =0 ( V S ) sing( n − ℓ,ℓ ) . Note that ( V S ) ( n − ℓ,ℓ ) and ( V S ) sing( n − ℓ,ℓ ) are also filtered C [ z , . . . , z n ] S -modules.3.2. Properties of V S and ( V S ) sing . In this section, we describe properties of V S and ( V S ) sing . Lemma 3.1. The space ( V S ) ( n − ℓ,ℓ ) is a free C [ z , . . . , z n ] S -module of rank (cid:0) nℓ (cid:1) . In particular, the space V S is a free C [ z , . . . , z n ] S -module of rank n .Proof. The lemma is proved in Section 3.3. (cid:3) Lemma 3.2. The space ( V S ) sing( n − ℓ,ℓ ) is a free C [ z , . . . , z n ] S -module of rank (cid:0) n − ℓ (cid:1) . In particular, the space ( V S ) sing is a free C [ z , . . . , z n ] S -module of rank n − .Proof. The statement is proved in Section 3.3. (cid:3) For a Z > -filtered space M with finite-dimensional graded components F r M/ F r − M . Let gr( M ) bethe Z > -grading on M induced from this filtration. Then the graded character of gr( M ) , see (2.2), is givenby ch(gr( M )) = ∞ X r =0 (dim( F r M/ F r − M )) q r . Set ( q ) r = Q ri =1 (1 − q i ) . Proposition 3.3. We have ch (cid:0) gr(( V S ) ( n − ℓ,ℓ ) ) (cid:1) = q ℓ ( ℓ − / ( q ) ℓ ( q ) n − ℓ , ch (cid:0) gr(( V S ) sing( n − ℓ,ℓ ) ) (cid:1) = q ℓ ( ℓ +1) / ( q ) ℓ ( q ) n − − ℓ (1 − q n ) . Proof. The statement is proved in Section 3.3. (cid:3) Consider C | ⊗ V = C | ⊗ ( C | ) ⊗ n and label the factors by , , , . . . , n . Define a polynomial L ( x ) ∈ End( C | ⊗ V ) ⊗ C [ x, z , . . . , z n ] by L ( x ) = ( x − z n + P (0 ,n ) ) · · · ( x − z + P (0 , ) . Consider L ( x ) as a × matrix with entries L ij ( x ) ∈ End( V ) ⊗ C [ x, z , . . . , z n ] , i, j = 1 , .Define the assignment Γ : T ij ( x ) L ij ( x ) n Y r =1 ( x − z r ) − , i, j = 1 , . (3.2)Here we consider L ij ( x ) Q nr =1 ( x − z r ) − as a formal power series in x − whose coefficients are in End( V ) ⊗ C [ z , . . . , z n ] . UPERSYMMETRIC XXX SPIN CHAINS 13 Lemma 3.4. The map Γ defines a Y( gl | ) -action on V . Moreover, the Y( gl | ) -action on V preserves thefiltration, F r Y( gl | ) × F s V → F r + s V for any r, s ∈ Z > .Proof. The first statement follows from the Yang-Baxter equation (2.3), c.f. [MTV14, Lemma 3.1]. Thesecond statement is clear. (cid:3) Lemma 3.5. The Y( gl | ) -action on V defined by Γ commutes with the modified S n -action (3.1) on V andwith multiplication by the elements of C [ z , . . . , z n ] .Proof. The first statement follows again from the Yang-Baxter equation (2.3), c.f. [MTV14, Lemma 3.3].The second statement is straightforward. (cid:3) Therefore, it follows from Lemma 3.5 that the space V S is a filtered Y( gl | ) -module. Lemma 3.6. The Y( gl | ) -module V S is a cyclic module generated by v ⊗ n = v ⊗ · · · ⊗ v .Proof. The lemma is proved in Section 3.3. (cid:3) Given a = ( a , . . . , a n ) ∈ C n , let I S a be the ideal of C [ z , . . . , z n ] S generated by σ i ( z ) − σ i ( a ) , i =1 , . . . , n . Then for any a , by Lemmas 3.1 and 3.5, the quotient space V S /I S a V S is a Y( gl | ) -module ofdimension n over C . Proposition 3.7. Assume that a is ordered such that a i = a j + 1 for i > j . Then the Y( gl | ) -module V S /I S a V S is isomorphic to Y( gl | ) -module V ( a ) = C | ( a ) ⊗ · · · ⊗ C | ( a n ) .Proof. Thanks to Lemmas 2.1, 3.1, and 3.6, the proof is similar to that of [MTV14, Proposition 3.5]. (cid:3) Proofs of properties. We start with Lemma 3.1.Define the modified S n -action on scalar functions in z , . . . , z n by the rule, see e.g. [GRTV12, formula(2.1)], c.f. also (3.1), ˆ s i : f ( z , . . . , z n ) f ( z , . . . , z i +1 , z i , . . . , z n ) − f ( z , . . . , z n ) − f ( z , . . . , z i +1 , z i , . . . , z n ) z i − z i +1 . Let f ( z , . . . , z n ) be a ( V ) ( n − ℓ,ℓ ) -valued function with coordinates { f I ( z , . . . , z n ) | I ∈ I ℓ } , f ( z , . . . , z n ) = X I ∈I ℓ f I ( z , . . . , z n ) v I . For σ ∈ S n and I = ( i , . . . , i n ) , define the natural S n -action on I ℓ σ : I ℓ → I ℓ , I = ( i , . . . , i n ) σ ( I ) = ( i σ − (1) , . . . , i σ − ( n ) ) . Lemma 3.8. The function f ( z , . . . , z n ) is invariant under modified S n -action (3.1) if and only if for anysimple reflection s j and I = ( i , . . . , i n ) , we have f s j ( I ) = ( − ˆ s j f I , if i j = i j +1 = 2 , ˆ s j f I , otherwise . (cid:3) A two-partition I ∈ I ℓ corresponds to a permutation σ I ∈ S n as follows: ( σ I ) − ( j ) = ( { r | r j, i r = 2 } , if i j = 2 ,ℓ + { r | r j, i r = 1 } , if i j = 1 . (3.3)Set I max = (2 , . . . , , , . . . , ∈ I ℓ . Then σ I ( I max ) = I . For any f ∈ C [ z , . . . , z n ] S ℓ × S n − ℓ , let ˇ f = f · Y r Define the map ϑ ℓ : C [ z , . . . , z n ] S ℓ × S n − ℓ → ( V S ) ( n − ℓ,ℓ ) , ϑ ℓ ( f ) = X I ∈I ℓ ˆ σ I ( ˇ f ) v I . We show that the map ϑ ℓ is a well-defined C [ z , . . . , z n ] S -module isomorphism.We first show that ϑ ℓ is well-defined. Clearly, ϑ ℓ ( f ) ∈ ( V ) ( n − ℓ,ℓ ) . Hence it suffices to show that ϑ ℓ ( f ) ∈V S . By Lemma 3.8, it reduces to show that for any j n − , ˆ σ s j ( I ) ( ˇ f ) = ( − ˆ s j ˆ σ I ( ˇ f ) , if i j = i j +1 = 2 , ˆ s j ˆ σ I ( ˇ f ) , otherwise . (3.4)If s j ( I ) = I , then i j = i j +1 and σ s j ( I ) = σ I . By (3.3) we have ( σ I ) − · s j · σ I = (( σ I ) − ( j ) , ( σ I ) − ( j + 1)) ∈ ( S ℓ × , if i j = i j +1 = 2 , × S n − ℓ , if i j = i j +1 = 1 . Since ˇ f is anti-symmetric with respect to the modified S ℓ × -action and symmetric with respect to themodified × S n − ℓ -action, the equation (3.4) follows. If s j ( I ) = I , then i j = i j +1 . Clearly, we have σ s j ( I ) = s j σ I . Hence the equation (3.4) also follows. Thus the map ϑ ℓ is well-defined.Let f ( z , . . . , z n ) ∈ ( V S ) ( n − ℓ,ℓ ) , f ( z , . . . , z n ) = X I ∈I ℓ f I ( z , . . . , z n ) v I . It follows from Lemma 3.8 that f is uniquely determined by f I max . Moreover, f I max is anti-symmetric withrespect to the modified S ℓ × -action and symmetric with respect to the modified × S n − ℓ -action. A directcomputation implies that f I max · Y r The space V is a gl | [ t ] -module where e ij [ r ] acts by e ij [ r ]( p ( z , . . . , z n ) w ⊗ · · · ⊗ w n )= p ( z , . . . , z n ) n X s =1 ( − ( | w | + ··· + | w s − | )( | i | + | j | ) z rs w ⊗ · · · ⊗ e ij w s ⊗ · · · ⊗ w n , for p ( z , . . . , z n ) ∈ C [ z , . . . , z n ] and w s ∈ C | .The gl | [ t ] -action on V commutes with the standard S n -action on V , therefore V S is a gl | [ t ] -module.The proof of the following lemma is similar to that of Lemma 3.1. Lemma 3.9. The space ( V S ) ( n − ℓ,ℓ ) is a free C [ z , . . . , z n ] S -module of rank (cid:0) nℓ (cid:1) . In particular, the space V S is a free C [ z , . . . , z n ] S -module of rank n . (cid:3) Lemma 3.10. The gl | [ t ] -module V S is a cyclic module generated by v ⊗ n = v ⊗ · · · ⊗ v .Proof. The proof is similar to that of [MTV09, Lemma 2.11]. (cid:3) Lemma 3.11. The set { e [ r ] e [ r ] · · · e [ r ℓ ] v + | r < r < · · · < r ℓ n − } (3.5) is a free generating set of ( V S ) ( n − ℓ,ℓ ) over C [ z , . . . , z n ] S . (cid:3) Proof. We use the shorthand notation v + for v ⊗ n . Clearly, we have e [ r ] v + = n X s =1 z rs v ⊗ · · · ⊗ v ⊗ · · · ⊗ v , where v is in the s -th factor. Note that z ns + n X i =1 ( − i σ i ( z ) z n − is = 0 and { , z s , . . . , z n − s } is linearly independent over C [ z , . . . , z n ] S , therefore e [ n ] v + = n X i =1 ( − i − σ i ( z ) e [ n − i ] v + and { e [0] v + , e [1] v + , . . . , e [ n − v + } is linearly independent over C [ z , . . . , z n ] S . Similarly, oneshows that each e [ r ] v + is spanned by e [0] v + , e [1] v + , . . . , e [ n − v + over C [ z , . . . , z n ] S . ByLemma 3.9, ( V S ) ( n − , is free over C [ z , . . . , z n ] S of rank n , therefore e [0] v + , e [1] v + , . . . , e [ n − v + are free generators of ( V S ) ( n − , over C [ z , . . . , z n ] S .Since e [ r ] e [ s ] = − e [ s ] e [ r ] , the case of general ℓ is proved similarly. (cid:3) Lemma 3.12. The space ( V S ) sing( n − ℓ,ℓ ) is a free C [ z , . . . , z n ] S -module of rank (cid:0) n − ℓ (cid:1) with a free generatingset given by { e [0] e [0] e [ r ] · · · e [ r ℓ ] v + , r < r < · · · < r ℓ n − } . (3.6) In particular, the space ( V S ) sing is a free C [ z , . . . , z n ] S -module of rank n − . Proof. Let w be a non-zero singular vector in ( V S ) sing( n − ℓ,ℓ ) , then e [0] e [0] w = ( e [0] + e [0]) w − e [0] e [0] w = ( e [0] + e [0]) w = nw. By Lemma 3.11, we have that w is a linear combination of vectors in (3.6) over C [ z , . . . , z n ] S .Now it suffices to show that the set given in (3.6) is linearly independent over C [ z , . . . , z n ] S . Note that e [0] e [0] e [ r ] · · · e [ r ℓ ] v + = ne [ r ] · · · e [ r ℓ ] v + + ℓ X i =1 ( − i p r i ( z ) e [0] e [ r ] · · · \ e [ r i ] · · · e [ r ℓ ] v + , where p j ( z ) = P ns =1 z js and \ e [ r i ] means the factor e [ r i ] is skipped. Therefore the statement followsfrom Lemma 3.11. (cid:3) We are ready to prove Proposition 3.3. Note that gr(( V S ) ( n − ℓ,ℓ ) ) = ( V S ) ( n − ℓ,ℓ ) , gr(( V S ) sing( n − ℓ,ℓ ) ) = ( V S ) sing( n − ℓ,ℓ ) . Proof of Proposition 3.3. By Lemma 3.9, the graded character ch (cid:0) ( V S ) ( n − ℓ,ℓ ) (cid:1) is equal to the graded char-acter of the space spanned by the set (3.5) over C multiplying with the graded character of C [ z , . . . , z n ] S .We have ch (cid:0) C [ z , . . . , z n ] S (cid:1) = 1 / ( q ) n , therefore ch (cid:0) gr(( V S ) ( n − ℓ,ℓ ) ) (cid:1) = ch (cid:0) ( V S ) ( n − ℓ,ℓ ) (cid:1) = q ℓ ( ℓ − / · ( q ) n ( q ) ℓ ( q ) n − ℓ · q ) n = q ℓ ( ℓ − / ( q ) ℓ ( q ) n − ℓ . Similarly, by Lemma 3.12, ch (cid:0) gr(( V S ) sing( n − ℓ,ℓ ) ) (cid:1) = ch (cid:0) ( V S ) sing( n − ℓ,ℓ ) (cid:1) = q ℓ ( ℓ +1) / · ( q ) n − ( q ) ℓ ( q ) n − − ℓ · q ) n = q ℓ ( ℓ +1) / ( q ) ℓ ( q ) n − − ℓ (1 − q n ) . (cid:3) 4. M AIN THEOREMS The algebra O l . Let Ω l be the n -dimensional affine space with coordinates f , . . . , f l , g , . . . , g n − l − and Σ n . Introduce two polynomials f ( x ) = x l + l X i =1 f i x l − i , g ( x ) = x n − l − + n − l − X i =1 g i x n − l − i − . (4.1)Denote by O l the algebra of regular functions on Ω l , namely O l = C [ f , . . . , f l , g , . . . , g n − l − , Σ n ] . Definethe degree function by deg f i = i, deg g j = j, deg Σ n = n, for all i = 1 , . . . , l and j = 1 , . . . , n − l − . The algebra O l is graded with the graded character given by ch( O l ) = 1( q ) l ( q ) n − l − (1 − q n ) . (4.2)Let F O l ⊂ F O l ⊂ · · · ⊂ O l be the increasing filtration corresponding to this grading, where F s O l consists of elements of degree at most s .Let Σ , . . . , Σ n − be the elements of O l such that nf ( x ) g ( x ) = (cid:16) ( x + 1) n + n − X i =1 ( − i Σ i ( x + 1) n − i (cid:17) − (cid:16) x n + n − X i =1 ( − i Σ i x n − i (cid:17) . (4.3)The homomorphism π l : C [ z , . . . , z n ] S → O l , σ i ( z ) Σ i , i = 1 , . . . , n. (4.4) UPERSYMMETRIC XXX SPIN CHAINS 17 is injective and induces a C [ z , . . . , z n ] S -module structure on O l .Express nf ( x − g ( x ) as follows, nf ( x − g ( x ) = nx n − + n − X i =1 G i x n − − i , (4.5)where G i ∈ O l . Lemma 4.1. The elements G i and Σ j , i = 1 , . . . , n − , j = 1 , . . . , n , generate the algebra O l . (cid:3) Lemma 4.2. We have G i ∈ F i O l \ F i − O l and Σ j ∈ F j O l \ F j − O l for i = 1 , . . . , n − , j =1 , . . . , n . (cid:3) The algebra O Ql . We rework Section 4.1 for the case of q = q .Let q = q . Let Ω Ql be the n -dimensional affine space with coordinates f , . . . , f l and g , . . . , g n − l .Introduce two polynomials f ( x ) = x l + l X i =1 f i x l − i , g ( x ) = x n − l + n − l X i =1 g i x n − l − i . Denote by O Ql the algebra of regular functions on Ω Ql , namely O Ql = C [ f , . . . , f l , g , . . . , g n − l ] . Let deg f i = i and deg g j = j for all i = 1 , . . . , l and j = 1 , . . . , n − l . The algebra O Ql is graded with thegraded character ch( O Ql ) = 1( q ) l ( q ) n − l . (4.6)Let F O Ql ⊂ F O Ql ⊂ · · · ⊂ O Ql be the increasing filtration corresponding to this grading, where F s O Ql consists of elements of degree no greater than s .Let Σ , . . . , Σ n be the elements of O Ql such that ( q − q ) f ( x ) g ( x ) = q (cid:16) ( x + 1) n + n X i =1 ( − i Σ i ( x + 1) n − i (cid:17) − q (cid:16) x n + n X i =1 ( − i Σ i x n − i (cid:17) . The homomorphism π Ql : C [ z , . . . , z n ] S → O Ql , σ i ( z ) Σ s , i = 1 , . . . , n. is injective and induces a C [ z , . . . , z n ] S -module structure on O Ql .Express ( q − q ) f ( x − g ( x ) as follows, ( q − q ) f ( x − g ( x ) = ( q − q ) x n + n X i =1 G Qi x n − i , where G Qi ∈ O Ql . Lemma 4.3. The elements G Qi and Σ i , i = 1 , . . . , n , generate the algebra O Ql . (cid:3) Lemma 4.4. We have G Qi ∈ F i O Ql \ F i − O Ql and Σ i ∈ F i O Ql \ F i − O Ql for i = 1 , . . . , n . (cid:3) Berezinian and Bethe algebra. We follow the convention of [MR14]. Let A be a superalgebra. Con-sider the operators of the form K = X i,j =1 ( − | i || j | + | j | E ij ⊗ K ij ∈ End( C | ) ⊗ A , where K ij are elements of A of parity | i | + | j | . We say that K is a Manin matrix if [ K ij , K rs ] = ( − | i || j | + | i || r | + | j || r | [ K rj , K is ] for all i, j, r, s = 1 , . In other words, K is Manin if and only if [ K , K ] = [ K , K ] = 0 , [ K , K ] = [ K , K ] . Assume that K , K , and K − K K − K are all invertible. Define the Berezinian of K by Ber( K ) = K ( K − K K − K ) − . Then it is a straightforward check, see also [MR14, Corollary 2.16], that we have three other ways to writethe Berezinian: Ber( K ) = ( K + K K − K ) − K = K − ( K − K K − K ) = ( K + K K − K ) K − . (4.7)Let τ be the difference operator, ( τ f )( x ) = f ( x − for any function f in x . Let A be the superalgebra Y( gl | )[ τ ] , where τ has parity ¯0 . Consider the operator Z Q ( x, τ ) , Z Q ( x, τ ) = T t ( x ) Q t τ = T t ( x ) Qτ ∈ End( C | ) ⊗ Y( gl | )[ τ ] . It follows from (2.4) or (2.5) that Z Q ( x, τ ) is a Manin matrix, see e.g. [MR14, Remark 2.12]. Note that ourgenerating series T ij ( x ) corresponds to z ji ( u ) in [MR14].Denote Ber Q ( x ) = Ber( Z Q ( x, τ )) . Then Ber Q ( x ) is a scalar operator (does not contain τ ), moreover, Ber Q ( x ) q /q does not depend on Q .Expand Ber Q ( x ) as a power series in x − with coefficients in Y( gl | ) . The following proposition wasconjectured by Nazarov [Naz91, Conjecture 2] and proved by Gow [Gow07, Theorem 4] for the general caseof Y( gl m | n ) . Proposition 4.5. The coefficients of Ber Q ( x ) generate the center of Y( gl | ) . (cid:3) We call the subalgebra of Y( gl | ) generated by the coefficients of Ber Q ( x ) and T Q ( x ) the Bethe algebraassociated to Q , c.f. Remark 6.12. We denote the Bethe algebra associated to Q by B Q . We simply write B for B I , where I is the identity matrix.Let λ = ( λ (1) , . . . , λ ( k ) ) be a sequence of polynomial gl | -weights. Let a = ( a , . . . , a n ) and b =( b , . . . , b k ) be two sequences of complex numbers. Recall that we have three kinds of modules, V S , V ( a ) = C | ( a ) ⊗ · · · ⊗ C | ( a n ) , and L ( λ , b ) = L λ (1) ( b ) ⊗ · · · ⊗ L λ ( k ) ( b k ) .Our main problem is to understand the spectrum of the Bethe algebra acting on L ( λ , b ) , when L ( λ , b ) isa cyclic Y( gl | ) -module.If q = q , let B Ql ( z ) , B Ql ( a ) , and B Ql ( λ , b ) denote, respectively, the images of the Bethe algebra B Q in End(( V S ) ( n − l,l ) ) , End(( V ( a )) ( n − l,l ) ) , and End(( L ( λ , b )) ( n − l,l ) ) . For any element X Q ∈ B Q , we denoteby X Q ( z ) , X Q ( a ) , X Q ( λ , b ) the respective linear operators.If q = q = 1 , let B l ( z ) , B l ( a ) , and B l ( λ , b ) denote, respectively, the images of the Bethe algebra B in End(( V S ) sing( n − l,l ) ) , End(( V ( a )) sing( n − l,l ) ) , and End(( L ( λ , b )) sing( n − l,l ) ) . For any element X ∈ B , we denote by X ( z ) , X ( a ) , X ( λ , b ) the respective linear operators.By abuse of language, we call the image of Bethe algebra again Bethe algebra. UPERSYMMETRIC XXX SPIN CHAINS 19 Since by Lemma 3.6 the Y( gl | ) -module V S is generated by v ⊗ n = v ⊗ · · · ⊗ v , the series Ber Q ( x ) acts on V S by multiplication by the series q q · ( x − z + 1) · · · ( x − z n + 1)( x − z ) · · · ( x − z n ) . Therefore there exist uniquely central elements C , . . . , C n of Y( gl | ) of minimal degrees such that each C i acts on V S by multiplication by σ i ( z ) .Define B Qi ∈ B Q by (cid:16) x n + n X i =1 ( − i C i x n − i (cid:17) T Q ( x ) = x n (cid:16) ( q − q ) + ∞ X i =1 B Qi x − i (cid:17) . (4.8)We write simply B i for B Ii , where I is the identity matrix.Recall the Y( gl | ) -action on V S from (3.2), we have B Qi ( z ) = 0 for i > n . When Q = I , we have B ( z ) = n . Lemma 4.6. The elements B Qi ( z ) and C i ( z ) , i = 1 , . . . , n , generate the algebra B Ql ( z ) . (cid:3) Lemma 4.7. We have C i ∈ F i B Q \ F i − B Q , B i ∈ F i − B \ F i − B , and B Qi ∈ F i B Q \ F i − B Q for i = 1 , . . . , n . (cid:3) Main theorems for the case q = q . Recall from Proposition 3.3 that there exists a unique vector (upto proportionality) of degree l ( l + 1) / in ( V S ) sing( n − l,l ) explicitly given by u l := T (1)21 T (1)12 T (2)12 · · · T ( l +1)12 v + ,see Lemma 3.12.Any commutative algebra A is a module over itself induced by left multiplication. We call it the regularrepresentation of A . The dual space A ∗ is naturally an A -module which is called the coregular represen-tation . A bilinear form ( ·|· ) : A ⊗ A → C is called invariant if ( ab | c ) = ( a | bc ) for all a, b, c ∈ A . Afinite-dimensional commutative algebra A admitting an invariant non-degenerate symmetric bilinear form ( ·|· ) : A ⊗ A → C is called a Frobenius algebra . The regular and coregular representations of a Frobeniusalgebra are isomorphic.Let M be an A -module and E : A → C a character, then the A -eigenspace associated to E in M isdefined by T a ∈A ker( a | M − E ( a )) . The generalized A -eigenspace associated to E in M is defined by T a ∈A (cid:0) S ∞ m =1 ker( a | M − E ( a )) m (cid:1) . Theorem 4.8. The action of the Bethe algebra B l ( z ) on ( V S ) sing( n − l,l ) has the following properties. (i) The map η l : G i B i +1 ( z ) , Σ j C j ( z ) , i = 1 , . . . , n − , j = 1 , . . . , n , extends uniquely to anisomorphism η l : O l → B l ( z ) of filtered algebras. Moreover, the isomorphism η l is an isomorphismof C [ z , . . . , z n ] S -modules. (ii) The map ρ l : O l ( V S ) sing( n − l,l ) , F η l ( F ) u l , is an isomorphism of filtered vector spaces identify-ing the B l ( z ) -module ( V S ) sing( n − l,l ) with the regular representation of O l . Theorem 4.8 is proved in Section 5.Let λ = ( λ (1) , . . . , λ ( k ) ) be a sequence of polynomial gl | -weights such that | λ | = n . Let b =( b , . . . , b k ) be a sequence of complex numbers. Suppose the Y( gl | ) -module L ( λ , b ) is cyclic. Theorem 4.9. The action of the Bethe algebra B l ( λ , b ) on ( L ( λ , b )) sing( n − l,l ) has the following properties. (i) The Bethe algebra B l ( λ , b ) is isomorphic to C [ w , . . . , w k − ] S l × S k − l − / h σ i ( w ) − ε i i i =1 ,...,k − , where ε i are given by ϕ λ , b ( x ) − ψ λ , b ( x ) = n ( x k − + P k − i =1 ( − i ε i x k − − i ) and σ i ( w ) are ele-mentary symmetric functions in w , . . . , w k − . Under this isomorphism, Q ks =1 ( x − b s ) T ( x ) corre-sponds to n Q li =1 ( x − w i − Q k − j = l +1 ( x − w j ) . (ii) The Bethe algebra B l ( λ , b ) is a Frobenius algebra. Moreover, the B l ( λ , b ) -module ( L ( λ , b )) sing( n − l,l ) is isomorphic to the regular representation of B l ( λ , b ) . (iii) The Bethe algebra B l ( λ , b ) is a maximal commutative subalgebra in End(( L ( λ , b )) sing( n − l,l ) ) of di-mension (cid:0) k − l (cid:1) . (iv) Every B -eigenspace in ( L ( λ , b )) sing( n − l,l ) has dimension one. (v) The B -eigenspaces in ( L ( λ , b )) sing( n − l,l ) bijectively correspond to the monic degree l divisors y ( x ) ofthe polynomial ϕ λ , b ( x ) − ψ λ , b ( x ) . Moreover, the eigenvalue of T ( x ) corresponding to the monicdivisor y is described by E Iy, λ , b ( x ) , see (2.22) . (vi) Every generalized B -eigenspace in ( L ( λ , b )) sing( n − l,l ) is a cyclic B -module. (vii) The dimension of the generalized B -eigenspace associated to E Iy, λ , b ( x ) is Y a ∈ C (cid:18) Mult a ( ϕ λ , b − ψ λ , b )Mult a ( y ) (cid:19) , where Mult a ( p ) is the multiplicity of a as a root of the polynomial p . Theorem 4.9 is proved in Section 5.4.5. Main theorems for the case q = q . Recall from Proposition 3.3 that there exists a unique vector (upto proportionality) of degree l ( l − / in ( V S ) ( n − l,l ) explicitly given by u Ql := T (1)12 T (2)12 · · · T ( l )12 v + , seeLemma 3.11. Theorem 4.10. The action of the Bethe algebra B Ql ( z ) on ( V S ) ( n − l,l ) has the following properties. (i) The map η Ql : G Qi B Qi ( z ) , Σ i C i ( z ) , i = 1 , . . . , n , extends uniquely to an isomorphism η Ql : O Ql → B Ql ( z ) of filtered algebras. Moreover, the isomorphism η Ql is an isomorphism of C [ z , . . . , z n ] S -modules. (ii) The map ρ Ql : O Ql ( V S ) ( n − l,l ) , F η l ( F ) u Ql , is an isomorphism of filtered vector spacesidentifying the B Ql ( z ) -module ( V S ) ( n − l,l ) with the regular representation of O Ql . Theorem 4.10 is proved in Section 5.Let λ = ( λ (1) , . . . , λ ( k ) ) be a sequence of polynomial gl | -weights such that | λ | = n . Let b =( b , . . . , b k ) be a sequence of complex numbers. Suppose the Y( gl | ) -module L ( λ , b ) is cyclic. Theorem 4.11. The action of the Bethe algebra B Ql ( λ , b ) on ( L ( λ , b )) ( n − l,l ) has the following properties. (i) The Bethe algebra B Ql ( λ , b ) is isomorphic to C [ w , . . . , w k ] S l × S k − l / h σ i ( w ) − ε i i i =1 ,...,k , where ε i are given by q ϕ λ , b ( x ) − q ψ λ , b ( x ) = ( q − q )( x k + P ki =1 ( − i ε i x k − i ) and σ i ( w ) are elementary symmetric functions in w , . . . , w k . Under this isomorphism, Q ks =1 ( x − b s ) T Q ( x ) corresponds to ( q − q ) Q li =1 ( x − w i − Q kj = l +1 ( x − w j ) . (ii) The Bethe algebra B Ql ( λ , b ) is a Frobenius algebra. Moreover, the B Ql ( λ , b ) -module ( L ( λ , b )) ( n − l,l ) is isomorphic to the regular representation of B Ql ( λ , b ) . (iii) The Bethe algebra B Ql ( λ , b ) is a maximal commutative subalgebra in End(( L ( λ , b )) ( n − l,l ) ) ofdimension (cid:0) kl (cid:1) . UPERSYMMETRIC XXX SPIN CHAINS 21 (iv) Every B Q -eigenspace in ( L ( λ , b )) ( n − l,l ) has dimension one. (v) The B Q -eigenspaces in ( L ( λ , b )) ( n − l,l ) bijectively correspond to the monic degree l divisors y ( x ) of the polynomial q ϕ λ , b ( x ) − q ψ λ , b ( x ) . Moreover, the eigenvalue of T Q ( x ) corresponding to themonic divisor y ( x ) is described by E Qy, λ , b ( x ) , see (2.22) . (vi) Every generalized B Q -eigenspace in ( L ( λ , b )) ( n − l,l ) is a cyclic B Q -module. (vii) The dimension of the generalized B Q -eigenspace associated to E Qy, λ , b ( x ) is Y a ∈ C (cid:18) Mult a ( q ϕ λ , b − q ψ λ , b )Mult a ( y ) (cid:19) , where Mult a ( p ) is the multiplicity of a as a root of the polynomial p . Theorem 4.11 is proved in Section 5.5. P ROOF OF MAIN THEOREMS In this section, we prove the main theorems.5.1. The first isomorphism. Proof of Theorem 4.8. We first show the homomorphism defined by η l is well-defined.Consider the tensor product V ( a ) = C | ( a ) ⊗ · · · ⊗ C | ( a n ) , where a i ∈ C , and the correspondingBethe ansatz equation associated to weight ( n − l, l ) . Let t be a solution with distinct coordinates and b B l ( t ) the corresponding on-shell Bethe vector. Denote E i, t the eigenvalues of B i acting on b B l ( t ) , see Theorem 2.4.Define a character π : O l → C by sending f ( x ) y t ( x ) , g ( x ) ny t ( x ) (cid:16) n Y i =1 ( x − a i + 1) − n Y i =1 ( x − a i ) (cid:17) , Σ n n Y i =1 a i . Then π ( Σ i ) = σ i ( a ) , π ( G i ) = E i, t , (5.1)by (4.3) and by (2.22), (4.5), respectively.Let now P ( G i , Σ j ) be a polynomial in G i , Σ j such that P ( G i , Σ j ) is equal to zero in O l . It suffices toshow P ( B i ( z ) , C j ( z )) is equal to zero in B l ( z ) .Note that P ( B i ( z ) , C j ( z )) is a polynomial in z , . . . , z n with values in End(( V ) sing( n − l,l ) ) . For any se-quence a of complex numbers, we can evaluate P ( B i ( z ) , C j ( z )) at z = a to an operator on V ( a ) sing( n − l,l ) . ByTheorem 2.9, the transfer matrix T ( x ) is diagonalizable and the Bethe ansatz is complete for ( V ( a )) sing( n − l,l ) when a ∈ C n is generic. Hence by (5.1) the value of P ( B i ( z ) , C j ( z )) at z = a is also equal to zero forgeneric a . Therefore P ( B i ( z ) , C j ( z )) is identically zero and the map η l is well-defined.Let us now show that the map η l is injective. Let P ( G i , Σ j ) be a polynomial in G i , Σ j such that P ( G i , Σ j ) is non-zero in O l . Then the value at a generic point of Ω l (e.g. the non-vanishing points of P ( G i , Σ j ) suchthat f and g are relatively prime and have only simple zeros) is not equal to zero. Moreover, at those pointsthe transfer matrix T ( x ) is diagonalizable and the Bethe ansatz is complete again by Theorem 2.9. Therefore,again by (5.1), the polynomial P ( B i ( z ) , C j ( z )) is a non-zero element in B l ( z ) . Thus the map η l is injective.The surjectivity of η l follows from Lemma 4.6. Hence η l is an isomorphism of algebras.The fact that η l is an isomorphism of filtered algebra respecting the filtration follows from Lemmas 4.2and 4.7. This completes the proof of part (i). The kernel of ρ l is an ideal of O l . Note that the algebra O l contains the algebra C [ z , . . . , z n ] S if weidentify σ i ( z ) with Σ i , see (4.4). The kernel of ρ l intersects C [ z , . . . , z n ] S trivially. Therefore the ker-nel of ρ l is trivial as well. Hence ρ l is an injective map. Comparing (4.2) and Proposition 3.3, we have ch (cid:0) gr(( V S ) sing( n − ℓ,ℓ ) ) (cid:1) = q l ( l +1) / ch( O l ) . Thus ρ l is an isomorphism of filtered vector spaces which shifts thedegree by l ( l + 1) / , completing the proof of part (ii). (cid:3) Proof of Theorem 4.10. The proof of Theorem 4.10 is similar to that of Theorem 4.8 with the help of (4.6). (cid:3) The second isomorphism. Let a = ( a , . . . , a n ) be a sequence of complex numbers such that a i = a j + 1 for i > j . Let I O l, a be the ideal of O l generated by the elements Σ i − σ i ( a ) , i = 1 , . . . , n , where Σ , . . . , Σ n − are defined in (4.3). Let O l, a be the quotient algebra O l, a = O l /I O l, a . Let I B l, a be the ideal of B l ( z ) generated by C i ( z ) − σ i ( a ) , i = 1 , . . . , n . Consider the subspace I M l, a = I B l, a ( V S ) sing( n − l,l ) = ( I S a V S ) sing( n − l,l ) , where I S a as before is the ideal of C [ z , . . . , z n ] S generated by σ i ( z ) − σ i ( a ) . Lemma 5.1. We have η l ( I O l, a ) = I B l, a , ρ l ( I O l, a ) = I M l, a , B l ( a ) = B l ( z ) /I B l, a , ( V ( a )) sing( n − l,l ) = ( V S ) sing( n − l,l ) /I M l, a . Proof. The lemma follows from Theorem 4.8 and Proposition 3.7. (cid:3) We prove part (ii) of Theorem 4.9 for the special case V ( a ) .By Lemma 5.1, the maps η l and ρ l induce the maps η l, a : O l, a → B l ( a ) , ρ l, a : O l, a → ( V ( a )) sing( n − l,l ) . The map η l, a is an isomorphism of algebras. Since B l ( a ) is finite-dimensional, by e.g. [MTV09, Lemma3.9], O l, a is a Frobenius algebra, so is B l ( a ) . The map ρ l, a is an isomorphism of vector spaces. Moverover,it follows from Theorem 4.8 and Lemma 5.1 that ρ l, a identifies the regular representation of O l, a with the B l ( a ) -module ( V ( a )) sing( n − l,l ) . Therefore part (ii) is proved for the case of V ( a ) .5.3. The third isomorphism. Recall from Section 2.5, that without loss of generality, we can assume that λ ( s )2 = 0 , s = 1 , . . . , k . Rearrange the sequences { b s , b s − , . . . , b s − λ ( s )1 + 1 } , s = 1 , . . . , k, to a single sequence in decreasing order with respect to the real parts. Denote this new sequence by a =( a , . . . , a n ) (recall that | λ | = n ). Then by Lemma 2.1, V ( a ) is cyclic. Lemma 5.2. If L ( λ , b ) is cyclic, then there exists a surjective Y( gl | ) -module homomorphism from V ( a ) to L ( λ , b ) which maps vacuum vector to vacuum vector.Proof. Rearrange the sequences { b s , b s − , . . . , b s − λ ( s )1 + 1 } , s = 1 , . . . , k, to a single sequence in the same order displayed as s runs from to k . Denote this new sequence by a .Clearly, we have a surjective Y( gl | ) -module homomorphism V ( a ) ։ L ( λ , b ) which maps vacuum vectorto vacuum vector. Therefore it suffices to show that there is a Y( gl | ) -module homomorphism V ( a ) → V ( a ) which preserves the vacuum vector. UPERSYMMETRIC XXX SPIN CHAINS 23 We have the Y( gl | ) -module homomorphism P ◦ R ( a − b ) : C | ( a ) ⊗ C | ( b ) → C | ( b ) ⊗ C | ( a ) . Here P is the graded flip operator, R ( a − b ) is the R-matrix, see (2.14), and a, b are complex numbers. Notethat if ℜ a − ℜ b > , then P ◦ R ( a − b ) is well-defined and preserves the vacuum vector. Since ℜ a i > ℜ a j for i < j n , we obtain a by permuting elements in a via a sequence of simple reflections whichmove numbers with larger real parts through numbers with smaller real parts from left to right. Therefore a Y( gl | ) -module homomorphism V ( a ) → V ( a ) which preserves the vacuum vector exists.Since L ( λ , b ) is cyclic, the map we constructed is surjective. (cid:3) By Proposition 3.7, the surjective Y( gl | ) -module homomorphism V ( a ) ։ L ( λ , b ) induces a surjective Y( gl | ) -module homomorphism V S ։ L ( λ , b ) . The second map then induces a projection of the Bethealgebras B l ( z ) ։ B l ( λ , b ) . We describe the kernel of this projection. We consider the corresponding idealin the algebra O l .Suppose l k − . Define the polynomial h ( x ) by h ( x ) = k Y s =1 λ ( s )1 − Y i =1 ( x − b s + i ) . If L ( λ , b ) is irreducible then h ( x ) is the greatest common divisor of Q ni =1 ( x − a i + 1) and Q ni =1 ( x − a i ) .Divide the polynomial g ( x ) in (4.1) by h ( x ) and let p ( x ) = x k − l − + p x k − l − + · · · + p k − l − x + p k − l − , (5.2) r ( x ) = r x n − k − + r x n − k − + · · · + r n − k − x + r n − k (5.3)be the quotient and the remainder, respectively. Clearly, p i , r j ∈ O l .Denote by I O l, λ , b the ideal of O l generated by r , . . . , r n − k , Σ n − a · · · a n , and the coefficients of poly-nomial k Y s =1 ( x − b s + λ ( s )1 ) − k Y s =1 ( x − b s ) − np ( x ) f ( x ) . Let O l, λ , b be the quotient algebra O l, λ , b = O l /I O l, λ , b . Clearly, if O l, λ , b is finite-dimensional, then it is a Frobenius algebra.Let I B l, λ , b be the image of I O l, λ , b under the isomorphism η l . Lemma 5.3. The ideal I B l, λ , b is contained in the kernel of the projection B l ( z ) ։ B l ( λ , b ) .Proof. We treat b = ( b , . . . , b k ) as variables. Note that the elements of I B l, λ , b act on ( L ( λ , b )) sing( n − l,l ) poly-nomially in b . Therefore it suffices to show it for generic b . Let f ( x ) be the image of f ( x ) under η l . Thecondition that I B l, λ , b vanishes is equivalent to the condition that ϕ λ , b ( x ) − ψ λ , b ( x ) is divisible by f ( x ) .By Theorems 2.9, there exists a common eigenbasis of the transfer matrix T ( x ) in ( L ( λ , b )) sing( n − l,l ) forgeneric b . Let ω = ( ω , . . . , ω ) consisting of n of ω = (1 , . Clearly, a solution of Bethe ansatzequation associated to λ , b , l is also a solution to Bethe ansatz equation associated to ω , a , l . Moreover,the expressions of corresponding on-shell Bethe vectors differ by a scalar multiple (with different vacuumvectors). By Lemma 5.2 and Theorem 2.4, ϕ λ , b ( x ) − ψ λ , b ( x ) is divisible by f ( x ) for generic b since theeigenvalue of f ( x ) corresponding to y t ( x ) in (2.22). Therefore I B l, λ , b vanishes for generic b , completing theproof. (cid:3) Therefore, we have the epimorphism O l, λ , b ∼ = B l ( z ) /I B l, λ , b ։ B l ( λ , b ) . (5.4)We claim that the surjection in (5.4) is an isomorphism by checking dim O l, λ , b = dim B l ( λ , b ) . Lemma 5.4. We have dim O l, λ , b = (cid:18) k − l (cid:19) .Proof. Note that C [ p , . . . , p k − l − , r , . . . , r n − k ] ∼ = C [ g , . . . , g n − l − ] , where p i and r j are defined in (5.2)and (5.3). It is not hard to check that O l, λ , b ∼ = C [ f , . . . , f l , p , . . . , p k − l − ] / ˜ I O l, λ , b , (5.5)where ˜ I O l, λ , b is the ideal of C [ f , . . . , f l , p , . . . , p k − l − ] generated by the coefficients of the polynomial Q ks =1 ( x − b s + λ ( s )1 ) − Q ks =1 ( x − b s ) − np ( x ) f ( x ) .Introduce new variables w = ( w , . . . , w k − ) such that f ( x ) = l Y i =1 ( x − w i ) , p ( x ) = k − l − Y i =1 ( x − w l + i ) . Let ε = ( ε , . . . , ε k − ) be complex numbers such that k Y s =1 ( x − b s + λ ( s )1 ) − k Y s =1 ( x − b s ) = n (cid:16) x k − + k − X i =1 ( − i ε i x k − − i (cid:17) . Then C [ f , . . . , f l , p , . . . , p k − l − ] / ˜ I O l, λ , b ∼ = C [ w , . . . , w k − ] S l × S k − l − / h σ i ( w ) − ε i i i =1 ,...,k − . (5.6)The lemma follows from the fact that C [ w , . . . , w k − ] S l × S k − l − is a free module over C [ w , . . . , w k − ] S of rank (cid:0) k − l (cid:1) . (cid:3) Note that we have the projection ( V S ) sing( n − l,l ) ։ ( L ( λ , b )) sing( n − l,l ) . Since by Theorem 4.8 the Bethe algebra B l ( z ) acts on ( V S ) sing( n − l,l ) cyclically, the Bethe algebra B l ( λ , b ) acts on ( L ( λ , b )) sing( n − l,l ) cyclically as well.Therefore we have dim B l ( λ , b ) = dim( L ( λ , b )) sing( n − l,l ) = (cid:18) k − l (cid:19) . Proof of Theorem 4.9. Part (i) follows from Lemma 5.4 and (5.4), (5.5), (5.6). Clearly, we have B l ( λ , b ) ∼ = O l, λ , b is a Frobenius algebra. Moreover, the map ρ l from Theorem 4.8 induces a map ρ l, λ , b : O l, λ , b → ( L ( λ , b )) sing( n − l,l ) which identifies the regular representation of O l, λ , b with the B l ( λ , b ) -module ( L ( λ , b )) sing( n − l,l ) . Thereforepart (ii) is proved.Since B l ( λ , b ) is a Frobenius algebra, the regular and coregular representations of B l ( λ , b ) are isomorphicto each other. Parts (iii)–(vi) follow from the general facts about the coregular representations, see e.g.[MTV09, Section 3.3].Due to part (iv), it suffices to consider the algebraic multiplicity of every eigenvalue. It is well knownthat roots of a polynomial depend continuously on its coefficients. Hence the eigenvalues of T ( x ) dependcontinuously on b . Part (vii) follows from deformation argument and Theorem 2.9. (cid:3) UPERSYMMETRIC XXX SPIN CHAINS 25 Proof of Theorem 4.11. It is parallel to that of Theorem 4.9 with the following minor modification.The degree of g ( x ) is l instead of l − . Divide the polynomial g ( x ) in (4.1) by h ( x ) and let p ( x ) = x k − l + p x k − l − + · · · + p k − l − x + p k − l ,r ( x ) = r x n − k − + r x n − k − + · · · + r n − k − x + r n − k be the quotient and remainder, respectively. Clearly, p i , r j ∈ O Ql .Denote by I Q, O l, λ , b the ideal of O Ql generated by r , . . . , r n − k and the coefficients of polynomial q k Y s =1 ( x − b s + λ ( s )1 ) − q k Y s =1 ( x − b s − λ ( s )2 ) − ( q − q ) p ( x ) f ( x ) . Let O Ql, λ , b be the quotient algebra O Ql, λ , b = O Ql /I Q, O l, λ , b . The rest of the proof is similar, we only note that dim( O Ql, λ , b ) = (cid:0) kl (cid:1) . (cid:3) Proof of Theorem 2.11. We give a proof for the case q = q . The case q = q is similar.By part (v) of Theorem 4.9, all eigenvalues of T ( x ) have the form (2.22) with a monic divisor y t of ϕ λ , b ( x ) − ψ λ , b ( x ) . Moreover, it follows from part (iv) of Theorem 4.9 that all eigenspaces of T ( x ) havedimension one. By Lemma 2.10, all on-shell Bethe vectors are non-zero and the theorem follows. (cid:3) 6. H IGHER TRANSFER MATRICES Higher transfer matrices. We have the standard action of symmetric group S m on the space ( C | ) ⊗ m where s i acts as the graded flip operator P ( i,i +1) . We denote by A m and H m the images of the normalizedanti-symmetrizer and symmetrizer, A m = 1 m ! X σ ∈ S m sign( σ ) · σ, H m = 1 m ! X σ ∈ S m σ. Define the m -th transfer matrix associated to the diagonal matrix Q = diag( q , q ) by T Qm ( x ) = str( A m Q (1) T (1 ,m +1) ( x ) Q (2) T (2 ,m +1) ( x − · · · Q ( m ) T ( m,m +1) ( x − m + 1)) . (6.1)Here the supertrace is taken over all copies of End( C | ) . Clearly, T Q ( x ) = T Q ( x ) .Expand T Qm ( x ) as a power series in x − , T Qm ( x ) = ∞ X s =0 B Qm,s x − s . We denote the unital subalgebra of Y( gl | ) generated by elements B Qm,s for all m > and s > by B Q .When q = q , we simple write B for B Q . The subalgebra B Q does not change if q , q are multiplied bythe same non-zero number. Therefore if q = q , we assume further that q = 1 .The following statements are standard. Proposition 6.1. The algebra B Q is commutative. If q = q , then B Q contains the algebra U( h ) and hencecommutes with U( h ) . If q = q , the algebra B Q commutes with the algebra U( gl | ) .Proof. The first statement follows from the RTT relation (2.4), c.f. [MTV06, Proposition 4.5]. The secondis a corollary of the formulas B Q , = q T (1)11 − q T (1)22 , B Q , = q T (1)11 + ( q − q ) T (1)22 . The last statement is obtained from (2.9), c.f. [MTV06, Proposition 4.7]. (cid:3) As a subalgebra of Y( gl | ) , the algebra B Q acts naturally on any Y( gl | ) -module M . Since B Q com-mutes with U( h ) , it preserves the weight spaces ( M ) λ . Moreover, B preserves the weight singular spaces ( M ) sing λ . Proposition 6.2. The algebra B Q is stable under the anti-automorphism ι in (2.16) , ι ( T Qm ( x )) = T Qm ( x ) .Proof. Note that ι ( T ( x )) = ( T ( x )) t , Q t = Q , and ( A m ) t = A m , the proof is parallel to that of [MTV06,Proposition 4.11]. Here t is the supertranspose and the supertransposition ( A m ) t is taken over all copies of End( C | ) in A m . (cid:3) Berezinian and rational difference operator. Define the rational difference operator D Q ( x, τ ) , D Q ( x, τ ) = Ber(1 − Z Q ( x, τ )) , where as before Z Q ( x, τ ) = T t ( x ) Qτ .Applying the supertransposition to all copies of End( C | ) and using cyclic property of supertrace, see(2.1), one has T Qm ( x ) = str( A m ( T t ) (1 ,m +1) ( x ) Q (1) ( T t ) (2 ,m +1) ( x − Q (2) · · · ( T t ) ( m,m +1) ( x − m + 1) Q ( m ) ) . Therefore by [MR14, Theorem 2.13] we have D Q ( x, τ ) = ∞ X m =0 ( − m T Qm ( x ) τ m . (6.2)By (4.7), we obtain D Q ( x, τ ) = (1 − q T ( x ) τ + q T ( x ) τ (1 − q T ( x ) τ ) − q T ( x ) τ )(1 − q T ( x ) τ ) − . Expand (1 − q T ( x ) τ ) − as a power series in τ , (1 − q T ( x ) τ ) − = ∞ X m =0 ( q T ( x ) τ ) m = ∞ X m =0 q m m Y i =1 T ( x − i + 1) τ m , and compare to (6.2). It gives T Q ( x ) = T Q ( x ) = q T ( x ) − q T ( x ) and for m > , e T Qm ( x ) := ( − m q − m T Qm ( x ) = − ( q T ( x ) − q T ( x )) m − Y i =1 T ( x − i )+ m − X s =1 q T ( x ) (cid:16) s − Y i =1 T ( x − i ) (cid:17) T ( x − s ) m − Y j = s +1 T ( x − j ) . (6.3) Remark . The expansion (6.3) (and other variations) of the higher transfer matrices T Qm ( x ) can also beobtained from [MR14, Proposition 2.3, Remark 2.4].Let, as in Section 2.5, b = ( b , . . . , b k ) be a sequence of complex numbers, λ = ( λ (1) , . . . , λ ( k ) ) asequence of gl | -weights. Let t = ( t , . . . , t l ) be a solution of the Bethe ansatz equation (2.19). Define tworational functions ζ ( x ) = k Y s =1 x − b s + λ ( s )1 x − b s , ζ ( x ) = k Y s =1 x − b s − λ ( s )2 x − b s . (6.4)We also use the following notation, f [ i ] := τ i ( f ) = f ( x − i ) for any function f in x . UPERSYMMETRIC XXX SPIN CHAINS 27 Let y = ( x − t ) · · · ( x − t l ) . In [HLM19], we associate a rational difference operator D Q t , λ , b ( x, τ ) (or D Qy, λ , b ( x, τ ) ) to each solution t of the Bethe ansatz equation, D Q t , λ , b ( x, τ ) = D Qy, λ , b ( x, τ ) = (cid:16) − q ζ y [1] y τ (cid:17)(cid:16) − q ζ y [1] y τ (cid:17) − . (6.5)The operator D Q t , λ , b ( x, τ ) describes the eigenvalues of the algebra B Q acting on the corresponding on-shellBethe vector b B l ( t ) . Theorem 6.4. Assume that t i = t j for i = j . We have D Q ( x, τ ) b B l ( t ) = D Q t , λ , b ( x, τ ) b B l ( t ) . We give the proof of this theorem in the next section.6.3. Proof of Theorem 6.4. Consider the expansion of the rational difference operator D Q t , λ , b ( x, τ ) , D Q t , λ , b ( x, τ ) = 1 − ∞ X m =1 q m − ( q ζ − q ζ ) y [ m ] y (cid:16) m − Y i =1 ζ [ i ]2 (cid:17) τ m . Therefore, it suffices to show that e T Qm ( x ) b B l ( t ) = − ( q ζ − q ζ ) y [ m ] y (cid:16) m − Y i =1 ζ [ i ]2 (cid:17)b B l ( t ) . (6.6)We split the proof into three steps. It would be convenient to work with the unrenormalized Bethe vector B l ( t ) .6.3.1. Actions of T ij ( x ) on Bethe vectors. We prepare several lemmas for the proof. Following [HLPRS16],we set µ ( x , x ) = x − x + 1 x − x , ν ( x , x ) = 1 x − x , κ ( x , x ) = x − x + 1 . Note that we have ν ( x , x ) κ ( x , x ) = µ ( x , x ) .For a sequence of complex numbers t = ( t , . . . , t l ) define sequences of complex numbers t i and t ij by t i = ( t , . . . , t i − , t i +1 , . . . , t l ) , i l, t ij = ( t , . . . , t i − , t i +1 , . . . , t j − , t j +1 , . . . , t l ) , i < j l. We use the shorthand notation as follows. Let u = ( u , . . . , u r ) and w = ( w , . . . , w s ) be sequences ofcomplex numbers. Set µ ( x, u ) = r Y i =1 µ ( x, u i ) , µ ( u , w ) = r Y i =1 s Y j =1 µ ( u i , w j ) . The same convention also applies to functions ν ( x , x ) , κ ( x , x ) , and currents T ii ( x ) , etc.By B l ( t i , z ) and B l − ( t ij , z ) , we mean the off-shell Bethe vectors (2.21) associated to the sequences t i ⊔ { z } and t ij ⊔ { z } , respectively. Lemma 6.5 ([HLPRS16]) . We have T ( z ) B l ( t ) = ζ ( z ) κ ( t , z ) B l +1 ( t , z ) , T ( z ) B l ( t ) = ζ ( z ) l X i =1 µ ( t i , z ) ν ( t i , t i ) ν ( t i , z ) (cid:16) ζ ( z ) ζ ( z ) − ζ ( t i ) ζ ( t i ) (cid:17) B l − ( t i )+ ζ ( z ) X i This is a particular case of [HLPRS16, equation (3.6)]. A number of simplifications occur for thespecial choice of z b . (cid:3) Strategy of computation. We aim at (6.6). Let t be a solution of Bethe ansatz equation. Let α s = { x − , . . . , x − s + 1 } and β s = { x − s − , . . . , x − m + 1 } , s = 1 , . . . , m − , where α = β m − = ∅ .Then e T Qm ( x ) = − ( q T ( x ) − q T ( x )) T ( α m ) + m − X s =1 q T ( x ) T ( α s ) T ( x − s ) T ( β s ) . We consider the action of each term in the summation above on the on-shell Bethe vector B l ( t ) .Consider the action of ( q T ( x ) − q T ( x )) T ( α m ) on B l ( t ) . By Lemma 6.6, T ( α m ) B l ( t ) is alinear combination of Bethe vectors B l ( t ) and B l ( t i , x − . Note that B l ( t ) is an eigenvector of q T ( x ) − q T ( x ) and T ii ( x ) T ( x − 1) = T ( x ) T ii ( x − , i = 1 , , it follows that ( q T ( x ) − q T ( x )) T ( α m ) B l ( t ) is a linear combination of B l ( t ) and B l ( t i , x ) .Consider the vector T ( x ) T ( α s ) T ( x − s ) T ( β s ) B l ( t ) . (6.7)Again by Lemma 6.6, T ( β s ) B l ( t ) is a linear combination of Bethe vectors B l ( t ) and B l ( t i , x − s − . Afterthe action of T ( x − s ) , it follows from Lemma 6.5 that we get Bethe vectors B l − ( t i ) and B l − ( t ij , x − s ) .Then the action of T ( α s ) on B l − ( t i ) gives B l − ( t i ) and B l − ( t ij , x − while the action of T ( α s ) on B l − ( t ij , x − s ) gives B l − ( t i , x − . Note that T ( x ) T ( x − 1) = 0 , hence the final result only involves B l ( t i , x ) for i = 1 , . . . , l . The vectors we are obtaining in (6.7) aredescribed by the following picture. UPERSYMMETRIC XXX SPIN CHAINS 29 B l ( t ) B l ( t ) B l − ( t i ) B l − ( t i ) B l − ( t ij , x − B l − ( t ij , x − s ) B l ( t i , x − s − 1) 0 B l ( t i , x ) T ( β s ) T ( β s ) T ( x − s ) T ( α s ) T ( x ) T ( x − s ) T ( x ) T ( α s ) More precisely, the picture describes the result of action of operators to linear combinations of variousBethe vectors. We apply the same operator to all vectors in the same column. This operator is indicated onthe top of the solid line in the first row and also at the bottom of the second row. Then the arrows show whichvectors are obtained in each case. Dashed arrows correspond to terms which eventually become zero. Solidlines correspond to terms which have a non-trivial contribution. For example, T ( x − s ) B l ( t ) is a linearcombination of Bethe vectors B l − ( t i ) and B l − ( t ij , x − s ) with i = 1 , . . . , l , j = i + 1 , . . . , l . The latterterms will be all annihilated by further applications of T ( α s ) and T ( x ) .6.3.3. End of proof. Note that q ζ ( t i ) = q ζ ( t i ) and ν ( x , x ) κ ( x , x ) = µ ( x , x ) . By Lemmas 6.5and 6.6, following the way described in Section 6.3.2, we obtain that T ( x ) T ( α s ) T ( x − s ) T ( β s ) B l ( t ) is equal to l X i =1 ζ ( x ) ζ ( α m ) ν ( t i , t i ) κ ( t i , x ) µ ( t i , α m ) (cid:18) µ ( t i , β s ) ν ( x − s, t i ) (cid:18) q ζ ( x − s ) q ζ ( x − s ) − (cid:19) (6.8) +(1 − δ s,m − ) µ ( x − s − , β s +1 ) ν ( x − s − , t i ) (cid:18) q ζ ( x − s ) q ζ ( x − s ) − q ζ ( x − s − q ζ ( x − s − (cid:19)(cid:19) B l ( t i , x ) . Similarly, ( q T ( x ) − q T ( x )) T ( α m ) B l ( t ) = ( q ζ ( x ) − q ζ ( x )) ζ ( α m ) µ ( t , x ) µ ( t , α m ) B l ( t ) (6.9) + l X i =1 q ζ ( x ) ζ ( α m ) ν ( t i , t i ) κ ( t i , x ) µ ( t i , α m ) µ ( x − , α ◦ m ) ν ( x − , t i ) (cid:18) q ζ ( x − q ζ ( x − − (cid:19) B l ( t i , x ) . Therefore, e T Qm ( x ) B l ( t ) is equal to the sum of (6.9) and the summation of (6.8) over s = 1 , . . . , m − .For fixed i ∈ { , . . . , l } and s ∈ { , . . . , m − } , we combine all terms containing ζ ( x ) ζ ( α m ) q ζ ( x − s ) / ( q ζ ( x − s )) in (6.8), (6.9) and consider the corresponding coefficient. To show (6.6), we first show thatthis coefficient vanishes. This follows from the following lemma. Lemma 6.7. We have µ ( t i , β s ) ν ( x − s, t i ) + µ ( x − s − , β s +1 ) ν ( x − s − , t i ) = µ ( x − s, β s ) ν ( x − s, t i ) ,for s = 1 , . . . , m − . (cid:3) If we set µ ( x − s − , β s +1 ) = 0 , then the lemma also holds for s = m − .We then combine the terms containing ζ ( x ) ζ ( α m ) in this sum. The next lemma asserts this coefficientis equal to zero. Lemma 6.8. We have m − X s =1 µ ( t i , β s ) ν ( x − s, t i ) = µ ( x − , α ◦ m ) ν ( x − , t i ) . Proof. We have m − X s =1 µ ( t i , β s ) ν ( x − s, t i ) = m − X s =1 m − Y j = s +1 t i − ( x − j ) + 1 t i − ( x − j ) 1 x − s − t i = m − X s =1 x − m − t i ( x − s − − t i )( x − s − t i )=( x − m − t i ) m − X s =1 (cid:18) x − s − − t i − x − s − t i (cid:19) = m − x − − t i . Clearly, µ ( x − , α ◦ m ) = m − . Therefore the lemma follows. (cid:3) Thus we have e T Qm ( x ) B l ( t ) = ( q ζ ( x ) − q ζ ( x )) ζ ( α m ) µ ( t , x ) µ ( t , α m ) B l ( t ) . Since µ ( t , x ) = l Y i =1 t i − x + 1 t i − x = y ( x − y ( x ) , we have ( q ζ ( x ) − q ζ ( x )) ζ ( α m ) µ ( t , x ) µ ( t , α m ) = ( q ζ − q ζ ) y [ m ] y m − Y i =1 ζ [ i ]2 , which completes the proof of Theorem 6.4.6.4. Relations between transfer matrices. Similar to (6.1), define H Qm ( x ) = str( H m Q (1) T (1 ,m +1) ( x ) Q (2) T (2 ,m +1) ( x − · · · Q ( m ) T ( m,m +1) ( x − m + 1)) . Using Theorem 6.4, we are able to express T Qm ( x ) and H Qm ( x ) in terms of the first transfer matrix T Q ( x ) andthe center Ber Q ( x ) .We start with the following technical lemma. Lemma 6.9. No non-zero element in Y( gl | ) acts by zero on L ( λ , b ) for generic λ and b .Proof. The lemma follows from the proof of [Naz99, Proposition 2.2]. (cid:3) Theorem 6.10. We have T Qm ( x ) m − Y i =1 (1 − Ber Q ( x − i )) = m Y i =1 T Q ( x − i + 1) , H Qm ( x ) m − Y i =1 (Ber Q ( x − i ) − 1) = m Y i =1 T Q ( x − i + 1) m − Y i =1 Ber Q ( x − i ) . Proof. We prove it for the case q = q . The case q = q is similar.Note that for an on-shell Bethe vector b B l ( t ) , where t = ( t , . . . , t l ) with t i = t j for i = j , by (6.6) and(2.22), we have T Qm ( x ) m − Y i =1 (1 − Ber Q ( x − i )) b B l ( t ) = m Y i =1 T Q ( x − i + 1) b B l ( t ) . By Theorem 2.9, the transfer matrix T Q ( x ) is diagonalizable and the Bethe ansatz is complete for generic λ and b , namely there exists a basis of L ( λ , b ) consisting of on-shell Bethe vectors. Therefore, the coefficientsof the formal series T Qm ( x ) m − Y i =1 (1 − Ber Q ( x − i )) − m Y i =1 T Q ( x − i + 1) UPERSYMMETRIC XXX SPIN CHAINS 31 act by zero on Y( gl | ) -modules L ( λ , b ) for generic λ and b . The first equality of the theorem follows fromLemma 6.9.By [MR14, Theorem 2.13], one has ( D Q ( x, τ )) − = (Ber Q ( x )) − = ∞ X m =0 H Qm ( x ) τ m . Therefore, the second equality is proved similarly. (cid:3) Remark . Equation (2.13) in Example 2.2 is the second equality of Theorem 6.10 on the representationlevel. We explain it in more detail for the case q = q . Following e.g. [FR99, Section 3.1], let R be the uni-versal R-matrix in the Y( gl | ) Yangian double Y ∗ ( gl | ) b ⊗ Y( gl | ) . For a finite-dimensional representation ( V, ρ V ) of the dual Yangian Y ∗ ( gl | ) , let T V ( x ) = str (cid:0) ( ρ V ( x ) ⊗ id) R (cid:1) ∈ Y( gl | )[[ x − ]] . Similar to [FR99, Lemma 2], one has T V ⊗ W ( x ) = T V ( x ) T W ( x ) . Moreover T W ( x ) = T V ( x ) + T U ( x ) for a short exact sequence V ֒ → W ։ U . We expect that after proper rescaling of the universal R-matrix R one has T L ω (0) ( x ) = T ( x ) , T L mω (0) ( x ) = H m ( x ) , and T C ¯1 ,ξm ( x ) is a certain rational function in Ber( x ) . Note that the modules here should be replaced withcorresponding Y ∗ ( gl | ) -modules. The first equality of Theorem 6.10 can be understood similarly. (cid:3) Remark . Recall that we call B Q the Bethe algebra. Often, it is the algebra B Q which is named theBethe algebra. Due to Theorem 6.10, the images of B Q and B Q acting on the modules V S , L ( λ , b ) with | λ | 6 = 0 coincide. (cid:3) Corollary 6.13. 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