Representations of the Super Yangians of Types A and C

We classify the finite-dimensional irreducible representations of the super Yangian associated with the orthosymplectic Lie superalgebra osp2|2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak {osp}_{2|2n}$\end{document}. The classification is given in terms of the highest weights and Drinfeld polynomials. We also include an R-matrix construction of the polynomial evaluation modules over the Yangian associated with the Lie superalgebra glm|n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak {gl}_{m|n}$\end{document}, as an appendix. This is a super-version of the well-known construction for the gln\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak {gl}_{n}$\end{document} Yangian and it relies on the Schur–Sergeev duality.


Introduction
The Yangian Y(osp M |2n ) associated with the orthosymplectic Lie superalgebra osp M |2n is a deformation of the universal enveloping algebra U(osp M |2n [u]) in the class of Hopf algebras.The original definition is due to Arnaudon et al. [1], where some basic properties of the Yangian were also described.Its gl m|n counterpart was introduced earlier by Nazarov [16], and the finitedimensional irreducible representations of the Yangian Y(gl m|n ) were classified by Zhang [23] in a way similar to the Drinfeld Yangians [6].Apart from the case of the Yangian Y(osp 1|2n ) considered in [13], the general classification problem for the orthosymplectic Yangians remains open.We show in this paper that the finite-dimensional irreducible representations of the Yangian Y(osp 2|2n ) are described in essentially the same way as those of the Yangian Y(gl m|n ), thus extending the similarity between the modules over the Lie superalgebras of types A and C as given by Kac [9].
In more detail, we will regard the Yangian Y(osp 2|2n ) as a quotient of the extended Yangian X(osp 2|2n ) by an ideal generated by central elements.The highest weight representation L(λ(u)) of X(osp 2|2n ) is defined as the irreducible quotient of the Verma module M(λ(u)) associated with an (n + 2)-tuple λ(u) = (λ 1 (u), . . ., λ n+2 (u)) of formal series in u −1 .The tuple is called the highest weight of the representation.
A standard argument shows that every finite-dimensional irreducible representation of the extended Yangian is of the form L(λ(u)) for a certain highest weight λ(u).The key step in the classification is to determine the necessary and sufficient conditions on the highest weight for the representation L(λ(u)) to be finite-dimensional.We will rely on reduction properties of the representations to establish necessary conditions, and their sufficiency will be verified by constructing a family of fundamental representations of the Yangian X(osp 2|2n ).This will prove the following theorem.
We will refer to the monic polynomials occurring in these conditions as the Drinfeld polynomials of the representation; cf.[6].
The Appendix is devoted to the polynomial evaluation modules over the Yangian Y(gl m|n ).Such modules over the Yangian Y(gl n ) proved to be useful in the analysis of the representations of the orthosymplectic Yangians; see Proposition 2.9 and also [2], [13].Many applications of the evaluation homomorphism rely on its R-matrix interpretation going back to Cherednik [4].In this interpretation, the polynomial evaluation modules arise from the Schur-Weyl duality and are constructed as submodules of the tensor products of the vector representation of the general linear Lie algebra.As another ingredient, the construction uses some key steps of the fusion procedure for the symmetric group originated in the work of Jucys [7].It was rediscovered in [4], with detailed arguments given in [17]; see also [12,Sec. 6.4].
A super-version of the Schur-Weyl duality involving the general linear Lie superalgebra and the symmetric group is due to Sergeev [20] and Berele and Regev [3].It is called the Schur-Sergeev duality in [5,Sec. 3.2].We will use this duality to give an R-matrix construction of the polynomial evaluation modules over the Yangian associated with the Lie superalgebra gl m|n .More precisely, for any Young diagram λ contained in the (m, n)-hook we consider a standard λ-tableau U and the associated primitive idempotent e U ∈ CS d , where d is the number of boxes in λ.By using the action of the symmetric group S d on the tensor product of the Z 2graded vector spaces C m|n , we show that the subspace e U (C m|n ) ⊗d is a representation of the Yangian Y(gl m|n ).We derive that both this representation and its twisted version are isomorphic to evaluation modules over the Yangian and identify them by calculating the highest weights.
Note that more general skew representations of the Yangian Y(gl m|n ) were investigated by Lu and Mukhin [11] with the use of the Schur-Sergeev duality and a super version of the Drinfeld functor.Those results provide an extension of the corresponding constructions of Nazarov [18] for representations of the Yangian Y(gl n ).

Definition and basic properties of the Yangian
Fix an integer n 1 and introduce the involution i → i ′ = 2n−i+3 on the set {1, 2, . . ., 2n+2}.Consider the Z 2 -graded vector space C 2|2n over C with the canonical basis e 1 , e 2 , . . ., e 2n+2 , where the vector e i has the parity ī mod 2 and The endomorphism algebra End C 2|2n gets a Z 2 -gradation with the parity of the matrix unit e ij found by ī +  mod 2. We will identify the algebra of even matrices over a superalgebra A with the tensor product algebra End C 2|2n ⊗ A, so that a matrix A = [a ij ] is regarded as the element The involutive matrix super-transposition t is defined by where we set This super-transposition is associated with the bilinear form on the space C 2|2n defined by the anti-diagonal matrix G = [δ ij ′ θ i ].We will also regard t as the linear map A standard basis of the general linear Lie superalgebra gl 2|2n is formed by elements E ij of the parity ī +  mod 2 for 1 i, j 2n + 2 with the commutation relations We will regard the orthosymplectic Lie superalgebra osp 2|2n associated with the bilinear from defined by G as the subalgebra of gl 2|2n spanned by the elements The symplectic Lie algebra osp 0|2n ∼ = sp 2n will be considered as the subalgebra of osp 2|2n spanned by the elements F ij with 2 i, j 2 ′ .Introduce the permutation operator P by The R-matrix associated with osp 2|2n is the rational function in u given by This is a super-version of the R-matrix originally found in [21].Following [1], we define the extended Yangian X(osp 2|2n ) as a Z 2 -graded algebra with generators t (r) ij of parity ī +  mod 2, where 1 i, j 2n + 2 and r = 1, 2, . . ., satisfying certain quadratic relations.To write them down, introduce the formal series and combine them into the matrix The defining relations for the algebra X(osp 2|2n ) take the form of the RT T -relation As shown in [1], the product T (u − κ) T t (u) is a scalar matrix with where c(u) is a series in u −1 .As with the Lie algebra case considered in [2], all its coefficients belong to the center ZX(osp 2|2n ) of X(osp 2|2n ) and generate the center.We will also use the extended Yangian X(osp 0|2n ) which is defined in the same way as X(osp 2|2n ), by using the subspace C 0|2n ⊂ C 2|2n with odd basis vectors e 2 , e 3 , . . ., e 2 ′ , instead of C 2|2n and the corresponding RT T -relation (2.4).The R-matrix is defined by (2.2) with the modified expressions e ij ⊗ e i ′ j ′ θ i θ j and with the value κ = −n − 1.It will be convenient to use the index set {2, 3, . . ., 2 ′ } to label the corresponding generating series which we will denote by tij (u).We have an isomorphism X(osp 0|2n ) ∼ = X(sp 2n ) so that the series tij (−u) satisfy the defining relations for the extended Yangian X(sp 2n ); see [1], [2].
The Yangian Y(osp 2|2n ) is defined as the subalgebra of X(osp 2|2n ) which consists of the elements stable under the automorphisms We have the tensor product decomposition The Yangian Y(osp 2|2n ) is isomorphic to the quotient of X(osp 2|2n ) by the relation c(u) = 1.
The defining relations (2.4) can be written with the use of super-commutator in terms of the series (2.3) as follows: Note that the mapping defines an automorphism of X(osp 2|2n ).
The universal enveloping algebra U(osp 2|2n ) can be regarded as a subalgebra of X(osp 2|2n ) via the embedding This fact relies on the Poincaré-Birkhoff-Witt theorem for the orthosymplectic Yangian which was pointed out in [1].It states that the associated graded algebra for Y(osp 2|2n ) is isomorphic to U(osp 2|2n [u]).This implies that the algebra X(osp 2|2n ) is generated by the coefficients of the series c(u) and t ij (u) with i+j 2n+3, excluding t 11 ′ (u) and t 1 ′ 1 (u).Moreover, given any total ordering on the set of the generators, the ordered monomials with the powers of odd generators not exceeding 1, form a basis of the algebra.The theorem follows by using essentially the same arguments as in the non-super case; see [2,Sec. 3].The extended Yangian X(osp 2|2n ) is a Hopf algebra with the coproduct defined by The coproduct on the algebra X(osp 0|2n ) is defined by the same formula for the series tij (u) instead of t ij (u), with the sum taken over k = 2, 3, . . ., 2n + 1.

Highest weight representations
The proof of the necessity of the conditions of the Main Theorem will rely on the following reduction property for representations of the extended Yangians X(osp 2|2n ) which is verified in the same way as for the case of X(osp 1|2n ) in [13, Prop.4.1]; cf.[2, Lemma 5.13].For an arbitrary X(osp 2|2n )-module V set (2.12) Proposition 2.1.The subspace V + is stable under the action of the operators t ij (u) subject to 2 i, j 2 ′ .Moreover, the assignment tij (u) → t ij (u) defines a representation of the extended Yangian X(osp 0|2n ) on V + .
A representation V of the algebra X(osp 2|2n ) is called a highest weight representation if there exists a nonzero vector ξ ∈ V such that V is generated by ξ, for some formal series The vector ξ is called the highest vector of V .
Proposition 2.2.The series λ i (u) associated with a highest weight representation V satisfy the consistency conditions

15)
for i = 1, . . ., n.Moreover, the coefficients of the series c(u) act in the representation V as the multiplications by scalars determined by c(u Proof.Introduce the subspace V + by (2.12) and note that the vector ξ belongs to V + .By applying Proposition 2.1 we find that the cyclic span X(osp 0|2n ) ξ is a highest weight submodule over X(osp 0|2n ) associated with the series λ 2 (u), . . ., λ 2 ′ (u).Due to the isomorphism X(osp 0|2n ) ∼ = X(sp 2n ), conditions (2.15) for i = 2, . . ., n follow from the consistency conditions for the highest weight modules over X(sp 2n ); see [2, Prop.5.14].Furthermore, using the defining relations (2.8), we get and so Finally, the last part of the proposition is obtained by using the expression for c(u) implied by taking the (1 ′ , 1 ′ ) entry in the matrix relation (2.5).
The Poincaré-Birkhoff-Witt theorem for the algebra X(osp 2|2n ) implies that the Verma module M(λ(u)) is nonzero, and we denote by The first part of the Main Theorem is implied by the following proposition whose proof is the same as in the non-super case; see [2, Thm 5.1].
Proposition 2.3.Every finite-dimensional irreducible representation of the algebra X(osp 2|2n ) is a highest weight representation.Moreover, it contains a unique, up to a constant factor, highest vector.
We will now suppose that the representation L(λ(u)) of X(osp 2|2n ) is finite-dimensional and derive the conditions on the highest weight λ(u) given in the Main Theorem.First consider the subalgebra Y 0 of X(osp 2|2n ) generated by the coefficients of the series t 11 (u), t 12 (u), t 21 (u) and t 22 (u).This subalgebra is isomorphic to the Yangian Y(gl 1|1 ), and the cyclic span Y 0 ξ of the highest vector ξ is a finite-dimensional module over Y 0 with the highest weight (λ 1 (u), λ 2 (u)).This implies that condition (1.1) must hold by [22,Thm. 4]; see also [13,Prop. 5.1].

Fundamental representations
Our next step in the proof of the Main Theorem is to show that the conditions (1.1), (1.2) and (1.3) are sufficient for the representation L(λ(u)) of X(osp 2|2n ) to be finite-dimensional.The tuple of Drinfeld polynomials determines the highest weight λ(u) of the representation up to a simultaneous multiplication of all components λ i (u) by a formal series f (u) ].This operation corresponds to twisting the action of the algebra X(osp 2|2n ) on L(λ(u)) by the automorphism (2.6).Hence, it suffices to prove that a particular module L(λ(u)) corresponding to a given tuple (2.16) is finite-dimensional.Suppose that L(ν(u)) and L(µ(u)) are the irreducible highest weight modules with the highest weights and µ(u) = µ 1 (u), . . ., µ n+2 (u) . (2.17) By the coproduct rule (2.11), the cyclic span X(osp 2|2n )(ξ ⊗ ξ ′ ) of the tensor product of the respective highest vectors ξ ∈ L(ν(u)) and ξ ′ ∈ L(µ(u)) is a highest weight module with the highest weight This observation implies the corresponding transition rule for the associated tuples of Drinfeld polynomials.Namely, if the highest weights in (2.17) are associated with the respective tuples then the highest weight (2.18) is associated with the tuple where d(u) is the monic polynomial in u defined as the greatest common divisor of the polynomials Therefore, it will be enough to show that the fundamental representations of X(osp 2|2n ) are finite-dimensional.They correspond to the tuples of Drinfeld polynomials of the form for α = β, where u + γ represents the polynomial P d (u) and d runs over the values 2, . . ., n + 1.
Note that by applying the shift automorphism (2.9), we can assume that β = 0 in the first tuple and use a particular value of γ in the second tuple.
We will begin with the first tuple in (2.20).The required property is implied by the following.
Proof.Let ξ denote the highest vector of L(λ(u)).We will show first that all vectors t (r) kl ξ with k, l ∈ {2, 3, . . ., 2 ′ } and r 1 are zero.This holds by definition for k l, while the defining relations (2.8) imply that for k > l we have the properties t 1j (u)t (r) kl ξ = 0 for j > 1 and t i1 ′ (u)t (r) kl ξ = 0 for i < 1 ′ .This means that the vector t (r) kl ξ belongs to the subspace L(λ(u)) + defined in (2.12).By Proposition 2.1, the subspace L(λ(u)) + is a representation of the algebra X(osp 0|2n ), and ξ generates a highest weight module over this algebra with the highest weight (1, . . ., 1).The irreducible quotient of this module is one-dimensional, which implies that the vector t (r) kl ξ ∈ L(λ(u)) is annihilated by all coefficients of the series t ij (u) with 1 i < j 1 ′ and therefore must be zero.
We now prove two lemmas providing the formulas for the action of some other elements of the extended Yangian on the highest vector ξ.Lemma 2.5.We have t (r) k1 ξ = 0 for all k = 2, 3, . . ., 2 ′ and r 2.
Proof.As in the above argument, we will check that the vectors t (r) k1 ξ are annihilated by all coefficients of the series t ij (u) with 1 i < j 1 ′ .The Poincaré-Birkhoff-Witt theorem and the defining relations imply that it is sufficient to check this property for the coefficients of the series t i,i+1 (u) with i = 1, . . ., n + 1. Suppose first that 2 k n + 1.Then (2.8) gives Since t k,i+1 (u)ξ = δ k,i+1 ξ, we derive Now proceed by induction on k.For k = 2, by (2.21) we have 21 ξ.Using (2.21) for the induction step, we can conclude that k1 ξ for all 2 k n + 1.Now let (n + 1) ′ k 2 ′ and use induction on k starting from k = (n + 1) ′ .Note that (2.21) holds for these values of k unless i = k ′ .By applying (2.8) to the super-commutator t (n+1) ′ ,1 (v), t n+1,(n+1) ′ (u) and using the assumptions λ i (u) = 1 for i = 2, 3, . . ., n + 2, we get By another application of (2.8) we derive Therefore, the previous expression can be written in the form On the other hand, by calculating the super-commutator [t n+1,1 (v), t 1 ′ 1 ′ (u)] we obtain By the consistency conditions of Proposition 2.2, we find that and so the residue at v = u − n − 1 gives As we showed above, n+1,1 ξ, so that (2.22) simplifies to n+1,1 ξ.
Hence we may conclude that the vectors t (r) (n+1) ′ ,1 ξ with r 2 are annihilated by all coefficients of the series t i,i+1 (u) with i = 1, . . ., n + 1 and so t (r) (n+1) ′ ,1 ξ = 0. Now suppose that k = l ′ with 2 l n.Calculating as above, we find that Furthermore, using the defining relations again, we derive the relation which by taking the residue at v = u − n − 1 also implies (l+1) ′ ,1 ξ by the induction hypothesis, relation (2.23) simplifies to thus proving that the vectors t (r) l ′ ,1 ξ with r 2 are annihilated by the coefficients of the series t l,l+1 (u).The proof is competed by taking into account (2.21) for the remaining values i = l.Lemma 2.6.We have Proof.Calculate the commutator [t 11 (v), t 1 ′ 1 (u)] by (2.8) and rearrange the terms to get By setting u = v + n + 1 we come to the relation Apply both sides of the relation to the highest vector ξ.By Lemma 2.4 we have k ′ 1 ξ.Hence, using the super-commutator [t k1 (v), t (1) This relation uniquely determines the series t 1 ′ 1 (v) ξ.It is easily seen to be satisfied by In particular, t (1) k 1 ] ξ = 0 yielding the required relation.Returning to the proof of the proposition, we can now derive that the representation L(λ(u)) is spanned by the vectors of the form t (1) with s 0. Indeed, by the first part of the proof, and the Poincaré-Birkhoff-Witt theorem applied to a suitable ordering of the generators, it is enough to verify that the span of these vectors is stable under the action of all generators of the form t (r) k 1 with k = 2, . . ., 2 ′ .Note that by taking the (1 ′ , 1) entry in the matrix relation (2.5), we find that t (1) 1 ′ 1 = 0 in the algebra X(osp 2|2n ).This implies that the elements t (1) k 1 with k = 2, . . ., 2 ′ pairwise anticommute.They also have the property k 1 ] = 0. Hence, applying the series t k 1 (v) to a basis vector (2.24), we get with the hat indicating the factor to be skipped.By Lemma 2.5 we have k 1 ξ, and together with the relation of Lemma 2.6 this proves that the span of the vectors (2.24) is stable under the action of all operators t (r) k1 with k = 2, . . ., 2 ′ .Since the representation L(λ(u)) is spanned by these vectors, its dimension does not exceed 4 n .Let V (µ) denote the irreducible highest weight representation of the Lie superalgebra osp 2|2n with the highest weight µ = (µ 1 , . . ., µ n+1 ) with respect to the upper-triangular Borel subalgebra.This means that F ii ζ = µ i ζ for i = 1, . . ., n + 1 and F ij ζ = 0 for i < j, where ζ is the highest vector of V (µ).The modules of the form V (α, 0, . . ., 0) are known to be typical, if and only if α does not belong to the set {0, 1, . . ., n − 1} ∪ {n + 1, . . ., 2n}; see [9].
We will now turn to the second family of fundamental representations in (2.20).We will show that they can be constructed as subquotients of tensor products of the vector representations of X(osp 2|2n ) and representations of the first family in (2.20).
The vector representation of X(osp 2|2n ) on C 2|2n is defined by (2.25) The homomorphism property follows from the RT T -relation (2.4) and the Yang-Baxter equation satisfied by R(u) as in (A.8); cf.[1].The mapping T (u) → R(u) defines an algebra homomorphism X(osp 2|2n ) → End C 2|2n , and to get (2.25) we take its composition with the automorphism t ij (u) → t i ′ j ′ (u + n) θ i θ j (−1) ī+ .Now use the coproduct (2.11) and the shift automorphism (2.9) to equip the tensor square V = (C 2|2n ) ⊗2 with the action of X(osp 2|2n ) by setting where the generators act in the respective copies of the vector space C 2|2n via the rule (2.25).
Introduce the vectors w k ∈ V by w k = e 1 ⊗ e k − e k ⊗ e 1 for k = 2, 3, . . ., 2 ′ and denote by W the subspace of V spanned by these vectors.The vector representation C 0|2n of the extended Yangian X(osp 0|2n ) is defined by the rule similar to (2.25); see [2]: The subspace W is invariant with respect to the action of the operators t ij (u) with 2 i, j 2 ′ .Moreover, the assignment tij (u) → t ij (u) defines a representation of the algebra X(osp 0|2n ) on W isomorphic to the vector representation.
(ii) The subspace W is annihilated by the operators t 1j (u) with j > 1 and t i 1 ′ (u) with i < 1 ′ .Moreover, the coefficients of the series t 11 (u) act on W as multiplications by scalars determined by t 11 (u) → 1 + u −1 .Proof.Assuming that 2 i, j 2 ′ , for the action of the operator t ij (u) we have Due to (2.25), a nonzero contribution can only come from the terms and the calculation yields the formula Taking onto account (2.27), we thus prove the first part of the proposition.The second part is verified by a similar calculation.
For any a ∈ C we will denote by V a the representation of the algebra X(osp 2|2n ) on the space V = (C 2|2n ) ⊗2 obtained by twisting the action (2.26) with the automorphism (2.9).Similarly, we let W a denote the representation of X(osp 0|2n ) on W twisted by the same shift automorphism on X(osp 0|2n ).For d = 1, . . ., n introduce the vectors where S ′ d denotes the group of permutations of the set {2, . . ., d + 1}.
Proposition 2.9.The cyclic span X(osp 2|2n ) ξ d is a highest weight module whose tuple of Drinfeld polynomials (2.16) has the form for d = 1, . . ., n − 1, while for d = n it has the form Proof.It follows from Proposition 2.8 and the coproduct rule (2.11) that for all 2 i, j 2 ′ the image t ij (u) ξ d coincides with tij (u) ξ d in the representation W 0 ⊗ W −1 ⊗ . . .⊗ W −d+1 of the algebra X(osp 0|2n ).Since the series tij (−u) satisfy the defining relations of the extended Yangian X(sp 2n ), we derive from the proof of [2, Thm.5.16] (see also Corollary A.3 below), that the vector ξ d has the properties where d = 1, . . ., n − 1.Moreover, the same relations hold for d = n, except for (2.30) with i = n + 2, which is replaced by Finally, Proposition 2.8(ii) and (2.11) imply that ξ d is annihilated by the action of t 1j (u) with j > 1, while Thus, the vector ξ d generates a highest weight module over the algebra X(osp 2|2n ), whose highest weight is found from the action of the series t ii (u).The formulas for the Drinfeld polynomials easily follow.
We are now in a position to complete the proof of the Main Theorem.Proposition 2.4 implies that the irreducible highest weight representations of X(osp 2|2n ), associated with any tuple of Drinfeld polynomials of the first type in (2.20) is finite-dimensional.Furthermore, Proposition 2.9 shows that the irreducible highest weight representations of X(osp 2|2n ) associated with the tuples of the form (2.28) and (2.29) are also finite-dimensional.On the other hand, by applying the shift automorphism (2.9) and using the transition rule (2.19), we can get all tuples of Drinfeld polynomials of the second type in (2.20) from certain tuples of the first type and those appearing in (2.28) and (2.29).This proves the sufficiency of the conditions of the Main Theorem for the representation L(λ(u)) to be finite-dimensional.The last part of the theorem is an easy consequence of the decomposition (2.7); cf.[2,Cor. 5.19].
A Polynomial evaluation modules over the Yangian Y(gl m|n )

A.1 Defining relations and representations
Given nonnegative integers m and n, we will use the notation ī = 0 for i = 1, . . ., m and ī = 1 for i = m + 1, . . ., m + n.Introduce the Z 2 -graded vector space C m|n over C with the basis e 1 , e 2 , . . ., e m+n , where the parity of the basis vector e i is defined to be ī mod 2. Accordingly, equip the endomorphism algebra End C m|n with the Z 2 -gradation, where the parity of the matrix unit e ij is found by ī +  mod 2.
A standard basis of the general linear Lie superalgebra gl m|n is formed by elements E ij of the parity ī +  mod 2 for 1 i, j m + n with the commutation relations The Yang R-matrix associated with gl m|n is the rational function in u given by where P is the permutation operator, Following [16] and [19], define the Yangian Y(gl m|n ) as the Z 2 -graded algebra with generators t (r) ij of parity ī +  mod 2, where 1 i, j m + n and r = 1, 2, . . ., satisfying the quadratic relations written in terms of the formal series Combining them into the matrix T (u) = [t ij (u)] and regarding it as the element we can write the defining relations in the standard RT T -form (2.4) with the R-matrix (A.1).The universal enveloping algebra U(gl m|n ) can be regarded as a subalgebra of Y(gl m|n ) via the embedding defines the evaluation homomorphism ev: Y(gl m|n ) → U(gl m|n ).
The Yangian Y(gl m|n ) is a Hopf algebra with the coproduct defined by A representation V of the algebra Y(gl m|n ) is called a highest weight representation if there exists a nonzero vector ξ ∈ V such that V is generated by ξ, t ij (u) ξ = 0 for 1 i < j m + n, and t ii (u) ξ = π i (u) ξ for i = 1, . . ., m + n, for some formal series The vector ξ is called the highest vector of V and the (m+n)-tuple π(u) = (π 1 (u), . . ., π m+n (u)) is called its highest weight.
Given an arbitrary tuple π(u) = (π 1 (u), . . ., π m+n (u)) of formal series of the form (A.4), the Verma module M(π(u)) is defined as the quotient of the algebra Y(gl m|n ) by the left ideal generated by all coefficients of the series t ij (u) with 1 i < j m + n, and t ii (u) − π i (u) for i = 1, . . ., m + n.We will denote by L(π(u)) its irreducible quotient.The isomorphism class of L(π(u)) is determined by π(u).Necessary and sufficient conditions on π(u) for the representation L(π(u)) to be finite-dimensional are known due to [23].Their extension to arbitrary parity sequences via odd reflections was given in [14]; see also [10].
Consider the irreducible highest weight representation V (π) of the Lie superalgebra gl m|n with the highest weight π = (π 1 , . . ., π m+n ), associated with the standard Borel subalgebra.This means that E ii ζ = π i ζ for i = 1, . . ., m + n and E ij ζ = 0 for i < j, where ζ is the highest vector of V (π).The representation V (π) is finite-dimensional if and only if the highest weight π satisfies the conditions π i − π i+1 ∈ Z + for all i = m; see [9].Use the evaluation homomorphism (A.2) to equip V (π) with a Y(gl m|n )-module structure.The evaluation module V (π) is isomorphic to the Yangian highest weight module L(π(u)), where the components of the highest weight π(u) are A.2 Schur-Sergeev duality and fusion procedure We will follow [5,Sec. 3.2] to recall a version of the Schur-Sergeev duality going back to [3] and [20].An (m, n)-hook partition λ = (λ 1 , λ 2 , . . . ) is a partition with the property λ m+1 n.This means that the Young diagram λ is contained in the (m, n)-hook as depicted below.The figure also illustrates the partitions µ = (µ 1 , . . ., µ m ) and ν = (ν 1 , . . ., ν n ) associated with λ.They are introduced by setting and where λ ′ denotes the conjugate partition so that λ ′ j is the length of column j in the diagram λ: We will associate two (m + n)-tuples of nonnegative integers with λ by The tensor product space (C m|n ) ⊗d is naturally a module over both gl m|n and the symmetric group S d .For the action of the basis elements of gl m|n we have while the transposition (a b) ∈ S d with a < b acts by (a b) → P ab with The images of U(gl m|n ) and CS d in the endomorphism algebra End (C m|n ) ⊗d satisfy the double centralizer property, which leads to the multiplicity-free decomposition summed over the (m, n)-hook partitions λ with d boxes, where S λ is the Specht module over S d associated with λ.By representing the group algebra CS d as the direct sum of matrix algebras we can think of S λ as the canonical irreducible module over Mat f λ (C) isomorphic to C f λ .The diagonal matrix units e U = e λ U U ∈ Mat f λ (C) parameterized by standard λ-tableaux U are primitive idempotents in CS d .We may conclude that if λ is an (m, n)-hook partition with d boxes, then the image e U (C m|n ) ⊗d is isomorphic to the gl m|n -module V (λ ♯ ).If λ is not contained in the (m, n)-hook, then the image is zero.
Explicit formulas for the idempotents e U can be derived with the use of the orthonormal Young basis of S λ via the Jucys-Murphy elements x 1 , . . ., x d of the group algebra CS d defined by x 1 = 0 and Given a standard λ-tableau U, denote by V the standard tableau obtained from U by removing the box α occupied by d.Then the shape of V is a diagram which we denote by λ − .A box outside λ − is called addable, if the union of λ − and the box is a Young diagram.We let c = j − i denote the content of the box α = (i, j) and let a 1 , . . ., a l be the contents of all addable boxes of λ − except for α.The Jucys-Murphy formula gives an inductive rule for the calculation of e U : On the other hand, by [5,Sec. 2.4], an application of a chain of odd reflections shows that the extremal weight of the gl m|n -module V (λ ♯ ) with respect to the opposite Borel subalgebra is λ ♭ .That is, there is a nonzero vector η ∈ V (λ ♯ ) of the weight λ ♭ such that E i,i+1 η = 0 for all i = m and E m+n,1 η = 0.By taking the lowest vector with respect to the action of gl m ⊕gl n we conclude that V (λ ♯ ) contains a nonzero vector ζ of the weight (µ m , . . ., µ 1 , λ ′ n , . . ., λ ′ 1 ) such that E ij ζ = 0 for all i > j.Thus, the vector ζ is the highest vector of the Yangian module L U and its weight is found by taking into account (A.2) and (A.16) so that this module is isomorphic to L(π ♭ (u)).
By using the coproduct (A.3) and the vector representation (A.12) instead of (A.9), for any complex parameters z a we get a representation of the Yangian on the space (C m|n ) ⊗d defined by For suitable parameters z a , the subspaces L U = e U (C m|n ) ⊗d associated with the standard tableaux U of shapes λ = (d) and λ = (1 d ), turn out to be invariant under this Yangian action as well.The primitive idempotents e U associated with the standard row and column tableaux are, respectively, the symmetrizer and anti-symmetrizer in CS d , Note that (A.6) and (A.7) yield multiplicative formulas for h (d) and a (d) .
Since ω h (d) = h (d) and ω a (d) = sgn ω •a (d) , both parts of the corollary follow from the particular cases of Theorem A.1 concerning the action (A.13) for the row and column tableaux U.
The following corollary in the case n = 0 was used in the construction of the fundamental modules over the Yangians of types B, C and D in [2,Sec. 5.3].It was also applied to the orthosymplectic Yangians in [13] and in the proof of Proposition 2.9 in the previous section.Equip the tensor product space (C m|n ) ⊗d with the action of Y(gl m|n ) by setting Proof.This is immediate from Corollary A.2(ii), because the vector ξ d is the highest vector of the Y(gl m|n )-module a (d) (C m|n ) ⊗d with the highest weight (A.17).