A quantum Sylvester theorem and skew representations of twisted Yangians

A. I. Molev

School of Mathematics and Statistics University of Sydney, NSW 2006, Australia

             alexm@maths.usyd.edu.au

Abstract

Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL_{N} and the Yangian for 𝔤𝔩_{N}. We prove a version of this theorem for the twisted Yangians Y(𝔤_{N}) associated with the orthogonal and symplectic Lie algebras 𝔤_{N} = 𝔬_{N} or 𝔰𝔭_{N}. This gives rise to representations of the twisted Yangian Y(𝔤_{N−M}) on the space of homomorphisms Hom_{𝔤_{M}}(W, V), where W and V are finite-dimensional irreducible modules over 𝔤_{M} and 𝔤_{N}, respectively. In the symplectic case these representations turn out to be irreducible and we identify them by calculating the corresponding Drinfeld polynomials. We also apply the quantum Sylvester theorem to realize the twisted Yangian as a projective limit of certain centralizers in universal enveloping algebras.

1 Introduction

Let 𝔤 be a complex reductive Lie algebra and 𝔞 ⊂ 𝔤 a reductive subalgebra. Suppose that V is a finite-dimensional irreducible 𝔤-module and consider its restriction to the subalgebra 𝔞. This restriction is isomorphic to a direct sum of irreducible finitedimensional 𝔞-modules W_{μ} with certain multiplicities m_{μ},

V |_{𝔞} ≅ ⊕_{μ} m_{μ} W_{μ}.

If each W_{μ} is provided with a basis and the decomposition is multiplicity-free (i.e., m_{μ} ⩽ 1 for all μ) then it can be used to get a basis of V as the union of the bases of the spaces W_{μ} which occur in the decomposition. This observation played a key role in the construction of the Gelfand–Tsetlin bases for the representations of the general linear and orthogonal Lie algebras. Although the restriction of an irreducible finitedimensional representation of the symplectic Lie algebra 𝔰𝔭_{2n} to the subalgebra 𝔰𝔭_{2n−2} is not multiplicity-free in general, this approach can be extended to the symplectic case with the use of the isomorphism

V ≅ ⊕_{μ} U_{μ} ⊗ W_{μ},  (1.1)

where

U_{μ} = Hom_{𝔞}(W_{μ}, V), dim U_{μ} = m_{μ}.

The space U_{μ} is an irreducible module over the algebra C(𝔤, 𝔞) = U(𝔤)^{𝔞}, the centralizer of 𝔞 in the universal enveloping algebra U(𝔤); see e.g. Dixmier [1, Section 9.1]. Now, if some bases of the spaces U_{μ} and W_{μ} are given then the decomposition (1.1) yields the natural tensor product basis of V. The general difficulty of this approach is the complicated structure of the algebra C(𝔤, 𝔞). For each pair of the classical Lie algebras

(𝔤, 𝔞) = (𝔤𝔩_{N}, 𝔤𝔩_{M}), (𝔬_{N}, 𝔬_{M}), (𝔰𝔭_{N}, 𝔰𝔭_{M}),

(with even N and M in the symplectic case), the centralizer C(𝔤, 𝔞) and its representations can be studied with the use of the quantum algebras called Yangians and twisted Yangians. The Yangian Y(𝔤𝔩_{N}) for the general linear Lie algebra 𝔤𝔩_{N} is a deformation of the universal enveloping algebra U(𝔤𝔩_{N} ⊗ ℂ[x]) in the class of Hopf algebras; see e.g. Drinfeld [2]. The twisted Yangian Y(𝔤_{N}) for the orthogonal or symplectic Lie algebra (𝔤_{N} = 𝔬_{N} or 𝔤_{N} = 𝔰𝔭_{N}) was introduced by Olshanski [18]. This is a subalgebra of Y(𝔤𝔩_{N}) and it can also be presented by generators and defining relations; see also [13]. Finite-dimensional irreducible representations of the algebras Y(𝔤𝔩_{N}) and Y(𝔤_{N}) admit a complete parametrization; see Drinfeld [3] and Tarasov [19] for the Yangian case, and the author’s work [9] for the twisted Yangian case. The Olshanski centralizer construction [17, 18] provides ‘almost surjective’ algebra homomorphisms

Y(𝔤𝔩_{N−M}) → C(𝔤𝔩_{N}, 𝔤𝔩_{M}),  Y(𝔤_{N−M}) → C(𝔤_{N}, 𝔤_{M})  (1.2)

which allow one to equip the corresponding C(𝔤, 𝔞)-module U_{μ} in (1.1) with the structure of a representation of the Yangian or twisted Yangian, respectively. In particular, in the case N −M = 2 this module over the twisted Yangian Y(𝔤_{2}) admits a natural basis which leads to a construction of weight bases of Gelfand–Tsetlin type for the representations of the orthogonal and symplectic Lie algebras; see [12] for a review of these results.

In this paper we exploit the relationship between the (twisted) Yangians and the classical Lie algebras in the reverse direction: we use the weight bases constructed in [12] to investigate the representations of the twisted Yangians Y(𝔤_{N−M}) emerging from the homomorphisms (1.2).

By the results of [3] and [9] the isomorphism class of each finite-dimensional irreducible representation V of the (twisted) Yangian is determined by its highest weight which is a tuple of formal series over ℂ in a formal parameter. Moreover, simultaneous multiplication of all components of the highest weight by a fixed invertible formal series corresponds to a representation obtained from V by the composition with a simple automorphism of the (twisted) Yangian. It is natural to combine these representations into a single similarity class. In the case of the Yangian Y(𝔤𝔩_{N}) these similarity classes correspond to finite-dimensional irreducible representations of the Yangian for the special linear Lie algebra 𝔰𝔩_{N}. Both in the case of the Yangian and the twisted Yangian the similarity classes are parameterized by families of the Drinfeld polynomials (P_{1}(u), …, P_{r}(u)) with some additional data in the twisted case. Each P_{i}(u) is a monic polynomial in u, and r is the rank of the corresponding Lie algebra.

Given partitions λ = (λ_{1}, …, λ_{N}) and μ = (μ_{1}, …, μ_{M}), let V (λ) and V (μ) be the finite-dimensional irreducible representation of 𝔤𝔩_{N} and 𝔤𝔩_{M} with the highest weights λ and μ, respectively. The space Hom_{𝔤𝔩_{M}} (V (μ), V (λ)) is then an irreducible representation of the Yangian Y(𝔤𝔩_{N−M}). Its Drinfeld polynomials were calculated by Nazarov and Tarasov [16]. The result is a simple combinatorial rule which allows one to ‘read off’ each polynomial P_{i}(u) from the contents of the cells of the skew diagram λ/μ. These skew representations of the Yangian (they were called elementary in [16]), may be regarded as building blocks for the class of tame representations. This class is characterized by the property that the action of a natural commutative subalgebra of the Yangian in such a representation is semisimple. By [16], each tame representation is isomorphic to a tensor product of skew representations.

A different way to define the homomorphism (1.2) in the case of 𝔤𝔩_{N} is provided by the quantum Sylvester theorem. Recall that the classical Sylvester theorem is the following identity for a numerical N × N matrix A = (a_{ij}):

det B = det A · (a _{m+1··· N}^{m+1··· N})^{m−1},

where B = (b_{ij}) is the m × m matrix formed by the minors b_{ij} = a _{j,m+1··· N}^{i,m+1··· N} of A. The sequences of top and bottom indices indicate the row and column numbers of the minor, respectively. The most general noncommutative analog of this identity was given by Gelfand and Retakh in the context of the theory of quasideterminants originated in their work [5]; see also [4] for a review of this theory. ‘Quantum’ versions of this identity apply to the matrices formed by the generators of certain quantum algebras, and the determinants are replaced by appropriate quantum determinants. In particular, such a version was given by Krob and Leclerc [7] for the quantized algebra of functions on GL_{N}. Their approach is also applicable to the Yangian Y(𝔤𝔩_{N}). A different proof for the Yangian case is given in [11] where the corresponding quantum Sylvester theorem was used to give a modified version of the Olshanski centralizer construction. This provided a new definition of the skew representations of the Yangian and the calculation of their Drinfeld polynomials.

In this paper we produce a quantum Sylvester theorem for the twisted Yangian Y(𝔤_{N}) with the use of the Sklyanin minors of the matrix of generators of Y(𝔤_{N}). We first obtain the theorem for the extended twisted Yangian X(𝔤_{N}) (Section 2), following the approach of [7]. The twisted Yangian Y(𝔤_{N}) is a quotient of X(𝔤_{N}) which yields the corresponding result for Y(𝔤_{N}) (Section 3). In Section 4 we apply the quantum Sylvester theorem to construct a new homomorphism (1.2) for the twisted Yangian and introduce the corresponding skew representations. We show that in the symplectic case each skew representation is irreducible and calculate its highest weight and the Drinfeld polynomials. The Drinfeld polynomials are found by the following simple combinatorial rule somewhat analogous to the Yangian case [16] (see Section 4 below for a detailed formulation). Given a partition ν = (ν_{1}, …, ν_{n}) we draw its diagram Γ(ν) as follows. First, place the row with ν_{n} unit cells on the plane in such a way that the center of the leftmost cell coincides with the origin. Then place the second row with ν_{n−1} − ν_{n} cells in such a way that the southwest corner of this row coincides with the northeast corner of the first row. Continuing in this manner, we complete this procedure by placing an infinite row of cells in such a way that its southwest corner coincides with the northeast corner of the row with ν_{1} − ν_{2} cells. The diagram Γ(ν) is obtained as the union of the rows just placed and their images under the central symmetry with respect to the southwest corner of the first row. The figure below represents the diagram for the partition ν = (7, 4), where the dot indicates the origin.

[figure]

To each cell of the diagram we attach its diagonal number, where by diagonals we mean the lines passing northwest-southeast through the integer points of the plane. The line on the figure indicates the 0-th diagonal and the diagonal numbers are consecutive integers increasing from right to left. For any nonnegative integer p denote by Γ(λ)^{(p)} the diagram Γ(λ) lifted p units up.

Suppose now that V (λ) and V (μ) are the irreducible finite-dimensional representations of 𝔰𝔭_{2n} and 𝔰𝔭_{2m} corresponding to partitions λ and μ having n and m parts, respectively. Then the Drinfeld polynomials P_{1}(u), …, P_{n−m}(u) for the skew representation Hom_{𝔰𝔭_{2m}}(V (μ), V (λ)) of the twisted Yangian Y(𝔰𝔭_{2n−2m}) can be calculated by the following rule: all roots of the polynomial P_{k}(u) are simple and they coincide with the diagonal numbers decreased by 1/2 of the cells of the intersection Γ(μ) ∩ Γ(λ)^{(k−1)} (see Theorem 4.9 and Example 4.10 below).

Finally, in Section 5 we give a realization of the twisted Yangian Y(𝔤_{N}) as a projective limit of centralizers in the universal enveloping algebras. This is a new version of the centralizer construction (cf. [18, 14]) which is based on the quantum Sylvester theorem.

The recent work of Nazarov [15] is also devoted to the skew representations of the twisted Yangians although from a different perspective. He uses the classical Weyl’s approach and gives a realization of the skew representations in the tensor powers of the vector representation by applying certain generalized Young symmetrizers.

2 Extended twisted Yangian

We start by stating and proving some auxiliary results about the extended twisted Yangian X(𝔤_{N}); see [13] for more details.

2.1 Preliminaries

We shall be considering the orthogonal and symplectic cases simultaneously, unless otherwise stated. Given a positive integer N, we number the rows and columns of N × N matrices by the indices {−n, …, −1, 0, 1, …, n} if N = 2n + 1, and by {−n, …, −1, 1, …, n} if N = 2n. Similarly, in the latter case the range of summation indices −n ⩽ i, j ⩽ n will usually exclude 0. It will be convenient to use the symbol θ_{ij} which is defined by

θ_{ij} = \begin{cases}1 & in the orthogonal case, \\ sgn i · sgn j & in the symplectic case. \end{cases}

Throughout the paper, whenever the double sign ± or ∓ occurs, the upper sign corresponds to the orthogonal case and the lower sign to the symplectic case. By A ↦ A^{t} we will denote the matrix transposition such that (A^{t})_{ij} = θ_{ij} A_{−j,−i}. Let the E_{ij} denote the standard basis vectors of the general linear Lie algebra 𝔤𝔩_{N}. These vectors may be also regarded as elements of the universal enveloping algebra U(𝔤𝔩_{N}). For this reason we want to distinguish the E_{ij} from the standard matrix units e_{ij} which are considered as basis elements of the endomorphism algebra End ℂ^{N}. Introduce the following elements of the Lie algebra 𝔤𝔩_{N}:

F_{ij} = E_{ij} − θ_{ij}E_{−j,−i}, −n ⩽ i, j ⩽ n.

The Lie subalgebra 𝔤_{N} of 𝔤𝔩_{N} spanned by the elements F_{ij} is isomorphic to the orthogonal Lie algebra 𝔬_{N} or the symplectic Lie algebra 𝔰𝔭_{N} (in the latter case N is even).

The extended twisted Yangian X(𝔤_{N}) corresponding to the Lie algebra 𝔤_{N} is the associative algebra with generators s_{ij}^{(1)}, s_{ij}^{(2)}, … where −n ⩽ i, j ⩽ n, subject to the defining relations written in terms of the generating series

s_{ij}(u) = δ_{ij} + s_{ij}^{(1)}u^{−1} + s_{ij}^{(2)}u^{−2} + ⋯ ∈ X(𝔤_{N})[[u^{−1}]]

as follows

\begin{matrix}(u^{2} − v^{2}) [s_{ij}(u), s_{kl}(v)] = (u + v) & (s_{kj}(u)s_{il}(v) − s_{kj}(v)s_{il}(u)) \\ − (u − v) & (θ_{k,−j}s_{i,−k}(u)s_{−j,l}(v) − θ_{i,−l}s_{k,−i}(v)s_{−l,j}(u)) \\ + θ_{i,−j} & (s_{k,−i}(u)s_{−j,l}(v) − s_{k,−i}(v)s_{−j,l}(u)), \end{matrix}

where u and v denote formal variables. The defining relations can also be presented in a convenient matrix form. Denote by S(u) the N × N matrix whose ij-th entry is s_{ij}(u). We may regard S(u) as an element of the algebra X(𝔤_{N})[[u^{−1}]] ⊗ End ℂ^{N} given by

S(u) = ∑_{i,j} s_{ij}(u) ⊗ e_{ij},

where the e_{ij} denote the standard matrix units. For any positive integer m we shall be using the algebras of the form

X(𝔤_{N})[[u^{−1}]] ⊗ End ℂ^{N} ⊗ ⋯ ⊗ End ℂ^{N},  (2.1)

with m copies of End ℂ^{N}. For any a ∈ {1, …, m} we denote by S_{a}(u) the matrix S(u) which acts on the a-th copy of End ℂ^{N}. That is, S_{a}(u) is an element of the algebra (2.1) of the form

S_{a}(u) = ∑_{i,j} s_{ij}(u) ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ e_{ij} ⊗ 1 ⊗ ⋯ ⊗ 1,

where the e_{ij} belong to the a-th copy of End ℂ^{N} and 1 is the identity matrix. Similarly, if

C = ∑_{i,j,k,l} c_{ijkl} e_{ij} ⊗ e_{kl} ∈ End ℂ^{N} ⊗ End ℂ^{N},

then for distinct indices a, b ∈ {1, …, m} we introduce the element C_{ab} of the algebra (2.1) by

C_{ab} = ∑_{i,j,k,l} c_{ijkl} 1 ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ e_{ij} ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ e_{kl} ⊗ 1 ⊗ ⋯ ⊗ 1,

where the e_{ij} and e_{kl} belong to the a-th and b-th copies of End ℂ^{N}, respectively. Consider now the permutation operator

P = ∑_{i,j} e_{ij} ⊗ e_{ji} ∈ End ℂ^{N} ⊗ End ℂ^{N}.

The rational function R(u) = 1 − P u^{−1} with values in the tensor product algebra End ℂ^{N} ⊗ End ℂ^{N} is called the Yang R-matrix. Introduce its transposed R^{t}(u) by

R^{t}(u) = 1 − Q u^{−1},  Q = ∑_{i,j} θ_{ij} e_{−j,−i} ⊗ e_{ji}.  (2.2)

The defining relations for the extended twisted Yangian X(𝔤_{N}) are equivalent to the quaternary relation

R(u − v)S_{1}(u)R^{t}(−u − v)S_{2}(v) = S_{2}(v)R^{t}(−u − v)S_{1}(u)R(u − v).  (2.3)

The matrix S(u) is invertible and we shall denote by S^{−1}(u) the inverse matrix. The mapping

S(u) ↦ S^{−1}(−u − N/2)  (2.4)

defines an involutive automorphism of the algebra X(𝔤_{N}); see [13, Proposition 6.5].

Let u_{1}, …, u_{k} be independent variables. For k ⩾ 2 consider the rational function R(u_{1}, …, u_{k}) with values in (End ℂ^{N})^{⊗k} defined by

R(u_{1}, …, u_{k}) = (R_{k−1,k})(R_{k−2,k}R_{k−2,k−1}) ⋯ (R_{1k} ⋯ R_{12}),

where we abbreviate R_{ij} = R_{ij}(u_{i} − u_{j}). Set

S_{i} = S_{i}(u_{i}), 1 ⩽ i ⩽ k and R_{ij}^{t} = R_{ji}^{t} = R_{ij}^{t}(−u_{i} − u_{j}), 1 ⩽ i < j ⩽ k.

For an arbitrary permutation (p_{1}, …, p_{k}) of the numbers 1, …, k, we abbreviate

⟨S_{p_{1}}, …, S_{p_{k}}⟩ = S_{p_{1}}(R_{p_{1}p_{2}}^{t} ⋯ R_{p_{1}p_{k}}^{t})S_{p_{2}}(R_{p_{2}p_{3}}^{t} ⋯ R_{p_{2}p_{k}}^{t}) ⋯ S_{p_{k}}.

The identity

R(u_{1}, …, u_{k})⟨S_{1}, …, S_{k}⟩ = ⟨S_{k}, …, S_{1}⟩R(u_{1}, …, u_{k})  (2.5)

can be deduced from the quaternary relation (2.3); see [13, Proposition 4.2]. Now specialize the variables u_{i} by setting

u_{i} = u − i + 1,  i = 1, …, k.  (2.6)

It is well known that under this specialization, R(u_{1}, …, u_{k}) coincides with the antisymmetrization operator A_{k} on (ℂ^{N})^{⊗k}, where

A_{k} = ∑_{σ∈𝔖_{k}} sgn σ · P_{σ},

and P_{σ} denotes the image of σ ∈ 𝔖_{k} under the natural action of 𝔖_{k} on (ℂ^{N})^{⊗k}; see e.g. [13, Proposition 2.3]. Hence specializing the variables in (2.5) we get

A_{k} ⟨S_{1}, …, S_{k}⟩ = ⟨S_{k}, …, S_{1}⟩ A_{k}.

This element of the tensor product X(𝔤_{N})[[u^{−1}]] ⊗ (End ℂ^{N})^{⊗k} can be written as

∑ s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u) ⊗ e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{k}b_{k}},

summed over the indices a_{i}, b_{i} ∈ {−n, …, n}. We also set s_{b}^{a}(u) = s_{ab}(u). We call the elements s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u) of X(𝔤_{N})[[u^{−1}]] the Sklyanin minors of the matrix S(u). Clearly, the Sklyanin minors are skew-symmetric with respect to permutations of the upper indices and of the lower indices:

s _{b_{1}··· b_{k}}^{a_{σ(1)}··· a_{σ(k)}}(u) = sgn σ · s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u) and s _{b_{σ(1)}··· b_{σ(k)}}^{a_{1}··· a_{k}}(u) = sgn σ · s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u)

for any σ ∈ 𝔖_{k}.

Proposition 2.1. We have the relations

\begin{matrix}(u^{2} − v^{2}) & [s_{pq}(u), s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(v)] \\ = & (u + v) ∑_{i=1}^{k} (s_{a_{i}q}(u) s _{b_{1} ··· b_{k}}^{a_{1}··· p ··· a_{k}}(v) − s _{b_{1}··· q ··· b_{k}}^{a_{1} ··· a_{k}}(v) s_{pb_{i}}(u)) \\ − & (u − v) ∑_{i=1}^{k} (θ_{a_{i},−q} s_{p,−a_{i}}(u) s _{b_{1} ··· b_{k}}^{a_{1}··· −q ··· a_{k}}(v) − θ_{p,−b_{i}} s _{b_{1}··· −p ··· b_{k}}^{a_{1} ··· a_{k}}(v) s_{−b_{i},q}(u)) \\ + & θ_{p,−q} ∑_{i=1}^{k} (s_{a_{i},−p}(u) s _{b_{1} ··· b_{k}}^{a_{1}··· −q ··· a_{k}}(v) − s _{b_{1}··· −p ··· b_{k}}^{a_{1} ··· a_{k}}(v) s_{−q,b_{i}}(u)) \\ + & ∑_{i≠j} (θ_{a_{j},−q} s_{a_{i},−a_{j}}(u) s _{b_{1} ··· b_{k}}^{a_{1}··· p ··· −q ··· a_{k}}(v) − θ_{p,−b_{i}} s _{b_{1}··· −p ··· q ··· b_{k}}^{a_{1} ··· a_{k}}(v) s_{−b_{i},b_{j}}(u)), \end{matrix}

where in the Sklyanin minors the indices p and q replace a_{i} and b_{i}, respectively, in the first sum; the indices −q and −p replace a_{i} and b_{i}, respectively, in the second and third sums; in the fourth sum p and −q replace a_{i} and a_{j}, respectively, and −p and q replace b_{i} and b_{j}, respectively.

Proof. By (2.5), we have the relation

R(u, v, v − 1, …, v − k + 1) ⟨S_{0}, …, S_{k}⟩ = ⟨S_{k}, …, S_{0}⟩ R(u, v, v − 1, …, v − m + 1), (2.7)

where we have used an extra copy of the algebra End ℂ^{N} labelled by 0 and the parameters are specialized as follows

u_{0} = u, and u_{i} = v − i + 1 for i = 1, …, k.

Then one easily verifies (see e.g. [11]) that the product of R-matrices in (2.7) simplifies to

R(u, v, v − 1, …, v − k + 1) = A_{k} (1 − \frac{1}{u − v} (P_{01} + ⋯ + P_{0k})).

Applying the transposition over the zeroth copy of End ℂ^{N} and replacing u by −u we also deduce that

A_{k} R_{01}^{t} ⋯ R_{0k}^{t} = A_{k} (1 + \frac{1}{u + v} (Q_{01} + ⋯ + Q_{0k})).  (2.8)

Hence (2.7) takes the form

(1 − \frac{1}{u − v} (P_{01} + ⋯ + P_{0k})) S_{0}(u) (1 + \frac{1}{u + v} (Q_{01} + ⋯ + Q_{0k})) A_{k} ⟨S_{1}, …, S_{k}⟩

= ⟨S_{k}, …, S_{1}⟩ A_{k} (1 + \frac{1}{u + v} (Q_{01} + ⋯ + Q_{0k})) S_{0}(u) (1 − \frac{1}{u − v} (P_{01} + ⋯ + P_{0k})).

It remains to apply both sides to the vector e_{q} ⊗ e_{b_{1}} ⊗ ⋯ ⊗ e_{b_{k}} and compare the coefficients at the vector e_{p} ⊗ e_{a_{1}} ⊗ ⋯ ⊗ e_{a_{k}}, where the e_{i} denote the canonical basis vectors of ℂ^{N}.

Corollary 2.2. Suppose that for some indices i, j, l, m ∈ {1, …, k} we have a_{i} = −b_{l} and b_{j} = −a_{m}. Then

[s_{a_{i}b_{j}}(u), s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(v)] = 0.

Proof. By the skew-symmetry property, the Sklyanin minor is zero if it has two repeated upper or lower indices. Hence we may assume that i = m if and only if j = l. Suppose first that i ≠ m. Then using the skew-symmetry of Sklyanin minors, we derive from Proposition 2.1 that

(u − v − 1)(u + v + 1) [s_{a_{i}b_{j}}(u), s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(v)] = θ_{a_{i},−b_{j}} [s_{−b_{j},−a_{i}}(u), s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(v)].

The same relation holds with i and j replaced by m and l, respectively, which proves the claim in the case under consideration. If a_{i} = −b_{j} then Proposition 2.1 immediately gives

(u − v − 1)(u + v + 1) [s_{a_{i}b_{j}}(u), s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(v)] = 0,

completing the proof.

The series

sdet S(u) = s_{−n ··· n}^{−n ··· n}(u) ∈ X(𝔤_{N})[[u^{−1}]]

is called the Sklyanin determinant of the matrix S(u). Corollary 2.2 implies that all the coefficients of this series belong to the center of the algebra X(𝔤_{N}); see also [13, Theorem 4.8] for a slightly different proof. The Sklyanin comatrix Ŝ(u) is defined by the relation

Ŝ(u) S(u − N + 1) = sdet S(u).  (2.9)

Due to (2.4), the mapping

S(u) ↦ Ŝ(−u + N/2 − 1)  (2.10)

defines a homomorphism of X(𝔤_{N}) into itself. Multiplying both sides of (2.9) by the inverse to S(u − N + 1) and taking the nn-th entry we get

ŝ_{nn}(u) = sdet S(u) (S^{−1}(u − N + 1))_{nn}.

Therefore,

sdet S(u) = ŝ_{nn}(u) · |S(u − N + 1)|_{nn},  (2.11)

where we have used the notation |A|_{ij} for the ij-th quasideterminant of a matrix A over a ring; see [5, 6]. By definition, |A|_{ij} equals the inverse of the matrix element (A^{−1})_{ji} provided that the matrix A and the element (A^{−1})_{ji} are invertible.

We shall also use the auxiliary minors š_{b_{1}··· b_{k−1},c}^{a_{1}··· a_{k}}(u) ∈ X(𝔤_{N})[[u^{−1}]] defined by

A_{k} ⟨S_{1}, …, S_{k−1}⟩ R_{1k}^{t} ⋯ R_{k−1,k}^{t}

= ∑ š_{b_{1}··· b_{k−1},c}^{a_{1}··· a_{k}}(u) ⊗ e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{k−1}b_{k−1}} ⊗ e_{a_{k}c}, (2.12)

summed over a_{i}, b_{i}, c ∈ {−n, …, n}. Since

A_{k} ⟨S_{1}, …, S_{k−1}⟩ R_{1k}^{t} ⋯ R_{k−1,k}^{t} S_{k} = A_{k} ⟨S_{1}, …, S_{k}⟩,

we immediately obtain the relation

s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u) = ∑_{c=−n}^{n} š_{b_{1}··· b_{k−1},c}^{a_{1}··· a_{k}}(u) s_{cb_{k}}(u − k + 1).  (2.13)

We obviously have

š_{b_{1}··· b_{k−1},c}^{a_{σ(1)}··· a_{σ(k)}}(u) = sgn σ · š_{b_{1}··· b_{k−1},c}^{a_{1}··· a_{k}}(u)

for any σ ∈ 𝔖_{k}. Also,

š_{b_{σ(1)}··· b_{σ(k−1)},c}^{a_{1}··· a_{k}}(u) = sgn σ · š_{b_{1}··· b_{k−1},c}^{a_{1}··· a_{k}}(u)

for any σ ∈ 𝔖_{k−1}; see [8]. Furthermore, it is straightforward to obtain the following property of the auxiliary minors from their definition (cf. [8, Proposition 4.4]): if c ∉ {a_{1}, …, a_{k−1}} and c ∉ {−b_{1}, …, −b_{k−1}} then

š_{b_{1}··· b_{k−1},c}^{a_{1}··· a_{k}}(u) = 0  (2.14)

if c ≠ a_{k}, while

š_{b_{1}··· b_{k−1}, c}^{a_{1}··· a_{k−1}, c}(u) = s _{b_{1}··· b_{k−1}}^{a_{1}··· a_{k−1}}(u).  (2.15)

Set (a_{1}, …, a_{N}) = (−n, …, n). Then the matrix elements ŝ_{a_{i}a_{j}}(u) of the Sklyanin comatrix Ŝ(u) are given by

ŝ_{a_{i}a_{j}}(u) = (−1)^{N−i} š_{a_{1}···â_{i} ··· a_{N} , a_{j}}^{a_{1}··· a_{N}}(u),  (2.16)

where the hat on the right hand side indicates the index to be omitted; see [8, Section 6].

2.2 Sylvester theorem

Fix a nonnegative integer m < n and set M = 2m or M = 2m + 1 if N = 2n or N = 2n + 1, respectively, so that N − M = 2n − 2m. Set B = {−m, …, m} and denote by S_{BB}(u) the submatrix of S(u) whose rows and columns are numbered by the elements of B. It follows from the defining relations for the extended twisted Yangian that the subalgebra of X(𝔤_{N}) generated by the elements s_{ij}^{(r)} with i, j ∈ B can be regarded as a natural homomorphic image of the extended twisted Yangian X(𝔤_{M}). The homomorphism takes the generators s_{ij}^{(r)} of X(𝔤_{M}) to the elements of X(𝔤_{N}) with the same name. This makes the Sklyanin determinant of the matrix S_{BB}(u) welldefined. Denote by A the complement of the subset B in the set {−n, …, n} and let X(𝔤_{N−M}) denote the extended twisted Yangian whose generator series s_{ab}(u) are enumerated by elements a, b ∈ A. For any such a, b set

s̃_{ab}(u) = s_{−m··· m,b}^{−m··· m,a} (u + M/2)  (2.17)

and denote by S̃(u) the (N − M) × (N − M) matrix whose ab-entry is s̃_{ab}(u).

Theorem 2.3. The mapping

s_{ab}(u) ↦ s̃_{ab}(u)  (2.18)

defines an algebra homomorphism X(𝔤_{N−M}) → X(𝔤_{N}). Moreover,

sdet S̃(u) = sdet S(u + M/2)

× sdet S_{BB}(u + M/2 − 1) ⋯ sdet S_{BB}(u + M/2 − N + M + 1).

Proof. We follow the approach of [7] based on complimentary minor identities. By the definition of the Sklyanin determinant,

A_{N} ⟨S_{1}, …, S_{N}⟩ = A_{N} sdet S(u).

This implies the relation

A_{N} ⟨S_{1}, …, S_{M+1}⟩ ∏_{i=1,...,M}^{→}(R_{i,M+2}^{t} ⋯ R_{iN}^{t})

= A_{N} sdet S(u) S_{N}^{−1}(R_{N−1,N}^{t})^{−1} S_{N−1}^{−1} ⋯ S_{M+2}^{−1}(R_{M+1,N}^{t})^{−1} ⋯ (R_{M+1,M+2}^{t})^{−1}. (2.19)

Note that since Q^{2} = NQ we find from (2.2) that

R^{t}(u)^{−1} = R^{t}(−u + N).

Therefore, the right hand side of (2.19) can be written as

A_{N} sdet S(u) S_{N}^{∘}R_{N−1,N}^{∘} S_{N−1}^{∘} ⋯ S_{M+2}^{∘}R_{M+1,N}^{∘} ⋯ R_{M+1,M+2}^{∘},

where we have used the notation S^{∘}(u) = S^{−1}(−u − N/2) and

S_{i}^{∘} = S_{i}^{∘}(u_{i}^{∘}),  R_{ij}^{∘} = R_{ij}^{t}(−u_{i}^{∘} − u_{j}^{∘}),  u_{i}^{∘} = −u_{i} − N/2  (2.20)

with the u_{i} are specialized as in (2.6). Let us set

(a_{1}, …, a_{N−M}) = (−n, …, −m − 1, m + 1, …, n)

and apply both sides of (2.19) to the vector

v_{ij} = e_{−m} ⊗ ⋯ ⊗ e_{m} ⊗ e_{a_{j}} ⊗ e_{a_{1}} ⊗ ⋯ ⊗ ê_{a_{i}} ⊗ ⋯ ⊗ e_{a_{N−M}} ,

where the e_{r} denote the canonical basis vectors of ℂ^{N} and i, j ∈ {1, …, N − M}. Comparing the coefficients at the vector A_{N} v_{11} we come to the relation

s_{−m ··· m, a_{j}}^{−m ··· m, a_{i}}(u) = (−1)^{N−M−i} sdet S(u) · š^{∘} _{a_{1}···â_{i} ··· a_{N−M} , a_{j}}^{a_{1}··· a_{N−M}}(u_{N}^{∘}),  (2.21)

where the auxiliary minors on the right hand side correspond to the matrix S^{∘}(u). They are well-defined, since by (2.4) S^{∘}(u) satisfies the quaternary relation (2.3). Now observe that we may regard these auxiliary minors as those for the submatrix S_{AA}^{∘}(u) of S^{∘}(u) whose row and column indices belong to the set A = {a_{1}, …, a_{N−M}}. Indeed, we have the following identity analogous to (2.8) which is verified in the same way:

A_{k−1} R_{1k}^{t} ⋯ R_{k−1,k}^{t} = A_{k−1}(1 + \frac{Q_{1k} + ⋯ + Q_{k−1,k}}{2u − k + 1} ).  (2.22)

Therefore the left hand side of (2.12) with S(u) replaced by S^{∘}(u) can be written as

A_{k} S_{1}^{∘} (1 + \frac{Q_{12}}{2u − 1}) S_{2}^{∘} (1 + \frac{Q_{13} + Q_{23}}{2u − 2} ) ⋯ S_{k−1}^{∘} (1 + \frac{Q_{1k} + ⋯ + Q_{k−1,k}}{2u − k + 1} ).

If we now put k = N − M and apply this operator to the vector

e_{a_{1}} ⊗ ⋯ ⊗ ê_{a_{i}} ⊗ ⋯ ⊗ e_{a_{N−M}} ⊗ e_{a_{j}}

then the coefficient at the vector e_{a_{1}} ⊗ ⋯ ⊗ e_{a_{N−M}} will be an expression involving only the entries of the submatrix S_{AA}^{∘}(u) of S^{∘}(u). Furthermore, by (2.16),

(−1)^{N−M−i} š^{∘} _{a_{1}···â_{i} ··· a_{N−M} , a_{j}}^{a_{1}··· a_{N−M}}(u_{N}^{∘}) = ŝ_{a_{i}a_{j}}^{∘}(u_{N}^{∘}),

where by ŝ_{ab}^{∘}(u) we denote the entries of the Sklyanin comatrix corresponding to the matrix S_{AA}^{∘}(u). Recalling that u_{N}^{∘} = −u + N/2 − 1 and N − M = 2n − 2m, we derive from (2.21) that

s_{−m ··· m, a_{j}}^{−m ··· m, a_{i}}(u + M/2) = sdet S(u + M/2) · ŝ_{a_{i}a_{j}}^{∘}(−u + n − m − 1).  (2.23)

By (2.10), the mapping s_{a_{i}a_{j}}(u) ↦ ŝ_{a_{i}a_{j}}^{∘}(−u + n − m − 1) defines a homomorphism X(𝔤_{N−M}) → X(𝔤_{N}), and hence so does the mapping (2.18) since the coefficients of the Sklyanin determinant are central in X(𝔤_{N}). This proves the first part of the theorem.

Applying the relation (2.23) to the matrix S^{∘}(u) we obtain for any elements a, b ∈ A

s^{∘}_{−m ··· m, b}^{−m ··· m, a}(u + M/2) = sdet S^{∘}(u + M/2) · ŝ′_{ab}(−u + n − m − 1),  (2.24)

where the ŝ′_{ab}(u) denote the entries of the Sklyanin comatrix corresponding to the matrix S_{AA}(u). Denote by S^{♯}(u) the inverse matrix to S_{AA}(−u − n + m). Due to (2.9) we have

Ŝ_{AA}(−u + n − m − 1) = sdet S_{AA}(−u + n − m − 1) · S^{♯}(u).

Combining this with (2.24) and using the notation (2.17) for the matrix S^{∘}(u) we come to

s̃_{ab}^{∘}(u) = sdet S^{∘}(u + M/2) · sdet S_{AA}(−u + n − m − 1) · s_{ab}^{♯}(u).

By [13, Theorem 7.6] applied to the matrix S^{∘}(u) we have

sdet S^{∘}(u) · sdet S_{AA}(−u + N/2 − 1) = sdet S_{BB}^{∘}(u).  (2.25)

Together with the previous relation this gives

s_{ab}^{♯}(u) = sdet S_{BB}^{∘}(u + M/2)^{−1} · s̃_{ab}^{∘}(u).  (2.26)

Note that sdet S_{BB}^{∘}(u) commutes with s̃_{ab}^{∘}(v). Indeed, this follows from Corollary 2.2 and the fact that the expansion of sdet S_{BB}^{∘}(u) in terms of the matrix elements of S^{∘}(u) only involves the series s_{ij}^{∘}(u) with i, j ∈ B. The latter is easily deduced from the definition of the Sklyanin minors and with the assistance of (2.22).

On the other hand, the particular case of (2.25) with B = ∅ yields

sdet S^{∘}(u) · sdet S(−u + N/2 − 1) = 1.

Hence,

sdet S^{♯}(u) · sdet S_{AA}(−u + n − m − 1) = 1

and so by (2.25),

sdet S^{♯}(u) = sdet S^{∘}(u + M/2) · sdet S_{BB}^{∘}(u + M/2)^{−1}.  (2.27)

Finally, write down the expansion of sdet S^{♯}(u) in terms of the matrix elements s_{ab}^{♯}(u) which is implied by the definition of the Sklyanin determinant. The expansion has the form of a linear combination of products

s_{a_{1}b_{1}}^{♯}(u) s_{a_{2}b_{2}}^{♯}(u − 1) ⋯ s_{a_{N−M} b_{N−M}}^{♯} (u − N + M + 1),

the coefficients being rational functions in u. Using the relation (2.26) we obtain

sdet S̃^{∘}(u) = sdet S^{♯}(u) · sdet S_{BB}^{∘}(u + M/2) ⋯ sdet S_{BB}^{∘}(u + M/2 − N + M + 1).

Taking into account (2.27) we derive the desired formula for the matrix S^{∘}(u). Since S(u) ↦ S^{∘}(u) is an involutive automorphism of X(𝔤_{N}), the formula holds for the matrix S(u) as well.

Denote by σ̂_{ab}(u) the entries of the Sklyanin comatrix corresponding to the matrix S̃(u).

Proposition 2.4. For any a, b ∈ A we have the relation

σ̂_{ab}(u) = ŝ_{ab}(u + M/2)

× sdet S_{BB}(u + M/2 − 1) ⋯ sdet S_{BB}(u + M/2 − N + M + 2).

Proof. The defining relations for X(𝔤_{N}) imply that

[s_{ab}^{(1)}, s_{ij}(u)] = δ_{ib} s_{aj}(u) − δ_{aj} s_{ib}(u) − θ_{i,−b} δ_{a,−i} s_{−b,j}(u) + θ_{a,−j} δ_{−j,b} s_{i,−a}(u). (2.28)

Hence, the elements s_{ab}^{(1)} with a, b ∈ A commute with sdet S_{BB}(v) (this also follows from Proposition 2.1). On the other hand, it is straightforward to verify with the use of (2.22) that s_{ab}^{(1)} with a ≠ b is stable under both the homomorphism (2.10) and the homomorphism which takes S_{AA}(u) to the Sklyanin comatrix [σ̂_{ab}(−u + n − m − 1)]. Therefore, it is sufficient to prove the relation in the particular case a = b = n. Taking subsequent commutators with appropriate elements s_{ab}^{(1)} will yield the relation in the general case.

Using the notation of the proof of Theorem 2.3 we can write the following relations implied by (2.11),

sdet S^{∘}(u + M/2) = ŝ_{nn}^{∘}(u + M/2) · |S^{∘}(u + M/2 − N + 1)|_{nn}

and

sdet S^{♯}(u) = ŝ_{nn}^{♯}(u) · |S^{♯}(u − N + M + 1)|_{nn},

where the ŝ_{ij}^{♯}(u) denote the entries of the Sklyanin comatrix corresponding to the matrix S^{♯}(u). The definition of quasideterminants implies that

|S^{∘}(u + M/2 − N + 1)|_{nn} = |S^{♯}(u − N + M + 1)|_{nn}.

Therefore, the above relations together with (2.27) yield

ŝ_{nn}^{♯}(u) = ŝ_{nn}^{∘}(u + M/2) · sdet S_{BB}^{∘}(u + M/2)^{−1}.  (2.29)

Now we conclude the argument in the same way as in the proof of Theorem 2.3. Using (2.16) write down the expansion of ŝ_{nn}^{♯}(u) in terms of the matrix elements s_{ab}^{♯}(u). It has the form of a linear combination of products

s_{a_{1}b_{1}}^{♯}(u) s_{a_{2}b_{2}}^{♯}(u − 1) ⋯ s_{a_{N−M} b_{N−M}}^{♯} (u − N + M + 2),

the coefficients being rational functions in u. Now (2.26) implies

σ̂_{nn}^{∘}(u) = ŝ_{nn}^{♯}(u) · sdet S_{BB}^{∘}(u + M/2) ⋯ sdet S_{BB}^{∘}(u + M/2 − N + M + 2),

where by σ̂_{ab}^{∘}(u) we denote the entries of the Sklyanin comatrix corresponding to the matrix S̃^{∘}(u). Using (2.29) we derive the desired formula for the matrix S^{∘}(u) and hence it holds for the matrix S(u) as well.

Interchanging the roles of the sets A and B in the above arguments one can easily derive the corresponding dual versions of Theorem 2.3 and Proposition 2.4. Here we only record the counterpart of the first part of Theorem 2.3 which will be used below.

Proposition 2.5. The mapping

s_{ij}(u) ↦ s_{−n ··· −m−1, j, m+1··· n}^{−n ··· −m−1, i, m+1··· n} (u + n − m), −m ⩽ i, j ⩽ m

defines an algebra homomorphism X(𝔤_{M}) → X(𝔤_{N}).

3 Sylvester theorem for the twisted Yangian

The twisted Yangian Y(𝔤_{N}) corresponding to the Lie algebra 𝔤_{N} is the quotient of the extended twisted Yangian X(𝔤_{N}) by the following symmetry relation

θ_{ij}s_{−j,−i}(−u) = s_{ij}(u) ± \frac{s_{ij}(u) − s_{ij}(−u)}{2u},  (3.1)

or, in the matrix form,

S^{t}(−u) = S(u) ± \frac{S(u) − S(−u)}{2u}.

From now on, we shall mainly work with the twisted Yangian and so we keep the same notation s_{ij}^{(r)} for the generators of the algebra Y(𝔤_{N}). Note that for any even series g(u) ∈ 1 + ℂ[[u^{−2}]] u^{−2} the mapping

s_{ij}(u) ↦ g(u) s_{ij}(u)  (3.2)

defines an automorphism of Y(𝔤_{N}).

As we shall see below, the homomorphism of Theorem 2.3 respects the symmetry relation in the orthogonal case, while in the symplectic case a minor correction is needed to obtain a corresponding homomorphism of the twisted Yangians. In order to treat both cases simultaneously, introduce the following notation

α_{p}(u) = \begin{cases}1 & in the orthogonal case \\ \frac{u + 1/2}{u − p + 1/2} & in the symplectic case. \end{cases}

The image of the Sklyanin determinant in the twisted Yangian Y(𝔤_{N}) acquires the following symmetry property

α_{n}(u)^{−1} · sdet S(u) = α_{n}(−u + N − 1)^{−1} · sdet S(−u + N − 1),  (3.3)

see [13, Section 4.11]. Moreover, the mapping

S(u) ↦ α_{n}(u) · Ŝ(−u + N/2 − 1)  (3.4)

defines a homomorphism of Y(𝔤_{N}) into itself; see [9, Proposition 2.1].

We can now prove a quantum Sylvester theorem for the twisted Yangian Y(𝔤_{N}). We use the notation of the previous section. In particular, recall that s̃_{ab}(u) denotes the quantum minor as in (2.17) for any a, b ∈ A.

Theorem 3.1. The mapping

s_{ab}(u) ↦ α_{−m}(u) s̃_{ab}(u)  (3.5)

defines an algebra homomorphism Y(𝔤_{N−M}) → Y(𝔤_{N}). Moreover,

sdet [α_{−m}(u) S̃(u)] = α(u) · sdet S(u + M/2) × sdet S_{BB}(u + M/2 − 1) ⋯ sdet S_{BB}(u + M/2 − N + M + 1),

where

α(u) = α_{−m}(u) α_{−m}(u − 1) ⋯ α_{−m}(u − N + M + 1).

Proof. Denote by S^{∗}(u) the matrix which occurs on the right hand side of (3.4). Then by (3.3) we have

S^{∘}(u) = \frac{α_{n}(u + N/2)}{c (u + N/2)} · S^{∗}(u),

where we have put c(u) = sdet S(u) for brevity. Due to (2.16) and the definition of the auxiliary minors, the relation (2.23) can be written as

s_{−m ··· m, a_{j}}^{−m ··· m, a_{i}}(u + M/2) = c (u + M/2)

× ∏_{i=1}^{N−M−1} \frac{α_{n}(−u + n − m + N/2 − i)}{c (−u + n − m + N/2 − i)} · ŝ_{a_{i}a_{j}}^{∗}(−u + n − m − 1), (3.6)

where we have also used the fact that the coefficients of the Sklyanin determinant are central in the twisted Yangian Y(𝔤_{N}), and by ŝ_{ab}^{∗}(u) we denote the entries of the Sklyanin comatrix corresponding to the matrix S_{AA}^{∗}(u). Observe that by (3.3) we have

c (u + M/2) · \frac{α_{n}(−u + n − m + N/2 − 1)}{c (−u + n − m + N/2 − 1)} = α_{n}(u + M/2).

Therefore, (3.6) takes the form

s_{−m ··· m, a_{j}}^{−m ··· m, a_{i}}(u + M/2) = α_{n}(u + M/2) · φ(u) · ŝ_{a_{i}a_{j}}^{∗}(−u + n − m − 1),

where

φ(u) = ∏_{i=2}^{N−M−1} \frac{α_{n}(−u + n − m + N/2 − i)}{c (−u + n − m + N/2 − i)}.

By the symmetry property (3.3) we have φ(u) = φ(−u), and so the multiplication of the generator series s_{ij}(u) by φ(u) preserves the twisted Yangian defining relations. Furthermore, by (3.4), the mapping

s_{ab}(u) ↦ α_{n−m}(u) ŝ_{ab}^{∗}(−u + n − m − 1), a, b ∈ A

defines a homomorphism Y(𝔤_{N−M}) ↦ Y(𝔤_{N−M}). Thus, we may conclude that the mapping

s_{ab}(u) ↦ α_{n−m}(u) α_{n}(u + M/2)^{−1} s_{−m ··· m, a_{j}}^{−m ··· m, a_{i}}(u + M/2)

defines a homomorphism Y(𝔤_{N−M}) ↦ Y(𝔤_{N−M}). To complete the proof, observe that α_{n−m}(u) α_{n}(u + M/2)^{−1} = α_{−m}(u).

The formula for the Sklyanin determinant of the matrix α_{−m}(u)S̃(u) is immediate from Theorem 2.3 and the definition of sdet S̃(u).

The corresponding version of Proposition 2.5 for the twisted Yangian has the following form.

Proposition 3.2. The mapping

s_{ij}(u) ↦ α_{m−n}(u) · s_{−n ··· −m−1, j, m+1··· n}^{−n ··· −m−1, i, m+1··· n} (u + n − m), −m ⩽ i, j ⩽ m

defines an algebra homomorphism Y(𝔤_{M}) → Y(𝔤_{N}).

Now we shall demonstrate that in the case of the twisted Yangian the entries of the Sklyanin comatrix can be expressed in terms of Sklyanin minors; cf. (2.16). We need the following lemma.

Lemma 3.3. For the twisted Yangian Y(𝔤_{N}) we have

A_{N} S_{1}(u) R_{12}^{t}(−2u + 1) ⋯ R_{1N}^{t}(−2u + N − 1) = \frac{2u + 1}{2u ± 1} A_{N} S_{1}^{t}(−u).  (3.7)

Proof. We have (N − 1)! A_{N} = A_{N} A′_{N−1}, where A′_{N−1} denotes the anti-symmetrizer corresponding to the subset of indices {2, …, N}. By (2.8),

A′_{N−1} R_{12}^{t}(−2u + 1) ⋯ R_{1N}^{t}(−2u + N − 1) = A′_{N−1} (1 + \frac{Q_{12} + ⋯ + Q_{1N}}{2u − 1} ).

Therefore, the left hand side of (3.7) takes the form

A_{N} S_{1}(u) (1 + \frac{Q_{12} + ⋯ + Q_{1N}}{2u − 1} ).

Apply this operator to a basis vector

v_{ij} = e_{a_{j}} ⊗ e_{a_{1}} ⊗ ⋯ ⊗ ê_{a_{i}} ⊗ ⋯ ⊗ e_{a_{N}} , i, j ∈ {1, …, N},

where (a_{1}, …, a_{N}) = (−n, …, n). The coefficient at A_{N} v_{ii} will be equal to s_{a_{i}a_{j}}(u) if a_{j} = −a_{i}, and equal to the expression

\frac{2u}{2u − 1}s_{a_{i}a_{j}}(u) ∓ \frac{1}{2u − 1}θ_{a_{i}a_{j}}s_{−a_{j},−a_{i}}(u)

if a_{j} ≠ −a_{i}. In both cases the coefficient coincides with

\frac{2u + 1}{2u ± 1} s_{a_{i}a_{j}}^{t}(−u)

due to the symmetry relation (3.1).

As before, we set (a_{1}, …, a_{N}) = (−n, …, n).

Proposition 3.4. For any i, j ∈ {1, …, N} we have the relation

ŝ_{a_{i}a_{j}}^{t}(u) = (−1)^{i+j} · α_{N−1}(u) · s _{a_{1} ··· â_{i}··· a_{N}}^{a_{1} ··· â_{j}··· a_{N}} (−u + N − 2).

Proof. The relations (2.9) and (2.19) with M = N − 2 imply that

A_{N} ⟨S_{1}, …, S_{N−1}⟩ = A_{N} Ŝ_{N}(u) (R_{N−1,N}^{t})^{−1} ⋯ (R_{1,N}^{t})^{−1}.  (3.8)

Using the notation (2.20) and the definition of the matrix S^{∗}(u) we can write Ŝ(u) = α_{n}(u) S^{∗}(u_{N}^{∘}) so that (3.8) becomes

A_{N} ⟨S_{1}, …, S_{N−1}⟩ = α_{n}(u) A_{N} S_{N}^{∗}(u_{N}^{∘}) R_{N−1,N}^{∘} ⋯ R_{1N}^{∘}.  (3.9)

However, S(u) ↦ S^{∗}(u) defines a homomorphism Y(𝔤_{N}) → Y(𝔤_{N}). Therefore, writing A_{N} = sgn σ · A_{N} P_{σ}, where σ = (1N)(2, N − 1) ⋯, and applying Lemma 3.3 we can simplify the right hand side of (3.9) as

α_{n}(u) A_{N} S_{N}^{∗}(u_{N}^{∘}) R_{N−1,N}^{∘} ⋯ R_{1N}^{∘} = α_{n}(u) \frac{2u_{N}^{∘} + 1}{2u_{N}^{∘} ± 1} A_{N} S^{∗}_{N}^{t}(−u_{N}^{∘}).

Hence, we come to the identity

A_{N} ⟨S_{1}, …, S_{N−1}⟩ = α_{n−1}(u) A_{N} S^{∗}_{N}^{t}(−u_{N}^{∘}).

Applying both sides to the basis vector e_{a_{1}} ⊗ ⋯ ⊗ ê_{a_{i}} ⊗ ⋯ ⊗ e_{a_{N}} ⊗ e_{a_{j}} we get

α_{n−1}(u) s_{a_{i}a_{j}}^{∗ t}(u − N/2 + 1) = (−1)^{i+j} s _{a_{1} ··· â_{i}··· a_{N}}^{a_{1} ··· â_{j}··· a_{N}} (u).

The argument is completed by using the definition of S^{∗}(u).

Suppose now that m = n − 1. As before, we identify the subalgebra of Y(𝔤_{N}) generated by the elements s_{ab}^{(r)} for a, b ∈ {−n, n} with the twisted Yangian Y(𝔤_{2}). The restriction of the homomorphism (3.4) to the subalgebra Y(𝔤_{2}) defines the homomorphism

ϕ : Y(𝔤_{2}) → Y(𝔤_{N}).

The defining relations of the twisted Yangian imply that the map

s_{ab}(u) ↦ sgn a · sgn b · s_{ab}(u), a, b ∈ {−n, n}

defines an automorphism ψ : Y(𝔤_{2}) → Y(𝔤_{2}).

Corollary 3.5. The homomorphism ϕ coincides with the homomorphism (3.5) in the symplectic case, while in the orthogonal case the homomorphism (3.5) coincides with the composition ϕ ∘ ψ.

Proof. This is immediate from Proposition 3.4 and the skew-symmetry of the Sklyanin minors.

The symmetry relation (3.1) allows one to obtain the following expansion of the auxiliary minors (see [8, Proposition 4.4]): if −b_{1} ∈ {a_{1}, …, a_{k−1}, c} and c ∉ {−b_{2}, …, −b_{k−1}} then

š_{b_{1}··· b_{k−1}, c}^{a_{1}··· a_{k−1}, c}(u) = \frac{2u + 1}{2u ± 1} ∑_{i=1}^{k−1}(−1)^{i−1} s_{a_{i}b_{1}}^{t}(−u) s _{b_{2} ··· b_{k−1}}^{a_{1}··· â_{i} ··· a_{k−1}}(u − 1).  (3.10)

Using this relation together with (2.13) and (2.14) one can derive explicit formulas for the Sklyanin determinant and some Sklyanin minors. The formulas use a special map

ω_{N} : 𝔖_{N} → 𝔖_{N},  p ↦ p′  (3.11)

from the symmetric group 𝔖_{N} into itself which is defined by the following inductive procedure. Given a set of positive integers c_{1} < ⋯ < c_{N} we regard 𝔖_{N} as the group of their permutations. If N = 2 we define ω_{2} as the map 𝔖_{2} → 𝔖_{2} whose image is the identity permutation. For N > 2 define a map from the set of ordered pairs (c_{k}, c_{l}) with k ≠ l into itself by the rule

\begin{matrix}(c_{k}, c_{l}) & ↦ (c_{l}, c_{k}), & k, l < N, \\ (c_{k}, c_{N}) & ↦ (c_{N−1}, c_{k}), & k < N − 1, \\ (c_{N}, c_{k}) & ↦ (c_{k}, c_{N−1}), & k < N − 1, \\ (c_{N−1}, c_{N}) & ↦ (c_{N−1}, c_{N−2}), \\ (c_{N}, c_{N−1}) & ↦ (c_{N−1}, c_{N−2}). \end{matrix}  (3.12)

Let p = (p_{1}, …, p_{N}) be a permutation of the indices c_{1}, …, c_{N}. Its image under the map ω_{N} is the permutation p′ = (p′_{1}, …, p′_{N−1}, c_{N}), where the pair (p′_{1}, p′_{N−1}) is the image of the ordered pair (p_{1}, p_{N}) under the map (3.12). Then the pair (p′_{2}, p′_{N−2}) is found as the image of (p_{2}, p_{N−1}) under the map (3.12) which is defined on the set of ordered pairs of elements obtained from (c_{1}, …, c_{N}) by deleting p_{1} and p_{N}. The procedure is completed in the same manner by determining consequently the pairs (p′_{i}, p′_{N−i}).

Now suppose that M is a positive integer and M = 2m or M = 2m + 1. For the proof of the following formula for the Sklyanin minor in the twisted Yangian Y(𝔤_{N}) see [8].

Proposition 3.6. Suppose that a_{1}, …, a_{M}, b_{M} are arbitrary indices from the set {−n, …, n}. We have

s _{a_{1} ··· a_{M−1},b_{M}}^{−a_{1} ··· −a_{M}} (u) = α_{m}(u) ∑_{p∈𝔖_{M}} sgn pp′ · s_{−a_{p(1)},a_{p′(1)}}^{t}(−u) ⋯ s_{−a_{p(m)},a_{p′(m)}}^{t}(−u + m − 1)

× s_{−a_{p(m+1)},a_{p′(m+1)}}(u − m) ⋯ s_{−a_{p(M)},b_{p′(M)}}(u − M + 1),

where the s_{ij}^{t}(u) denote the entries of the matrix S^{t}(u).

Remark 3.7. Although this determinant-like formula does not apply to arbitrary Sklyanin minors s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u), it provides explicit formulas for the Sklyanin determinant sdet S(u) and the matrix elements ŝ_{ij}(u) of the Sklyanin comatrix; see Proposition 3.4.

For the use in Section 5 we shall prove the following simple property of the map (3.11).

Lemma 3.8. The map 𝔖_{N} → 𝔖_{N} defined by p ↦ p (p′)^{−1}is bijective.

Proof. Suppose that p and q are two elements of 𝔖_{N} such that p (p′)^{−1} = q (q′)^{−1}. It suffices to show that p = q. By definition of the map ω_{N} we have p′_{N} = q′_{N} = N which implies that p_{N} = q_{N}. Then, due to the formulas (3.12), we have p′_{1} = q′_{1}. Hence, p_{1} = q_{1}. Now, since the pairs (p_{1}, p_{N}) and (q_{1}, q_{N}) coincide, so do their images under the map (3.12). In particular, p′_{N−1} = q′_{N−1}. This implies that p_{N−1} = q_{N−1} and the proof is completed by repeating this argument for the pairs (p_{i+1}, p_{N−i}) and (q_{i+1}, q_{N−i}) with i = 1, 2, . . ..

4 Skew representations

As before, we suppose that N = 2n or N = 2n + 1 so that

𝔤_{N} = 𝔬_{2n+1}, 𝔰𝔭_{2n}, or 𝔬_{2n}.  (4.1)

The finite-dimensional irreducible representations of 𝔤_{N} are in a one-to-one correspondence with n-tuples λ = (λ_{1}, …, λ_{n}) where the numbers λ_{i} satisfy the conditions

λ_{i} − λ_{i+1} ∈ ℤ_{+} for i = 1, …, n − 1,

and

−2 λ_{1} ∈ ℤ_{+} for 𝔤_{N} = 𝔬_{2n+1},
−λ_{1} ∈ ℤ_{+} for 𝔤_{N} = 𝔰𝔭_{2n},
−λ_{1} − λ_{2} ∈ ℤ_{+} for 𝔤_{N} = 𝔬_{2n}.

Such an n-tuple λ is called the highest weight^{1} of the corresponding representation which we shall denote by V (λ). It contains a unique, up to a constant factor, nonzero vector ξ (the highest vector) such that

\begin{aligned}F_{ii} ξ & = λ_{i} ξ & for & i = 1, …, n, \\ F_{ij} ξ & = 0 & for & − n ⩽ i < j ⩽ n. \end{aligned}

Let M be a nonnegative integer such that N −M is even and positive. So, M = 2m or M = 2m+1 for some m < n. We shall identify the Lie algebra 𝔤_{M} with the subalgebra of 𝔤_{N} spanned by the elements F_{ij} with the indices satisfying −m ⩽ i, j ⩽ m. Denote by V (λ)^{+} the subspace of 𝔤_{M}-highest vectors in V (λ):

1In a more common notation, the highest weight is the n-tuple (−λ_{n}, …, −λ_{1}). In particular, in the symplectic case this n-tuple is a partition.

V (λ)^{+} = {η ∈ V (λ) | F_{ij} η = 0, −m ⩽ i < j ⩽ m}.

Given a 𝔤_{M}-weight μ = (μ_{1}, …, μ_{m}) we denote by V (λ)_{μ}^{+} the corresponding weight subspace in V (λ)^{+}:

V (λ)_{μ}^{+} = {η ∈ V (λ)^{+} | F_{ii} η = μ_{i} η, i = 1, …, m}.

We have a natural vector space isomorphism V (λ)_{μ}^{+} ≅ Hom_{𝔤_{M}}(V (μ), V (λ)).

For any i, j ∈ {−n, …, n} introduce the series in u^{−1} with coefficients in the universal enveloping algebra U(𝔤_{N}) by

f_{ij}(u) = δ_{ij} + F_{ij}(u ± \frac{1}{2})^{−1}.

The mapping

π : s_{ij}(u) ↦ f_{ij}(u)  (4.2)

defines a surjective homomorphism Y(𝔤_{N}) → U(𝔤_{N}) called the evaluation homomorphism; see [18] and [13, Proposition 3.11]. Let F (u) denote the N × N matrix whose ij-th entry is the series f_{ij}(u). We may introduce the Sklyanin minors f _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u) of this matrix as the images of the corresponding minors of the matrix S(u) with respect to the evaluation homomorphism,

π : s _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u) ↦ f _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u).

By Theorem 3.1, we have a homomorphism Y(𝔤_{N−M}) → U(𝔤_{N}) given by

ρ : s_{ab}(u) ↦ α_{−m}(u) f _{−m··· m, b}^{−m··· m, a} (u + M/2).  (4.3)

Due to Corollary 2.2, the image of this homomorphism is contained in the centralizer U(𝔤_{N})^{𝔤_{M}} of the subalgebra 𝔤_{M} in the universal enveloping algebra U(𝔤_{N}). On the other hand, the vector space V (λ)_{μ}^{+} is obviously a representation of U(𝔤_{N})^{𝔤_{M}}. Thus, V (λ)_{μ}^{+} becomes equipped with the Y(𝔤_{N−M})-module structure defined via the homomorphism ρ. We call this module the skew representation of Y(𝔤_{N−M}). In the particular case M = 0 (with even N) the skew representation is just the evaluation module V (λ) over Y(𝔤_{N}) defined via the evaluation homomorphism (4.2).

The universal enveloping algebra U(𝔤_{N−M}) can be identified with a subalgebra of the twisted Yangian Y(𝔤_{N−M}) via the embedding F_{ab} ↦ s_{ab}^{(1)}, see [13, Proposition 3.12]. The elements F_{ab} are stable under the composition of this embedding with the homomorphism ρ. Indeed, if a ≠ b then this is verified directly from the definition of the Sklyanin minors with the use of (2.22). If a = b then we may assume without loss of generality that m = n − 1 and a = ±n. So, the claim follows from Corollary 3.5 and (3.3). In other words, the restriction of the Y(𝔤_{N−M})-module V (λ)_{μ}^{+} to the subalgebra U(𝔤_{N−M}) coincides with its natural action defined by the U(𝔤_{N})-action on V (λ).

We shall now concentrate on the symplectic case 𝔤_{N} = 𝔰𝔭_{N} where N = 2n. Our next goal is to prove that the skew representations of Y(𝔰𝔭_{N−M}) are irreducible. First we show the following.

Proposition 4.1. The centralizer U(𝔰𝔭_{N})^{𝔰𝔭^{M}} is generated by the image of the homomorphism ρ and the center of U(𝔰𝔭_{N}).

Proof. For a different homomorphism ρ′ : Y(𝔰𝔭_{N−M}) → U(𝔰𝔭_{N})^{𝔰𝔭^{M}} given by

s_{ab}(u) ↦ α_{n}(u) f̂_{ab}(−u + n − 1),  a, b ∈ A,  (4.4)

where f̂_{ab}(u) is the image of ŝ_{ab}(u) under the evaluation homomorphism (4.2), this statement was proved in [14, Section 4]. Denote by U′ the subalgebra of U(𝔰𝔭_{N})^{𝔰𝔭^{M}} generated by the center of U(𝔰𝔭_{N}) and the coefficients of the series f _{−m··· m, b}^{−m··· m, a} (u) with a, b ∈ A. It is sufficient to prove that all the coefficients of the series f̂_{ab}(u) belong to U′. However, the center of U(𝔰𝔭_{N}) is generated by the coefficients of the Sklyanin determinant sdet F (u), i.e., the image of sdet S(u) under the evaluation homomorphism (4.2); see [8, Theorem 5.2]. Then by Theorem 3.1 the coefficients of the series sdet F_{BB}(u) belong to U′. Due to Proposition 2.4, the coefficients of the series f̂_{ab}(u) also belong to U′.

Corollary 4.2. The skew representation V (λ)_{μ}^{+} of the twisted Yangian Y(𝔰𝔭_{N−M}) is irreducible.

Proof. Since the representation V (λ)_{μ}^{+} of U(𝔰𝔭_{N})^{𝔰𝔭^{M}} is irreducible [1, Section 9.1], the statement follows from Proposition 4.1 as the central elements of U(𝔰𝔭_{N}) act on V (λ)_{μ}^{+} by scalar operators.

Remark 4.3. In general, the corresponding statement for the orthogonal twisted Yangian Y(𝔬_{N−M}) is false; see [12] for the particular case N − M = 2. If N is even, then the Y(𝔬_{2})-module V (λ)_{μ}^{+} is still irreducible. If N is odd, then for general parameters λ and μ the Y(𝔬_{2})-module V (λ)_{μ}^{+} is isomorphic to the direct sum of two irreducibles. It looks plausible that, in general, the Y(𝔬_{N−M})-module V (λ)_{μ}^{+} is completely reducible. It would be interesting to obtain its irreducible decomposition.

Now recall the classification results for representations of the twisted Yangian Y(𝔰𝔭_{N}); see [9]. If V is a finite-dimensional irreducible representation of Y(𝔰𝔭_{N}) then V contains a unique, up to a scalar factor, vector ξ ≠ 0 such that

\begin{aligned}s_{ii}(u) ξ & = μ_{i}(u) ξ & for & i = 1, …, n, \\ s_{ij}(u) ξ & = 0 & for & − n ⩽ i < j ⩽ n, \end{aligned}

where each μ_{i}(u) is a formal series in u^{−1} with coefficients in ℂ. Moreover, there exist monic polynomials P_{1}(u), …, P_{n}(u) in u with P_{1}(u) = P_{1}(−u + 1) such that

\frac{μ_{i−1}(u)}{μ_{i}(u)} = \frac{P_{i}(u + 1)}{P_{i}(u)},  i = 2, …, n  (4.5)

and

\frac{μ_{1}(−u)}{μ_{1}(u)} = \frac{P_{1}(u + 1)}{P_{1}(u)}.  (4.6)

The n-tuple μ(u) = (μ_{1}(u), …, μ_{n}(u)) is called the highest weight and the P_{i}(u) are called the Drinfeld polynomials of the representation V. Furthermore, given an ntuple (P_{1}(u), …, P_{n}(u)) of monic polynomials with P_{1}(u) = P_{1}(−u+1) there exists a finite-dimensional irreducible representation V of Y(𝔰𝔭_{N}) having this n-tuple as the family of its Drinfeld polynomials. The isomorphism class of such representation V is determined uniquely, up to the twisting by an automorphism of Y(𝔰𝔭_{N}) of the form

s_{ij}(u) ↦ g(u) s_{ij}(u),

where g(u) is a series in u^{−2} with constant term 1.

Since the Sklyanin determinant sdet S(u) is central in Y(𝔰𝔭_{N}), it acts on V by scalar multiplication. The scalar can be calculated with the use of Proposition 3.6; see also [8]. We have

sdet S(u)|_{V} = α_{n}(u) ∏_{i=1}^{n} μ_{i}(−u + i − 1) μ_{i}(u − N + i).  (4.7)

Proposition 4.4. With the above notation, for any k = 1, …, n in the representation V we have

s _{−k+1 ··· k}^{−k+1 ··· k}(u) ξ = μ_{k}(u − 2k + 2) s _{−k+1 ··· k−1}^{−k+1 ··· k−1}(u) ξ.

Proof. By (2.13) we can write

s _{−k+1 ··· k}^{−k+1 ··· k}(u) = ∑_{c=−n}^{n} š_{−k+1 ··· k−1,c}^{−k+1 ··· k}(u) s_{ck}(u − 2k + 2).

Apply both sides to the highest vector ξ. We have s_{ck}(u − 2k + 2) ξ = 0 if c < k. On the other hand, if c > k then š_{−k+1 ··· k−1,c}^{−k+1 ··· k}(u) = 0 by (2.14). Finally, if c = k then (2.15) gives

š_{−k+1 ··· k−1,c}^{−k+1 ··· k}(u) = s _{−k+1 ··· k−1}^{−k+1 ··· k−1}(u)

completing the proof.

Our aim now is to identify the representation V (λ)_{μ}^{+} of Y(𝔰𝔭_{N−M}) by calculating its highest weight and Drinfeld polynomials. Note that for the evaluation module V (λ) over Y(𝔰𝔭_{N}) these can be immediately found from (4.2). In particular, the i-th component of the highest weight is given by

\frac{u + λ_{i} − 1/2}{u − 1/2},  i = 1, …, n.  (4.8)

In the case M > 0 we employ the basis in the 𝔰𝔭_{N}-module V (λ) constructed in [10]. This basis is parameterized by the patterns Λ which are arrays of non-positive integers of the form

λ_{n1} λ_{n2}  ⋯  λ_{nn}
λ′_{n1} λ′_{n2}  ⋯  λ′_{nn}
λ_{n−1,1}  ⋯  λ_{n−1,n−1}
λ′_{n−1,1}  ⋯  λ′_{n−1,n−1}
⋯  ⋯
λ_{11}
λ′_{11}

where λ = (λ_{n1}, …, λ_{nn}) is the top row of Λ and the following betweenness conditions hold

0 ⩾ λ′_{k1} ⩾ λ_{k1} ⩾ λ′_{k2} ⩾ λ_{k2} ⩾ ⋯ ⩾ λ′_{k,k−1} ⩾ λ_{k,k−1} ⩾ λ′_{kk} ⩾ λ_{kk}

for k = 1, …, n, and

0 ⩾ λ′_{k1} ⩾ λ_{k−1,1} ⩾ λ′_{k2} ⩾ λ_{k−1,2} ⩾ ⋯ ⩾ λ′_{k,k−1} ⩾ λ_{k−1,k−1} ⩾ λ′_{kk}

for k = 2, …, n. The representation V (λ) admits a basis ζ_{Λ} parameterized by all patterns Λ. The formulas for the matrix elements of a family of generators of the Lie algebra 𝔰𝔭_{N} in this basis can be explicitly written down; see [10]. We shall only need these formulas for the generators F_{kk}. We have

F_{kk} ζ_{Λ} = (2 ∑_{i=1}^{k} λ′_{ki} − ∑_{i=1}^{k} λ_{ki} − ∑_{i=1}^{k−1} λ_{k−1,i}) ζ_{Λ},  k = 1, …, n.  (4.9)

Given a highest weight μ = (μ_{1}, …, μ_{m}) for 𝔰𝔭_{M}, a basis of the space V (λ)_{μ}^{+} is formed by those vectors ζ_{Λ} for which the row (λ_{m1}, …, λ_{mm}) of Λ coincides with μ and

λ_{ki} = λ′_{ki} = μ_{i}, 1 ⩽ i ⩽ k ⩽ m.

Omitting the same triangle part below the m-th row of all such patterns Λ we get trapezium-like patterns (still denoted by Λ) with the top row λ and the bottom row μ, as illustrated:

λ_{1} λ_{2}  ⋯  λ_{n}
λ′_{n1} λ′_{n2}  ⋯  λ′_{nn}
⋯  ⋯
λ′_{m+1,1} λ′_{m+1,2}  ⋯  λ′_{m+1,m+1}
μ_{1}  μ_{2}  ⋯  μ_{m}

Due to the betweenness conditions, the space V (λ)_{μ}^{+} is nonzero if and only if

μ_{i} ⩾ λ_{i+n−m}, i = 1, …, m,

and

λ_{i} ⩾ μ_{i+n−m}, i = 1, …, n,

assuming μ_{i} = −∞ for i > m. We also set μ_{i} = 0 for i ⩽ 0. In what follows we suppose that V (λ)_{μ}^{+} is nonzero.

Denote by 𝔥 the diagonal Cartan subalgebra of 𝔰𝔭_{N−M} spanned by the basis vectors F_{kk} with k = m + 1, …, n. Let the ε_{k} ∈ 𝔥^{∗} with k = m + 1, …, n be the dual basis vectors of 𝔥^{∗}. Consider the root system for 𝔰𝔭_{N−M} with respect to 𝔥 where the basis vectors F_{ij} with i < j are positive root vectors so that the positive roots are −2ε_{k} with k = m + 1, …, n and ± ε_{i} − ε_{j} with m + 1 ⩽ i < j ⩽ n. By (4.9), the weight w(Λ) = (w_{m+1}, …, w_{n}) of a trapezium pattern Λ with respect to 𝔥 is given by

w_{k} = 2 ∑_{i=1}^{k} λ′_{ki} − ∑_{i=1}^{k} λ_{ki} − ∑_{i=1}^{k−1} λ_{k−1,i}, k = m + 1, …, n.

We shall need the standard partial ordering on the set of weights of an 𝔰𝔭_{N−M}- module V. We shall write w ≼ w′ for two weights w and w′ of V if w′ − w is a linear combination of positive roots with nonnegative integral coefficients. Since the vectors ζ_{Λ} corresponding to the trapezium patterns Λ form a basis of the 𝔰𝔭_{N−M}- module V (λ)_{μ}^{+}, the set of weights of this module is comprised by the weights w(Λ) for all possible patterns Λ.

Given three integers i, j, k we shall denote by mid{i, j, k} that of the three which is between the two others. If one of the indices, say, k is the symbol −∞ then mid{i, j, k} is understood as min{i, j}. Consider the trapezium array Λ_{0} whose entries are determined by

λ_{ki} = mid{λ_{i}, μ_{i+k−m}, μ_{i+m−k}}  (4.10)

and

λ′_{ki} = mid{λ_{i}, μ_{i+k−m−1}, μ_{i+m−k}}

for all possible values of i and k. One easily verifies that Λ_{0} is a pattern.

Proposition 4.5. The 𝔰𝔭_{N−M}-module V (λ)_{μ}^{+} has a unique maximal weight. This weight coincides with w(Λ_{0}).

Proof. Suppose that w(Λ) is a maximal weight of V (λ)_{μ}^{+} for some pattern Λ. Then the entries of Λ should satisfy

λ′_{ki} = max{λ_{ki}, λ_{k−1,i}}  (4.11)

for all k = m + 1, …, n − 1 and i = 1, …, k, where we assume λ_{ki} = −∞ for i > k. Indeed, if the equality is not attained for some k and i then by the betweenness conditions we have λ′_{ki} > max{λ_{ki}, λ_{k−1,i}}. Therefore, decreasing the entry λ′_{ki} by 1 we get a pattern of greater weight than w(Λ) which contradicts the maximality of w(Λ). Thus, the weight w(Λ) = (w_{m+1}, …, w_{n}) can now be written as

w_{k} = ∑_{i=1}^{k−1} |λ_{ki} − λ_{k−1,i}| + λ_{kk}, k = m + 1, …, n.

We shall argue by induction on n − m to show that the λ_{ki} must be given by (4.10). This will imply the statement since the entries of Λ_{0} do satisfy (4.11) which is easily seen. In the case n − m = 1 there is nothing to prove. Suppose that n − m = 2. Omitting the primed entries, we can depict Λ as

λ_{1} λ_{2} ⋯ λ_{n−2} λ_{n−1} λ_{n}
ρ_{1} ρ_{2} ⋯ ρ_{n−2} ρ_{n−1}
μ_{1} μ_{2} ⋯ μ_{n−2}

where we have put ρ_{i} = λ_{n−1,i}. The weight w(Λ) = (w_{n−1}, w_{n}) is given by

\begin{aligned}w_{n−1} & = ∑_{i=1}^{n−2} |ρ_{i} − μ_{i}| + ρ_{n−1}, \\ w_{n} & = ∑_{i=1}^{n−1} |λ_{i} − ρ_{i}| + λ_{n}. \end{aligned}  (4.12)

Suppose that (4.10) is violated for some i so that ρ_{i} ≠ mid{λ_{i}, μ_{i+1}, μ_{i−1}}. Observe that by the betweenness conditions we have

ρ_{i+1} ⩽ mid{λ_{i}, μ_{i+1}, μ_{i−1}} ⩽ ρ_{i−1}.

Hence, if ρ_{i} < mid{λ_{i}, μ_{i+1}, μ_{i−1}} (respectively, ρ_{i} > mid{λ_{i}, μ_{i+1}, μ_{i−1}}) then we can increase (respectively, decrease) the value of ρ_{i} by 1 without violating the betweenness conditions and thus to get another pattern Λ′. Due to the formulas (4.12), we have w(Λ) ≺ w(Λ′) which contradicts the maximality of w(Λ). This proves the statement in the case under consideration.

Suppose now that n − m > 2. Let us set ρ_{i} = λ_{n−1,i} as above, and consider the set of patterns having ρ = (ρ_{1}, …, ρ_{n−1}) as the top row and μ as the bottom row. Then, by the maximality of w(Λ), the trapezium subpattern of Λ with the top row ρ and bottom row μ will clearly be of a maximal weight amongst all patterns of this set. By the induction hypothesis, we must have

λ_{ki} = mid{ρ_{i}, μ_{i+k−m}, μ_{i+m−k}}  (4.13)

for all k = m + 1, …, n − 2 and i = 1, …, k. Similarly, the subpattern of Λ having the top row λ and the bottom row σ = (σ_{1}, …, σ_{n−2}) is of a maximal weight amongst all patterns of this form, where we have put σ_{i} = λ_{n−2,i}. By the statement for the case n − m = 2, we must have

ρ_{i} = mid{λ_{i}, σ_{i+1}, σ_{i−1}}, i = 1, …, n − 1.

Combining this with the relations σ_{i} = mid{ρ_{i}, μ_{i+n−m−2}, μ_{i+m−n+2}} implied by (4.13), we get the desired relation for the row ρ of Λ,

ρ_{i} = mid{λ_{i}, μ_{i+n−m−1}, μ_{i+m−n+1}}, i = 1, …, n − 1.

Indeed, this is easily verified by looking at all possible combinations of the three values of each of σ_{i−1} and σ_{i+1}. Finally, substituting these values of ρ_{i} into (4.13) we conclude that (4.10) holds for all possible k and i. Thus, Λ coincides with Λ_{0}.

Corollary 4.6. The vector ζ_{Λ_{0}} is the highest vector of the Y(𝔰𝔭_{N−M})-module V (λ)_{μ}^{+}.

Proof. It suffices to demonstrate that ζ_{Λ_{0}} is annihilated by all generators s_{ab}(u) of Y(𝔰𝔭_{N−M}) with a < b, since by [9, Remark 4.4] the vector ζ_{Λ_{0}} will then have to be an eigenvector for all s_{aa}(u) with a = m + 1, …, n.

Recall that s_{ab}(u) acts on V (λ)_{μ}^{+} as a Sklyanin minor which is a series in u^{−1} with coefficients in the universal enveloping algebra U(𝔰𝔭_{N}); see (4.3). By (2.28), the weight of all these coefficients coincides with the weight of the element F_{ab} with respect to the adjoint action of the Cartan subalgebra 𝔥 of 𝔰𝔭_{N−M}. Therefore, if a < b then the vector s_{ab}(u) ζ_{Λ_{0}} has the weight w(Λ_{0}) + α for a positive root α. Now the claim follows from Proposition 4.5.

Corollary 4.6 implies that the highest weight μ(u) = (μ_{m+1}(u), …, μ_{n}(u)) of the Y(𝔰𝔭_{N−M})-module V (λ)_{μ}^{+} is determined by the relations

s_{aa}(u) ζ_{Λ_{0}} = μ_{a}(u) ζ_{Λ_{0}}, a = m + 1, …, n.

In order to calculate μ(u) we shall use the results of [10], where the particular case m = n − 1 was considered. In [10] the vector space V (λ)_{μ}^{+} was endowed with the Y(𝔰𝔭_{2})- module structure defined by the composition of an automorphism of the type (3.2) and the homomorphism (4.4). Therefore, using Corollary 3.5, we can reformulate the result for the Y(𝔰𝔭_{2})-module structure on V (λ)_{μ}^{+} defined by the homomorphism (4.3) to obtain the following.

Proposition 4.7. For m = n − 1 in the Y(𝔰𝔭_{2})-module V (λ)_{μ}^{+} we have

α_{−n+1}(u) f _{−n+1 ··· n}^{−n+1 ··· n} (u + n − 1) ζ_{Λ_{0}}

= ∏_{i=2}^{n} \frac{u − min\{λ_{i−1}, μ_{i−1}\} + i − 1/2}{u + i − 1/2} · ∏_{i=1}^{n} \frac{u + max\{λ_{i}, μ_{i}\} − i + 1/2}{u − i + 1/2} ζ_{Λ_{0}}.

In the case of arbitrary m < n introduce the following notation:

ν(u) = ∏_{i=1}^{m} \frac{(u + μ_{i} − i + 1/2) (u − μ_{i} + i + 1/2)}{(u − i + 1/2) (u + i + 1/2)}.  (4.14)

Theorem 4.8. The highest weight μ(u) = (μ_{m+1}(u), …, μ_{n}(u)) of the Y(𝔰𝔭_{N−M})- module V (λ)_{μ}^{+} is given by the formulas

μ_{k}(u) = ν(u) · ∏_{{i=1 \atop λ_{i}<μ_{i+k−m−1}}}^{k−1} \frac{u − max\{λ_{i}, μ_{i+k−m}\} + k − m + i − 1/2}{u − μ_{i+k−m−1} + k − m + i − 1/2}

× ∏_{{i=1 \atop λ_{i}>μ_{i+m−k+1}}}^{k−1} \frac{u + min\{λ_{i}, μ_{i+m−k}\} + k − m − i − 1/2}{u + μ_{i+m−k+1} + k − m − i − 1/2}

× \frac{u + min\{λ_{k}, μ_{m}\} − m − 1/2}{u − m − 1/2},

where k = m + 1, …, n.

Proof. Let us denote by φ_{ab}(u) the image of the series s_{ab}(u) under the homomorphism ρ defined in (4.3), that is,

φ_{ab}(u) = α_{−m}(u) f _{−m··· m, b}^{−m··· m, a} (u + M/2).  (4.15)

Combine these series into the (N − M) × (N − M) matrix Φ(u). We shall use the usual notation for the Sklyanin comatrix and the Sklyanin minors of Φ(u). Applying Proposition 2.4, we obtain the following expression for the nn-th entry of the Sklyanin comatrix Φ̂(u),

φ̂_{nn}(u) = α_{−m}(u) f̂_{nn}(u + M/2) · f_{B}(u − 1) ⋯ f_{B}(u − N + M + 2),  (4.16)

where we have put f_{B}(u) = α_{−m}(u) sdet F_{BB}(u + M/2). On the other hand, by Proposition 3.4, we have

φ̂_{nn}(u) = α_{N−M−1}(u) φ _{−n+1 ··· −m−1, m+1··· n}^{−n+1 ··· −m−1, m+1··· n} (−u + N − M − 2)

and

f̂_{nn}(u) = α_{N−1}(u) f _{−n+1 ··· n}^{−n+1 ··· n} (−u + N − 2).

Therefore, by (4.16),

α_{N−M−1}(u) φ _{−n+1 ··· −m−1, m+1··· n}^{−n+1 ··· −m−1, m+1··· n} (−u + N − M − 2) =

α_{N−1}(u + M/2) f _{−n+1 ··· n}^{−n+1 ··· n} (−u + N − M/2 − 2) × α_{−m}(u) f_{B}(u − 1) ⋯ f_{B}(u − N + M + 2). (4.17)

Similarly, applying Theorem 3.1, we derive the identity

α_{N−M}(u) φ _{−n ··· −m−1, m+1··· n}^{−n ··· −m−1, m+1··· n} (−u + N − M − 1) =

α_{N}(u + M/2) f _{−n ··· n}^{−n ··· n} (−u + N − M/2 − 1) × α_{−m}(u) f_{B}(u − 1) ⋯ f_{B}(u − N + M + 1).

Replacing here n by n − 1 and u by u − 1 we get

α_{N−M−2}(u − 1) φ _{−n+1 ··· −m−1, m+1··· n−1}^{−n+1 ··· −m−1, m+1··· n−1} (−u + N − M − 2) =

α_{N−2}(u + M/2 − 1) f _{−n+1 ··· n−1}^{−n+1 ··· n−1} (−u + N − M/2 − 2) × α_{−m}(u − 1) f_{B}(u − 2) ⋯ f_{B}(u − N + M + 2). (4.18)

Due to Proposition 4.4, in V (λ)_{μ}^{+} we have

φ _{−n+1 ··· −m−1, m+1··· n}^{−n+1 ··· −m−1, m+1··· n} (v) ζ_{Λ_{0}} = μ_{n}(v − N + M + 2) φ _{−n+1 ··· −m−1, m+1··· n−1}^{−n+1 ··· −m−1, m+1··· n−1} (v) ζ_{Λ_{0}}.

Hence, comparing (4.17) and (4.18) we come to the relation

f_{B}(u−1) f _{−n+1 ··· n}^{−n+1 ··· n} (−u+2n−m−2) ζ_{Λ_{0}} = μ_{n}(−u) f _{−n+1 ··· n−1}^{−n+1 ··· n−1} (−u+2n−m−2) ζ_{Λ_{0}}.

Therefore, for each k = m + 1, …, n the corresponding component of the highest weight of the Y(𝔰𝔭_{N−M})-module V (λ)_{μ}^{+} can be found from the relation

f_{B}(−u − 1) f _{−k+1 ··· k}^{−k+1 ··· k} (u + 2k − m − 2) ζ_{Λ_{0}} = μ_{k}(u) f _{−k+1 ··· k−1}^{−k+1 ··· k−1} (u + 2k − m − 2) ζ_{Λ_{0}}. (4.19)

Observe that each of the Sklyanin minors f _{−k+1 ··· k}^{−k+1 ··· k} (v) and f _{−k+1 ··· k−1}^{−k+1 ··· k−1} (v) commutes with all elements of the subalgebra 𝔰𝔭_{2k−2} by Corollary 2.2. Therefore, since the basis {ζ_{Λ}} of V (λ) is consistent with the embeddings 𝔰𝔭_{2k−2} ⊂ 𝔰𝔭_{2k} [10], the vector ζ_{Λ_{0}} is an eigenvector for each of these minors. The corresponding eigenvalue for the first minor can be calculated from Proposition 4.7, which gives

α_{−k+1}(u) f _{−k+1 ··· k}^{−k+1 ··· k} (u + k − 1) ζ_{Λ_{0}}

= ∏_{i=2}^{k} \frac{u − min\{λ_{k,i−1}, λ_{k−1,i−1}\} + i − 1/2}{u + i − 1/2} · ∏_{i=1}^{k} \frac{u + max\{λ_{ki}, λ_{k−1,i}\} − i + 1/2}{u − i + 1/2} ζ_{Λ_{0}}, (4.20)

where the λ_{ki} are the entries of Λ_{0}. The second minor coincides with the image of the Sklyanin determinant for Y(𝔰𝔭_{2k−2}) under the evaluation homomorphism (4.2). Hence the corresponding eigenvalue can be found from (4.7) and (4.8) which gives

α_{−k+1}(u) f _{−k+1 ··· k−1}^{−k+1 ··· k−1} (u + k − 1) ζ_{Λ_{0}}

= ∏_{i=2}^{k} \frac{u − λ_{k−1,i−1} + i − 1/2}{u + i − 1/2} · ∏_{i=1}^{k−1} \frac{u + λ_{k−1,i} − i + 1/2}{u − i + 1/2} ζ_{Λ_{0}};

see also [8]. Note that since f_{B}(u) = α_{−m}(u) f _{−m ··· m}^{−m ··· m} (u + m), the last formula with k = m + 1 also applies for the calculation of f_{B}(u) ζ_{Λ_{0}}. Hence,

f_{B}(−u − 1) ζ_{Λ_{0}} = ν(u) ζ_{Λ_{0}}  (4.21)

with ν(u) defined in (4.14). Furthermore, the formulas (4.10) for the entries of Λ_{0} imply that

max{λ_{ki}, λ_{k−1,i}} = mid{λ_{i}, μ_{i+k−m−1}, μ_{i+m−k}}  (4.22)

and

min{λ_{ki}, λ_{k−1,i}} = mid{λ_{i}, μ_{i+k−m}, μ_{i+m−k+1}}.  (4.23)

Using (4.19) we obtain the following expression for μ_{k}(u):

μ_{k}(u) = ν(u) ∏_{i=2}^{k} \frac{u − mid\{λ_{i−1}, μ_{i+k−m−1}, μ_{i+m−k}\} + k − m + i − 3/2}{u − mid\{λ_{i−1}, μ_{i+k−m−2}, μ_{i+m−k}\} + k − m + i − 3/2}

× ∏_{i=1}^{k−1} \frac{u + mid\{λ_{i}, μ_{i+k−m−1}, μ_{i+m−k}\} + k − m − i − 1/2}{u + mid\{λ_{i}, μ_{i+k−m−1}, μ_{i+m−k+1}\} + k − m − i − 1/2}

× \frac{u + min\{λ_{k}, μ_{m}\} − m − 1/2}{u − m − 1/2}.

Finally, replace the index i in the first product by i + 1 and note that if λ_{i} ⩾ μ_{i+k−m−1} then the corresponding factor equals 1. Similarly, if λ_{i} ⩽ μ_{i+m−k+1} then the corresponding factor in the second product equals 1. This brings the expression for μ_{k}(u) to the required form.

We can now compute the Drinfeld polynomials for the Y(𝔰𝔭_{N−M})-module V (λ)_{μ}^{+} by using Theorem 4.8. Given any 𝔰𝔭_{2n}-highest weight λ = (λ_{1}, …, λ_{n}), set λ_{−i} = −λ_{i} for i = 1, …, n. We also assume that λ_{0} = 0 while λ_{k} = −∞ and λ_{−k} = +∞ for k > n. Introduce the diagram Γ(λ) as a certain infinite set of unit squares (cells) on the plane whose centers have integer coordinates. The coordinates (i, j) of a cell are interpreted as the row and column number so that i increases from top to bottom and j increases from left to the right. With these assumptions^{2},

Γ(λ) = {(i, j) ∈ ℤ^{2} | −n ⩽ i ⩽ n + 1, λ_{i} ⩽ j < λ_{i−1}}.

The diagram has a central symmetry, as illustrated below for λ = (−4, −7) and n = 2:

[figure: i; j; 2; 3; −1; −2; 2; 3; 4; 5; 6; 7; −3; −4; −5; −6; −7; −8]

Note that in the case n = 0 the definition of the diagram formally makes sense with λ considered to be empty. Thus, Γ(∅) consists of two infinite rows of cells:

2This definition of Γ(λ) corresponds to the one outlined in the Introduction for the partition (λ_{−n}, …, λ_{−1}).

[figure: i; j; 2; −1; 2; 3; 4; 5; 6; 7; −3; −4; −5; −6; −7; −8]

By the content of a cell α = (i, j) with coordinates i and j we shall mean the number c(α) = j − i. For any nonnegative integer p we shall denote by Γ(λ)^{(p)} the image of Γ(λ) with respect to the shift operator (i, j) ↦ (i − p, j). In other words, Γ(λ)^{(p)} is obtained from the diagram Γ(λ) by lifting each cell p units up.

We let P_{1}(u), …, P_{n−m}(u) denote the Drinfeld polynomials corresponding to the Y(𝔰𝔭_{N−M})-module V (λ)_{μ}^{+}.

Theorem 4.9. For each k = 1, …, n − m the Drinfeld polynomial P_{k}(u) is given by

P_{k}(u) = ∏_{α}(u + c(α) + 1/2),

where α runs over the cells of the intersection Γ(μ) ∩ Γ(λ)^{(k−1)}.

Example 4.10. Let λ = (−2, −8, −10, −13) and μ = (−4, −7) so that n = 4 and m = 2. (In a more standard notation, λ and μ can be thought of as the partitions (13, 10, 8, 2) and (7, 4), respectively). The polynomial P_{1}(u) is calculated from the figure:

[figure: i; j; 2; 3; −1; −2; −4; 3; 9; −4; −7; −10; −13]

The horizontal and vertical shadings indicate the diagrams Γ(λ) and Γ(μ), respectively. The cells belonging to the intersection Γ(μ) ∩ Γ(λ) have the coordinates (3, −10), (3, −9), (2, −7), (2, −6), (2, −5), (1, −2), (1, −1), (0, 0), (0, 1), (−1, 4), (−1, 5), (−1, 6), (−2, 8), (−2, 9). Hence,

P_{1}(u) = (u − 25/2)(u − 23/2)(u − 17/2)(u − 15/2)(u − 13/2)(u − 5/2)(u − 3/2) (u + 1/2)(u + 3/2)(u + 11/2)(u + 13/2)(u + 15/2)(u + 21/2)(u + 23/2).

Note that the property P_{1}(u) = P_{1}(−u + 1) is implied by the central symmetry of the set Γ(μ) ∩ Γ(λ).

The polynomial P_{2}(u) is calculated from the figure:

[figure: i; j; 2; 3; −4; −2; −1; 3; 7; −4; −8; −13]

The cells which belong to the intersection Γ(μ)∩Γ(λ)^{(1)} have the coordinates (3, −13), (3, −12), (3, −11), (1, −4), (1, −3), (−2, 7). Hence,

P_{2}(u) = (u − 31/2)(u − 29/2)(u − 27/2)(u − 9/2)(u − 7/2)(u + 19/2).

Example 4.11. In the case m = 0 the vector space V (λ)_{μ}^{+} can be identified with V (λ) and the corresponding Y(𝔰𝔭_{2n})-module coincides with the evaluation module defined by the evaluation homomorphism (4.2). Applying Theorem 4.9 to the diagrams Γ(λ) and Γ(∅) we obtain

P_{1}(u) = (u + λ_{1} − 1/2)(u + λ_{1} + 1/2) ⋯ (u − 3/2) × (u + 1/2)(u + 3/2) ⋯ (u − λ_{1} − 1/2)

and

P_{k}(u) = (u + λ_{k} − 1/2)(u + λ_{k} + 1/2) ⋯ (u + λ_{k−1} − 3/2), k = 2, …, n.

On the other hand, the highest weight of this module is given by (4.8). Due to (4.5) and (4.6) this obviously agrees with the above calculation of the P_{k}(u).

Proof of Theorem 4.9. We shall derive the statement from Theorem 4.8 and the definition of the Drinfeld polynomials (4.5) and (4.6). In order to calculate P_{1}(u) observe that the component μ_{m+1}(u) of the highest weight can also be found from the formula

φ_{m+1,m+1}(u) ζ_{Λ_{0}} = μ_{m+1}(u) ζ_{Λ_{0}};

see (4.15). Hence, applying (4.20) with k = m + 1 and using (4.22) and (4.23) we get

μ_{m+1}(u) = ∏_{i=1}^{m+1} \frac{u − mid\{λ_{i−1}, μ_{i−1}, μ_{i}\} + i − 1/2}{u + i − 1/2}

× ∏_{i=1}^{m+1} \frac{u + mid\{λ_{i}, μ_{i−1}, μ_{i}\} − i + 1/2}{u − i + 1/2}.

Therefore, by (4.6),

\frac{P_{1}(u + 1)}{P_{1}(u)} = ∏_{i=1}^{m+1} \frac{u + mid\{λ_{i−1}, μ_{i−1}, μ_{i}\} − i + 1/2}{u + mid\{λ_{i}, μ_{i−1}, μ_{i}\} − i + 1/2}

× ∏_{i=1}^{m+1} \frac{u − mid\{λ_{i}, μ_{i−1}, μ_{i}\} + i − 1/2}{u − mid\{λ_{i−1}, μ_{i−1}, μ_{i}\} + i − 1/2}.

This gives P_{1}(u) = Q(u) Q(−u + 1) (−1)^{deg Q}, where

Q(u) = ∏_{i=1}^{m+1}(u + β_{i})(u + β_{i} + 1) ⋯ (u + α_{i} − 1)  (4.24)

with

α_{i} = mid{λ_{i−1}, μ_{i−1}, μ_{i}} − i + 1/2 and β_{i} = mid{λ_{i}, μ_{i−1}, μ_{i}} − i + 1/2.

Thus, the factor corresponding to the index i in the product in (4.24) can be interpreted as the product ∏_{α}(u + c(α) + 1/2), where α runs over the cells of the intersection of the diagrams Γ(λ) ∩ Γ(μ) whose first coordinate is i. Taking into account the central symmetry of Γ(λ)∩Γ(μ), we derive the desired formula for P_{1}(u).

Now using (4.5) and replacing the index k by r = k − m in Theorem 4.8 we derive the following expression for the Drinfeld polynomial P_{r+1}(u) with 1 ⩽ r < n − m:

\frac{P_{r+1}(u + 1)}{P_{r+1}(u)} = ∏_{{i=1 \atop λ_{i}<μ_{i+r}}}^{r+m−2} \frac{u − max\{λ_{i+1}, μ_{i+r+1}\} + r + i + 1/2}{u − max\{λ_{i}, μ_{i+r+1}\} + r + i + 1/2}

× ∏_{{i=1 \atop λ_{i}<μ_{i+r−1}⩽λ_{i−1}}}^{r+m−1} \frac{u − max\{λ_{i}, μ_{i+r}\} + r + i − 1/2}{u − μ_{i+r−1} + r + i − 1/2}

× ∏_{{i=2 \atop λ_{i}>μ_{i−r}}}^{r+m+1} \frac{u + min\{λ_{i−1}, μ_{i−r−1}\} + r − i + 1/2}{u + min\{λ_{i}, μ_{i−r−1}\} + r − i + 1/2}

× ∏_{{i=1 \atop λ_{i+1}⩽μ_{i−r+1}<λ_{i}}}^{r+m−1} \frac{u + min\{λ_{i}, μ_{i−r}\} + r − i − 1/2}{u + μ_{i−r+1} + r − i + 1/2}.

(4.25)

Note that the expression has the form

\frac{P_{r+1}(u + 1)}{P_{r+1}(u)} = ∏_{j} \frac{u + α_{j}}{u + β_{j}}

for some parameters α_{j} and β_{j} with α_{j} − β_{j} ∈ ℤ_{+}. This implies that P_{r+1}(u) is given by

P_{r+1}(u) = ∏_{j}(u + β_{j})(u + β_{j} + 1) ⋯ (u + α_{j} − 1).  (4.26)

It is straightforward to verify that this product coincides with ∏_{α}(u + c(α) + 1/2), where α runs over the cells of the intersection Γ(μ) ∩ Γ(λ)^{(r−1)}. Indeed, changing the product index i respectively by i − r and by i − r + 1 in the third and fourth products in (4.25) we bring these products to the form

∏_{{i=1 \atop λ_{i+r}>μ_{i}}}^{m+1} \frac{u + min\{λ_{i+r−1}, μ_{i−1}\} − i + 1/2}{u + min\{λ_{i+r−1}, μ_{i−1}\} − i + 1/2}

× ∏_{{i=1 \atop λ_{i+r}⩽μ_{i}<λ_{i+r−1}}}^{m} \frac{u + min\{λ_{i+r−1}, μ_{i−1}\} − i + 1/2}{u + μ_{i} − i + 1/2}. (4.27)

Therefore, the factor corresponding to the index i contributes into (4.26) the product ∏_{α}(u + c(α) + 1/2), where α runs over the cells of the intersection Γ(μ) ∩ Γ(λ)^{(r−1)} whose first coordinate is i. Note that writing

− max{λ_{i}, μ_{i+r+1}} = min{λ_{−i}, μ_{−i−r−1}}

we can similarly bring the first and second products in (4.25) to the form (4.27), where the product is taken over negative indices. This contributes ∏_{α}(u + c(α) + 1/2) into (4.26), where α runs over the cells of the intersection Γ(μ) ∩ Γ(λ)^{(r−1)} whose first coordinate is non-positive.

5 Centralizer construction

We now return to our general situation so that 𝔤_{N} denotes either the orthogonal or symplectic Lie algebra; see (4.1). We start by recalling the construction of the Olshanski algebra A_{M}; see [18, 14]. Fix a nonnegative integer M such that N − M is even. So, if N = 2n or N = 2n + 1 then M = 2m or M = 2m + 1, respectively, for some m ⩽ n. Denote by 𝔤′_{N−M} the subalgebra of 𝔤_{N} spanned by the elements F_{ij} subject to the condition m + 1 ≤ |i|, |j| ≤ n. Let A_{M}(N) denote the centralizer of 𝔤′_{N−M} in the universal enveloping algebra U(𝔤_{N}). Let U(𝔤_{N})^{0} denote the centralizer of F_{nn} in U(𝔤_{N}) and let I(N) be the left ideal in U(𝔤_{N}) generated by the elements F_{in}, i = −n, …, n. Then I(N)^{0} = I(N)∩U(𝔤_{N})^{0} is a two-sided ideal in U(𝔤_{N})^{0} which coincides with the intersection J(N) ∩ U(𝔤_{N})^{0}, where J(N) is the right ideal in U(𝔤_{N}) generated by the elements F_{ni}, i = −n, …, n. One has a vector space decomposition

U(𝔤_{N})^{0} = I(N)^{0} ⊕ U(𝔤_{N−2}).

Therefore the projection of U(𝔤_{N})^{0} onto U(𝔤_{N−2}) with the kernel I(N)^{0} is an algebra homomorphism. Its restriction to the subalgebra A_{M}(N) defines a filtrationpreserving homomorphism

π_{N} : A_{M}(N) → A_{M}(N − 2)  (5.1)

so that one can define the algebra A_{M} as the projective limit with respect to this sequence of homomorphisms in the category of filtered algebras; see [14] for more details.

Taking the composition of the homomorphism Y(𝔤_{M}) → Y(𝔤_{N}) defined in Proposition 3.2 and the evaluation homomorphism (4.2) we obtain another homomorphism ψ_{N} : Y(𝔤_{M}) → U(𝔤_{N}) which takes s_{ij}(u) to the series

α_{m−n}(u) · f_{−n ··· −m−1, j, m+1··· n}^{−n ··· −m−1, i, m+1··· n} (u + n − m),  −m ⩽ i, j ⩽ m,  (5.2)

where we have used the notation of Section 4 for the images of the Sklyanin minors with respect to (4.2). Observe now that by Corollary 2.2 the image of ψ_{N} is contained in the centralizer A_{M}(N), so that ψ_{N} : Y(𝔤_{M}) → A_{M}(N).

Proposition 5.1. The sequence of homomorphisms (ψ_{N}| N = M + 2k, k = 0, 1, …) defines a homomorphism

ψ : Y(𝔤_{M}) → A_{M}.

Proof. We have to verify that the homomorphisms ψ_{N} are compatible with the sequence of homomorphisms (5.1), that is, the following diagram is commutative:

Y(𝔤_{M})  Y(𝔤_{M})  ⋯  Y(𝔤_{M})  ⋯
ψ_{M} ↓  ^{ψ_{M+2}}↓  ^{ψ_{N}} ↓
A_{M}(M) ⟵_{π_{M+2}} A_{M}(M + 2) ⟵ ⋯ ⟵_{π_{N}} A_{M}(N) ⟵ · · · .

Let us calculate the image of the series ψ_{N}(s_{ij}(u)) under the homomorphism π_{N}. Applying (2.13) we obtain

f_{−n ··· −m−1, j, m+1··· n}^{−n ··· −m−1, i, m+1··· n} (u+n−m) = ∑_{c} f̌_{−n ··· −m−1, j, m+1··· n−1,c}^{−n ··· −m−1, i, m+1··· n} (u+n−m) f_{cn}(u−n+m),

where by f̌_{b_{1}··· b_{k−1},c}^{a_{1}··· a_{k}}(u) we denote the image of the auxiliary minor š_{b_{1}··· b_{k−1},c}^{a_{1}··· a_{k}}(u) under the evaluation homomorphism (4.2). Now observe that f_{cn}(u − n + m) belongs to the left ideal I(N) unless c = n. In this case f_{nn}(u−n+m) ≡ 1 mod I(N). Furthermore, by (3.10),

f̌_{−n ··· −m−1, j, m+1··· n}^{−n ··· −m−1, i, m+1··· n} (v)

= \frac{2v + 1}{2v ± 1} ∑_{i=1}^{2n−2m}(−1)^{i−1} f_{a_{i},−n}^{t}(−v) f _{−n+1 ··· −m−1, j, m+1··· n−1}^{a_{1}··· â_{i} ··· a_{2n−2m}}(v − 1),

where we have set (a_{1}, …, a_{2n−2m}) = (−n ⋯ − m − 1, i, m + 1 ⋯ n − 1) and v = u + n − m. Now f_{a_{i},−n}^{t}(−v) = θ_{−n,a_{i}} f_{n,−a_{i}}(−v) belongs to the right ideal J(N) unless a_{i} = −n, that is, i = 1. In this case f_{nn}(−v) ≡ 1 mod J(N). Note that

α_{m−n}(u) \frac{2v + 1}{2v ± 1} = α_{m−n+1}(u).

Hence, π_{N} takes ψ_{N}(s_{ij}(u)) to the series

α_{m−n+1}(u) · f_{−n+1 ··· −m−1, j, m+1··· n−1}^{−n+1 ··· −m−1, i, m+1··· n−1} (u + n − m − 1)

which coincides with ψ_{N−2}(s_{ij}(u)). Thus, the sequence of coefficients at each power of u^{−1} in ψ_{N}(s_{ij}(u)) with N = M + 2k, k = 0, 1, … defines an element of A_{M}.

Corollary 2.2 implies that all the coefficients of the series

α_{m−n}(u) · f_{−n ··· −m−1, m+1··· n}^{−n ··· −m−1, m+1··· n} (u + n − m) = 1 + c_{1}^{(n)}u^{−1} + c_{2}^{(n)}u^{−2} + ⋯  (5.3)

belong to the center of the universal enveloping algebra U(𝔤′_{N−M}). Hence, for any i the coefficient c_{i}^{(n)} is an element of the centralizer A_{M}(N). The argument of the proof of Proposition 5.1 shows that the image of the series (5.3) under the homomorphism π_{N} is

α_{m−n+1}(u) · f_{−n+1 ··· −m−1, m+1··· n−1}^{−n+1 ··· −m−1, m+1··· n−1} (u + n − m − 1).

Therefore, for each i the sequence c_{i} = (c_{i}^{(n)} | n ⩾ m + 1) determines an element of the projective limit algebra A_{M}. Denote by C_{M} the subalgebra of A_{M} generated by all c_{i} with i ≥ 1. The subalgebra C_{M} is studied in detail in [14, Section 3] where it was identified with the algebra of virtual Laplace operators. Up to an obvious change of notation, the series (5.3) coincides with the Sklyanin minor f_{B}(u) introduced in the proof of Theorem 4.8. So its Harish-Chandra image can be found from (4.21) which shows that the elements c_{2i} with even indices are algebraically independent and generate the algebra C_{M}; cf. [14, Section 3].

Theorem 5.2. The homomorphism ψ : Y(𝔤_{M}) ↪ A_{M} is injective. Moreover, one has an isomorphism

A_{M} = C_{M} ⊗ Y(𝔤_{M}),

where Y(𝔤_{M}) is identified with its image under the embedding ψ.

Proof. Our argument is similar to the proof of [14, Theorem 4.17]. Consider the canonical filtration of the universal enveloping algebra U(𝔤_{N}). The corresponding graded algebra gr U(𝔤_{N}) is isomorphic to the symmetric algebra S(𝔤_{N}) of the space 𝔤_{N}. Elements of S(𝔤_{N}) can be naturally identified with polynomials in matrix elements of an N × N matrix x = (x_{ij})_{i,j=−n}^{n} such that x^{t} = −x. Denote by P_{M}(N) the subalgebra of the elements of S(𝔤_{N}) which are invariant under the adjoint action of the Lie algebra 𝔤′_{N−M}. The algebra A_{M} possesses a natural filtration induced by the canonical filtrations on the centralizers A_{M}(N). The corresponding graded algebra gr A_{M} is naturally isomorphic to the projective limit P_{M} of the commutative algebras P_{M}(N) with respect to homomorphisms P_{M}(N) → P_{M}(N − 2) analogous to (5.1); see [14, Section 4]. The images in P_{M}(N) of the coefficients the series (5.2) and (5.3) can be found from the explicit formulas for the Sklyanin minors; see Proposition 3.6. Indeed, apply the proposition to the Sklyanin minor s _{−n ··· −m−1, m+1··· n}^{−n ··· −m−1, m+1··· n} (u + n − m) then replace each series s_{ij}(u) by its image f_{ij}(u) under the evaluation homomorphism (4.2). Observe that the image of s_{ij}^{t}(−u) coincides with f_{ij}(u ∓ 1). Since we are only interested in the highest degree component of the coefficient at each power of u^{−1}, we may replace each expression of type f_{ij}(u + c) by δ_{ij} + F_{ij}u^{−1}. Hence, denoting the elements of the set A = {−n, …, −m − 1, m + 1, …, n} by a_{1}, …, a_{N−M} we can write, modulo lower degree terms at each power of u,

f _{a_{1} ··· a_{N−M}}^{−a_{1} ··· −a_{N−M}} (u + n − m) ≡ α_{n−m}(u + n − m)

× ∑_{p∈𝔖_{M}} sgn pp′ · (1 + F u^{−1})_{−a_{p(1)},a_{p′(1)}} ⋯ (1 + F u^{−1})_{−a_{p(N−M)},a_{p′(N−M)}},

where F denotes the (N − M) × (N − M) matrix whose rows and columns are enumerated by the elements of the set A and whose ij-th entry is F_{ij}. Since these matrix elements commute modulo lower degree terms, taking into account the relation

α_{m−n}(u) α_{n−m}(u + n − m) = 1,

we can conclude from Lemma 3.8 that the image of the series (5.3) in P_{M}(N) is the determinant det(1 + xu^{−1})_{AA}. The same argument shows that the image of the series (5.2) in P_{M}(N) is det(1 + xu^{−1})_{A_{i}A_{j}}, where

A_{i} = {−n, …, −m − 1, i, m + 1, …, n}.

Our next step is to show that every element ϕ of the algebra P_{M}(N) such that deg ϕ < n − m can be represented as a polynomial in the coefficients of the series det(1+xu^{−1})_{AA} and det(1+xu^{−1})_{A_{i}A_{j}}. However, it was proved in [14, Section 4.9] that ϕ can be presented as a polynomial in the elements tr (x_{AA})^{k} and Λ_{ij}^{(k)} with −m ⩽ i, j ⩽ m and k ⩾ 1, where Λ_{ij}^{(k)} = ∑ x_{ic_{1}}x_{c_{1}c_{2}} ⋯ x_{c_{k−1}j}, summed over the indices c_{r} ∈ A. On the other hand, each element tr (x_{AA})^{k} is a polynomial in the coefficients of the series det(1 + xu^{−1})_{AA}. This follows from the fact that the coefficients of the characteristic polynomial det(u+x)_{AA} form a complete set of invariants of the matrix x_{AA}. (Explicit expressions for the elements tr (x_{AA})^{k} in terms of the coefficients of the characteristic polynomial can be derived e.g. from the Liouville formula; cf. [13, Remark 5.8]). The claim now follows from the well known identity

det(1 + xu^{−1})_{A_{i}A_{j}} = det(1 + xu^{−1})_{AA} · (δ_{ij} + ∑_{k=1}^{∞}(−1)^{k−1}Λ_{ij}^{(k)}u^{−k});  (5.4)

see e.g. [5, 6]. Indeed, the identity implies that the elements Λ_{ij}^{(k)} are polynomials in the coefficients of the series det(1 + xu^{−1})_{A_{i}A_{j}} and det(1 + xu^{−1})_{AA}. This allows us to conclude that the algebra A_{M} is generated by the subalgebra C_{M} and the image of the homomorphism ψ.

Observe that since the matrix x satisfies x^{t} = −x we have the relations

det(1 + xu^{−1})_{AA} = det(1 − xu^{−1})_{AA}

and

Λ_{ij}^{(k)} = (−1)^{k} θ_{ij} Λ_{−j,−i}^{(k)}.

Hence, we can write

det(1 + xu^{−1})_{AA} = 1 + ∑_{r=1}^{∞} Λ^{(2r)} u^{−2r}

for some polynomials Λ^{(2r)} in the matrix elements of x. For the elements Λ_{ij}^{(k)} we shall impose the following restrictions on i, j, k:

i + j < 0 for k odd, i + j ≤ 0 for k even

in the orthogonal case, and

i + j < 0 for k even, i + j ≤ 0 for k odd

in the symplectic case. Fix a positive integer K and assume that the index k satisfies 1 ⩽ k ⩽ K. It follows from the argument of [14, Section 4.10] that there exists a large enough value of N such that the polynomials Λ^{(k)} for even k, and Λ_{ij}^{(k)} with the above restrictions on i, j, k are algebraically independent. Due to the identity (5.4), the same statement will hold if each polynomial Λ_{ij}^{(k)} is replaced by the coefficient at u^{−k} of the series det(1 + xu^{−1})_{A_{i}A_{j}}. This proves the injectivity of ψ and the tensor product decomposition for the algebra A_{M}.

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