Weight bases of Gelfand–Tsetlin type for representations of classical Lie algebras

A. I. Molev

School of Mathematics and Statistics University of Sydney, NSW 2006, Australia alexm@maths.usyd.edu.au

Abstract

This paper completes a series devoted to explicit constructions of finitedimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers dealt with C and D types). A weight basis for each representation of the Lie algebra 𝔬(2n + 1) is constructed. The basis vectors are parametrized by Gelfand–Tsetlin-type patterns. Explicit formulas for the matrix elements of generators of 𝔬(2n + 1) in this basis are given. The construction is based on the representation theory of the Yangians. A similar approach is applied to the A type case where the well-known formulas due to Gelfand and Tsetlin are reproduced.

1 Introduction

In this paper we give an explicit construction of each finite-dimensional irreducible representation V of an odd orthogonal Lie algebra 𝔬(2n + 1) (i.e. a simple complex Lie algebra of type B). A weight basis in V is obtained by the application of certain elements of the enveloping algebra (the lowering operators) to the highest weight vector. Explicit formulas for the matrix elements of generators of the Lie algebra 𝔬(2n + 1) in this basis are given. We follow an approach applied in the previous papers [13] and [14] where similar results were obtained for the C and D type Lie algebras. We also reproduce a slightly modified version of the well-known construction of the Gelfand–Tsetlin bases for the A type Lie algebras.

Let 𝔤_{n} denote the rank n simple Lie algebra of type A, B, C, or D. The restriction of a finite-dimensional irreducible representation V of 𝔤_{n} to the subalgebra 𝔤_{n−1} is multiplicity-free for the A type case, and it is not necessarily so for the B, C, D types. Gelfand and Tsetlin [5] used the chain of subalgebras

𝔤_{1} ⊂ 𝔤_{2} ⊂ ⋯ ⊂ 𝔤_{n}  (1.1)

to parametrize basis vectors in V and give formulas for the matrix elements of generators for the A type case. Different approaches to derive these formulas are used e.g. in [24, 17, 25, 7, 18, 11].

Analogous results for representations of the orthogonal Lie algebra are obtained by Gelfand and Tsetlin in [6]; see also [21, 23, 8]. Here the chain (1.1) is replaced with the one which involves both the odd and even orthogonal Lie algebras. However, the corresponding basis vectors lose the weight property, i.e. they are not eigenvectors for the elements of the Cartan subalgebra. To get a weight basis we propose to use the chain (1.1) for the B, C, D types as well. We “separate” the multiplicities occurring in the reduction 𝔤_{n} ↓ 𝔤_{n−1} by applying the representation theory of the Yangians. Namely, the subspace V_{μ}^{+} of 𝔤_{n−1}-highest vectors of weight μ in V possesses a natural structure of a representation of the twisted Yangian Y^{+}(2) or Y^{−}(2), in the orthogonal and symplectic case, respectively. The twisted Yangians are introduced and studied by Olshanski [20]; see also [16] for a detailed exposition. The action of Y^{±}(2) in the space V_{μ}^{+} arises from his centralizer construction [20]. Finite-dimensional irreducible representations of the twisted Yangians are classified in [12]. In particular, it turns out that the representation V_{μ}^{+} of Y^{±}(2) can be extended to a larger algebra, the Yangian Y(2) for the Lie algebra 𝔤𝔩(2). The algebra Y(2) and its representations are very well studied; see [22], [2]. In particular, a large class of representation of Y(2) admits Gelfand–Tsetlin-type bases associated with the inclusion Y(1) ⊂ Y(2); see [11, 19]. This allows us to get a natural basis in the space V_{μ}^{+}, and then by induction to get a basis in the entire space V.

Note that in the case of C or D type the Y(2)-module V_{μ}^{+} is irreducible while in the B type case it is a direct sum of two irreducible submodules. This does not lead, however, to major differences in the constructions, and the final formulas are similar in all the three cases.

Our calculations of the matrix elements of the generators of 𝔤_{n} are based on the relationship between the twisted Yangian Y^{±}(2) and the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}) (it is also called the Mickelsson algebra or S-algebra). It is generated by the raising and lowering operators which preserve the subspace V^{+} of 𝔤_{n−1}-highest vectors in V. The algebraic structure of the transvector algebras is studied in detail in [26] with the use of the extremal projections for reductive Lie algebras [1]. We construct an algebra homomorphism Y^{±}(2) → Z(𝔤_{n}, 𝔤_{n−1}) which allows us to express the generators of the twisted Yangian, as operators in V_{μ}^{+}, in terms of the raising and lowering operators. This plays a key role in the calculation of the matrix elements of the generators of 𝔤_{n} in the basis of V.

Explicit combinatorial constructions of the fundamental representations of the symplectic and odd orthogonal Lie algebras were recently given by Donnelly [3]. He also showed that in the symplectic case the basis of [13] for the fundamental representations coincides, up to a scaling, with a basis of his [4]. It is likely that a similar connection exists in the odd orthogonal case.

2 Gelfand–Tsetlin basis for 𝔤𝔩(n)

Let E_{ij}, i, j = 1, …, n denote the standard basis of the general linear Lie algebra 𝔤_{n} = 𝔤𝔩(n) over the field of complex numbers. The subalgebra 𝔤_{n−1} is spanned by the basis elements E_{ij} with i, j = 1, …, n − 1. Denote by 𝔥 = 𝔥_{n} the diagonal Cartan subalgebra in 𝔤_{n}. The elements E_{11}, …, E_{nn} form a basis of 𝔥.

Finite-dimensional irreducible representations of 𝔤_{n} are in a one-to-one correspondence with n-tuples of complex numbers λ = (λ_{1}, …, λ_{n}) such that

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

Such an n-tuple λ is called the highest weight of the corresponding representation which we shall denote by L(λ). It contains a unique, up to a multiple, nonzero vector ξ (the highest vector) such that E_{ii} ξ = λ_{i} ξ for i = 1, …, n and E_{ij} ξ = 0 for 1 ≤ i < j ≤ n. Denote by L(λ)^{+} the subspace of 𝔤_{n−1}-highest vectors in L(λ):

L(λ)^{+} = {η ∈ L(λ) | E_{ij} η = 0, 1 ≤ i < j < n}.

Given a 𝔤_{n−1}-highest weight μ = (μ_{1}, …, μ_{n−1}) we denote by L(λ)_{μ}^{+} the corresponding weight subspace in L(λ)^{+}:

L(λ)_{μ}^{+} = {η ∈ L(λ)^{+} | E_{ii} η = μ_{i} η, i = 1, …, n − 1}.

It is well-known [24] that the space L(λ)_{μ}^{+} is either trivial or one-dimensional. Moreover, dim L(λ)_{μ}^{+} = 1 if and only if

λ_{i} − μ_{i} ∈ ℤ_{+} and μ_{i} − λ_{i+1} ∈ ℤ_{+} for i = 1, …, n − 1.  (2.1)

In other words, the restriction of L(λ) to the subalgebra 𝔤_{n−1} is multiplicity-free:

L(λ)|_{𝔤_{n−1}} ≃ ⨁ L′(μ),

where L′(μ) is the irreducible 𝔤_{n−1}-module with the highest weight μ satisfying the conditions (2.1). A parameterization of basis vectors in L(λ) is obtained by using its further restrictions to the subalgebras of the chain (1.1). A Gelfand–Tsetlin pattern Λ associated with λ is an array of row vectors

λ_{n1} λ_{n2}  ⋯  λ_{nn}
λ_{n−1,1}  ⋯  λ_{n−1,n−1}
⋯  ⋯  ⋯
λ_{21} λ_{22}
λ_{11}

such that the upper row coincides with λ and the following conditions hold

λ_{ki} − λ_{k−1,i} ∈ ℤ_{+},  λ_{k−1,i} − λ_{k,i+1} ∈ ℤ_{+},  i = 1, …, k − 1  (2.2)

for each k = 2, …, n.

Remark. If the highest weight λ is a partition then there is a natural bijection between the patterns associated with λ and semistandard λ-tableaux with entries in {1, …, n}. A pattern can be viewed as a sequence of partitions

λ^{(1)} ⊆ λ^{(2)} ⊆ ⋯ ⊆ λ^{(n)} = λ,

with λ^{(k)} = (λ_{k1}, …, λ_{kk}). Conditions (2.2) mean that the skew diagram λ^{(k)}/λ^{(k−1)} is a horizontal strip; see e.g. [10].

Let us set l_{ki} = λ_{ki} − i + 1.

Theorem 2.1 There exists a basis {ξ_{Λ}} in L(λ) parametrized by all patterns Λ such that the action of generators of 𝔤_{n} is given by the formulas

E_{kk} ξ_{Λ} = (∑_{i=1}^{k} λ_{ki} − ∑_{i=1}^{k−1} λ_{k−1,i}) ξ_{Λ},  (2.3)
E_{k,k+1} ξ_{Λ} = − ∑_{i=1}^{k} \frac{(l_{ki} − l_{k+1,1}) ⋯ (l_{ki} − l_{k+1,k+1})}{(l_{ki} − l_{k1}) ⋯ ∧ ⋯ (l_{ki} − l_{kk})} ξ_{Λ+δ_{ki}},  (2.4)
E_{k+1,k} ξ_{Λ} = ∑_{i=1}^{k} \frac{(l_{ki} − l_{k−1,1}) ⋯ (l_{ki} − l_{k−1,k−1})}{(l_{ki} − l_{k1}) ⋯ ∧ ⋯ (l_{ki} − l_{kk})} ξ_{Λ−δ_{ki}}.  (2.5)

The arrays Λ±δ_{ki} are obtained from Λ by replacing λ_{ki} by λ_{ki} ±1. It is supposed that ξ_{Λ} = 0 if the array Λ is not a pattern; the symbol ∧ indicates that the zero factor in the denominator is skipped.

Proof. Consider the extension of the universal enveloping algebra U(𝔤_{n})

U′(𝔤_{n}) = U(𝔤_{n}) ⊗_{U(𝔥)} R(𝔥),

where R(𝔥) is the field of fractions of the commutative algebra U(𝔥). Let J denote the left ideal in U′(𝔤_{n}) generated by the elements E_{ij} with 1 ≤ i < j < n. Introduce the normalizer of J in U′(𝔤_{n}):

Norm J = {x ∈ U′(𝔤_{n}) | J x ⊆ J}.

Note that J is a two-sided ideal in the algebra Norm J. We define the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}) as the quotient

Z(𝔤_{n}, 𝔤_{n−1}) = Norm J/J;

see [25, 26]. Equivalently, Z(𝔤_{n}, 𝔤_{n−1}) can be defined by using the extremal projection p = p_{n−1} for the Lie algebra 𝔤_{n−1} [1]. The projection p is, up to a factor from R(𝔥_{n−1}), a unique element of an extension of U′(𝔤_{n−1}) to an algebra of formal series, satisfying the conditions

E_{ij} p = p E_{ji} = 0 for 1 ≤ i < j ≤ n − 1.  (2.6)

The element p is of zero weight (with respect to the adjoint action of 𝔥_{n−1}) and it can be normalized to satisfy the condition p^{2} = p. The projection p is a welldefined operator in the quotient U′(𝔤_{n})/J which allows one to naturally identify the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}) with the image of p [26]:

Z(𝔤_{n}, 𝔤_{n−1}) = p (U′(𝔤_{n})/J) .

An analog of the Poincaré–Birkhoff–Witt theorem holds for the algebra Z(𝔤_{n}, 𝔤_{n−1}) so that ordered monomials in the elements pE_{in} and pE_{ni} with i = 1, …, n − 1 form a basis of Z(𝔤_{n}, 𝔤_{n−1}) as a left or right R(𝔥)-module [26]. These elements are called the raising and lowering operators and can be given by

pE_{in} = ∑_{i>i_{1}>···>i_{s}≥1} E_{ii_{1}}E_{i_{1}i_{2}} ⋯ E_{i_{s−1}i_{s}}E_{i_{s}n} \frac{1}{(h_{i} − h_{i_{1}}) ⋯ (h_{i} − h_{i_{s}})},
pE_{ni} = ∑_{i<i_{1}<···<i_{s}<n} E_{i_{1}i}E_{i_{2}i_{1}} ⋯ E_{i_{s}i_{s−1}}E_{ni_{s}} \frac{1}{(h_{i} − h_{i_{1}}) ⋯ (h_{i} − h_{i_{s}})},  (2.7)

where s = 0, 1, … and h_{i} = E_{ii} − i + 1. We shall also use normalized operators defined by

z_{in} = pE_{in} (h_{i} − h_{i−1}) ⋯ (h_{i} − h_{1}),
z_{ni} = pE_{ni} (h_{i} − h_{i+1}) ⋯ (h_{i} − h_{n−1}).  (2.8)

These can be viewed as elements of the enveloping algebra U(𝔤_{n}); cf. [24, 17]. We have the following relations [25, 26]

z_{ni}z_{nj} = z_{nj}z_{ni} for all i, j,  (2.9)
z_{in}z_{nj} = z_{nj}z_{in} for i ≠ j.  (2.10)

Indeed, assume that i < j. Then (2.6) and (2.7) imply that in Z(𝔤_{n}, 𝔤_{n−1})

pE_{ni} pE_{nj} = pE_{ni}E_{nj}, pE_{nj} pE_{ni} = pE_{ni}E_{nj} \frac{h_{i} − h_{j} + 1}{h_{i} − h_{j}}.

Now (2.9) follows from (2.8). The proof of (2.10) is similar.

Due to (2.6) the operators z_{in} and z_{ni} preserve the space L(λ)^{+}: for i = 1, …, n−1

z_{in} : L(λ)_{μ}^{+} → L(λ)_{μ+δ_{i}}^{+},  z_{ni} : L(λ)_{μ}^{+} → L(λ)_{μ−δ_{i}}^{+},  (2.11)

where μ ± δ_{i} is obtained from μ by replacing μ_{i} with μ_{i} ± 1.

The following is a key lemma in the derivation of the Gelfand–Tsetlin formulas.

Lemma 2.2 Given μ satisfying (2.1) the vector

ξ_{μ} = z_{n1}^{λ_{1}−μ_{1}} ⋯ z_{n,n−1}^{λ_{n−1}−μ_{n−1}} ξ

spans the subspace L(λ)_{μ}^{+} and for each i = 1, …, n − 1 we have

z_{in} ξ_{μ} = −(m_{i} − l_{1}) ⋯ (m_{i} − l_{n}) ξ_{μ+δ_{i}},  (2.12)

where m_{i} = μ_{i} − i + 1, l_{i} = λ_{i} − i + 1. It is supposed that ξ_{μ+δ_{i}} = 0 if λ_{i} = μ_{i}.

Proof. By (2.11) the vector ξ_{μ} belongs to the subspace L(λ)_{μ}^{+}. We need to show that it is nonzero. This will follow from relations (2.12). We shall outline a proof of these relations which involves the use of the Yangians; cf. [25]. Consider the n × n-matrix E whose ij-th entry is E_{ij} and let u be a formal variable. Introduce the polynomial T (u) with coefficients in the universal enveloping algebra U(𝔤_{n}):

T (u) = ∑_{σ∈S_{n}} sgn σ (u + E)_{σ(1),1} ⋯ (u + E − n + 1)_{σ(n),n}.

It is well known that all its coefficients belong to the center of U(𝔤_{n}) (and generate the center); see e.g. [9]. This also easily follows from the properties the quantum determinant of the Yangian for the Lie algebra 𝔤𝔩(n); see e.g. [16]. Therefore, these coefficients act in L(λ) as scalars which can be easily found by applying T (u) to the highest vector ξ:

T (u)|_{L(λ)} = (u + l_{1}) ⋯ (u + l_{n}).  (2.13)

On the other hand, the center of U(𝔤_{n}) is a subalgebra in the normalizer Norm J. We shall keep the same notation for the image of T (u) in the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}). To get explicit expressions of the coefficients of T (u) in terms of the raising and lowering operators we consider T (u) modulo the ideal J and apply the projection p. The details can be found in [15, Theorem 3.1]. We have

T (u) = (u + E_{nn}) ∏_{i=1}^{n−1}(u + h_{i} − 1) − ∑_{i=1}^{n−1} z_{in}z_{ni} ∏_{j=1, j≠i}^{n−1} \frac{u + h_{j} − 1}{h_{i} − h_{j}}.

In particular, T (−h_{i} + 1) = (−1)^{n−1}z_{in}z_{ni}.

Now, (2.10) implies that z_{in} ξ_{μ} = 0 unless λ_{i} − μ_{i} ≥ 1. In this case using (2.9) we obtain

z_{in} ξ_{μ} = z_{in}z_{ni} ξ_{μ+δ_{i}} = (−1)^{n−1}T (−h_{i} + 1) ξ_{μ+δ_{i}} = (−1)^{n−1}T (−m_{i}) ξ_{μ+δ_{i}},

where we have used h_{i} ξ_{μ} = m_{i} ξ_{μ}. The relation (2.12) now follows from (2.13).

Applying appropriate raising operators to the vector ξ_{μ} we can obtain the highest vector ξ of L(λ) with a nonzero coefficient. This proves that ξ_{μ} ≠ 0.

Given a Gelfand–Tsetlin pattern Λ introduce the vector ξ_{Λ} ∈ L(λ) by

ξ_{Λ} = ∏_{k=2,...,n}^{→}(z_{k1}^{λ_{k1}−λ_{k−1,1}} ⋯ z_{k,k−1}^{λ_{k,k−1}−λ_{k−1,k−1}}) ξ.

Lemma 2.2 implies

Corollary 2.3 The vectors ξ_{Λ} parametrized by the patterns Λ form a basis of the representation L(λ).

We now briefly outline a derivation of formulas (2.3)–(2.5) which is standard; see e.g. [25]. First, since E_{nn} z_{ni} = z_{ni} (E_{nn} + 1) for any i, we have

E_{nn} ξ_{μ} = (∑_{i=1}^{n} λ_{i} − ∑_{i=1}^{n−1} μ_{i}) ξ_{μ},

which implies (2.3). To prove (2.4) is suffices to find E_{n−1,n} ξ_{μν} where

ξ_{μν} = z_{n−1,1}^{μ_{1}−ν_{1}} ⋯ z_{n−1,n−2}^{μ_{n−2}−ν_{n−2}} ξ_{μ},

and the ν_{i} satisfy

μ_{i} − ν_{i} ∈ ℤ_{+} and ν_{i} − μ_{i+1} ∈ ℤ_{+} for i = 1, …, n − 2.

Since E_{n−1,n} commutes with the z_{n−1,i},

E_{n−1,n} ξ_{μν} = z_{n−1,1}^{μ_{1}−ν_{1}} ⋯ z_{n−1,n−2}^{μ_{n−2}−ν_{n−2}} E_{n−1,n} ξ_{μ}.

Now use the following identity in U′(𝔤_{n}) modulo the ideal J [25]

E_{n−1,n} = ∑_{i=1}^{n−1} z_{n−1,i} z_{in} \frac{1}{(h_{i} − h_{1}) ⋯ ∧ ⋯ (h_{i} − h_{n−1})},  (2.14)

where z_{n−1,n−1} := 1. Applying (2.12) we find that

E_{n−1,n} ξ_{μν} = − ∑_{i=1}^{n−1} \frac{(m_{i} − l_{1}) ⋯ (m_{i} − l_{n})}{(m_{i} − m_{1}) ⋯ ∧ ⋯ (m_{i} − m_{n−1})} ξ_{μ+δ_{i},ν}  (2.15)

which proves (2.4). To prove (2.5) we use a contravariant bilinear form ⟨ , ⟩ on L(λ) uniquely determined by the conditions:

⟨ξ, ξ⟩ = 1,  ⟨E_{ij} η, ζ⟩ = ⟨η, E_{ji} ζ⟩, η, ζ ∈ L(λ).  (2.16)

The basis {ξ_{Λ}} is orthogonal with respect to this form. This follows from (2.12) and the fact that the operators pE_{in} and pE_{ni} are adjoint to each other with respect to the restriction of the form ⟨ , ⟩ to the space L(λ)^{+}. In particular, (2.15) implies that

E_{n,n−1}ξ_{μν} = ∑_{i=1}^{n−1} c_{i}(μ, ν) ξ_{μ−δ_{i},ν}

for some coefficients c_{i}(μ, ν). Apply the operator z_{j,n−1} to both sides of this relation. Since z_{j,n−1} commutes with E_{n,n−1} we obtain from (2.12) a recurrence relation for the c_{i}(μ, ν): if μ_{j} − ν_{j} ≥ 1 then

c_{i}(μ, ν + δ_{j}) = c_{i}(μ, ν) \frac{m_{i} − γ_{j} − 1}{m_{i} − γ_{j}},

where γ_{j} = ν_{j} − j + 1. This proves (2.5) by induction.

Note that the original Gelfand–Tsetlin basis [5] is orthonormal. The basis vectors in [5] coincide with the ξ_{Λ} up to a norm factor which can be explicitly calculated; see e.g. [25].

3 A basis for odd orthogonal Lie algebras

We shall enumerate the rows and columns of (2n + 1) × (2n + 1)-matrices over ℂ by the indices −n, …, −1, 0, 1, …, n.

3.1 Main theorem

We keep the notation E_{ij}, i, j = −n, …, n for the standard basis of the Lie algebra 𝔤𝔩(2n + 1). Introduce the elements

F_{ij} = E_{ij} − E_{−j,−i}.  (3.1)

We have F_{−j,−i} = −F_{ij}. In particular, F_{−i,i} = 0 for all i. The orthogonal Lie algebra 𝔤_{n} := 𝔬(2n + 1) can be identified with the subalgebra in 𝔤𝔩(2n + 1) spanned by the elements F_{ij}, i, j = −n, …, n. The subalgebra 𝔤_{n−1} is spanned by the elements (3.1) with the indices i, j running over the set {−n + 1, …, n − 1}. Denote by 𝔥 = 𝔥_{n} the diagonal Cartan subalgebra in 𝔤_{n}. The elements F_{11}, …, F_{nn} form a basis of 𝔥.

The finite-dimensional irreducible representations of 𝔤_{n} are in a one-to-one correspondence with n-tuples λ = (λ_{1}, …, λ_{n}) where all the entries λ_{i} are simultaneously integers or half-integers (elements of the set \frac{1}{2} + ℤ) and the following inequalities hold:

0 ≥ λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n}.

Such an n-tuple λ is called the highest weight of the corresponding representation which we shall denote by V (λ). It contains a unique, up to a multiple, nonzero vector ξ (the highest vector) such that F_{ii} ξ = λ_{i} ξ for i = 1, …, n and F_{ij} ξ = 0 for −n ≤ i < j ≤ n. Denote by V (λ)^{+} the subspace of 𝔤_{n−1}-highest vectors in V (λ):

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

Given a 𝔤_{n−1}-highest weight μ = (μ_{1}, …, μ_{n−1}) 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, …, n − 1}.

By the branching rule for the reduction 𝔤_{n} ↓ 𝔤_{n−1} [24] we have

V (λ)|_{𝔤_{n−1}} ≃ ⨁ c(μ) V′(μ),  (3.2)

where V′(μ) is the irreducible finite-dimensional representation of 𝔤_{n−1} with the highest weight μ, and c(μ) equals the number of n-tuples ρ = (ρ_{1}, …, ρ_{n}) satisfying the inequalities

\begin{matrix}− λ_{1} ≥ ρ_{1} ≥ λ_{1} ≥ ρ_{2} ≥ λ_{2} ≥ ⋯ ≥ ρ_{n−1} ≥ λ_{n−1} ≥ ρ_{n} ≥ λ_{n}, \\ − μ_{1} ≥ ρ_{1} ≥ μ_{1} ≥ ρ_{2} ≥ μ_{2} ≥ ⋯ ≥ ρ_{n−1} ≥ μ_{n−1} ≥ ρ_{n} \end{matrix}  (3.3)

with all the ρ_{i} and μ_{i} being simultaneously integers or half-integers together with the λ_{i}. Any nonzero vector in V (λ)_{μ}^{+} generates a 𝔤_{n−1}-submodule in V (λ) isomorphic to V′(μ). We obviously have dim V (λ)_{μ}^{+} = c(μ). Basis vectors in V (λ)_{μ}^{+} can be parametrized by the n-tuples ρ. We shall be using an equivalent parameterization by (n + 1)-tuples ν = (σ, ν_{1}, …, ν_{n}), where ν_{i} = ρ_{i} for i ≥ 2, and

(σ, ν_{1}) = \begin{cases}(0, ρ_{1}) & if ρ_{1} ≤ 0, \\ (1, −ρ_{1}) & if ρ_{1} > 0. \end{cases}

A parameterization of basis vectors in V (λ) is obtained by using its subsequent restrictions to the subalgebras of the chain 𝔤_{1} ⊂ 𝔤_{2} ⊂ ⋯ ⊂ 𝔤_{n−1} ⊂ 𝔤_{n}. Define a pattern Λ associated with λ as an array of the form

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

such that λ = (λ_{n1}, …, λ_{nn}), each σ_{k} is 0 or 1, the remaining entries are all nonpositive integers or non-positive half-integers together with the λ_{i}, and the following inequalities hold

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

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; in addition, in the case of integer λ_{i} the condition

λ′_{k1} ≤ −1 if σ_{k} = 1

should hold for all k = 1, …, n. Let us set l_{k0} = −1/2 for all k, and

l_{ki} = λ_{ki} − i + 1/2,  l′_{ki} = λ′_{ki} − i + 1/2,  1 ≤ i ≤ k ≤ n.

Given a pattern Λ set for i = 0, 1, …, k − 1

A_{ki} = ∏_{a=1, a≠i}^{k−1} \frac{1}{l_{k−1,i} − l_{k−1,a}} · ∏_{a=1}^{k−1} \frac{1}{l_{k−1,i} + l_{k−1,a}} .

Furthermore, introduce polynomials B_{ki}(x) by

B_{ki}(x) = ∏_{a=1, a≠i}^{k} \frac{(x + l′_{ka} + 1)(x − l′_{ka})}{l′_{ka} − l′_{ki}},

and define the numbers C_{ki} by

C_{ki} = l′_{ki} (1 − 2 σ_{k} − 2 l′_{ki}) ∏_{a=1}^{k}(l_{ka} − l′_{ki}) ∏_{a=1}^{k−1}(l_{k−1,a} − l′_{ki}) ∏_{a=1, a≠i}^{k} \frac{1}{l′_{ka} − l′_{ki}} .

We denote by Λ ± δ_{ki} and Λ + δ′_{ki} the arrays obtained from Λ by replacing λ_{ki} and λ′_{ki} by λ_{ki} ± 1 and λ′_{ki} + 1 respectively.

The following is our main theorem which will be proved in Sections 3.4 and 3.5.

Theorem 3.1 There exists a basis {ζ_{Λ}} of V (λ) parametrized by the patterns Λ such that the action of the generators of 𝔤_{n} is given by the formulas

F_{kk} ζ_{Λ} = (σ_{k} + 2 ∑_{i=1}^{k} λ′_{ki} − ∑_{i=1}^{k} λ_{ki} − ∑_{i=1}^{k−1} λ_{k−1,i}) ζ_{Λ},
F_{k−1,−k} ζ_{Λ} = A_{k0} ζ_{Λ}(k, 0) + ∑_{i=1}^{k−1} A_{ki} (\frac{1}{l_{k−1,i} + 1/2} ζ_{Λ}^{+}(k, i) − \frac{1}{l_{k−1,i} − 1/2} ζ_{Λ}^{−}(k, i)) .

Here the following notation has been used

ζ_{Λ}^{−}(k, i) = ζ_{Λ−δ_{k−1,i}},
ζ_{Λ}^{+}(k, i) = ∑_{j=1}^{k} ∑_{m=1}^{k−1} B_{kj}(l_{k−1,i})B_{k−1,m}(l_{k−1,i}) ζ_{Λ+δ′_{kj}+δ_{k−1,i}+δ′_{k−1,m}},

and ζ_{Λ} := 0 if Λ is not a pattern. Furthermore,

ζ_{Λ}(k, 0) = (−1)^{k} ζ_{Λ̄}  if σ_{k} = σ_{k−1} = 0,
= ∑_{j=1}^{k} B_{kj}(l_{k−1,0}) ζ_{Λ̄+δ′_{kj}}  if σ_{k} = 1, σ_{k−1} = 0,
= − ∑_{m=1}^{k−1} B_{k−1,m}(l_{k−1,0}) ζ_{Λ̄+δ′_{k−1,m}}  if σ_{k} = 0, σ_{k−1} = 1,
= (−1)^{k−1} ∑_{j=1}^{k} ∑_{m=1}^{k−1} B_{kj}(l_{k−1,0})B_{k−1,m}(l_{k−1,0}) ζ_{Λ̄+δ′_{kj}+δ′_{k−1,m}}
if σ_{k} = σ_{k−1} = 1,

where Λ̄ is obtained from Λ by replacing σ_{k} and σ_{k−1} respectively with σ_{k} + 1 and σ_{k−1} + 1 (modulo 2). The action of F_{k−1,k} is found from the relation

F_{k−1,k} = [Φ_{k−1,−k}(u + 2) Φ_{−k,k} − Φ_{−k,k}Φ_{k−1,−k}(u)]_{u=0},

where the operator Φ_{−k,k} acts on the basis elements by the rule

Φ_{−k,k} ζ_{Λ} = ∑_{i=1}^{k} C_{ki} (F_{kk} − l′_{ki} + 1) ζ_{Λ−δ′_{ki}}

while the action of Φ_{k−1,−k}(u) is given by

Φ_{k−1,−k}(u) ζ_{Λ} = \frac{A_{k0}}{u + F_{kk} − 3/2} ζ_{Λ}(k, 0)

+ ∑_{i=1}^{k−1} A_{ki}(\frac{1}{(l_{k−1,i} + 1/2)(u + l_{k−1,i} + F_{kk} − 1)} ζ_{Λ}^{+}(k, i)

− \frac{1}{(l_{k−1,i} − 1/2)(u − l_{k−1,i} + F_{kk} − 1)} ζ_{Λ}^{−}(k, i)).

Remark. The image of ζ_{Λ} under the operator Φ_{k−1,−k}(u + 2) Φ_{−k,k} − Φ_{−k,k}Φ_{k−1,−k}(u) at u = 0 may be undefined for some patterns Λ. To get the action of F_{k−1,k}, one should first calculate its matrix elements in a “generic” representation V (λ) and then specialize the parameters; see Section 3.5. For example, consider the case n = 1. It will be shown in Section 3.4 that the basis vectors in V (λ) are given by

ζ_{Λ} = F_{10}^{σ_{1}} (F_{10}F_{0,−1})^{λ′_{11}−λ_{11}} ξ.

Furthermore, the operators Φ_{−1,1} and Φ_{0,−1}(u) are defined by

Φ_{−1,1} = −\frac{1}{2} F_{01}^{2},
Φ_{0,−1}(u) = F_{0,−1} \frac{1}{u + F_{11} − 1/2};

see Section 3.5. In the case λ = (−1/2) the basis of V (λ) consists of two vectors ξ and ξ′ = F_{10} ξ. Therefore, Φ_{−1,1} is the zero operator in V (λ) while the image Φ_{0,−1}(0)ξ′ is not defined. On the other hand, we find directly that F_{01} ξ′ = 1/2 ξ.

3.2 Transvector algebra Z(𝔤_{n}, 𝔤_{n−1})

Consider the extension of the universal enveloping algebra U(𝔤_{n})

U′(𝔤_{n}) = U(𝔤_{n}) ⊗_{U(𝔥)} R(𝔥),

where R(𝔥) is the field of fractions of the commutative algebra U(𝔥). Let J denote the left ideal in U′(𝔤_{n}) generated by the elements F_{ij} with −n < i < j < n. The transvector algebra Z(𝔤_{n}, 𝔤_{n−1}) is the quotient algebra of the normalizer

Norm J = {x ∈ U′(𝔤_{n}) | Jx ⊆ J}

modulo the two-sided ideal J [26]. It is an algebra over ℂ and an R(𝔥)-bimodule. Let p = p_{n−1} denote the extremal projection for the Lie algebra 𝔤_{n−1} [1, 26]; cf. Section 2. It satisfies the following (characteristic) relations

F_{ij} p = p F_{ji} = 0 for − n < i < j < n.

The projection p naturally acts in the space U′(𝔤_{n})/J and its image coincides with Z(𝔤_{n}, 𝔤_{n−1}). The elements

pF_{ia} = −pF_{−a,−i},  a = −n, n, i = −n + 1, …, n − 1  (3.4)

are generators of Z(𝔤_{n}, 𝔤_{n−1}) [26]. Set

f_{0} = −1/2,  f_{i} = F_{ii} − i + 1/2,  f_{−i} = −f_{i}

for i = 1, …, n; and set f′_{i} = −f_{−i} for all i. The elements (3.4) can be given by the following explicit formulas (modulo J):

pF_{ia} = ∑_{i>i_{1}>···>i_{s}>−n} F_{ii_{1}}F_{i_{1}i_{2}} ⋯ F_{i_{s−1}i_{s}}F_{i_{s}a} \frac{1}{(f_{i} − f_{i_{1}}) ⋯ (f_{i} − f_{i_{s}})},

where s = 0, 1, . . .. We shall use normalized generators of Z(𝔤_{n}, 𝔤_{n−1}) defined by

z_{ia} = pF_{ia} (f_{i} − f_{i−1}) ⋯ (f_{i} − f_{−n+1}),  (3.5)
z_{ai} = pF_{ai} (f′_{i} − f′_{i+1}) ⋯ (f′_{i} − f′_{n−1}).  (3.6)

We obviously have z_{ai} = (−1)^{n−i} z_{−i,−a}. The elements z_{ia} satisfy certain quadratic relations [26]. We shall use the following ones below (cf. (2.9) and (2.10)): for a, b ∈ {−n, n} and i + j ≠ 0 one has

z_{aj}z_{bi}(f′_{i} − f′_{j} + 1) = z_{bi}z_{aj}(f′_{i} − f′_{j}) + z_{ai}z_{bj}.  (3.7)

In particular, z_{ai} and z_{aj} commute for i + j ≠ 0. One easily verifies that z_{ai} and z_{bi} also commute for i ≠ 0 and all a, b.

The elements z_{ia} and z_{ai} naturally act in the space V (λ)^{+} and are called the raising and lowering operators. One has for i = 1, …, n − 1:

z_{ia} : V (λ)_{μ}^{+} → V (λ)_{μ+δ_{i}}^{+}, z_{ai} : V (λ)_{μ}^{+} → V (λ)_{μ−δ_{i}}^{+},

where μ±δ_{i} is obtained from μ by replacing μ_{i} with μ_{i}±1. The operators z_{0a} preserve each subspace V (λ)_{μ}^{+}.

We shall need the following element which can be checked to belong to the normalizer Norm J, and so it can be regarded as an element of the algebra Z(𝔤_{n}, 𝔤_{n−1}):

z_{n,−n} = ∑_{n>i_{1}>···>i_{s}>−n} F_{ni_{1}}F_{i_{1}i_{2}} ⋯ F_{i_{s},−n} (f_{n} − f_{j_{1}}) ⋯ (f_{n} − f_{j_{k}}),  (3.8)

where s = 1, 2, … and {j_{1}, …, j_{k}} is the complement to the subset {i_{1}, …, i_{s}} in {−n + 1, …, n − 1}. The following relation is proved exactly as its C and D series counterparts [13, 14] (cf. (2.14)): for a = −n, n

F_{n−1,a} = ∑_{i=−n+1}^{n−1} z_{n−1,i} z_{ia} \frac{1}{(f_{i} − f_{−n+1}) ⋯ ∧ ⋯ (f_{i} − f_{n−1})},  (3.9)

where z_{n−1,n−1} := 1 and the equalities are considered in U′(𝔤_{n}) modulo the ideal J.

3.3 Yangians and twisted Yangians

Let us introduce the 𝔤𝔩(2)-Yangian Y(2) and the (orthogonal) twisted Yangian Y^{+}(2); see [16] for more details. The Yangian Y(2) is the complex associative algebra with the generators t_{ab}^{(1)}, t_{ab}^{(2)}, … where a, b ∈ {−n, n}, and the defining relations

[t_{ab}(u), t_{cd}(v)] = \frac{1}{u − v} (t_{cb}(u)t_{ad}(v) − t_{cb}(v)t_{ad}(u)),  (3.10)

where

t_{ab}(u) := δ_{ab} + t_{ab}^{(1)}u^{−1} + t_{ab}^{(2)}u^{−2} + ⋯ ∈ Y(2)[[u^{−1}]].  (3.11)

Introduce the series s_{ab}(u), a, b ∈ {−n, n} by

s_{ab}(u) = t_{an}(u)t_{−b,−n}(−u) + t_{a,−n}(u)t_{−b,n}(−u).

Write s_{ab}(u) = δ_{ab}+s_{ab}^{(1)}u^{−1}+s_{ab}^{(2)}u^{−2}+· · · . The twisted Yangian Y^{+}(2) is defined as the subalgebra of Y(2) generated by the elements s_{ab}^{(1)}, s_{ab}^{(2)}, … where a, b ∈ {−n, n}. Also, Y^{+}(2) can be viewed as an abstract algebra with generators s_{ab}^{(r)} and the following defining relations (see [16, Section 3]):

[s_{ab}(u), s_{cd}(v)] = \frac{1}{u − v} (s_{cb}(u)s_{ad}(v) − s_{cb}(v)s_{ad}(u))
− \frac{1}{u + v} (s_{a,−c}(u)s_{−b,d}(v) − s_{c,−a}(v)s_{−d,b}(u))  (3.12)

+ \frac{1}{u^{2} − v^{2}} (s_{c,−a}(u)s_{−b,d}(v) − s_{c,−a}(v)s_{−b,d}(u))

and

s_{−b,−a}(−u) = \frac{2u + 1}{2u} s_{ab}(u) − \frac{1}{2u} s_{ab}(−u).  (3.13)

The Yangian Y(2) is a Hopf algebra with the coproduct

Δ(t_{ab}(u)) = t_{an}(u) ⊗ t_{nb}(u) + t_{a,−n}(u) ⊗ t_{−n,b}(u).  (3.14)

The twisted Yangian Y^{+}(2) is a left coideal in Y(2) with

Δ(s_{ab}(u)) = ∑_{c,d∈{−n,n}} t_{ac}(u)t_{−b,−d}(−u) ⊗ s_{cd}(u).  (3.15)

Given a pair of complex numbers (α, β) such that α − β ∈ ℤ_{+} we denote by L(α, β) the irreducible representation of the Lie algebra 𝔤𝔩(2) with the highest weight (α, β) with respect to the upper triangular Borel subalgebra; see Section 2. We have dim L(α, β) = α − β + 1. We may regard L(α, β) as a Y(2)-module by using the algebra homomorphism Y(2) → U(𝔤𝔩(2)) given by

t_{ab}(u) ↦ δ_{ab} + E_{ab}u^{−1}, a, b ∈ {−n, n}.

The coproduct (3.14) allows one to construct representations of Y(2) of the form

L = L(α_{1}, β_{1}) ⊗ ⋯ ⊗ L(α_{n}, β_{n}).

Note that the generators t_{ab}^{(r)} with r > n act as zero operators in L. Therefore, the operators T_{ab}(u) = u^{n} t_{ab}(u) are polynomials in u:

T_{ab}(u) = δ_{ab}u^{n} + t_{ab}^{(1)}u^{n−1} + ⋯ + t_{ab}^{(n)}.

For any γ ∈ ℂ denote by W (γ) the one-dimensional representation of Y^{+}(2) spanned by a vector w such that

s_{nn}(u) w = \frac{u + γ}{u + 1/2} w, s_{−n,−n}(u) w = \frac{u − γ + 1}{u + 1/2} w,

and s_{a,−a}(u) w = 0 for a = −n, n. By (3.15) we can regard the tensor product L ⊗ W (γ) as a representation of Y^{+}(2). Representations of this type essentially exhaust all finite-dimensional irreducible representations of Y^{+}(2) [12]. The vector space isomorphism

L ⊗ W (γ) → L,  v ⊗ w ↦ v,  v ∈ L  (3.16)

provides L ⊗ W (γ) with an action of Y(2).

3.4 Construction of the basis

Introduce the following polynomials in u with coefficients in the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}): for a, b ∈ {−n, n}

Z_{ab}(u) = −(δ_{ab}(u − n + 1) + F_{ab}) ∏_{i=−n+1}^{n−1}(u + g_{i}) + ∑_{i=−n+1}^{n−1} z_{ai}z_{ib} ∏_{j=−n+1, j≠i}^{n−1} \frac{u + g_{j}}{g_{i} − g_{j}} , (3.17)

where g_{i} := f_{i} + 1/2 for all i.

Proposition 3.2 The mapping

s_{ab}(u) ↦ −u^{−2n} Z_{ab}(u),  a, b ∈ {−n, n}  (3.18)

defines an algebra homomorphism Y^{+}(2) → Z(𝔤_{n}, 𝔤_{n−1}).

Proof. One of the possible ways to prove the claim is to check directly that the relations (3.12) and (3.13) are satisfied with the s_{ab}(u) respectively replaced with Z_{ab}(u). Here one needs to use the quadratic relations in the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}). In addition to (3.7) the relations which express z_{ia}z_{bi} in terms of the z_{bj}z_{ja} are needed; see [26].

Alternatively, we can follow the approach of [13, Section 5] to construct first a homomorphism from Y^{+}(2) to the centralizer C_{n} of 𝔤_{n−1} in U(𝔤_{n}) and then calculate the images of the centralizer elements in the algebra Z(𝔤_{n}, 𝔤_{n−1}). The calculation is similar to that in the symplectic case [13]; see also [14]. We shall only give a few key formulas here. Introduce the (2n + 1) × (2n + 1)-matrix F = (F_{ij}) whose ijth entry is the element F_{ij} ∈ 𝔤_{n} and set

F (u) = 1 + \frac{F}{u + 1/2}.

Denote by F̂(u) the corresponding Sklyanin comatrix; see [12, Section 2]. The mapping

s_{ab}(u) ↦ c(u) F̂(−u + n − 1/2)_{ab},  a, b ∈ {−n, n},  (3.19)

where c(u) = (1 − u^{−2})(1 − 4 u^{−2}) ⋯ (1 − (n − 1)^{2} u^{−2}), defines an algebra homomorphism Y^{+}(2) → C_{n} [12, Proposition 2.1]; cf. [20]. Its composition with the natural homomorphism C_{n} → Z(𝔤_{n}, 𝔤_{n−1}) gives (3.18).

As it follows from the branching rule (3.2), the space V (λ)_{μ}^{+} is nonzero only if there exists ν such that the inequalities (3.3) hold. We shall be assuming that this condition is satisfied. Proposition 3.2 allows one to equip V (λ)_{μ}^{+} with a structure of a Y^{+}(2)-module defined via the homomorphism (3.18). The next theorem provides an identification of this module.

Theorem 3.3 The Y^{+}(2)-module V (λ)_{μ}^{+} is isomorphic to the direct sum of two irreducible submodules, V (λ)_{μ}^{+} ≃ U ⊕ U′, where

U = L(0, β_{1}) ⊗ L(α_{2}, β_{2}) ⊗ ⋯ ⊗ L(α_{n}, β_{n}) ⊗ W (1/2),  (3.20)
U′ = L(−1, β_{1}) ⊗ L(α_{2}, β_{2}) ⊗ ⋯ ⊗ L(α_{n}, β_{n}) ⊗ W (1/2),  (3.21)

if the λ_{i} are integers (it is supposed that U′ = {0} if β_{1} = 0); or

U = L(−1/2, β_{1}) ⊗ L(α_{2}, β_{2}) ⊗ ⋯ ⊗ L(α_{n}, β_{n}) ⊗ W (0),  (3.22)
U′ = L(−1/2, β_{1}) ⊗ L(α_{2}, β_{2}) ⊗ ⋯ ⊗ L(α_{n}, β_{n}) ⊗ W (1),  (3.23)

if the λ_{i} are half-integers, and the following notation is used

α_{i} = min{λ_{i−1}, μ_{i−1}} − i + 1,  i = 2, …, n,
β_{i} = max{λ_{i}, μ_{i}} − i + 1,  i = 1, …, n,

with μ_{n} := −∞. In particular, each of U and U′ (and hence V (λ)_{μ}^{+}) is equipped with an action of Y(2) defined by (3.16).

Proof. Consider the following two vectors in V (λ)_{μ}^{+}

ξ_{μ} = ∏_{i=1}^{n−1} (z_{ni}^{max{λ_{i},μ_{i}}−μ_{i}}z_{i,−n}^{max{λ_{i},μ_{i}}−λ_{i}}) ξ,  ξ′_{μ} = z_{n0} ξ_{μ}.  (3.24)

Repeating the arguments of the proof of Theorem 5.2 in [13] we can show that both ξ_{μ} and ξ′_{μ} are eigenvectors for s_{nn}(u) and are annihilated by s_{−n,n}(u). Namely,

s_{nn}(u)ξ_{μ} = μ(u)ξ_{μ},  s_{nn}(u)ξ′_{μ} = (1 + u^{−1})μ(u)ξ′_{μ},  (3.25)

where

μ(u) = (1 − α_{2}u^{−1}) ⋯ (1 − α_{n}u^{−1})(1 + β_{1}u^{−1}) ⋯ (1 + β_{n}u^{−1}).

This is proved simultaneously with the following relations by induction on the degree of the monomial in (3.24): for i = 1, …, n − 1

z_{in} ξ_{μ} = −(m_{i} + α_{1}) ⋯ (m_{i} + α_{n})(m_{i} − β_{1}) ⋯ (m_{i} − β_{n}) ξ_{μ+δ_{i}},  (3.26)

and

z_{−ni} ξ_{μ} = −(m_{i} − α_{1} − 1) ⋯ (m_{i} − α_{n} − 1)(m_{i} + β_{1} − 1) ⋯ (m_{i} + β_{n} − 1) ξ_{μ−δ_{i}}, (3.27)

where α_{1} = 0 and m_{i} = μ_{i} − i + 1 for i = 1, …, n − 1. Indeed, we note that if μ_{i} ≥ λ_{i} then z_{in} ξ_{μ} = 0 which is implied by (3.7). This agrees with (3.26) because in this case β_{i} = m_{i}. Now assume that μ_{i} < λ_{i}. We have z_{in} ξ_{μ} = z_{in}z_{ni} ξ_{μ+δ_{i}} by (3.7). Formula (3.17) gives z_{in}z_{ni} = z_{−n,−i}z_{−i,−n} = Z_{−n,−n}(−g_{−i}). Further,

Z_{−n,−n}(−g_{−i}) ξ_{μ+δ_{i}} = Z_{−n,−n}(m_{i}) ξ_{μ+δ_{i}}.

By Proposition 3.2 and the symmetry relation (3.13) we can write

Z_{−n,−n}(m_{i}) = \frac{2m_{i} − 1}{2m_{i}}Z_{nn}(−m_{i}) + \frac{1}{2m_{i}} Z_{nn}(m_{i}).

By induction, Z_{nn}(u) ξ_{μ+δ_{i}} can be found from (3.18) and (3.25) which gives (3.26). The proof of (3.27) is very similar. To prove (3.25) we apply the induction hypotheses to (3.26) and (3.27) and also use the relation

z_{0n} ξ_{μ} = 0  (3.28)

which is a consequence of (3.7). The relations

Z_{−n,n}(u) ξ_{μ} = 0,  Z_{−n,n}(u) ξ′_{μ} = 0  (3.29)

follow from (3.17), (3.26) and (3.27).

Both vectors ξ_{μ} and ξ′_{μ} are nonzero, except for the case β_{1} = 0 where ξ′_{μ} = 0. Indeed, applying appropriate operators z_{in} to ξ_{μ} or ξ′_{μ} repeatedly, we can obtain the highest vector ξ of V (λ) with a nonzero coefficient. It follows from [12, Corollary 6.6] that the tensor products (3.20)–(3.23) are irreducible representations of Y^{+}(2). An easy calculation shows that the highest weights of the Y^{+}(2)-modules U and U′ respectively coincide with the Y^{+}(2)-weights of the vectors ξ_{μ} and ξ′_{μ}. So, U and U′ are respectively isomorphic to quotients of the Y^{+}(2)-submodules in V (λ)_{μ}^{+} generated by ξ_{μ} and ξ′_{μ}. On the other hand, the branching rule (3.2) implies that

dim V (λ)_{μ}^{+} = dim U + dim U′.

Therefore, to complete the proof of the theorem we need to show that the Y^{+}(2)- submodules generated by ξ_{μ} and ξ′_{μ} are disjoint. For this we employ a contravariant bilinear form ⟨ , ⟩ on V (λ) uniquely determined by the conditions:

⟨ξ, ξ⟩ = 1, ⟨F_{ij} η, ζ⟩ = ⟨η, F_{ji} ζ⟩, η, ζ ∈ V (λ),

cf. (2.16). One easily shows that its restriction to the subspace V (λ)_{μ}^{+} is nondegenerate. Therefore, our claim will follow from the fact that the submodules generated by ξ_{μ} and ξ′_{μ} are orthogonal to each other with respect to ⟨ , ⟩. Given an operator A in V (λ)^{+} we denote by A^{∗} its adjoint operator with respect to the form:

⟨A η, ζ⟩ = ⟨η, A^{∗} ζ⟩.

Since the extremal projection p is stable with respect to the anti-involution F_{ij} ↦ F_{ji} [26] we derive that (pF_{ia})^{∗} = pF_{ai} for a = −n, n and i = −n+1, …, n−1. Therefore, z_{ia}^{∗} = z_{ai} · c where c is an element of R(𝔥_{n−1}) which can be found from (3.5) and (3.6). This also implies that (z_{ai}z_{ib})^{∗} = z_{bi}z_{ia} and hence

Z_{ab}(u)^{∗} = Z_{ba}(u),  (3.30)

see (3.17). By Proposition 3.2 and the Poincaré–Birkhoff–Witt theorem for the twisted Yangians [16, Remark 3.14], every element of the Y^{+}(2)-submodules generated by ξ_{μ} and ξ′_{μ} can be written as a linear combination of vectors of the following form, respectively:

Z_{n,−n}(u_{1}) ⋯ Z_{n,−n}(u_{k}) ξ_{μ} or Z_{n,−n}(v_{1}) ⋯ Z_{n,−n}(v_{l}) ξ′_{μ},

where the u_{i} and v_{i} are complex parameters. Therefore, by (3.29) and (3.30) to prove that the submodules are orthogonal it now suffices to show that ⟨ξ_{μ}, ξ′_{μ}⟩ = 0. But this follows from (3.28).

Remark. Using Weyl’s formula for the dimension of V (λ) one can slightly modify the proof of Theorem 3.3 so that the branching rule (3.2) would not be used but follow from the theorem; cf. [13, 14].

It follows from (3.13) that the series s_{n,−n}(u) is even in u, and so is the polynomial Z_{n,−n}(u); see Proposition 3.2. On the other hand, (3.17) implies that Z_{n,−n}(−g_{i}) = z_{ni}z_{i,−n} for i = 1, …, n − 1. Moreover, Z_{n,−n}(−g_{n}) = z_{n,−n} which follows from (3.8). Since Z_{n,−n}(u) is a polynomial in u^{2} of degree n − 1, by the Lagrange interpolation formula Z_{n,−n}(u) can also be given by

Z_{n,−n}(u) = ∑_{i=1}^{n} z_{ni}z_{i,−n} ∏_{j=1, j≠i}^{n} \frac{u^{2} − g_{j}^{2}}{g_{i}^{2} − g_{j}^{2}} .  (3.31)

Remark. To make the above evaluation Z_{n,−n}(−g_{i}) well-defined we agree to consider the series Z_{ab}(u) with a, b ∈ {−n, n} as elements of the right module over the field of rational functions in g_{1}, …, g_{n}, u generated by monomials in the z_{ia}.

Given ν such that the conditions (3.3) are satisfied, set

γ_{i} = ν_{i} − i + 1,  l_{i} = λ_{i} − i + 1,  i ≥ 1

and introduce the vectors

ξ_{νμ} = \begin{cases}∏_{i=1}^{n} Z_{n,−n}(γ_{i} − 1) ⋯ Z_{n,−n}(β_{i} + 1)Z_{n,−n}(β_{i}) ξ_{μ} & if σ = 0, \\ ∏_{i=1}^{n} Z_{n,−n}(γ_{i} − 1) ⋯ Z_{n,−n}(β_{i} + 1)Z_{n,−n}(β_{i}) ξ′_{μ} & if σ = 1. \end{cases}

Using (3.31) and (3.7) we get an equivalent expression; cf. [13, Section 6]:

ξ_{νμ} = z_{n0}^{σ} ∏_{i=1}^{n−1} z_{ni}^{ν_{i}−μ_{i}}z_{i,−n}^{ν_{i}−λ_{i}} · ∏_{k=l_{n}}^{γ_{n}−1} Z_{n,−n}(k) ξ.

Proposition 3.4 The vectors ξ_{νμ} with ν satisfying (3.3) form a basis of V (λ)_{μ}^{+}.

Proof. Due to Theorem 3.3 it suffices to show that the vectors ξ_{νμ} with σ = 0 form a basis of the subspace U of V (λ)_{μ}^{+} while those with σ = 1 form a basis in U′. Let us write each of the tensor products in (3.20)–(3.23) in the form

L(α_{1}, β_{1}) ⊗ L(α_{2}, β_{2}) ⊗ ⋯ ⊗ L(α_{n}, β_{n}) ⊗ W (−α_{0}).  (3.32)

Regarding this as a Y(2)-module defined by (3.16) we can construct a Gelfand– Tsetlin-type basis in this module as follows. Set

ζ̃_{νμ} = ∏_{i=1}^{n} T_{n,−n}(−γ_{i} + 1) ⋯ T_{n,−n}(−β_{i} − 1)T_{n,−n}(−β_{i}) z_{n0}^{σ} ξ_{μ}.

The vectors ζ̃_{νμ} with ν satisfying (3.3) form a basis in the Y(2)-module (3.32); see [22, 11, 19]. Furthermore, we have

\begin{aligned}T_{nn}(u) ζ̃_{νμ} & = (u + γ_{1}) ⋯ (u + γ_{n}) ζ̃_{νμ}, \\ T_{n,−n}(−γ_{i}) ζ̃_{νμ} & = ζ̃_{ν+δ_{i},μ}. \end{aligned}  (3.33)

We have the following equality of operators in the space (3.32):

Z_{n,−n}(u) = \frac{(u − α_{0})T_{n,−n}(−u)T_{nn}(u) + (u + α_{0})T_{n,−n}(u)T_{nn}(−u)}{(−1)^{n+1} u}

which is easily derived from (3.10), (3.15) and (3.18). Therefore, by (3.33)

Z_{n,−n}(γ_{i}) ζ̃_{νμ} = −2(α_{0} − γ_{i}) ∏_{a=1, a≠i}^{n}(−γ_{a} − γ_{i}) ζ̃_{ν+δ_{i},μ}.

This shows that for each ν the vectors ξ_{νμ} and ζ̃_{νμ} coincide up to a nonzero factor.

We shall use the following normalized basis vectors

ζ_{νμ} = ∏_{1≤i<j≤n}(−γ_{i} − γ_{j})! ξ_{νμ}.

The following formulas for the action of the generators of the Yangian Y(2) in the basis {ζ_{νμ}} follow from the above proof: for i = 1, …, n

T_{nn}(u) ζ_{νμ} = (u + γ_{1}) ⋯ (u + γ_{n}) ζ_{νμ},
T_{n,−n}(−γ_{i}) ζ_{νμ} = \frac{1}{2(γ_{i} − α_{0})} ζ_{ν+δ_{i},μ},  (3.34)
T_{−n,n}(−γ_{i}) ζ_{νμ} = 2 ∏_{k=0}^{n}(α_{k} − γ_{i} + 1) ∏_{k=1}^{n}(β_{k} − γ_{i}) ζ_{ν−δ_{i},μ};

cf. [13] and [14].

Given a pattern Λ (see Section 3.1) introduce the vector

ξ_{Λ} = ∏_{k=1,...,n}^{→} (z_{k0}^{σ_{k}} · ∏_{i=1}^{k−1} z_{ki}^{λ′_{ki}−λ_{k−1,i}}z_{i,−k}^{λ′_{ki}−λ_{ki}} · ∏_{q=l_{kk}}^{l′_{kk}−1} Z_{k,−k}(q + \frac{1}{2})) ξ

and set

ζ_{Λ} = N_{Λ} ξ_{Λ}, N_{Λ} = ∏_{k=2}^{n} ∏_{1≤i<j≤k}(−l′_{ki} − l′_{kj} − 1)!

The following proposition is implied by the branching rule (3.2) and Proposition 3.4.

Proposition 3.5 The vectors ζ_{Λ} parametrized by the patterns Λ form a basis of the representation V (λ).

3.5 Matrix element formulas

Introduce the following elements of U(𝔤_{n}):

Φ_{−k,k} = ∑_{i=1}^{k−1} F_{−k,i}F_{ik} − \frac{1}{2} F_{0k}^{2}, k = 1, …, n.

We shall find the action of Φ_{−k,k} in the basis {ζ_{Λ}}, which will be used later on. Since Φ_{−k,k} commutes with the subalgebra 𝔤_{k−1} it suffices to consider the case k = n. The image of 2Φ_{−n,n} under the natural homomorphism π : C_{n} → Z(𝔤_{n}, 𝔤_{n−1}) coincides with the coefficient at u^{2n−2} of the polynomial Z_{−n,n}(u); see the proof of Proposition 3.2. The following equality of operators in (3.32) is obtained from (3.10), (3.15) and (3.18):

Z_{−n,n}(u) = \frac{(u − α_{0})T_{−n,−n}(−u)T_{−n,n}(u) + (u + α_{0})T_{−n,−n}(u)T_{−n,n}(−u)}{(−1)^{n+1} u}.

Therefore,

Φ_{−n,n} = −t_{−n,n}^{(2)} + t_{−n,n}^{(1)}t_{−n,−n}^{(1)} + (1 + α_{0}) t_{−n,n}^{(1)}.  (3.35)

The image of s_{nn}^{(1)} under the homomorphism (3.19) is F_{nn}. On the other hand, by (3.15) we have

s_{nn}^{(1)} = t_{nn}^{(1)} − t_{−n,−n}^{(1)} − α_{0} − 1/2,

as operators in the space (3.32). Therefore, (3.35) can be written as

Φ_{−n,n} = −t_{−n,n}^{(2)} + t_{−n,n}^{(1)}t_{nn}^{(1)} − (F_{nn} + 3/2) t_{−n,n}^{(1)}.

Finally, relations (3.34) imply that

Φ_{−n,n}ζ_{νμ} = ∑_{i=1}^{n} θ_{i} (F_{nn} − γ_{i} + 3/2) ζ_{ν−δ_{i},μ},  (3.36)

where

θ_{i} = −2 ∏_{k=0}^{n}(α_{k} − γ_{i} + 1) ∏_{k=1}^{n}(β_{k} − γ_{i}) ∏_{j=1, j≠i}^{n}(γ_{j} − γ_{i})^{−1}.

Using the notation in (3.32) we can also write this as

θ_{i} = (2γ_{i} − 1)(1 − σ − γ_{i}) ∏_{k=1}^{n}(l_{k} − γ_{i}) ∏_{k=1}^{n−1}(m_{k} − γ_{i}) ∏_{j=1, j≠i}^{n}(γ_{j} − γ_{i})^{−1}.

The action of F_{nn} in V (λ)_{μ}^{+} is immediately found so that

F_{nn} ξ_{νμ} = (σ + 2 ∑_{i=1}^{n} ν_{i} − ∑_{i=1}^{n} λ_{i} − ∑_{i=1}^{n−1} μ_{i}) ξ_{νμ}.

The operator F_{n−1,−n} preserves the subspace of 𝔤_{n−2}-highest vectors in V (λ). Therefore it suffices to calculate its action on the basis vectors of the form

ξ_{νμν′} = X_{μν′} ξ_{νμ},

where X_{μν′} denotes the operator

X_{μν′} = z_{n−1,0}^{σ′} ∏_{i=1}^{n−2} z_{n−1,i}^{ν′_{i}−μ′_{i}} z_{i,−n+1}^{ν′_{i}−μ_{i}} · ∏_{a=m_{n−1}}^{γ′_{n−1}−1} Z_{n−1,−n+1}(a).

Here we assume that the conditions (3.3) are satisfied with λ, ν, μ respectively replaced by μ, ν′, μ′; we have used the notation γ′_{i} = ν′_{i} − i + 1. The operator F_{n−1,−n} is permutable with the elements z_{n−1,i}, z_{i,−n+1} and Z_{n−1,−n+1}(u) which follows from their explicit formulas. Hence, we can write

F_{n−1,−n} ξ_{νμν′} = X_{μν′} F_{n−1,−n} ξ_{νμ}.

Let us apply (3.9) with a = −n. We have

f_{i} ξ_{νμ} = (m_{i} − 1/2) ξ_{νμ}, f_{−i} ξ_{νμ} = (−m_{i} + 1/2) ξ_{νμ}

for i ≥ 1. Recall also that f_{0} = −1/2. We now need to express

X_{μν′} z_{n−1,i}z_{i,−n} ξ_{νμ}, i = −n + 1, …, n − 1

as a linear combination of the vectors ξ_{νμν′}. Suppose first that i ≥ 1. Assuming that ν_{i} − μ_{i} ≥ 1 we obtain from (3.7)

z_{i,−n} ξ_{νμ} = z_{i,−n}z_{ni} ξ_{ν,μ+δ_{i}}.  (3.37)

By (3.17) this equals

z_{i,−n}z_{ni} ξ_{ν,μ+δ_{i}} = z_{n,−i}z_{−i,−n} ξ_{ν,μ+δ_{i}} = Z_{n,−n}(−g_{−i}) ξ_{ν,μ+δ_{i}}.

We have −g_{−i} ξ_{ν,μ+δ_{i}} = m_{i} ξ_{ν,μ+δ_{i}}. Since Z_{n,−n}(γ_{p}) ξ_{ν,μ+δ_{i}} = ξ_{ν+δ_{p},μ+δ_{i}} for each p = 1, …, n, we obtain from the Lagrange interpolation formula that (3.37) takes the form

Z_{n,−n}(m_{i}) ξ_{ν,μ+δ_{i}} = ∑_{p=1}^{n} ∏_{a=1, a≠p}^{n} \frac{m_{i}^{2} − γ_{a}^{2}}{γ_{p}^{2} − γ_{a}^{2}} ξ_{ν+δ_{p},μ+δ_{i}}.

Furthermore, for i ≥ 1

z_{−i,−n} ξ_{νμ} = (−1)^{n−i} z_{ni} ξ_{νμ} = (−1)^{n−i} ξ_{ν,μ−δ_{i}}.

Consider now the vector z_{0,−n} ξ_{νμ}. If σ = 0 then it equals

z_{0,−n} ξ_{νμ} = (−1)^{n} z_{n0} ξ_{νμ} = (−1)^{n} ξ_{\overline{ν}μ},

where \overline{ν} = (σ + 1, ν_{1}, …, ν_{n}) (addition modulo 2). If σ = 1 then

z_{0,−n} ξ_{νμ} = z_{n0}z_{0,−n} ξ_{\overline{ν}μ},

which coincides with Z_{n,−n}(−g_{0}) ξ_{\overline{ν}μ}, where g_{0} = 0. Using again the Lagrange interpolation formula we find that this equals

Z_{n,−n}(m_{0}) ξ_{\overline{ν}μ} = ∑_{p=1}^{n} ∏_{a=1, a≠p}^{n} \frac{m_{0}^{2} − γ_{a}^{2}}{γ_{p}^{2} − γ_{a}^{2}} ξ_{\overline{ν}+δ_{p},μ}

with m_{0} = 0. The operator X_{μν′} z_{n−1,i} is transformed in exactly the same manner; cf. [13]. Combining the results we obtain

F_{n−1,−n} ξ_{νμν′} = A_{0} ξ(0) + ∑_{i=1}^{n−1} A_{i} (\frac{1}{m_{i}} ξ^{+}(i) − \frac{1}{m_{i} − 1} ξ^{−}(i)) ,

where

A_{i} = ∏_{a=1, a≠i}^{n−1} \frac{1}{m_{i} − m_{a}} ∏_{a=1}^{n−1} \frac{1}{m_{i} + m_{a} − 1}.

Furthermore,

ξ^{−}(i) = ξ_{ν,μ−δ_{i},ν′},
ξ^{+}(i) = ∑_{p=1}^{n} ∑_{q=1}^{n−1} ∏_{a=1, a≠p}^{n} \frac{m_{i}^{2} − γ_{a}^{2}}{γ_{p}^{2} − γ_{a}^{2}} ∏_{a=1, a≠q}^{n−1} \frac{m_{i}^{2} − γ′_{a}^{2}}{γ′_{q}^{2} − γ′_{a}^{2}} ξ_{ν+δ_{p},μ+δ_{i},ν′+δ_{q}},

and

ξ(0) = (−1)^{n}ξ_{\overline{ν}μ\overline{ν}′}  if σ = σ′ = 0,
= ∑_{p=1}^{n} ∏_{a=1, a≠p}^{n} \frac{m_{0}^{2} − γ_{a}^{2}}{γ_{p}^{2} − γ_{a}^{2}} ξ_{\overline{ν}+δ_{p},μ,\overline{ν}′}  if σ = 1, σ′ = 0,
= − ∑_{q=1}^{n−1} ∏_{a=1, a≠q}^{n−1} \frac{m_{0}^{2} − γ′_{a}^{2}}{γ′_{q}^{2} − γ′_{a}^{2}} ξ_{\overline{ν},μ,\overline{ν}′+δ_{q}}  if σ = 0, σ′ = 1,
= (−1)^{n−1} ∑_{p=1}^{n} ∑_{q=1}^{n−1} ∏_{a=1, a≠p}^{n} \frac{m_{0}^{2} − γ_{a}^{2}}{γ_{p}^{2} − γ_{a}^{2}} ∏_{a=1, a≠q}^{n−1} \frac{m_{0}^{2} − γ′_{a}^{2}}{γ′_{q}^{2} − γ′_{a}^{2}} ξ_{\overline{ν}+δ_{p},μ,\overline{ν}′+δ_{q}}
if σ = σ′ = 1,

with \overline{ν}′ = (σ′ + 1, ν′_{1}, …, ν′_{n−1}) (addition modulo 2). We now compute the action of F_{n−1,n}. In the formula (3.9) with a = n replace the operators z_{in} by with following expression: for i = −n + 1, …, n − 1

z_{in} = [z_{i,−n}, Φ_{−n,n}] \frac{1}{f_{i} + F_{nn}}  (3.38)

and then use the formulas for the action of z_{i,−n} and Φ_{−n,n}; see (3.36). More precisely, we regard (3.38) as a relation in the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}) which can be proved as follows. First, we calculate the commutator [F_{i,−n}, Φ_{−n,n}] in U(𝔤_{n}) then consider it modulo the ideal J and apply the extremal projection p (see Section 2).

We have Φ_{−n,n}F_{nn} = (F_{nn} + 2) Φ_{−n,n} and so, (3.9) and (3.38) imply that

F_{n−1,n} ξ_{νμν′} = X_{μν′} (Φ_{n−1,−n}(2) Φ_{−n,n} − Φ_{−n,n}Φ_{n−1,−n}(0)) ξ_{νμ},  (3.39)

where

Φ_{n−1,−n}(u) = ∑_{i=−n+1}^{n−1} z_{n−1,i} z_{i,−n} ∏_{a=−n+1, a≠i}^{n−1} \frac{1}{f_{i} − f_{a}} · \frac{1}{u + f_{i} + F_{nn}} .  (3.40)

The action of Φ_{n−1,−n}(u) is found exactly as that of F_{n−1,−n}. Formula (3.39) is valid provided the denominators in (3.40) do not vanish. However, since (3.38) holds in the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}), the relation (3.39) holds for generic parameters ν, μ and ν′ which allows one to get explicit formulas for all matrix elements of F_{n−1,n}.

Finally, the proof of Theorem 3.1 is completed by rewriting the above formulas for the action of the generators in terms of the parameters σ_{k}, l_{ki} and l′_{ki} of the patterns Λ. The parameters l_{i}, γ_{i}, m_{i} are replaced by

l_{i} ↦ l_{ki} + 1/2,  γ_{i} ↦ l′_{ki} + 1/2,  m_{i} ↦ l_{k−1,i} + 1/2.

Remark. There is another way of calculating the matrix elements of F_{k−1,k} based on the formulas for the action of T_{−n,−n}(u) in the basis {ζ_{νμ}}; see [14].

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