A weight basis for representations of even orthogonal Lie algebras

A. I. Molev

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

Abstract

A weight basis for each finite-dimensional irreducible representation of the orthogonal Lie algebra 𝔬(2n) is constructed. The basis vectors are parametrized by the D-type Gelfand–Tsetlin patterns. The basis is consistent with the chain of subalgebras 𝔤_{1} ⊂ ⋯ ⊂ 𝔤_{n}, where 𝔤_{k} = 𝔬(2k). Explicit formulas for the matrix elements of generators of 𝔬(2n) in this basis are given. The construction is based on the representation theory of the Yangians and extends our previous results for the symplectic Lie algebras.

1 Introduction

In their pioneering works [3] and [4] Gelfand and Tsetlin proposed a combinatorial method to explicitly construct representations of the classical Lie algebras. For each finite-dimensional irreducible representation of the general linear Lie algebra 𝔤𝔩(N) and the orthogonal Lie algebra 𝔬(N) they gave a parametrization of basis vectors and provided explicit formulas for the matrix elements of generators of the Lie algebras in the basis. Derivations of the matrix element formulas in the orthogonal case are given in [14, 16]; see also [5]. A number of different approaches to the problem of constructing representation bases for simple Lie algebras has been developed; see [9] for more references. Note also recent results by Donnelly [2] and Littelmann [6]. In [2] explicit combinatorial constructions of the fundamental representations of the B and C series Lie algebras and of their q-analogs are given; in [6] monomial bases parametrized by patterns of Gelfand–Tsetlin type are constructed for all simple complex Lie algebras. In [9] an analog of the Gelfand–Tsetlin basis for the symplectic Lie algebras 𝔰𝔭(2n) is constructed and explicit formulas for the matrix elements of generators of 𝔰𝔭(2n) in this basis are given. Bases for the finite-dimensional irreducible representations of the classical Lie algebras of B, C, and D series can be constructed in a uniform manner with the use of the representation theory of the Yangians, as in [9]. Here we extend the results of [9] to the case of the D series and hope to treat the remaining B series case in a forthcoming publication.

Our basis for 𝔬(2n) is different from that of Gelfand and Tsetlin [4]. Their basis is consistent with the chain of subalgebras

𝔬(2) ⊂ 𝔬(3) ⊂ ⋯ ⊂ 𝔬(N).

The reductions 𝔬(k) ↓ 𝔬(k − 1) are multiplicity-free which makes the basis orthogonal with respect to a natural contravariant bilinear form. However, the basis vectors are not weight vectors with respect to the Cartan subalgebra of 𝔬(N). To get a weight (although non-orthogonal) basis we consider the following chain instead:

𝔬(2) ⊂ 𝔬(4) ⊂ ⋯ ⊂ 𝔬(2n)

so that all the subalgebras belong to the D series.

The reduction 𝔬(2n) ↓ 𝔬(2n − 2) is not multiplicity free. This means that the subspace V (λ)_{μ}^{+} of 𝔬(2n−2)-highest vectors of a weight μ in an 𝔬(2n)-module V (λ) is not necessarily one-dimensional. However, this space turns out to possess a natural structure of an irreducible representation of a large associative algebra Y^{+}(2) called the twisted Yangian (introduced by Olshanski in [13]) and can also be equipped with an action of the 𝔤𝔩(2)-Yangian Y(2). This allows us to construct a Yangian Gelfand– Tsetlin basis in V (λ)_{μ}^{+} associated with an inclusion Y(1) ⊂ Y(2); see [7, 11, 12].

Our calculations are based on the relationship between the twisted Yangian Y^{+}(2) and the transvector algebra Z(𝔤_{n}, 𝔤_{n−1}), 𝔤_{n} = 𝔬(2n). The transvector algebras (they are sometimes called the Mickelsson algebras or S-algebras) are studied in detail in [17, 18].

Although the constructions of the bases are very similar for the orthogonal and symplectic cases, there is a slight difference in the calculation of the matrix elements of generators of the Lie algebras in the basis. The Lie algebra 𝔰𝔭(2n) contains the “second diagonal” generators F_{−k,k} = 2E_{−k,k} where the E_{ij} denote the standard generators of 𝔤𝔩(2n) (see Section 2 below). The action of these elements in the basis is rather simple and can be easily found. However, their counterparts do not exist in the orthogonal case. Instead, there are second degree elements Φ_{−k,k} of the universal enveloping algebra U(𝔤_{n}) which belong to the centralizer of 𝔤_{k−1} in U(𝔤_{k}) and play the role similar to that of the elements F_{−k,k} in the symplectic case.

2 Notations and preliminary results

We shall enumerate the rows and columns of 2n × 2n-matrices over ℂ by the indices −n, …, −1, 1, …, n. We let the E_{ij}, i, j = −n, …, n denote the standard basis of the Lie algebra 𝔤𝔩(2n). We shall also assume throughout the paper that the index 0 is skipped in a sum or in a product. Introduce the elements

F_{ij} = E_{ij} − E_{−j,−i}.  (2.1)

We have F_{−j,−i} = −F_{ij}. In particular, F_{−i,i} = 0 for all i. The orthogonal Lie algebra 𝔤_{n} := 𝔬(2n) can be identified with the subalgebra in 𝔤𝔩(2n) spanned by the elements F_{ij}, i, j = −n, …, n.

The subalgebra 𝔤_{n−1} is spanned by the elements (2.1) 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:

−|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}.

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. Set

Z(𝔤_{n}, 𝔤_{n−1}) = {x ∈ U′(𝔤_{n})/J | F_{ij} x ≡ 0, −n < i < j < n}.

Then Z(𝔤_{n}, 𝔤_{n−1}) is an algebra with the multiplication inherited from U′(𝔤_{n}). We call it the transvector algebra; see [17, 18] for further details.

Set

f_{i} = F_{ii} − i + 1, f_{−i} = −f_{i}

for i = 1, …, n. Let p denote the extremal projection for the Lie algebra 𝔤_{n−1}; see [1, 18]. 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},  a = −n, n,  i = −n + 1, …, n − 1

are generators of Z(𝔤_{n}, 𝔤_{n−1}) [18]. They 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, … (it is assumed that index 0 is excluded in the sum). We shall use the normalized generators of Z(𝔤_{n}, 𝔤_{n−1}) defined by

z_{ia} = pF_{ia}(f_{i} − f_{i−1}) ⋯ { \atop (f_{i} − f_{−i})} ⋯ (f_{i} − f_{−n+1}),  (2.2)
z_{ai} = pF_{ai}(f_{i} − f_{i+1}) ⋯ { \atop (f_{i} − f_{−i})} ⋯ (f_{i} − f_{n−1}),

where the hats indicate the factors to be omitted if they occur. We obviously have z_{ai} = (−1)^{n−i} z_{−i,−a}. The following equivalent formula holds for z_{ai}:

z_{ai} = (f_{i} − f_{i+1}) ⋯ { \atop (f_{i} − f_{−i})} ⋯ (f_{i} − f_{n−1})

× ∑_{n>i_{1}>···>i_{s}>i} \frac{1}{(f_{i} − f_{i_{1}}) ⋯ (f_{i} − f_{i_{s}})}F_{ai_{1}}F_{i_{1}i_{2}} ⋯ F_{i_{s−1}i_{s}}F_{i_{s}i}. (2.3)

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.

Note the following relations between these operators; cf. [18]. 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}.

In particular, z_{ai} and z_{aj} commute for i + j ≠ 0. One easily verifies that z_{ai} and z_{bi} also commute for all a, b. We shall use the following element which can be checked to belong to 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} \frac{(f_{n} − f_{j_{1}}) ⋯ (f_{n} − f_{j_{k}})}{2f_{n}},  (2.4)

where s = 1, 2, … and {j_{1}, …, j_{k}} is the complement to the subset {i_{1}, …, i_{s}} in {−n + 1, …, n − 1}. There is an equivalent formula for z_{n,−n} which can either be proved directly (cf. [9, Section 2]), or can be deduced from (4.6) below (use the fact that Z_{n,−n}(g_{n}) = Z_{n,−n}(−g_{n})):

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

This formula together with (2.3) is used in the derivation of the following relations from (2.2) (cf. [9, Proposition 2.1]): for a = −n, n

F_{n−1,a} = ∑_{i=−n+1}^{n−1} z_{n−1,i} z_{ia} ∏_{j=−n+1, j≠±i}^{n−1} \frac{1}{f_{i} − f_{j}} ,  (2.5)

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

Let us now introduce the 𝔤𝔩(2)-Yangian Y(2) and the (orthogonal) twisted Yangian Y^{+}(2); see [10] 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)),  (2.6)

where

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

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}. Note that Y^{+}(2) can be equivalently defined as an abstract algebra with these generators and certain linear and quadratic defining relations; see [10, Section 3].

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).  (2.7)

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).  (2.8)

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. 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}.  (2.9)

The coproduct (2.7) allows one to construct representations of Y(2) of the form

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

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 (2.8) 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) [8]. The vector space isomorphism

L ⊗ W (γ) → L,  v ⊗ w ↦ v,  v ∈ L  (2.10)

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

3 Construction of the basis

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

Z_{ab}(u) = ( − (δ_{ab}(u − n + 3/2) + F_{ab}) ∏_{i=−n+1}^{n−1}(u + g_{i})

+ ∑_{i=−n+1}^{n−1} z_{ai}z_{ib} (u + g_{−i}) ∏_{j=−n+1, j≠±i}^{n−1} \frac{u + g_{j}}{g_{i} − g_{j}} )\frac{1}{2u + 1}, (3.1)

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

As we shall see below (Corollary 3.3) the space V (λ)_{μ}^{+} is nonzero only if there exist ν_{1}, …, ν_{n−1} such that the inequalities (3.10) hold. We shall be assuming that this condition is satisfied.

Proposition 3.1 (i) The mapping

s_{ab}(u) ↦ −2u^{−2n+2} Z_{ab}(u),  a, b ∈ {−n, n}  (3.2)

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

(ii) The representation of Y^{+}(2) in the space V (λ)_{μ}^{+} defined via the homomorphism (3.2) is irreducible.

Proof. We use the same arguments as for the proof of the corresponding statements in the symplectic case; see [9, Section 5]. So we shall only give a few key formulas; the details can be restored by using [9].

Introduce the 2n × 2n-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 [8, Section 2]. The mapping

s_{ab}(u) ↦ c(u) F̂(−u + n − 1)_{ab},  a, b ∈ {−n, n},  (3.3)

where

c(u) = ∏_{k=1}^{n−1}(1 − (k − 1/2)^{2} u^{−2}),

defines an algebra homomorphism from Y^{+}(2) to the centralizer C_{n} of 𝔤_{n−1} in U(𝔤_{n}) [8, Proposition 2.1]; cf. [13]. Further, a slight generalization of [9, Proposition 3.1] implies the following expression for the ab-entries of the matrix F̂(u + 1/2):

F̂(u + 1/2)_{ab} = (δ_{ab} − ∑_{k=1}^{∞} F_{ab}^{(k)} u^{−k}) · sdet F^{(n−1)}(u − 1/2).

Here sdet F^{(n−1)}(u) is the Sklyanin determinant of the matrix obtained from F (u) by deleting the (±n)th rows and columns [8, Section 2], and

F_{ab}^{(k)} = ∑ F_{ai_{1}}F_{i_{1}i_{2}} ⋯ F_{i_{k−1}b},

summed over the indices i_{m} ∈ {−n + 1, …, n − 1}. Finally, calculating the images of F_{ab}^{(k)} and sdet F^{(n−1)}(u) with respect to the natural homomorphism π : C_{n} → Z(𝔤_{n}, 𝔤_{n−1}) (cf. [9, Section 5]), we find that the composition of π and (3.3) yields (3.2).

The next theorem provides an identification of the Y^{+}(2)-module V (λ)_{μ}^{+}.

Theorem 3.2 We have an isomorphism of Y^{+}(2)-modules

V (λ)_{μ}^{+} ≃ L(α_{1}, β_{1}) ⊗ ⋯ ⊗ L(α_{n−1}, β_{n−1}) ⊗ W (−α_{0})  (3.4)

where α_{1} = min{−|λ_{1}|, −|μ_{1}|} − 1/2, α_{0} = α_{1} + |λ_{1} + μ_{1}|,

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

with μ_{n} := −∞. In particular, V (λ)_{μ}^{+} is equipped with an action of Y(2) defined by (2.10).

Proof. Consider the following vector 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.  (3.5)

Repeating the arguments of the proof of Theorem 5.2 in [9] we show that ξ_{μ} is the highest vector of the Y^{+}(2)-module V (λ)_{μ}^{+}. That is, ξ_{μ} is annihilated by s_{−n,n}(u), and ξ_{μ} is an eigenvector for s_{nn}(u). Namely, s_{nn}(u)ξ_{μ} = μ(u)ξ_{μ}, where the highest weight μ(u) is given by

μ(u) = (1 − α_{0}u^{−1}) ⋯ (1 − α_{n−1}u^{−1})(1 + β_{1}u^{−1}) ⋯ (1 + β_{n−1}u^{−1})(1 + \frac{1}{2}u^{−1})^{−1}. (3.6)

This is proved simultaneously with the following relations by induction on the degree of the monomial in (3.5): for i = 1, …, n − 1

z_{in} ξ_{μ} = −(m_{i} + α̃_{1} + 1) ⋯ { \atop (m_{i} + α_{i} + 1)} ⋯ (m_{i} + α_{n−1} + 1)

× (m_{i} − β_{0} + 1) ⋯ (m_{i} − β_{n−1} + 1) ξ_{μ+δ_{i}},

and

z_{−ni} ξ_{μ} = −(m_{i} − α̃_{1}) ⋯ (m_{i} − α_{n−1})(m_{i} + β_{0}) ⋯ { \atop (m_{i} + β_{i−1})} ⋯ (m_{i} + β_{n−1}) ξ_{μ−δ_{i}},

where we have used the notation

m_{i} = μ_{i} − i + 1/2,  i = 1, …, n − 1,
α̃_{1} = min{λ_{1}, μ_{1}} − 1/2,  β_{0} = max{λ_{1}, μ_{1}} + 1/2,

(note that {α̃_{1}, −β_{0}} = {α_{0}, α_{1}}). On the other hand, it follows from [8, Corollary 6.6] that the tensor product in (3.4) is an irreducible representation of Y^{+}(2). Its highest weight can be easily calculated and is given by the same formula (3.6).

Set T_{ab}(u) = u^{n−1} t_{ab}(u) for a, b ∈ {−n, n}. By (2.7) and (2.9), T_{ab}(u), as an operator in V (λ)_{μ}^{+}, is a polynomial in u:

T_{ab}(u) = δ_{ab}u^{n−1} + t_{ab}^{(1)}u^{n−2} + ⋯ + t_{ab}^{(n−1)}.  (3.7)

By (2.6), (2.8) and (3.2) we have an equality of operators in V (λ)_{μ}^{+}:

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} 2u}.  (3.8)

Therefore, Z_{n,−n}(u) is a polynomial in u^{2} of degree n − 2. On the other hand, we find from (3.1) that Z_{n,−n}(−g_{i}) = z_{ni}z_{i,−n}. Thus, by the Lagrange interpolation formula, Z_{n,−n}(u) can also be given by

Z_{n,−n}(u) = ∑_{i=1}^{n−1} z_{ni}z_{i,−n} ∏_{j=1, j≠i}^{n−1} \frac{u^{2} − g_{j}^{2}}{g_{i}^{2} − g_{j}^{2}} .  (3.9)

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 𝔥 and u generated by monomials in the z_{ia}.

Theorem 3.2 implies that basis vectors of V (λ)_{μ}^{+} can be naturally parametrized by (n − 1)-tuples (ν_{1}, …, ν_{n−1}), where all the entries are simultaneously integers or half-integers together with the λ_{i} and the μ_{i}, and the following inequalities hold:

\begin{aligned}−|λ_{1}| & ≥ ν_{1} ≥ λ_{2} ≥ ν_{2} ≥ λ_{3} ≥ ⋯ ≥ λ_{n−1} ≥ ν_{n−1} ≥ λ_{n}, \\ −|μ_{1}| & ≥ ν_{1} ≥ μ_{2} ≥ ν_{2} ≥ μ_{3} ≥ ⋯ ≥ μ_{n−1} ≥ ν_{n−1}. \end{aligned}  (3.10)

For i ≥ 1 set

γ_{i} = ν_{i} − i + 1/2, l_{i} = λ_{i} − i + 1/2.

Introduce the vectors

ξ_{νμ} = ∏_{i=1}^{n−1} Z_{n,−n}(γ_{i} − 1) ⋯ Z_{n,−n}(β_{i} + 1)Z_{n,−n}(β_{i}) ξ_{μ}.

Using (3.9) we can write an equivalent expression; cf. [9, Section 6]:

ξ_{νμ} = ∏_{i=1}^{n−1} z_{ni}^{ν_{i−1}−μ_{i}}z_{i,−n}^{ν_{i−1}−λ_{i}} · ∏_{k=l_{n}+1}^{γ_{n−1}−1} Z_{n,−n}(k) ξ,  (3.11)

where ν_{0} := max{λ_{1}, μ_{1}}. The vectors ξ_{νμ} with ν satisfying (3.10) form a basis of the space V (λ)_{μ}^{+}; see [9, Proposition 6.1]. We shall use the following normalized basis vectors

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

The generators of the Yangian Y(2) act in the basis {ζ_{νμ}} by the rule: for i = 1, …, n − 1

T_{nn}(u) ζ_{νμ} = (u + γ_{1}) ⋯ (u + γ_{n−1}) ζ_{νμ},
T_{n,−n}(−γ_{i}) ζ_{νμ} = \frac{1}{γ_{i} − α_{0}} ζ_{ν+δ_{i},μ},  (3.12)
T_{−n,n}(−γ_{i}) ζ_{νμ} = ∏_{k=0}^{n−1}(α_{k} − γ_{i} + 1) ∏_{k=1}^{n−1}(β_{k} − γ_{i}) ζ_{ν−δ_{i},μ};

cf. [9, Proposition 4.2]. The action of T_{−n,−n}(u) can be found by using the quantum determinant

d(u) = T_{−n,−n}(u + 1)T_{nn}(u) − T_{n,−n}(u + 1)T_{−n,n}(u)  (3.13)
= T_{−n,−n}(u)T_{nn}(u + 1) − T_{−n,n}(u)T_{n,−n}(u + 1);  (3.14)

see, e.g. [10, Section 2]. The coefficients of the quantum determinant belong to the center of Y(2) and so, d(u) acts in V (λ)_{μ}^{+} as a scalar which can be found by the application of (3.13) to the highest weight vector ξ_{μ}. So, we have

d(u) ζ_{νμ} = (u + α_{1} + 1) ⋯ (u + α_{n−1} + 1)(u + β_{1}) ⋯ (u + β_{n−1}) ζ_{νμ}.

Now, using (3.12) and (3.14) we obtain

T_{−n,−n}(u) ζ_{νμ} = ∏_{i=1}^{n−1} \frac{(u + α_{i} + 1)(u + β_{i})}{u + γ_{i} + 1} ζ_{νμ}

+ ∏_{i=1}^{n−1} \frac{1}{u + γ_{i} + 1} T_{−n,n}(u)T_{n,−n}(u + 1) ζ_{νμ}. (3.15)

The operators T_{−n,n}(u) and T_{n,−n}(u) are polynomials in u of degree ≤ n−2; see (3.7). Therefore, their action can be found from (3.12) by using the Lagrange interpolation formula.

The following branching rule for the reduction 𝔤_{n} ↓ 𝔤_{n−1} is implied by Theorem 3.2; cf. [9, Corollary 5.3].

Corollary 3.3 The restriction of V (λ) to the subalgebra 𝔤_{n−1} is isomorphic to the direct sum ⨁ c(μ)V′(μ) of finite-dimensional irreducible representations V′(μ) of 𝔤_{n−1} where the multiplicity c(μ) equals the number of (n − 1)-tuples ν satisfying the inequalities (3.10).

Proof. We have c(μ) = dim V (λ)_{μ}^{+}. By Theorem 3.2,

dim V (λ)_{μ}^{+} = ∏_{i=1}^{n−1}(α_{i} − β_{i} + 1),

if there exists ν satisfying (3.10). Otherwise, the space V (λ)_{μ}^{+} is trivial. This is proved by comparison of the dimensions of V (λ) and ⨁ c(μ)V′(μ) with the use of [15, Chapter VII, Section 9].

Applying the above construction of the vectors ζ_{νμ} to the subalgebras of the chain

𝔤_{1} ⊂ 𝔤_{2} ⊂ ⋯ ⊂ 𝔤_{n}, 𝔤_{k} = 𝔬(2k)

we obtain a basis of V (λ) parametrized by the D-type Gelfand–Tsetlin patterns (cf. [6]) which we denote by Λ:

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_{21} λ_{22}
λ′_{11}
λ_{11}

Here the upper row coincides with λ, all the entries are simultaneously integers or half-integers and the following inequalities hold

−|λ_{k1}| ≥ λ′_{k−1,1} ≥ λ_{k2} ≥ λ′_{k−1,2} ≥ ⋯ ≥ λ_{k,k−1} ≥ λ′_{k−1,k−1} ≥ λ_{kk},
−|λ_{k−1,1}| ≥ λ′_{k−1,1} ≥ λ_{k−1,2} ≥ λ′_{k−1,2} ≥ ⋯ ≥ λ_{k−1,k−1} ≥ λ′_{k−1,k−1}

for k = 2, …, n. Set

l_{ki} = λ_{ki} − i + 1,  l′_{ki} = λ′_{ki} − i + 1,  1 ≤ i ≤ k ≤ n  (3.16)

and introduce the vectors

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

with λ′_{k−1,0} := max{λ_{k1}, λ_{k−1,1}}. Finally, set

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

The following proposition is implied by Corollary 3.3.

Proposition 3.4 The vectors ζ_{Λ} parametrized by the Gelfand–Tsetlin patterns Λ form a basis of the representation V (λ).

4 Matrix element formulas

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

Φ_{−k,k} = ∑_{i=1}^{k−1} F_{−k,i}F_{ik}, k = 2, …, 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 Φ_{−n,n} under the natural homomorphism π : C_{n} → Z(𝔤_{n}, 𝔤_{n−1}) coincides with the coefficient at u^{2n−4} of the polynomial Z_{−n,n}(u); see the proof of Proposition 3.1. The following analog of (3.8) is obtained from (2.6), (2.8) and (3.2):

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} 2u}.

Therefore, we have an equality of operators in V (λ)_{μ}^{+}:

Φ_{−n,n} = −t_{−n,n}^{(2)} + t_{−n,n}^{(1)}t_{−n,−n}^{(1)} + (1 + α_{0}) t_{−n,n}^{(1)}.  (4.1)

The image of s_{nn}^{(1)} under the homomorphism (3.3) is F_{nn}. On the other hand, by (2.8) we have

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

as operators in V (λ)_{μ}^{+}. Therefore, (4.1) 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 (3.12) imply that

Φ_{−n,n}ζ_{νμ} = ∑_{i=1}^{n−1} θ_{i} (F_{nn} − γ_{i} + 3/2) ζ_{ν−δ_{i},μ},  (4.2)

where

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

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

F_{nn} ζ_{νμ} = (2 ∑_{i=0}^{n−1} ν_{i} − ∑_{i=1}^{n} λ_{i} − ∑_{i=1}^{n−1} μ_{i}) ζ_{νμ}.

Remark. One can introduce the elements Φ_{k,−k} by

Φ_{k,−k} = ∑_{i=1}^{k−1} F_{ki}F_{i,−k}.

The action of Φ_{n,−n} on V (λ)_{μ}^{+} is found in the same way as that of Φ_{−n,n}:

Φ_{n,−n}ζ_{νμ} = ∑_{i=1}^{n−1} ∏_{j=1, j≠i}^{n−1} \frac{1}{γ_{j} − γ_{i}} ζ_{ν+δ_{i},μ},

although this will not be used.

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},  (4.3)

where X_{μν′} denotes the operator

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

ν′ and μ′ are (n−2)-tuples of integers or half-integers such that the inequalities (3.10) are satisfied with λ, ν, μ respectively replaced by μ, ν′, μ′; we set γ′_{i} = ν′_{i} − i + 1/2 and ν′_{0} = max{μ_{1}, μ′_{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}.  (4.4)

By (2.5) (with a = −n) we need to express

X_{μν′} z_{n−1,i}z_{i,−n} ξ_{νμ},  i = −n + 1, …, n − 1  (4.5)

as a linear combination of the vectors ξ_{νμν′}. If i ≠ ±1 then the calculation is exactly the same as in [9, Section 6] where one uses the relations

Z_{n,−n}(−g_{n}) = z_{n,−n},  Z_{n,−n}(−g_{i}) = z_{ni}z_{i,−n},  (4.6)

which follow from (2.4) and (3.9). Now consider (4.5) with i = −1. We have

X_{μν′} z_{n−1,−1}z_{−1,−n} ξ_{νμ} = −X_{μν′} z_{1,−n+1}z_{n1} ξ_{νμ}.

If λ_{1} ≥ μ_{1} then z_{n1} ξ_{νμ} = ξ_{ν,μ−δ_{1}} while for λ_{1} < μ_{1} we derive from (4.6) that

z_{n1} ξ_{νμ} = ∑_{i=1}^{n−1} ∏_{a=1, a≠i}^{n−1} \frac{m_{1}^{2} − γ_{a}^{2}}{γ_{i}^{2} − γ_{a}^{2}} ξ_{ν+δ_{i},μ−δ_{1}}.

Similarly, if μ′_{1} ≥ μ_{1} then X_{μν′} z_{1,−n+1} = X_{μ−δ_{1},ν′} while for μ′_{1} < μ_{1} one has

X_{μν′} z_{1,−n+1} = ∑_{r=1}^{n−2} ∏_{a=1, a≠r}^{n−2} \frac{m_{1}^{2} − γ′_{a}^{2}}{γ′_{r}^{2} − γ′_{a}^{2}} X_{μ−δ_{1},ν′+δ_{r}}.

Finally, take i = 1 in (4.5). If λ_{1} ≤ μ_{1} then z_{1,−n} ξ_{νμ} = ξ_{ν,μ+δ_{1}}, and if λ_{1} > μ_{1} then

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

Similarly, if μ′_{1} ≤ μ_{1} then X_{μν′} z_{n−1,1} = X_{μ+δ_{1},ν′} and if μ′_{1} > μ_{1} then

X_{μν′} z_{n−1,1} = ∑_{r=1}^{n−2} ∏_{a=1, a≠r}^{n−2} \frac{(m_{1} + 1)^{2} − γ′_{a}^{2}}{γ′_{r}^{2} − γ′_{a}^{2}} X_{μ+δ_{1},ν′+δ_{r}}.

The action of the elements F_{n−1,n} on the vectors (4.3) can be expressed in two different ways. First we sketch a calculation similar to the one used above which leads to (rather complicated) explicit formulas for the matrix elements. Then we give slightly less explicit but more convenient formulas where F_{n−1,n} is represented by a commutator-like expression of simpler operators.

We have the following analog of (4.4):

F_{n−1,n} ξ_{νμν′} = X_{μν′} F_{n−1,n} ξ_{νμ}.  (4.7)

Now use (2.5) with a = n. Here we need to calculate z_{in} ξ_{νμ} instead of z_{i,−n} ξ_{νμ} in the previous case. Suppose that i > 1. We have

z_{in} ξ_{νμ} = z_{in}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}};

see (3.1). However,

−g_{−i} ξ_{ν,μ+δ_{i}} = (m_{i} + 1) ξ_{ν,μ+δ_{i}}.

To calculate Z_{−n,−n}(m_{i}+1) ξ_{ν,μ+δ_{i}} we use the following equality of operators in V (λ)_{μ}^{+}:

Z_{−n,−n}(u) = \frac{(u − α_{0})T_{−n,n}(u)T_{n,−n}(−u) + (u + α_{0} + 1)T_{−n,−n}(u)T_{nn}(−u)}{(−1)^{n} (2u + 1)},

see (2.6), (2.8) and (3.2); and then apply formulas (3.12) and (3.15).

To calculate z_{−i,n} ξ_{νμ} we first permute z_{−i,n} with the generators z_{nj} and z_{j,−n} with j = 1, …, i − 1 in (3.11). Further, we use the relation

z_{−i,n}z_{i,−n} = (−1)^{n−i}z_{−n,i}z_{i,−n} = (−1)^{n−i}Z_{−n,−n}(−g_{i})

and complete the calculation in a similar manner. To find z_{±1,n} ξ_{νμ} we need to consider a few different cases which depend on the relationship between the parameters λ_{1}, μ_{1} and μ′_{1} and then proceed exactly as above in the calculation of the action of F_{n−1,−n}.

We now give an alternative way of computing the action of F_{n−1,n}. The basic idea is to replace the operator z_{in} in the above calculation of z_{in} ξ_{νμ} by the following expression: for i = −n + 1, …, n − 1

z_{in} = [z_{i,−n}, Φ_{−n,n}] \frac{1}{f_{i} + F_{nn}}  (4.8)

and then use the formulas for the action of z_{i,−n} and Φ_{−n,n}; see (4.2). More precisely, we regard (4.8) 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, (2.5), (4.7) and (4.8) 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},  (4.9)

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}} .

The action of Φ_{n−1,−n}(u) is found exactly as that of F_{n−1,−n}. Note that the operators X_{μν′} and Φ_{−n,n} commute. Remark. The operator Φ_{n−1,−n}(u) is a rational function in u which can have singularities at the values u = 0 and u = 2 in (4.9). However, the operator

Φ_{n−1,−n}(u + 2) Φ_{−n,n} − Φ_{−n,n}Φ_{n−1,−n}(u)

is regular at u = 0 and coincides with F_{n−1,n}. Note the similarity with the symplectic case [9], where we have 2F_{n−1,n} = [F_{n−1,−n}, F_{−n,n}].

The elements F_{k−1,−k}, F_{k−1,k} with k = 2, …, n and F_{21}, F_{−2,1} generate 𝔤_{n} as a Lie algebra. Summarizing the above calculations we obtain the following formulas for the matrix elements of the generators. Given a pattern Λ we use the notation (3.16) and set for 1 ≤ i < k ≤ n:

A_{ki} = ∏_{a=1, a≠i}^{k−1} \frac{1}{l_{k−1,i}^{2} − l_{k−1,a}^{2}} ,
B_{ki}(x) = ∏_{a=1, a≠i}^{k−1} \frac{(x + l′_{k−1,a})(x − l′_{k−1,a} + 1)}{l′_{k−1,a} − l′_{k−1,i}},

and

C_{ki} =(max{λ_{k1}, λ_{k−1,1}} + l′_{k−1,i} − 1)(min{λ_{k1}, λ_{k−1,1}} − l′_{k−1,i} + 1)
× ∏_{a=2}^{k}(l_{ka} − l′_{k−1,i} + 1) ∏_{a=2}^{k−1}(l_{k−1,a} − l′_{k−1,i} + 1) ∏_{a=1, a≠i}^{k−1} \frac{1}{l′_{k−1,a} − l′_{k−1,i}} .

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. Consider the basis {ζ_{Λ}} of the representation V (λ); see Proposition 3.4. We shall suppose that ζ_{Λ} = 0 if the array Λ is not a pattern.

Theorem 4.1 The action of the generators of the Lie algebra 𝔬(2n) in the vasis {ζ_{Λ}} is given by the following formulas.

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

Here

ζ_{Λ}^{+}(k, i) = ∑_{j=1}^{k−1} ∑_{m=1}^{k−2} B_{kj}(l_{k−1,i})B_{k−1,m}(l_{k−1,i}) ζ_{Λ+δ′_{k−1,j}+δ_{k−1,i}+δ′_{k−2,m}}

for i = 2, …, k − 1; and for i = 1 if λ_{k−1,1} < λ_{k1}, λ_{k−2,1}. Otherwise,

ζ_{Λ}^{+}(k, 1) = ζ_{Λ+δ_{k−1,1}}  if λ_{k−1,1} ≥ λ_{k1}, λ_{k−2,1},
= ∑_{j=1}^{k−1} B_{kj}(l_{k−1,1}) ζ_{Λ+δ′_{k−1,j}+δ_{k−1,1}}  if λ_{k−2,1} ≤ λ_{k−1,1} < λ_{k1},
= ∑_{m=1}^{k−2} B_{k−1,m}(l_{k−1,1}) ζ_{Λ+δ_{k−1,1}+δ′_{k−2,m}}  if λ_{k1} ≤ λ_{k−1,1} < λ_{k−2,1}.

Furthermore,

ζ_{Λ}^{−}(k, i) = ζ_{Λ−δ_{k−1,i}}

for i = 2, …, k − 1; and for i = 1 if λ_{k−1,1} ≤ λ_{k1}, λ_{k−2,1}. Otherwise,

ζ_{Λ}^{−}(k, 1) = ∑_{j=1}^{k−1} B_{kj}(l_{k−1,1} − 1) ζ_{Λ+δ′_{k−1,j}−δ_{k−1,1}}  if λ_{k1} < λ_{k−1,1} ≤ λ_{k−2,1},
= ∑_{m=1}^{k−2} B_{k−1,m}(l_{k−1,1} − 1) ζ_{Λ−δ_{k−1,1}+δ′_{k−2,m}}  if λ_{k−2,1} < λ_{k−1,1} ≤ λ_{k1},
= ∑_{j=1}^{k−1} ∑_{m=1}^{k−2} B_{kj}(l_{k−1,1} − 1)B_{k−1,m}(l_{k−1,1} − 1)ζ_{Λ+δ′_{k−1,j}−δ_{k−1,1}+δ′_{k−2,m}}
if λ_{k−1,1} > λ_{k1}, λ_{k−2,1}.

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

Φ_{−k,k} ζ_{Λ} = ∑_{i=1}^{k−1} C_{ki} (F_{kk} − l′_{k−1,i} + 2) ζ_{Λ−δ′_{k−1,i}}

and

Φ_{k−1,−k}(u) ζ_{Λ} =

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

Example. Let n = 2. We have z_{21} = F_{21}, z_{1,−2} = F_{1,−2} and Z_{2,−2}(u) = F_{21}F_{1,−2}. Therefore, the basis vectors are given by

ζ_{Λ} = F_{21}^{λ′_{10}−λ_{11}}F_{1,−2}^{λ′_{10}−λ_{21}}(F_{21}F_{1,−2})^{λ′_{11}−λ_{22}} ξ,

where λ′_{10} = max{λ_{21}, λ_{11}}. The Lie algebra 𝔬(4) is isomorphic to the direct sum of two copies of 𝔰𝔩(2) and the action of their generators in the basis {ζ_{Λ}} is easily found. The resulting formulas also hold for the action of the elements of the subalgebra 𝔤_{2} ⊂ 𝔤_{n} in the basis {ζ_{Λ}} of the 𝔤_{n}-module V (λ).

Acknowledgements

The author wishes to thank R. Donnelly, M. Gould, M. Moshinsky, and R. Proctor for stimulating discussions of the results of [9].

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