Combinatorial Identities Related to Representations of U_{q}(𝔤̃𝔩_{2})

V.Tarasov

October 16, 1998

Introduction

Recently N.Jing discovered in [J] the following combinatorial identity:

(𝕀) ∑_{k=0}^{ℓ} ∏_{s=0}^{k−1} \frac{η^{ℓ} − η^{s}}{1 − η^{s+1}} ∑_{σ∈S_{ℓ}}( ∏_{1⩽a⩽k}(t_{σ_{a}} − 1) ∏_{k<b⩽ℓ}(t_{σ_{b}} − η^{ℓ−1}) ∏_{1⩽a<b⩽ℓ} \frac{t_{σ_{a}} − ηt_{σ_{b}}}{t_{σ_{a}} − t_{σ_{b}}} ) = 0 .

In his paper the identity comes from validity of the Serre relations in some vertex representations of quantum Kac-Moody algebras.

In this note we are going to generalize this identity, see Theorems 1.1, 4.1. The obtained identities are equivalent to existence of a singular vector in certain tensor products of evaluation representations of the quantum loop algebra U_{q}(𝔤̃𝔩_{2}) , see Proposition 3.1, or more generally, of the elliptic quantum group E_{ρ,γ}(𝔰𝔩_{2}) .

There are two directions for generalizing the identity (𝕀). First one can replace linear functions of t_{1}, …, t_{ℓ} by polynomials of larger degree. This is done in Section 1, see (1.3).

Furthermore, it is possible to take elliptic theta-functions instead of polynomials, see Section 4. The resulting identities depend on two extra parameters: the elliptic modulus p and the dynamical parameter α. In the limit p → 0 the elliptic functions degenerate into polynomials and we get a family of polynomial identities depending on α and turning into the identities (1.3) if either α → 0 or α → ∞.

The author is grateful to the Max-Planck-Institut für Mathematik in Bonn, where the basic part of the paper had been written, for hospitality.

1. Polynomial identities

Given nonnegative integers ℓ and n let P_{ℓ,n} be a set of partitions λ = (λ_{1}, …, λ_{ℓ}) such that n ⩾ λ_{1} ⩾ … ⩾ λ_{ℓ} ⩾ 1 . For a partition λ let ω_{k,λ} = #{j | λ_{j} = k}.

Introduce indeterminates t_{1}, …, t_{ℓ}, x_{1}, …, x_{n}, y_{1}, …, y_{n}, η. In the paper we use the following compact notations:

t = (t_{1}, …, t_{ℓ}) ,  x = (x_{1}, …, x_{n}) ,  y = (y_{1}, …, y_{n}) .

E-mail: vt@pdmi.ras.ru For any m = 1, …, n set

X_{m}(u; x; y) = u ∏_{1⩽j<m}(u − y_{j}) ∏_{m<k⩽n}(u − x_{k}) ,

X′_{m}(u; x; y) = ∏_{1⩽j<m}(u − x_{j}) ∏_{m<k⩽n}(u − y_{k}) ,

and for any λ ∈ P_{ℓ,n} set

r_{λ}(η) = ∏_{m=1}^{n} ∏_{s=1}^{ω_{m,λ}} \frac{1 − η}{1 − η^{s}} ,

P_{λ}(t; x; y; η) = r_{λ}(η) ∑_{σ∈S_{ℓ}}( ∏_{a=1}^{ℓ} X_{λ_{a}}(t_{σ_{a}}; x; y) ∏_{1⩽a<b⩽ℓ} \frac{t_{σ_{a}} − ηt_{σ_{b}}}{t_{σ_{a}} − t_{σ_{b}}} ) , (1.1)

P′_{λ}(t; x; y; η) = r_{λ}(η) ∑_{σ∈S_{ℓ}}( ∏_{a=1}^{ℓ} X′_{λ_{a}}(t_{σ_{a}}; x; y) ∏_{1⩽a<b⩽ℓ} \frac{ηt_{σ_{a}} − t_{σ_{b}}}{t_{σ_{a}} − t_{σ_{b}}} ) . (1.2)

Notice that

X_{m}(u; x; y) = uX′_{m}(u; y; x)

and

P_{λ}(t; x; y; η) = η^{ℓ(ℓ−1)/2 − ∑_{m=1}^{n}ω_{m,λ}(ω_{m,λ}−1)/2} t_{1} … t_{ℓ} P′_{λ}(t; y; x; η^{−1}) .

Notice also that both P_{λ} and P′_{λ} are polynomials in all the indeterminates involved. Theorem 1.1. Let x_{j} = η^{ℓ−1}y_{i} for some i < j. Then

∑_{λ∈P_{ℓ}^{i,j}}c_{λ}^{(i,j)}(x; y; η) P_{λ}(t; x; y; η) = 0 (1.3)

where P_{ℓ}^{i,j} = {λ = (λ_{1}, …, λ_{ℓ}) | j ⩾ λ_{1} ⩾ … ⩾ λ_{ℓ} ⩾ i} and

\begin{matrix}c_{λ}^{(i,j)}(x; y; η) = (−1)^{ω_{i,λ}} η^{ω_{j,λ}(ω_{j,λ}−1)/2} ∏_{i<k<j} ∏_{s=0}^{ω_{k,λ}−1}(x_{k} − η^{s}y_{k}) × \\ × ∏_{a=1}^{ℓ}( ∏_{i<k<λ_{a}}(η^{ℓ−a}y_{i} − x_{k}) ∏_{λ_{a}<m<j}(η^{ℓ−a}y_{i} − y_{m})) & . \end{matrix}

The theorem is proved in the next section. Example. Identity (1.3) for i = 1 , j = n = 2 is equivalent to identity (𝕀). Remark. Multiplied by a certain polynomial in x, y, η the identity (1.3) becomes more transparent and understandable, see (2.6).

Remark. All over the paper we assume integers ℓ and n to be fixed. There are two natural embeddings of P_{ℓ,n−1} into P_{ℓ,n} given by either λ ↦ λ or λ ↦ λ′ = (λ_{1} + 1, …, λ_{ℓ} + 1) . Indicating for a while dependence of polynomials defined by (1.1) on n explicitly, that is, writing P_{n,λ} instead of P_{λ}, we have

P_{n,λ}(t; x; y; η) = P_{n−1,λ}(t; x^{(n)}; y^{(n)}; η) ∏_{a=1}^{ℓ} (t_{a} − x_{n}) ,

= P_{n−1,λ′}(t; x^{(1)}; y^{(1)}; η) ∏_{a=1}^{ℓ} (t_{a} − y_{1}) ,

where x^{(n)} = (x_{1}, …, x_{n−1}) , y^{(n)} = (y_{1}, …, y_{n−1}) , x^{(1)} = (x_{2}, …, x_{n}) , y^{(1)} = (y_{2}, …, y_{n}) . Therefore, the claim of Theorem 1.1 for i = 1 , j = n implies the claim of the theorem for general i < j.

2. Proof of Theorem 1.1

In this section we think x_{1}, …, x_{n}, y_{1}, …, y_{n}, η being complex variables rather than indeterminates and consider them as parameters which functions of t_{1}, …, t_{ℓ} can in addition depend on.

Let f, g be polynomials in t_{1}, …, t_{ℓ}. We define below their scalar product ⟨f, g⟩_{S}. Let

S(t_{1}, …, t_{ℓ}) = ∏_{a=1}^{ℓ} ∏_{m=1}^{n} (t_{a} − x_{m})(t_{a} − y_{m}) ∏_{{a,b=1 \atop a≠b}}^{ℓ} \frac{t_{a} − ηt_{b}}{t_{a} − t_{b}} .

If |η| > 1 and |x_{m}| < 1 , |y_{m}| > 1 for all m = 1, …, n, then we set

⟨f, g⟩_{S} = \frac{1}{(2πi)^{ℓ} ℓ!} ∫_{𝕋^{ℓ}} \frac{f(t) g(t)}{S(t)} (dt/t)^{ℓ}

where (dt/t)^{ℓ} = ∏_{a=1}^{ℓ} dt_{a}/t_{a} and 𝕋^{ℓ} = {t ∈ ℂ^{ℓ} | |t_{1}| = 1 , …, |t_{ℓ}| = 1} each circle

oriented counterclockwise. Then one can show that ⟨f, g⟩_{S} is a rational function of x_{1}, …, x_{n}, y_{1}, …, y_{n}, η.

The polynomials defined by (1.1) and (1.2) are biorthogonal with respect to the introduced scalar product.

Theorem 2.1. [TV1, Theorem C.9] ⟨P′_{λ}, P_{μ}⟩_{S} = N_{λ}^{−1} δ_{λμ} where

(2.1)  N_{λ} = ∏_{m=1}^{n} ∏_{s=1}^{ω_{m,λ}} \frac{(1 − η^{s})(x_{m} − η^{s−1}y_{m})}{1 − η} .

Proof. The proof is based on Lemmas 2.2 and 2.3. The first equality in (2.3) implies that ⟨P′_{λ}, P_{μ}⟩_{S} = 0 unless λ ⩽ μ, while the second equality in (2.3) gives ⟨P′_{λ}, P_{μ}⟩_{S} = 0 unless λ ⩾ μ and ⟨P′_{λ}, P_{λ}⟩_{S} = P_{λ}(y ◁ λ)P′_{λ}(y ◁ λ) . The rest of the proof is straightforward. □

Remark. Notations here and in [TV1] are not always the same though we try to keep them consistent whenever possible. The identification of labels should be mentioned: a partition λ here corresponds to a label 𝔩 = (ω_{1,λ}, …, ω_{n,λ}) in [TV1], the polynomial P_{λ} being a numerator of the trigonometric weight function w_{𝔩} in [TV1] up to a simple factor.

For any λ ∈ P_{ℓ,n} introduce points x ▷ λ , y ◁ λ ∈ ℂ^{ℓ} as follows:

x ▷ λ = (η^{1−ω_{1,λ}}x_{1}, …, x_{1}, η^{1−ω_{2,λ}}x_{2}, …, x_{2}, …, η^{1−ω_{n,λ}}x_{n}, …, x_{n}) ,
y ◁ λ = (η^{ω_{1,λ}−1}y_{1}, …, y_{1}, η^{ω_{2,λ}−1}y_{2}, …, y_{2}, …, η^{ω_{n,λ}−1}y_{n}, …, y_{n}) .

For partitions λ, μ ∈ P_{ℓ,n} say that λ ⩾ μ if λ_{a} ⩾ μ_{a} for any a = 1, …, ℓ.

Lemma 2.2. P_{λ}(x ▷ μ) = 0 and P′_{λ}(y ◁ μ) = 0 unless λ ⩾ μ. P_{λ}(y ◁ μ) = 0 and P′_{λ}(x ▷ μ) = 0 unless λ ⩽ μ. Besides, only the terms in (1.1), (1.2) corresponding to the identity permutation contribute into the values P_{λ}(x ▷ λ) , P_{λ}(y ◁ λ) , P′_{λ}(x ▷ λ) , P′_{λ}(y ◁ λ) .

The proof is straightforward.

For a function f(t_{1}, …, t_{ℓ}) and a point t^{⋆} = (t1^{⋆}, …, tℓ^{⋆}) define a multiple residue Res(f(t)(dt/t)^{ℓ})|_{t=t^{⋆}} by

Res(f(t)(dt/t)^{ℓ})|_{t=t^{⋆}} = Res( … Res(f(t_{1}, …, t_{ℓ})(dt_{ℓ}/t_{ℓ}))|_{t_{ℓ}=t_{ℓ}^{⋆}} … (dt_{1}/t_{1}))|_{t_{1}=t_{1}^{⋆}}

and set

x ▷Res (f) = ∑_{λ∈P_{ℓ,n}}Res(f(t_{1}, …, t_{ℓ})(dt/t)^{ℓ})|_{t=x ▷λ} , (2.2)

y ◁ Res (f) = ∑_{λ∈P_{ℓ,n}}Res(f(t_{1}, …, t_{ℓ})(dt/t)^{ℓ})|_{t=y ◁ λ} .

Lemma 2.3. [TV1, Lemma C.8] Let polynomials f, g be such that their product is a symmetric polynomial in t_{1}, …, t_{ℓ} of degree less than 2n in each of the indeterminates and divisible by t_{1} … t_{n}. Then

(2.3)  ⟨f, g⟩_{S} = x ▷Res (fg/S) = (−1)^{ℓ} y ◁ Res (fg/S) .

Let Q_{λ} be a monomial symmetric polynomial:

Q_{λ}(t_{1}, …, t_{ℓ}) = \frac{1}{ω_{1,λ}! … ω_{n,λ}!} ∑_{σ∈S_{ℓ}}t_{σ_{1}}^{λ_{1}} … t_{σ_{ℓ}}^{λ_{ℓ}} .

Introduce transition coefficients A_{λμ}, λ, μ ∈ P_{ℓ,n}:

(2.4)  P_{λ} = ∑_{μ∈P_{ℓ,n}}A_{λμ} Q_{μ} .

Then by Theorem 2.1 for any ν ∈ P_{ℓ,n}

∑_{μ∈P_{ℓ,n}}A_{λμ} ⟨P′_{ν}, Q_{μ}⟩ = N_{λ}^{−1} δ_{λν} ,

and inverting this relation we get

∑_{λ∈P_{ℓ,n}}N_{λ} ⟨P′_{λ}, Q_{μ}⟩ A_{λμ} = δ_{μν} .

Calculating the scalar products by Lemma 2.3 we have

(2.5)  ∑_{κ,λ∈P_{ℓ,n}}M_{κ}^{−1} Q_{μ}(x ▷ κ)P′_{λ}(x ▷ κ)N_{λ} A_{λμ} = δ_{μν} ,

where M_{κ}^{−1} = Res(S(t)^{−1}(dt/t)^{ℓ})|_{t=x ▷κ}. Notice that M_{κ} is a rational function in x_{1}, …, x_{n}, η and a polynomial in y_{1}, …, y_{n}.

The matrix [Q_{λ}(x ▷κ)]_{κ,λ∈P_{ℓ,n}} is invertible, see e.g. (2.7), and we denote by B the inverse matrix, which is a rational function of x_{1}, …, x_{n}, η and, trivially, does not depend on y_{1}, …, y_{n}. Finally, the relation (2.5) is transformed to

∑_{λ∈P_{ℓ,n}}P′_{λ}(x ▷ κ)N_{λ} A_{λμ} = M_{κ} B_{κμ} .

Proof of Theorem 1.1. Fix i < j. Let κ^{(j)} = (j, …, j) . Assume that y_{i} = η^{1−ℓ}x_{j} . Then M_{κ^{(j)}} = 0 , and taking into account the definition (2.4) of A_{λμ} we have an identity

(2.6)  ∑_{λ∈P_{ℓ,n}}P′_{λ}(x ▷ κ^{(j)})N_{λ} P_{λ} = 0 .

Notice that x ▷ κ^{(j)} = (η^{1−ℓ}x_{j}, …, x_{j}) , so only the term in (1.2) corresponding to the identity permutation contribute into the value P′_{λ}(x▷κ^{(j)}) , and P′_{λ}(x▷κ^{(j)}) = 0 unless λ ∈ P_{ℓ}^{i,j}. Calculating P′_{λ}(x ▷ κ^{(j)})N_{λ} explicitly and removing all factors which does not depend on λ we get the identity (1.3). Theorem 1.1 is proved. □

Remark. Let us mention two determinant formulae though they are not actually used in the proof:

det[Q_{λ}(x ▷ μ)]_{λ,μ∈P_{ℓ,n}} = η^{−n(n+1)/2·\binom{n+ℓ−1}{n+1}} ∏_{m=1}^{n} x_{m}^{\binom{n+ℓ−1}{n}} × (2.7)

× ∏_{s=1−ℓ}^{ℓ−1} ∏_{1⩽j<k⩽n}(η^{s}x_{k} − x_{j})^{D(n,ℓ,s)}

where D(n, ℓ, s) = ∑_{{r∈ℤ_{⩾0} \atop 2r⩽ℓ−|s|−1}}\binom{n + ℓ − |s| − 2r − 3}{n − 2} , see formula (A.14) in [TV1],

and

(2.8)  det[A_{λμ}]_{λ,μ∈P_{ℓ,n}} = ∏_{s=0}^{ℓ−1} ∏_{1⩽j<k⩽n}(η^{s}y_{j} − x_{k})^{\binom{n+ℓ−s−2}{n−1}},

see formula (A.10) in [TV1]. Formula (2.7) is a deformation of a symmetric power of the Vandermonde determinant. Formula (2.8) implies that the polynomials P_{λ}, λ ∈ P_{ℓ,n} are linear independent if x, y, η are generic, and there are linear relations between them if x_{k} = η^{s}y_{j} for some j < k and s ∈ {0, …, ℓ − 1}. For s = ℓ − 1 there is just one linear relation given by (2.6). Relations for s < ℓ − 1 can be obtained in a similar manner. They also can be derived from the relation (2.6) written for polynomials in less number of indeterminates.

3. Tensor products of evaluation modules over U_{q}(𝔤̃𝔩_{2})

Let q be a nonzero complex number which is not a root of unity. Consider the quantum group U_{q}(𝔰𝔩_{2}) with generators E, F, q^{H} and relations

q^{H}E = qE q^{H} ,  q^{H}F = q^{−1}F q^{H} ,  [E,F] = \frac{q^{2H} − q^{−2H}}{q − q^{−1}} ,

and the quantum loop algebra U_{q}(𝔤̃𝔩_{2}) with generators L_{ij}^{(+0)}, L_{ji}^{(−0)}, 1 ⩽ j ⩽ i ⩽ 2 , and L_{ij}^{(s)}, i, j = 1, 2 , s = ±1, ±2, …, subject to relations (3.1).

Let e_{ij} , i, j = 1, 2 , be the 2×2 matrix with the only nonzero entry 1 at the intersection of the i-th row and j-th column. Set

R(u) = (uq − q^{−1})(e_{11} ⊗ e_{11} + e_{22} ⊗ e_{22}) +
+ (u − 1)(e_{11} ⊗ e_{22} + e_{22} ⊗ e_{11}) + u(q − q^{−1})e_{12} ⊗ e_{21} + (q − q^{−1})e_{21} ⊗ e_{12} .

Introduce the generating series L_{ij}^{±}(u) = L_{ij}^{(±0)}+ ∑_{s=1}^{∞} L_{ij}^{(±s)}u^{±s}. The relations in U_{q}(𝔤̃𝔩_{2})

have the form:

\begin{aligned}& R(u/z)L_{(1)}^{ν}(u)L_{(2)}^{ν}(z) = L_{(2)}^{ν}(z)L_{(1)}^{ν}(u)R(u/z) , ν = ± , \\ (3.1) \end{aligned} ; R(u/z)L_{(1)}^{+}(u)L_{(2)}^{−}(z) = L_{(2)}^{−}(z)L_{(1)}^{+}(u)R(u/z) ,

L_{ii}^{(+0)}L_{ii}^{(−0)} = 1 ,  L_{ii}^{(−0)}L_{ii}^{(+0)} = 1 ,  i = 1, 2 ,

where L_{(1)}^{ν}(u) = ∑_{ij} e_{ij} ⊗ 1 ⊗ L_{ij}^{ν}(u) and L_{(2)}^{ν}(u) = ∑_{ij} 1 ⊗ e_{ij} ⊗ L_{ij}^{ν}(u) .

The quantum loop algebra U_{q}(𝔤̃𝔩_{2}) is a Hopf algebra with a coproduct

Δ : L_{ij}^{±}(u) ↦ ∑_{k} L_{ik}^{±}(u) ⊗ L_{kj}^{±}(u) .

There is a one-parametric family of automorphisms ρ_{z} : L_{ij}^{±}(u) ↦ L_{ij}^{±}(u/z) and the evaluation homomorphism ϵ : U_{q}(𝔤̃𝔩_{2}) → U_{q}(𝔰𝔩_{2}) :

ϵ : L^{±}(u) ↦ ∓ u^{ϑ_{±}} ( \begin{matrix}uq^{H} − q^{−H} & uF (q − q^{−1}) \\ E (q − q^{−1}) & uq^{−H} − q^{H} \end{matrix} )

where ϑ_{+} = 0 , ϑ_{−} = −1 . For any U_{q}(𝔰𝔩_{2})-module V denote by V (z) the U_{q}(𝔤̃𝔩_{2}) module obtained from V via the homomorphism ϵ ∘ ρ_{z}. The module V (z) is called the evaluation module. In a tensor product V_{1}(z_{1})⊗…⊗V_{n}(z_{n}) of evaluation modules actions of the series L^{+}(u) and (−u)^{n}L^{−}(u) coincide, so it is enough to look at the action of only one of them. Besides, this implies that the series L^{+}(u) acts as a polynomial in u, so it can be evaluated for u being a complex number.

Let V_{1}, …, V_{n} be U_{q}(𝔰𝔩_{2}) Verma modules with highest weights q^{Λ_{1}}, …, q^{Λ_{n}} and generating vectors v_{1}, …, v_{n}, respectively. Using commutation relations in U_{q}(𝔤̃𝔩_{2}) it is rather straightforward to obtain the following statements.

Proposition 3.1. Consider U_{q}(𝔤̃𝔩_{2}) evaluation modules V_{1}(z_{1}), …, V_{n}(z_{n}) . Let z_{i} = q^{2Λ_{i}+2Λ_{j}−2ℓ}z_{j} for some i < j and ℓ ∈ ℤ_{⩾0} . Then

a) The vector v_{1} ⊗ … ⊗ v_{n} ∈ V_{1}(z_{1}) ⊗ … ⊗ V_{n}(z_{n}) generates a U_{q}(𝔤̃𝔩_{2})-submodule, which is annihilated by the action of a product L_{21}^{+}(q^{2Λ_{j}}z_{j}) … L_{21}^{+}(q^{2Λ_{j}−2ℓ}z_{j}) .

b) A vector ṽ = L_{12}^{+}(q^{2Λ_{j}}z_{j}) … L_{12}^{+}(q^{2Λ_{j}−2ℓ}z_{j}) v_{n} ⊗ … ⊗ v_{1} ∈ V_{n}(z_{n}) ⊗ … ⊗ V_{1}(z_{1})

is singular with respect to the U_{q}(𝔤̃𝔩_{2})-action, that is, L_{21}^{±}(u) ṽ = 0 .

Corollary 3.2. Let z_{i} = q^{2Λ_{i}+2Λ_{j}−2ℓ}z_{j} for some i < j and ℓ ∈ ℤ_{⩾0} . Then for arbitrary t_{1}, …, t_{ℓ}

(3.2)  L_{21}^{+}(q^{2Λ_{j}}z_{j}) … L_{21}^{+}(q^{2Λ_{j}−2ℓ}z_{j})L_{12}^{+}(t_{1}) … L_{12}^{+}(t_{ℓ}) v_{1} ⊗ … ⊗ v_{n} = 0

in V_{1}(z_{1}) ⊗ … ⊗ V_{n}(z_{n}) and

(3.3)  L_{21}^{+}(t_{1}) … L_{21}^{+}(t_{ℓ})L_{21}^{+}(q^{2Λ_{j}}z_{j}) … L_{12}^{+}(q^{2Λ_{j}−2ℓ}z_{j}) v_{n} ⊗ … ⊗ v_{1} = 0

in V_{n}(z_{n}) ⊗ … ⊗ V_{1}(z_{1}) .

Both relations (3.2) and (3.3) are equivalent to the identitiy (2.6) and, hence, the identity (1.3). For (3.2) it follows from Proposition 3.3 and formula (2.1), while for (3.3) an analogue of Proposition 3.3 for the tensor product V_{n}(z_{n}) ⊗ … ⊗ V_{1}(z_{1}) is required.

Remark. It is rather easy to see that Corollary 3.2, Proposition 3.3 and the determinant formula (2.8) together imply Proposition 3.1.

Let parameters q, q^{Λ_{1}}, …, q^{Λ_{n}} , z_{1}, …, z_{n} used in this section be related to the previously used parameters η, x_{1}, …, x_{n}, y_{1}, …, y_{n} as follows:

η = q^{2} ,  x_{m} = q^{2Λ_{m}}z_{m} ,  y_{m} = q^{−2Λ_{m}}z_{m} .

Proposition 3.3. [KBI], [TV1, Lemma 4.3], [TV2, Lemma 4.18]

L_{12}^{+}(t_{1}) … L_{12}^{+}(t_{ℓ}) v_{1} ⊗ … ⊗ v_{n} = (q − q^{−1})^{ℓ} ∏_{m=1}^{n}(−z_{m})^{−ℓ} ∑_{λ∈P_{ℓ,n}}P_{λ}(t_{1}, …, t_{ℓ}) × ; \begin{matrix}× ∏_{1⩽j<k⩽n} \\ q^{Λ_{j}ω_{k,λ}−Λ_{k}ω_{j,λ}− ω_{j,λ}ω_{k,λ}} F^{ω_{1,λ}}v_{1} ⊗ … ⊗ F^{ω_{n,λ}}v_{n} . \end{matrix}

L_{21}^{+}(t_{1}) … L_{21}^{+}(t_{ℓ}) F^{ω_{1,λ}}v_{1} ⊗ … ⊗ F^{ω_{n,λ}}v_{n} =

= ∏_{m=1}^{n}(−z_{m})^{ω_{m,λ}−ℓ} P′_{λ}(t_{1}, …, t_{ℓ}) ∏_{m=1}^{n} ∏_{s=1}^{ω_{m,λ}} \frac{(q^{s} − q^{−s})(q^{2Λ_{m}−s+1} − q^{−2Λ_{m}+s−1})}{q − q^{−1}} ×

× ∏_{1⩽j<k⩽n}q^{Λ_{k}ω_{j,λ}−Λ_{j}ω_{k,λ}− ω_{j,λ}ω_{k,λ}} v_{1} ⊗ … ⊗ v_{n} .

4. Elliptic identities

In this section we extend Theorem 1.1 to the elliptic case, see Theorem 4.1. The obtained identities have the same relation to representations of the elliptic quantum group E_{ρ,γ}(𝔰𝔩_{2}) as the identities (1.3) have to representations of the quantum loop algebra U_{q}(𝔤̃𝔩_{2}) .

Fix a nonzero complex number p such that |p| < 1 . Let

(u)_{∞} = (u; p)_{∞} = ∏_{s=0}^{∞}(1 − p^{s}u)

and let θ(u) = θ(u; p) = (u)_{∞}(pu^{−1})_{∞}(p)_{∞} be the Jacobi theta-function.

In addition to t_{1}, …, t_{ℓ}, x_{1}, …, x_{n}, y_{1}, …, y_{n} and η introduce one more variable α called the dynamical parameter.

Remark. The parameter α is related to the dynamical variable in the elliptic quantum group E_{ρ,γ}(𝔰𝔩_{2}) .

For any m = 1, …, n set

Z_{m}(u; x; y; α) = θ(α_{m}^{−1}u/x_{m}) ∏_{1⩽j<m}θ(u/y_{j}) ∏_{m<k⩽n}θ(u/x_{k}) ,

Z′_{m}(u; x; y; α) = θ(α_{m}u/y_{m}) ∏_{1⩽j<m}θ(u/x_{j}) ∏_{m<k⩽n}θ(u/y_{k}) ,

where α_{m} = α ∏_{1⩽j<m}x_{j}/y_{j} , and for any λ ∈ P_{ℓ,n} set

ρ_{λ}(η) = ∏_{m=1}^{n} ∏_{s=1}^{ω_{m,λ}} \frac{θ(η)}{θ(η^{s})} ,

Ξ_{λ}(t; x; y; η; α) = ρ_{λ}(η) ∑_{σ∈S_{ℓ}}( ∏_{a=1}^{ℓ} Z_{λ_{a}}(t_{σ_{a}}; x; y; αη^{2a−2ℓ}) ∏_{1⩽a<b⩽ℓ} \frac{θ(ηt_{σ_{b}}/t_{σ_{a}})}{θ(t_{σ_{b}}/t_{σ_{a}})} ) , (4.1)

Ξ′_{λ}(t; x; y; η; α) = ρ_{λ}(η) ∑_{σ∈S_{ℓ}}( ∏_{a=1}^{ℓ} Z′_{λ_{a}}(t_{σ_{a}}; x; y; αη^{2ℓ−2a}) ∏_{1⩽a<b⩽ℓ} \frac{θ(ηt_{σ_{a}}/t_{σ_{b}})}{θ(t_{σ_{a}}/t_{σ_{b}})} ) . (4.2)

Notice that

Z_{m}(u; x; y; α) = Z′_{m}(u; y; x; α^{−1})

and

Ξ_{λ}(t; x; y; η; α) = η^{ℓ(ℓ−1)/2 − ∑_{m=1}^{n}ω_{m,λ}(ω_{m,λ}−1)/2} Ξ′_{λ}(t; y; x; η^{−1}; α^{−1}) .

Theorem 4.1. Let x_{j} = η^{ℓ−1}y_{i} for some i < j. Then

∑_{λ∈P_{ℓ}^{i,j}}C_{λ}^{(i,j)}(x; y; η; α) Ξ_{λ}(t; x; y; η; α) = 0 (4.3)

where P_{ℓ}^{i,j} = {λ = (λ_{1}, …, λ_{ℓ}) | j ⩾ λ_{1} ⩾ … ⩾ λ_{ℓ} ⩾ i},

C_{λ}^{(i,j)}(x; y; η; α) =
= (α_{i}x_{i}/y_{i})^{ω_{i,λ}} η^{−ω_{i,λ}(ω_{i,λ}−1)} ∏_{i<k<j} ∏_{s=0}^{ω_{k,λ}−1} \frac{θ(η^{−s}x_{k}/y_{k})}{θ(η^{s}α_{k,λ}^{−1})θ(η^{1−s−ω_{k,λ}}α_{k,λ} x_{k}/y_{k})} ×

× ∏_{s=0}^{ω_{i,λ}−1} \frac{1}{θ(η^{1−s−ω_{i,λ}}α_{i,λ} x_{i}/y_{i})} ∏_{s=0}^{ω_{j,λ}−1} \frac{1}{θ(η^{s}α_{j,λ}^{−1})} ∏_{a=ω_{j,λ}+1}^{ℓ−ω_{i,λ}}θ(α_{λ_{a}}η^{a−ℓ}y_{i}/y_{λ_{a}}) ×

∏_{a=1}^{ℓ}( ∏_{i<k<λ_{a}}θ(η^{ℓ−a}y_{i}/x_{k}) ∏_{λ_{a}<m<j} θ(η^{ℓ−a}y_{i}/y_{m})) ,

α_{k,λ} = α ∏_{1⩽j<k}η^{−2ω_{j,λ}} x_{j}/y_{j} and α_{k} = α ∏_{1⩽j<k}x_{j}/y_{j} .

Example. Identity (4.3) for i = 1 , j = n = 2 takes the form

∑_{k=0}^{ℓ} (−1)^{k} θ(η^{2k}β) ∏_{s=0}^{k−1} \frac{η^{s} θ(η^{ℓ−s})θ(η^{s}β)}{θ(η^{s+1})θ(η^{s+ℓ+1}β)} × (4.4)

× ∑_{σ∈S_{ℓ}}( ∏_{1⩽a⩽k} θ(t_{σ_{a}})θ(η^{2−2a−ℓ}t_{σ_{a}}/β) ∏_{k<b⩽ℓ} θ(η^{1−ℓ}t_{σ_{b}})θ(η^{1−2b}t_{σ_{b}}/β) ×

× ∏_{1⩽a<b⩽ℓ} \frac{θ(ηt_{σ_{b}}/t_{σ_{a}})}{θ(t_{σ_{b}}/t_{σ_{a}})}) = 0 .

Here β = η^{1−2ℓ}αx_{1}/y_{1} . If p → 0 and then either β → 0 or β → ∞, then (4.4) transforms into (𝕀).

Proof of Theorem 4.1. The proof of Theorem 4.1 is very similar to the proof of Theorem 1.1. We descripe the main steps below. We do not indicate dependence on x_{1}, …, x_{n}, y_{1}, …, y_{n}, η, α explicitly.

Denote by 𝔼 the space of functions f(t_{1}, …, t_{ℓ}) holomorphic outside the coordinate hyperplanes t_{a} = 0 , a = 1, …, ℓ, and such that

f(t_{1}, …, pt_{a}, …, t_{ℓ}) = t_{a}^{−2n} ∏_{m=1}^{n}(x_{m} y_{m}) f(t_{1}, …, t_{ℓ}) .

Let

Ω(t_{1}, …, t_{ℓ}) = ∏_{a=1}^{ℓ} ∏_{m=1}^{n} θ(t_{a}/x_{m})θ(t_{a}/y_{m}) ∏_{{a,b=1 \atop a≠b}}^{ℓ} \frac{θ(ηt_{a}/t_{b})}{θ(t_{a}/t_{b})} .

There is an analogue of Lemma 2.3.

Lemma 4.2. [TV1, Lemma C.11] For any function f ∈ 𝔼 we have

x ▷Res (f/Ω) = (−1)^{ℓ} y ◁ Res (f/Ω) .

Define a scalar product ⟨f, g⟩_{Ω} = x ▷Res (fg/Ω) . The functions (4.1) and (4.2) are biorthogonal with respect to the introduced scalar product.

Theorem 4.3. [TV1, Theorem C.9] ⟨Ξ′_{λ}, Ξ_{μ}⟩_{Ω} = D_{λ}^{−1} δ_{λμ} where

(4.5)  D_{λ} = (−1)^{ℓ} ∏_{m=1}^{n} ∏_{s=0}^{ω_{m,λ}−1} \frac{(p)_{∞}^{3} θ(η^{s+1})θ(η^{−s}x_{m}/y_{m})}{θ(η)θ(η^{s}α_{m,λ}^{−1})θ(η^{1−s−ω_{m,λ}}α_{m,λ} x_{m}/y_{m})}

and α_{m,λ} = α ∏_{1⩽j<m}η^{−2ω_{λ,j}} x_{j}/y_{j} .

Let ϑ_{1}, …, ϑ_{n} be the following functions in one variable:

(4.6)  ϑ_{m}(u) = u^{m−1} θ(−p^{m−1}η^{ℓ−1}α^{−1} ∏_{m=1}^{n}x_{m} (−u)^{n}; p^{n}) (p^{n}; p^{n})_{∞}^{−1} (p; p)_{∞}^{n} .

They are linearly independent, since ϑ_{m}(εu) = ε^{m−1} ϑ(u) where ε = e^{2πi/n}. For any λ ∈ P_{ℓ,n} set

(4.7)  Θ_{λ}(t_{1}, …, t_{ℓ}) = \frac{1}{ω_{1,λ}! … ω_{n,λ}!} ∑_{σ∈S_{ℓ}}ϑ_{λ_{1}}(t_{σ_{1}}) … ϑ_{λ_{n}}(t_{σ_{ℓ}}) .

The functions Θ_{λ}, λ ∈ P_{ℓ,n}, are linearly independent, and the functions Ξ_{μ} are linear combinations of the functions Θ_{λ}, see Proposition 4.4.

We proceed further like in Section 2 and under the assumption of Theorem 4.1: y_{i} = η^{1−ℓ}x_{j} for some i < j, we get an identity

∑_{λ∈P_{ℓ,n}}Ξ′_{λ}(x ▷ κ^{(j)})D_{λ} Ξ_{λ} = 0 ,

and then the identity (4.3). Theorem 4.1 is proved. □

Remark. There are elliptic analogues of determinant formulae (2.7) and (2.8), see formulae (B.11) and (B.8) in [TV1]:

(4.8) det[Θ_{λ}(x ▷ μ)]_{λ,μ∈P_{ℓ,n}} = K η^{n(1−n)/2·\binom{n+ℓ−1}{n+1}} ∏_{m=1}^{n} (−x_{m})^{(m−1)\binom{n+ℓ−1}{n}} ×

× ∏_{s=0}^{ℓ−1} θ(η^{s}α^{−1})^{\binom{n+s−1}{n−1}} ∏_{s=1−ℓ}^{ℓ−1} ∏_{1⩽j<k⩽n}θ(η^{s}x_{j}/x_{k})^{D(n,ℓ,s)}

where K = [(p)_{∞}^{n^{2}−1} ∏_{m=1}^{n−1} ( \frac{θ(e^{2πim/n})}{e^{2πim/n} − 1} )^{m−n} ]^{\binom{n+ℓ−1}{n}}, and

(4.9) det[Aλμ^{ell}]_{λ,μ∈P_{ℓ,n}} = K^{−1} ∏_{s=1−ℓ}^{ℓ−1} ∏_{m=1}^{n−1} θ(η^{s+ℓ−1}α^{−1} ∏_{1⩽j⩽m}y_{j}/x_{j})^{d(n,m,ℓ,s)} ×

× ∏_{m=1}^{n} y_{m}^{(m−n)\binom{n+ℓ−1}{n}} ∏_{s=0}^{ℓ−1} ∏_{1⩽j<k⩽n}θ(η^{s}y_{j}/x_{k})^{\binom{n+ℓ−s−2}{n−1}}

where Ξ_{λ} = ∑_{μ∈P_{ℓ,n}}Aλμ^{ell} Θ_{μ} and d(n, m, ℓ, s) = ∑_{i,j⩾0 ; {i+j<ℓ \atop i−j=s}} \binom{m − 1 + i}{m − 1} \binom{n − m − 1 + j}{n − m − 1} .

Remark. Consider a limit p → 0 in all formulae of this section, which essentially amounts to replacing the theta-function θ(u) by the linear function 1 − u. The functions ϑ_{1}, …, ϑ_{n} tend to power functions as p → 0 :

ϑ_{1}(u) → 1 + η^{ℓ−1}α^{−1} ∏_{m=1}^{n}x_{m} (−u)^{n} ,  ϑ_{m}(u) → u^{m−1} ,  m = 2, …, n .

The limit p → 0 of the identities (4.3) and the determinant formulae (4.8) and (4.9) deliver one-parametric deformations of the identities (1.3) and the determinant formulae (2.7) and (2.8), which can be recovered in the limit either α → 0 or α → ∞. The limit α → 0 is straightforward while the limit α → ∞ produces formulae (1.3), (2.7), (2.8) up to a certain change of variables.

Proposition 4.4. For any λ ∈ P_{ℓ,n} the function Ξ_{λ}, cf. (4.1), is a linear combination of the functions Θ_{μ}, μ ∈ P_{ℓ,n}, cf. (4.7).

Proof. Let E be a space of functions f(u) holomorphic for u ≠ 0 and such that

f(pu) = αη^{1−ℓ} ∏_{m=1}^{n}x_{m} (−u)^{n} f(u) .

Let E_{ℓ} be a space of symmetric functions f(t_{1}, …, t_{ℓ}) which considered as functions of one variable t_{a} belong to E for any a = 1, …, ℓ. In particular, E_{1} = E.

It is clear that dim E = n, say by Fourier series. A basis in E is given by the functions ϑ_{1}, …, ϑ_{n}. Hence, the functions Θ_{μ}, μ ∈ P_{ℓ,n}, form a basis in the space E_{ℓ}.

Since for any λ ∈ P_{ℓ,n} the function Ξ_{λ} belongs to the space E_{ℓ}, the proposition follows. □

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