On irreducibility of tensor products of Yangian modules associated with skew Young diagrams

MAXIM NAZAROV^{1} and VITALY TARASOV^{2}

1 Department of Mathematics, University of York, York YO1 5DD, England E-mail: mln1@york.ac.uk 2 Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, Russia E-mail: vt@pdmi.ras.ru

Introduction

In this article we continue our study [NT1,NT2] of the finite-dimensional modules over the Yangian Y(𝔤𝔩_{N}) of the general linear Lie algebra 𝔤𝔩_{N} . The Yangian Y(𝔤𝔩_{N}) is a canonical deformation of the enveloping algebra U(𝔤𝔩_{N}[u]) in the class of Hopf algebras [D1]. The unital associative algebra Y(𝔤𝔩_{N}) has an infinite family of generators T_{ij}^{(s)} where s = 1, 2, … and i,j = 1, …,N . In Section 4 we recall the defintion of the Hopf algebra Y(𝔤𝔩_{N}) in terms of the generating series

(0.1)  T_{ij}(u) = δ_{ij} + T_{ij}^{(1)}u^{−1} + T_{ij}^{(2)}u^{−2} + ….

The classification of the irreducible finite-dimensional Y(𝔤𝔩_{N})-modules has been given by V. Drinfeld [D2] by generalizing the results of the second author [T1,T2]. But the structure of these modules still needs better understanding. For instance, the dimensions of these modules are not explicitly known in general.

There is a distinguished family of irreducible finite-dimensional Y(𝔤𝔩_{N})-modules [C2,O]. We call these modules elementary. Every elementary Y(𝔤𝔩_{N})-module is determined by a complex number z, and by two partitions λ = (λ_{1}, λ_{2} , …) and μ = (μ_{1}, μ_{2} , …) such that λ_{i} ⩾ μ_{i} for any i = 1, 2, …. This module depends on λ and μ through the skew Young diagram ω = λ/μ , see Section 1. We denote the corresponding Y(𝔤𝔩_{N})-module by V_{ω}(z) , its explicit construction will be given in Section 4. The vector spaces of the modules V_{ω}(z) are the same for all values of z, we denote this vector space by V_{ω}. Suppose that for some positive integer M the numbers of non-zero parts of λ and μ do not exceed M + N and M respectively. Then the dimension of V_{ω} equals the multiplicity of the irreducible 𝔤𝔩_{M} -module of highest weight (μ_{1} , …,μ_{M}) in the restriction to 𝔤𝔩_{M} of the the irreducible 𝔤𝔩_{M+N} - module of highest weight (λ_{1} , …,λ_{M+N}) . In particular, the space V_{ω} is non-zero if and only if λ_{i}^{∗} − μ_{i}^{∗} ⩽ N for any index i = 1, 2, …. Here λ^{∗} = (λ_{1}^{∗}, λ_{2}^{∗} , …) and μ^{∗} = (μ_{1}^{∗}, μ_{2}^{∗} , …) are the partitions conjugate to λ and μ respectively.

For any formal series f(u) ∈ ℂ[[u^{−1}]] with the leading term 1, the assignment T_{ij}(u) ↦ f(u) · T_{ij}(u) determines an automorphism of the algebra Y(𝔤𝔩_{N}) . Let us denote by α_{f} this automorphism. Further, there is a canonical chain of algebras Y(𝔤𝔩_{1}) ⊂ … ⊂ Y(𝔤𝔩_{N}) . The elementary modules are distinguished amongst all irreducible finite-dimensional Y(𝔤𝔩_{N})-modules V by the following fact. Consider the commutative subalgebra in Y(𝔤𝔩_{N}) generated by the centres of all algebras in that chain. This subalgebra is maximal commutative [C2,NO]. The action of this subalgebra in V is semi-simple if and only if the module V is obtained by pulling back through some automorphism α_{f} from the tensor product

(0.2)  V_{ω_{1}}(z_{1}) ⊗ … ⊗ V_{ω_{k}}(z_{k})

of the elementary Y(𝔤𝔩_{N})-modules, for some skew Young diagrams ω_{1} , …,ω_{k} and some complex numbers z_{1} , …,z_{k} such that z_{i} − z_{j} ∉ ℤ when i ≠ j. This fact was conjectured by I. Cherednik, and proved by him in [C2] under certain extra conditions on the module V . In full generality this fact has been proved in [NT1].

Furtermore, if z_{i} − z_{j} ∉ ℤ for all i ≠ j, the tensor product of Y(𝔤𝔩_{N})-modules (0.2) is always irreducible. Given the skew Young diagrams ω_{1} , …,ω_{k} it is natural to ask, for which values of z_{1} , …,z_{k} ∈ ℂ the Y(𝔤𝔩_{N})-module (0.2) is reducible. In [NT2] we answered this question in the case when each of the the diagrams ω_{1} , …,ω_{k} has a rectangular shape. In that particular case, our answer confirmed the following general conjecture. For any indices i and j consider the tensor product V_{ω_{i}}(z_{i}) ⊗ V_{ω_{j}}(z_{j}) . Consider also the tensor product V_{ω_{i}}(z_{i}) ˜⊗ V_{ω_{j}}(z_{j}) obtained via the opposite comultiplication on Y(𝔤𝔩_{N}) . If z_{i} − z_{j} ∉ ℤ, these two tensor products are irreducible and equivalent. Hence there is an invertible intertwining operator

R_{ij}(z_{i},z_{j}) : V_{ω_{i}}(z_{i}) ⊗ V_{ω_{j}}(z_{j}) ⟶ V_{ω_{i}}(z_{i}) ˜⊗ V_{ω_{j}}(z_{j}) ,

unique up to a non-zero multiplier from ℂ. These multipliers can be chosen so that R_{ij}(z_{i},z_{j}) depends on z_{i} and z_{j} as a rational function of z_{i} − z_{j} , see Section 4. Now for any z_{i}−z_{j} ∈ ℂ denote by I_{ij} the leading coefficient of the expansion of the rational function R_{ij}(z_{i}+u,z_{j}) into Laurent series in u near the origin u = 0 . This coefficient is always an interwining operator between the two tensor products. If z_{i} − z_{j} ∉ ℤ, we simply have I_{ij} = R_{ij}(z_{i},z_{j}) . It has been conjectured that the Y(𝔤𝔩_{N})-module (0.2) is irreducible if and only if for all i < j the operators I_{ij} are invertible; see for instance [CP]. We prove this conjecture in the present article, see Theorem 4.8.

This theorem implies, in particular, that the Y(𝔤𝔩_{N})-module (0.2) is irreducible if and only if for all i < j the pairwise tensor products V_{ω_{i}}(z_{i}) ⊗ V_{ω_{j}}(z_{j}) are irreducible. Given the skew Young diagrams ω_{i} and ω_{j} , it would be interesting to describe explicitly the set of all differences z_{i} − z_{j} ∈ ℤ, such that the module V_{ω_{i}}(z_{i}) ⊗ V_{ω_{j}}(z_{j}) is reducible. In the special case when both ω_{i} and ω_{j} are usual Young diagrams, this description was recently given in [LNT] for sufficiently large N , and independently in [M] in general.

The only if part of Theorem 4.8 is an easy and well known fact. Our proof of the if part is based on the following general observation. Let A be any unital associative algebra over ℂ, and let W be any finite-dimensional A-module. Denote by ρ the corresponding homomorphism A → End(W) . Suppose that the subalgebra

ρ(A) ⊗ End(W) ⊂ End(W) ⊗ End(W) = End(W ⊗ W)

contains the flip map x ⊗ y ↦ y ⊗ x in W ⊗ W . Then ρ(A) = End(W) and the A-module W is irreducible, see Lemma 4.10. We employ this observation when A = Y(𝔤𝔩_{N}) , and W is the tensor product (0.2). Moreover, to prove the irreducibility of (0.2) under the condition that all the operators I_{ij} with i < j are invertible, it suffices to show that I_{ii} is proportional to the flip map in V_{ω_{i}} ⊗ V_{ω_{i}} for each index i. For the details of this argument see Proposition 4.14. We took the idea of this argument from the recent physical literature [KMT,MT]. This argument extends to the quantum affine algebra U_{q}(𝔤̂𝔩_{N}) in a straightforward way. We work with the Yangian Y(𝔤𝔩_{N}) in this article only to simplify the exposition. Moreover, this argument extends to other quantum affine algebras due to the existence of the universal R-matrix; see [FM,K] for the precise formulation of the problem and known results in the general case.

In this article, we use the explicit realization of the elementary Y(𝔤𝔩_{N})-module V_{ω}(z) , introduced in [C2]. Denote λ_{1} − μ_{1} + λ_{2} − μ_{2} + … = n. Let n > 0 . Consider the action of the symmetric group S_{n} in (ℂ^{N})^{⊗n} by permutations of the n tensor factors. Denote by π_{n} the corresponding homomorphism ℂ · S_{n} → End((ℂ^{N})^{⊗n}). Then V_{ω} is the subspace in (ℂ^{N})^{⊗n} defined as the image of the operator π_{n}(F_{ω}) for a certain element F_{ω} of the group ring ℂ·S_{n}. This element is constructed by a limiting process, called the fusion procedure [C1]. Note that in [C1] most of the results were given without proofs. In Section 2 we give proofs of all properties of the element F_{ω} which we need in this article. These proofs are new, cf. [N].

This choice of the realization of the Y(𝔤𝔩_{N})-module V_{ω}(z) allows us to write down explicitly, for any z′ ∈ ℂ such that z − z′ ∉ ℤ, an invertible intertwining operator

R(z,z′) : V_{ω}(z) ⊗ V_{ω}(z′) ⟶ V_{ω}(z) ˜⊗ V_{ω}(z′) .

This operator comes with a natural normalization. In partucular, it is a rational function in z − z′ with values in End(V_{ω} ⊗ V_{ω}) . Denote by P_{ω} the flip map in V_{ω} ⊗ V_{ω}. We prove that the leading coefficient of the Laurent expansion in u near the origin u = 0 of the function R(z + u,z) equals c(ω)P_{ω} u^{−d(ω)} for a certain non-zero rational number c(ω) and a positive integer d(ω) , see Proposition 4.7. The proof of this proposition is based on the results of Section 3. We call d(ω) the Durfee rank of the skew Young diagram ω. When ω is a usual Young diagram, d(ω) is the number of boxes on its main diagonal. In Section 1 we study the remarkable combinatorial properties of the numbers d(ω) for all skew Young diagrams ω.

1. Durfee rank of a skew Young diagram

Let λ = (λ_{1}, λ_{2} , …) and μ = (μ_{1}, μ_{2} , …) be any two partitions. As usual, here the parts are arranged in the non-increasing order: we have λ_{1} ⩾ λ_{2} ⩾ … ⩾ 0 and μ_{1} ⩾ μ_{2} ⩾ … ⩾ 0 . We will always suppose that λ_{i} ⩾ μ_{i} for any i = 1, 2, …. Consider the skew Young diagram ω = λ/μ. It can be defined as the set of pairs

{ (i,j) ∈ ℤ^{2} | i ⩾ 1, λ_{i} ⩾ j > μ_{i} }

When μ = (0,0, …) , this is the Young diagram of the partition λ. We will employ the standard graphic representation of Young diagrams on the plane ℝ^{2} with the matrix style coordinates (x,y) . Here the first coordinate x increases from top to bottom, while the second coordinate y increases from left to right. The element (i,j) ∈ ω is represented by the unit box with the bottom right corner at the point (i,j) ∈ ℝ^{2} . For example, below we represent the skew Young diagram ω for the partitions λ = (9,9,9,7,7,3,3,3,3,0,0, …) and μ = (5,5,3,3,3,3,2,0,0, …) .

⟶ y |↓ x

We will identify the elements of the set ω with the corresponding unit boxes on ℝ^{2} .

Let us call an element (i,j) ∈ ω left-convex if (i−1,j) ∉ ω and (i,j −1) ∉ ω. An element (i,j) ∈ ω will be called left-concave if both (i−1,j) ∈ ω and (i,j −1) ∈ ω, but (i−1,j−1) ∉ ω. Each of these two definitions has a natural interpretation in terms of boxes of the Young diagram ω. If λ ≠ μ = (0,0, …) then there is only one leftconvex element (1,1) ∈ ω and no left-concave elements. Now consider those diagonals of ω, which contain a left-convex box. Let d be the total number of boxes on those diagonals. Let d′ be the total number of boxes on the diagonals of ω which contain a left-concave box. The difference d − d′ will be called the Durfee rank of the skew diagram ω and denoted by d(ω) . If μ = (0,0, …) then d(ω) = d is the number of boxes on the main diagonal { (i,i) ∈ ℤ^{2} | λ_{i} ⩾ i ⩾ 1 } of the Young diagram of λ, and our definition coincides with the usual one [ A, Section 2.3 ]. Below for our exemplary skew Young diagram ω we mark by the + or − signs the diagonals, which contain a left-convex or a left-concave box respectively.

+

+

+ − + + − +

+ + − +

In this example the Durfee rank d(ω) equals 9 − 3 = 6 . Our first observation is

Proposition 1.1. If the diagram ω is not empty then d(ω) > 0 .

Proof. For each i = 1,2, … let d_{i} be the number of boxes on that diagonal of ω which contains the leftmost box of the i-th row. If the i-th row of ω is empty, we set d_{i} = 0 . Let d′_{i} be the number of boxes on the next diagonal down. Then

(1.1)  d(ω) = d_{1} − d′_{1} + d_{2} − d′_{2} + ….

By the definition of a skew Young diagram we have d_{i} − d′_{i} ⩾ 0 for every index i = 1,2, …. But if i is the index of the last non-empty row of ω, then d_{i} = 1 and d′_{i} = 0 □

Denote by ℓ(ω) the number of non-empty rows in ω. Here is a formula for d(ω) .

Proposition 1.2 : d(ω) = ℓ(ω) − #{ (i,j) | λ_{j} −j = μ_{i} −i, i < j } .

Proof. We will keep to the notation from the proof of Proposition 1.1. Further, for each i = 1,2, … put e_{i} = 0 or e_{i} = 1 depending on whether the i-th row of ω is empty or not. By the definition of a skew Young diagram we have the equalities

d_{i} = #{ j | λ_{j} −j ⩾ μ_{i} −i+1 ⩾ μ_{j} −j +1 }
= #{ j | λ_{j} −j ⩾ μ_{i} −i+1, i ⩽ j }
= #{ j | λ_{j} −j ⩾ μ_{i} −i+1, i < j } + e_{i} ,
d′_{i} = #{ j | λ_{j} −j ⩾ μ_{i} −i ⩾ μ_{j} −j +1 }
= #{ j | λ_{j} −j ⩾ μ_{i} −i, i < j } .

Therefore,

d_{i} − d′_{i} = e_{i} − #{ j | λ_{j} −j = μ_{i} −i, i < j } .

Hence we obtain from (1.1) that

d(ω) = (e_{1} + e_{2} + …) − #{ (i,j) | λ_{j} −j = μ_{i} −i, i < j } .

By observing that e_{1} + e_{2} + … = ℓ(ω) we now complete the proof □

Let λ^{∗} = (λ_{1}^{∗}, λ_{2}^{∗} , …) and μ^{∗} = (μ_{1}^{∗}, μ_{2}^{∗} , …) be the partitions conjugate to λ and μ respectively. Consider the corresponding skew Young diagram ω^{∗} = λ^{∗}/μ^{∗}. It is obtained from the diagram ω ⊂ ℤ^{2} by transposition (i,j) ↦ (j,i) . Evidently, d(ω^{∗}) = d(ω) . Therefore, Proposition 1.2 can be reformulated as

Proposition 1.3 : d(ω) = ℓ(ω^{∗}) − #{ (i,j) | λ_{j}^{∗} −j = μ_{i}^{∗} −i, i < j } .

By our definition, the Durfee rank d(ω) depends only on the shape of the diagram ω. That is, if a skew Young diagram ω^{♭} is obtained from ω ⊂ ℤ^{2} by translation (i,j) ↦ (i+k,j+l) for a certain (k,l) ∈ ℤ^{2} , then d(ω^{♭}) = d(ω) . Furthermore, suppose that a skew Young diagram ω^{♮} is obtained from ω via the map (i,j) ↦ (k−i+1,l−j+1) . The boxes of ω^{♮} on the plane ℝ^{2} are obtained from those of ω via rotation by 180^{∘} around the point (\frac{k}{2} , \frac{l}{2}) . The main result of this section is

Theorem 1.4 : d(ω^{♮}) = d(ω) .

Proof. We have the inequalities λ_{1} − 1 > λ_{2} − 2 > … and μ_{1} − 1 > μ_{2} − 2 > …. So under the condition λ_{j} −j = μ_{i} −i in Proposition 1.2 we have the equivalencies

i < j ⇔ λ_{i} −i > λ_{j} −j ⇔ λ_{i} −i > μ_{i} −i ⇔ λ_{i} > μ_{i} ,
i < j ⇔ μ_{i} −i > μ_{j} −j ⇔ λ_{j} −j > μ_{j} −j ⇔ λ_{j} > μ_{j} .

Denote by L(ω) the set of differences μ_{i}−i such that the i-th row of ω is not empty. The set of differences λ_{j} −j such that the j-th row of ω is not empty, will be denoted by R(ω) . By the above equivalencies, Proposition 1.2 can be restated as

(1.2)  d(ω) = ℓ(ω) − #(L(ω) ∩ R(ω)) .

We have ℓ(ω^{♮}) = ℓ(ω) . We also have #(L(ω^{♮}) ∩ R(ω^{♮})) = #(L(ω) ∩ R(ω)) since

L(ω^{♮}) = l −k −R(ω) −1, R(ω^{♮}) = l −k −L(ω) −1,

by the defintion of ω^{♮}. Thus we obtain Corollary 1.4 from the formula (1.2) □

Let us call an element (i,j) ∈ ω right-convex if (i+1,j) ∉ ω and (i,j +1) ∉ ω. An element (i,j) ∈ ω is called right-concave if both (i +1,j) ∈ ω and (i,j +1) ∈ ω, but (i +1,j +1) ∉ ω. Each of these two definitions has a natural interpretation in terms of boxes of the Young diagram ω. We could define the Durfee rank of ω using the right-convex and right-concave boxes instead of the left-convex and left-concave, respectively. Theorem 1.4 ensures that we would then get the same number d(ω) . Note that although the Young diagram of a non-empty partition always has only one left-convex box and no left-concave boxes, it may have several right-convex and rightconcave boxes. So our definition of the Durfee rank of ω is quite natural.

2. Fusion procedure for a skew Young diagram

Take a non-empty skew Young diagram ω. Let n be the total number of boxes in ω. In this section we will introduce a certain element F_{ω} of the symmetric group ring ℂ · S_{n}. Under some extra conditions on the diagram ω, this element will be used in Section 4 to determine a family of irredicible modules V_{ω}(z) over the Yangian Y(𝔤𝔩_{N}) . Here the parameter z is ranging over the complex field ℂ. For the skew Young diagram ω^{♭} obtained from ω by translation (i,j) ↦ (i + k,j + l) for a certain (k,l) ∈ ℤ^{2}, we will have V_{ω^{♭}}(z) = V_{ω}(z − k + l) .

Consider the column tableau of shape ω. It is obtained by filling the boxes of the diagram ω with the numbers 1, …,n consecutively by columns from left to right, downwards in every column. We will denote this tableau by Ω . Further, for each p = 1, …,n put c_{p} = j − i if the box (i, j) ∈ ω is filled with the number p in the column tableau Ω . The difference j − i is called the content of the box (i, j) of the diagram ω. Our choice of the tableau Ω provides an ordering of the collection of all contents of ω. Below on the left for the partitions λ = (5,3,3,3,3,0,0, …) and μ = (3,3,3,2,0,0, …) we show the column tableau Ω . On the right we indicate the contents of all boxes of the diagram ω.

⟶ j | ↓

i

8 9 3 4

5 0 1 3 6 -3 -2 -1 2 4 7 -4 -3 -2

Here n = 9 and the sequence of contents (c_{1} , …,c_{9}) is (-3,-4,-2,-3,0,-1,-2,3,4) .

For any two distinct numbers p,q ∈ {1, …,n} let (pq) be the transposition in the symmetric group S_{n}. Consider the rational functions of two complex variables u,v with values in the group ring ℂ ·S_{n}

(2.1)  f_{pq}(u,v) = 1 − \frac{(pq)}{u − v} .

As a direct calculation shows, these rational functions satisfy the equations

(2.2)  f_{pq}(u,v)f_{pr}(u,w)f_{qr}(v,w) = f_{qr}(v,w)f_{pr}(u,w)f_{pq}(u,v)

for all pairwise distinct indices p,q,r. Evidently, for all pairwise distinct p,q,s,t

(2.3)  f_{pq}(u,v)f_{st}(z,w) = f_{st}(z,w)f_{pq}(u,v) .

For all distinct p, q we also have

(2.4)  f_{pq}(u,v) f_{qp}(v,u) = 1 − \frac{1}{(u − v)^{2}} .

Our construction of the element F_{ω} ∈ ℂ · S_{n} will be based on the following simple observation. Consider the rational function of u,v ,w defined by the product at either side of (2.2). The factor f_{pr}(u,w) in (2.2) has a pole at u = w. However, we have

Lemma 2.1. The restriction of the rational function (2.2) to the set of (u,v ,w) such that u = v ± 1 , is regular at u = w.

Proof. Under the condition u = v ± 1 the product on the left hand side of (2.2) can be written as

(1 ∓ (pq)) · (1 − \frac{(pr) + (q r)}{v − w}).

The latter rational function of v,w is manifestly regular at w = v ± 1 □

Now introduce n complex variables z_{1} , …,z_{n}. Equip the set of all pairs (p,q) where 1 ⩽ p < q ⩽ n, with the lexicographical ordering. Take the ordered product

(2.5)  ∏_{(p,q)}^{⟶} f_{pq}(c_{p} + z_{p} , c_{q} + z_{q})

over this set. Consider (2.5) as a rational function of the variables z_{1} , …,z_{n} with values in ℂ · S_{n}. Denote this rational function by F_{ω}(z_{1} , …,z_{n}) . Let D_{ω} be the vector subspace in ℂ^{n} consisting of all tuples (z_{1} , …,z_{n}) such that z_{p} = z_{q} whenever the numbers p and q appear in the same column of the tableau Ω . The origin (0, …,0) ∈ ℂ^{n} belongs to D_{ω}. The following statement goes back to [C1].

Proposition 2.2. The restriction of the rational function F_{ω}(z_{1} , …,z_{n}) to the subspace D_{ω} ⊂ ℂ^{n} is regular at the point (0, …,0) .

Proof. We shall provide an expression for the restriction of F_{ω}(z_{1} , …,z_{n}) to D_{ω} which is manifestly regular at (0, …,0) . The factor f_{pq}(c_{p} + z_{p} , c_{q} + z_{q}) has a pole at z_{p} = z_{q} if and only if the numbers p and q stand on the same diagonal of the tableau Ω . We shall then call the pair (p,q) singular.

Let any singular pair (p,q) be fixed. We will write (p,q) ≺ (s,t) if the pair (p,q) precedes (s,t) in the lexicographical ordering, that is, if p < s or if p = s, q < t. In this ordering, the pair immediately before (p,q) is (p,q − 1) . Moreover, the number q − 1 stands just above q in the tableau Ω . Therefore, z_{q−1} = z_{q} on D_{ω}. We also have (p,q) ≺ (q−1,q) , since p < q. By (2.2) and (2.3) the product over the pairs (s,t)

∏_{(p,q)≺(s,t)}^{⟶} f_{st}(c_{s} + z_{s} , c_{t} + z_{t})

is divisible on the left by the factor f_{q−1,q}(c_{q−1}+z_{q−1} , c_{q}+z_{q}) . Restriction of this factor to z_{q−1} = z_{q} is just 1− (q − 1,q) since c_{q−1} = c_{q} +1 .

The element (1 − (q − 1,q))/2 ∈ ℂ · S_{n} is an idempotent. Now for every singular pair (p,q) let us replace the two adjacent factors in (2.5)

f_{p,q−1}(c_{p} + z_{p} , c_{q−1} + z_{q−1})f_{pq}(c_{p} + z_{p} , c_{q} + z_{q})

by

f_{p,q−1}(c_{p} + z_{p} , c_{q−1} + z_{q−1})f_{pq}(c_{p} + z_{p} , c_{q} + z_{q})f_{q−1,q}(c_{q−1} + z_{q−1}, c_{q} + z_{q})/2 .

This does not affect the values of the restriction of (2.5) to D_{ω}. But the restriction to z_{q−1} = z_{q} of the replacement product is regular at z_{p} = z_{q} by Lemma 2.1 □

Due to Proposition 2.2, an element F_{ω} ∈ ℂ·S_{n} can be now defined as the value at the point (0, …,0) of the restriction to D_{ω} of the function F_{ω}(z_{1} , …,z_{n}) . Note that for n = 1 we have F_{ω} = 1 . For any n ⩾ 1 we have the following fact.

Proposition 2.3. The coefficient of F_{ω} ∈ ℂ ·S_{n} at the unit element of S_{n} is 1 .

Proof. For each r = 1, …,n − 1 let g_{r} ∈ S_{n} be the adjacent transposition (r, r +1) . Let g_{0} ∈ S_{n} be the element of the maximal length. Let us multiply the ordered product (2.5) by the element g_{0} on the right. Using the reduced decomposition

(2.6)  g_{0} = ∏_{(p,q)}^{⟶} g_{q−p} ,

we get the product

∏_{(p,q)}^{⟶} (g_{q−p} − \frac{1}{c_{p} +z_{p} −c_{q} −z_{q}} ) .

It expands as a sum of the elements g ∈ S_{n} with the coefficients from the field of rational functions of z_{1} , …,z_{n} valued in ℂ. Since the decomposition (2.6) is reduced, the coefficient at g_{0} is 1 . By the definition of F_{ω} this implies that the coefficient of F_{ω} g_{0} ∈ ℂ ·S_{n} at g_{0} ∈ S_{n} is also 1 □

In particular, Proposition 2.3 shows that F_{ω} ≠ 0 for any non-empty diagram ω. Let us now denote by α the involutive antiautomorphism of the group ring ℂ ·S_{n} defined by α(g) = g^{−1} for every g ∈ S_{n}.

Proposition 2.4. The element F_{ω} ∈ ℂ ·S_{n} is α-invariant.

Proof. Any element of the group ring ℂ · S_{n} of the form f_{pq}(u,v) is α-invariant . Applying the antiautomorphism α to an element of the form (2.5) just reverses the ordering of the factors corresponding to the pairs (p,q) . However, the initial ordering can be then restored by using the relations (2.2) and (2.3). Therefore, any value of the function F_{ω}(z_{1} , …,z_{n}) is α-invariant. So is the element F_{ω} □

The next property of the element F_{ω} ∈ ℂ ·S_{n} also easily follows from its definition.

Theorem 2.5. Suppose the numbers r and r + 1 stand in the same column of Ω . Then the element F_{ω} ∈ ℂ ·S_{n} is divisible by f_{r,r+1}(c_{r},c_{r+1}) = 1−(r, r+1) on the left and right.

Proof. By Proposition 2.4 the divisibility of F_{ω} by the element 1− (r, r +1) on the left is equivalent to the divisibility by the same element on the right. Let us prove the divisibility on the left. Using the definition (2.5) along with the relations (2.2) and (2.3), we can write

(2.7)  F_{ω}(z_{1} , …,z_{n}) = f_{r,r+1}(c_{r} +z_{r},c_{r+1} +z_{r+1} ) F (z_{1} , …,z_{n})

for some rational function F (z_{1} , …,z_{n}) valued in ℂ · S_{n}. By Proposition 2.2, the restriction of the function F_{ω}(z_{1} , …,z_{n}) to D_{ω} is regular at (0, …,0) . But the restriction to D_{ω} of the factor f_{r,r+1}(c_{r} +z_{r},c_{r+1} +z_{r+1} ) in (2.7) is f_{r,r+1}(c_{r},c_{r+1}) . In particular, the value at (0, …,0) of the restriction of F_{ω}(z_{1} , …,z_{n}) to D_{ω} is divisible on the left by f_{r,r+1}(c_{r},c_{r+1}) = 1−(r, r+1) □

Let l = 1, …,n − 1 . Denote by σ_{l} the embedding ℂ · S_{n−l} → ℂ · S_{n} defined by σ_{l} : (pq) ↦ (p + l,q + l) for all distinct p, q = 1, …,n − l. Define a skew Young diagram φ as the shape of the tableau obtained from Ω by removing each of the numbers 1, …,l. Consider the corresponding element F_{φ} ∈ ℂ ·S_{n−l}.

Proposition 2.6. The element F_{ω} is divisible by σ_{l}(F_{φ}) on the left and right.

Proof. Due to Proposition 2.4 the divisibility of F_{ω} by the element σ_{l}(F_{φ}) on the left is equivalent to the divisibility by the same element on the right. We will actually prove the divisibility on the right. By the definition (2.5) we have

(2.8)  F_{ω}(z_{1} , …,z_{n}) = ∏_{(p,q)}^{⟶} f_{pq}(c_{p} + z_{p} , c_{q} + z_{q}) · σ_{l}(F_{φ}(z_{l+1} , …,z_{n}))

where the pairs (p,q) are ordered lexicigraphically, and precede the pair (l + 1,l + 2) . According to our proof of Proposition 2.2, the product over the pairs (p,q) in (2.8) maybe replaced by a certain rational function of z_{1} , …,z_{n} with the restriction to D_{ω} regular at (0, …,0) , without affecting the values of the restriction. Now the divisibility of F_{ω} by σ_{l}(F_{φ}) on the right follows from the decomposition (2.8) and the definition of the element F_{φ} ∈ ℂ ·S_{n−l} □

Let m = 1, …,n − 1 . Regard the group ring ℂ · S_{m} as a subalgebra in ℂ · S_{n} using the standard embedding S_{m} → S_{n}. Define a skew Young diagram ψ as the shape of the tableau obtained from Ω by removing each of the numbers m+1, …,n. Consider the corresponding element F_{ψ} ∈ ℂ ·S_{m}. The following property of the element F_{ω} is obtained by combining the arguments of Propositions 2.2 and 2.6.

Proposition 2.7. The element F_{ω} ∈ ℂ ·S_{n} is divisible by F_{ψ} on the left and right.

Proof. Let us introduce another expression for restriction of F_{ω}(z_{1} , …,z_{n}) to D_{ω} which is again manifestly regular at (0, …,0) . Reorder the pairs (p,q) in the product (2.5) as follows; this reordering will not affect the values of the product due to (2.2) and (2.3). The pair (s,t) will precede the pair (p,q) if t < q or if t = q, s < p. Let us now write (s,t) ≺ (p,q) referring to this new ordering.

Let any singular pair (p,q) be fixed. In our new ordering, the pair immediately after (p,q) is (p+1,q) . Moreover, the number p+1 stands just below p in the tableau Ω . Therefore, z_{p} = z_{p+1} on D_{ω}. We also have (p,p+1) ≺ (p,q) , since p < q. By (2.2) and (2.3) the product

∏_{(s,t)≺(p,q)}^{⟶} f_{st}(c_{s} + z_{s} , c_{t} + z_{t})

over the pairs (s,t) is divisible on the right by the factor f_{p,p+1}(c_{p} + z_{p} , c_{p+1} + z_{p+1}) . Restriction of this factor to z_{p} = z_{p+1} is just 1− (p,p + 1) since c_{p+1} = c_{p} −1 .

The element (1 − (p,p + 1))/2 ∈ ℂ · S_{n} is an idempotent. Now for every singular pair (p,q) let us replace the two adjacent factors in the reordered product (2.5)

f_{pq}(c_{p} + z_{p} , c_{q} + z_{q})f_{p+1,q}(c_{p+1} + z_{p+1} , c_{q} + z_{q})

by

f_{p,p+1}(c_{p} +z_{p} , c_{p+1} +z_{p+1}) f_{pq}(c_{p} + z_{p} , c_{q} + z_{q}) f_{p+1,q}(c_{p+1} + z_{p+1} , c_{q} + z_{q})/2 .

This does not affect the values of the restriction of (2.5) to D_{ω}. But the restriction to z_{p} = z_{p+1} of the replacement product is regular at z_{p} = z_{q} by Lemma 2.1.

Due to Proposition 2.4 the divisibility of F_{ω} by the element F_{ψ} on the left is equivalent to the divisibility by the same element on the right. Let us now prove the divisibility on the left. In our new ordering of the factors in (2.5), we have

(2.9)  F_{ω}(z_{1} , …,z_{n}) = F_{ψ}(z_{1} , …,z_{m}) · ∏_{(p,q)}^{⟶} f_{pq}(c_{p} + z_{p} , c_{q} + z_{q})

where (p, q) ≻ (m−1,m) . By the above argument, in (2.9) the product over the pairs (p,q) may be replaced by a certain rational function of z_{1} , …,z_{n} with the restriction to D_{ω} regular at (0, …,0) , without affecting the values of the restriction. Now the divisibility of F_{ω} by F_{ψ} on the left follows from the decomposition (2.9) and the definition of the element F_{ψ} ∈ ℂ ·S_{m} □

We can now give a relatively short proof of the main result of this section, cf. [N].

Theorem 2.8. Suppose the numbers p < q stand next to each other in the same row of the tableau Ω . Let r be the number at the bottom of the column of Ω containing p. Then the element F_{ω} is divisible on the left by the product

(2.10)  ∏_{s=p,...,r}^{⟵}( ∏_{t=r+1,...,q}^{⟶} f_{st}(c_{s},c_{t})) .

Proof. It suffices to prove Theorem 2.8 only in the particular case q = n. For q < n the required statement would then follow by Proposition 2.7 applied to m = q. Furthermore, it suffices to consider only the case p = 1 . For p > 1 the required statement would then follow by Proposition 2.6 applied to l = p − 1 .

Assume that p = 1 and q = n. Then the skew Young diagram ω consists of two columns only. Further, every row of ω but one consists of a single box. There is also a row of two boxes, which in the tableau Ω are filled with numbers 1 and n. So there is only one box on every diagonal of ω. The function F_{ω}(z_{1} , …,z_{n}) is then regular at (0, …,0) and takes the value

∏_{s=1,...,n−1}^{⟶}( ∏_{t=s+1,...,n}^{⟶} f_{st}(c_{s},c_{t})) = ∏_{s=1,...,r}^{⟵}( ∏_{t=r+1,...,n}^{⟶} f_{st}(c_{s},c_{t})) ×

∏_{s=1,...,r−1}^{⟶}( ∏_{t=s+1,...,r}^{⟶} f_{st}(c_{s},c_{t})) · ∏_{s=r+1,...,n}^{⟶}( ∏_{t=s+1,...,n}^{⟶} f_{st}(c_{s},c_{t}));

the last equality was obtained by using the relations (2.2) and (2.3). So the element F_{ω} is indeed divisible on the left by the product (2.10) with p = 1 and q = n □

3. Leading term near the origin

Take any two non-empty skew Young diagrams ω and ω′. Let n and n′ be the numbers of boxes in ω and ω′ respectively. In this section we will introduce a certain rational function F_{ωω′}(u) of one complex variable u with the values in the symmetric group ring ℂ · S_{n+n′} . Let z and z′ be any two complex numbers. In Section 4, the leading term of the Laurent expansion of the function F_{ωω′}(u) at the point u = z −z′ will determine an intertwining operator of Y(𝔤𝔩_{N})-modules

V_{ω}(z) ⊗ V_{ω′}(z′) ⟶ V_{ω}(z) ˜⊗ V_{ω′}(z′) .

Here the tilde refers to the opposite comultiplication on the Hopf algebra Y(𝔤𝔩_{N}) . In the present section we will compute the leading term of the Laurent expansion of F_{ωω′}(u) with ω = ω′ near the origin u = 0 . This result will be used in Section 4.

Let us regard ℂ · S_{n} as a subalgebra in ℂ · S_{n+n′} via the standard embedding S_{n} → S_{n+n′} . In particular, F_{ω} ∈ ℂ · S_{n} may be regarded as an element of ℂ · S_{n+n′} . Consider also the embedding ℂ · S_{n′} → ℂ · S_{n+n′} defined by (pq) ↦ (p + n,q + n) for distinct p, q = 1, …,n′. In this section, the image in ℂ · S_{n+n′} of any element F ∈ ℂ ·S_{n′} under the latter embedding will be denoted by F^{∨}.

Let c_{1} , …,c_{n} and c′_{1} , …,c′_{n′} be the sequences of contents corresponding to the skew diagrams ω and ω′. Here we order the contents using the column tableaux. Define the function F_{ωω′}(u) as the ordered product

(3.1)  F_{ω} · ∏_{p=1,...,n}^{⟶}( ∏_{q=1,...,n′}^{⟶} f_{p,q+n}(c_{p} +u, c′_{q})) · F_{ω′}^{∨} .

Note that the so defined rational function F_{ωω′}(u) may have poles only at u ∈ ℤ.

Proposition 3.1. The function F_{ωω′}(u) can be written as either of the products

∏_{p=1,...,n}^{⟵}( ∏_{q=1,...,n′}^{⟶} f_{p,q+n}(c_{p} +u, c′_{q})) · F_{ω} F_{ω′}^{∨} , (3.2)

F_{ω} F_{ω′}^{∨} · ∏_{p=1,...,n}^{⟶}( ∏_{q=1,...,n′}^{⟵} f_{p,q+n}(c_{p} +u, c′_{q})) . (3.3)

Proof. Using the definition (2.5) along with the relations (2.2) and (2.3), we obtain the equality of the rational functions in z_{1} , …,z_{n} and u

F_{ω}(z_{1} , …,z_{n}) · ∏_{p=1,...,n}^{⟶}( ∏_{q=1,...,n′}^{⟶} f_{p,q+n}(c_{p} +z_{p} +u, c′_{q})) =

∏_{p=1,...,n}^{⟵}( ∏_{q=1,...,n′}^{⟶} f_{p,q+n}(c_{p} +z_{p} +u, c′_{q})) · F_{ω}(z_{1} , …,z_{n}) .

By the definition of the element F_{ω} ∈ ℂ · S_{n}, this equality implies that the products (3.1) and (3.2) are also equal to each other. Similarly, we can prove the equality of the products (3.1) and (3.3) □

Now consider the function F_{ωω}(u) . For any diagram ω^{♭} obtained from ω ⊂ ℤ^{2} by a translation, we have F_{ω^{♭}}_{ω^{♭}}(u) = F_{ωω}(u) . Our preliminary result on F_{ωω}(u) is

Proposition 3.2. The order of the pole of F_{ωω}(u) at u = 0 does not exceed d(ω) .

Proof. We will use the expression for d(ω) given by Proposition 1.3. Our argument will be similar to that of Proposition 2.2, but more elaborate, cf. [N,Theorem 4.1].

The function F_{ωω}(u) is defined as the ordered product

(3.4)  F_{ω} · ∏_{p=1,...,n}^{⟶}( ∏_{q=1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) · F_{ω}^{∨} .

The factor f_{p,q+n}(c_{p} + u, c_{q}) in (3.4) has a pole at u = 0 if and only if the numbers p and q stand on the same diagonal of the tableau Ω . This pole is simple. Again, we will then call the pair (p,q) singular. Any singular pair (p,q) belongs to one of the following three types:

(i) the number p is not at the bottom of its column in Ω ; (ii) p is at the bottom of its column in Ω but q is not leftmost in its row; (iii) p is at the bottom of its column in Ω and q is leftmost in its row.

Observe that the total number of singular pairs of type (iii) is exactly d(ω) . Indeed, let p be at the bottom of the j-th column. Then c_{p} = j − λ_{j}^{∗}. Consider the diagonal of the tableau Ω containing p. Any number q on this diagonal, except maybe for the first number, has in Ω a neighbour on the left. As usual, here we are reading the diagonal from top left to bottom right. Suppose q is the first on the diagonal. The total number of such pairs (p, q) equals the number of the columns, which is ℓ(ω^{∗}) . But if q has a neighbour on the left, say in the i-th row, then i − μ_{i}^{∗} ⩾ c_{q}. If we had i − μ_{i}^{∗} > c_{q}, the number q could not be the first on its diagonal. So we must have i−μ_{i}^{∗} = c_{q} = c_{p} = j −λ_{j}^{∗}. Here we also have i < j. Thus the total number of singular pairs (iii) equals

ℓ(ω^{∗}) − #{ (i,j) | λ_{j}^{∗} −j = μ_{i}^{∗} −i, i < j } = d(ω) .

We shall prove that when we estimate the order of pole of (3.4) at u = 0 from above, the singular pairs (i) and (ii) do not count. Proposition 3.2 will then follow.

I. Using only the relations (2.3), we can rewrite the ordered product (3.4) as

(3.5)  F_{ω} · ∏_{q=1,...,n}^{⟶}( ∏_{p=1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) · F_{ω}^{∨} .

Consider any singular pair (p,q) of type (i). In the ordered product (3.5) the factor f_{p,q+n}(c_{p} +u, c_{q}) is immediately followed by factor f_{p+1,q+n}(c_{p+1} +u, c_{q}) . Note that here c_{p} = c_{q}. The numbers p and p+1 stand in the same column of the tableau Ω . By Theorem 2.5, the element F_{ω} is divisible on the right by

f_{p,p+1}(c_{p},c_{p+1}) = 1−(p, p+1) = f_{p,p+1}(c_{p} +u,c_{p+1} +u) .

Again using the relations (2.2) and (2.3) we obtain, that the product of all factors in (3.5) preceding f_{p,q+n}(c_{p} + u, c_{q}) is also divisible by f_{p,p+1}(c_{p} + u,c_{p+1} + u) on the right. But the product

f_{p,p+1}(c_{p} +u,c_{p+1} +u) f_{p,q+n}(c_{p} +u, c_{q}) f_{p+1,q+n}(c_{p+1} +u, c_{q})

has no pole at u = 0 by Lemma 2.1. Thus when we estimate the order of pole of (3.5) at u = 0 from above, the singular pairs (i) do not count.

II. Now consider the diagonal of Ω containing the number n. Let q_{1} , …,q_{d} be all entries of Ω on that diagonal. We assume that q_{1} < … < q_{d} so that q_{d} = n. The singular pairs (n,q_{2}), …,(n,q_{d}) are all of type (ii). The singular pair (n,q_{1}) is of type (ii) or (iii). Depending on this, let the index k range over 1, …,d or over 2, …,d respectively. For every such k let p_{k} be the number standing in Ω next to the left of q_{k}. Of course, for k > 1 we have p_{k} = q_{k−1}+1 . However, to include the possible case k = 1 in our argument, we will still use the notation p_{k}.

Let r_{k} be the number standing in the tableau Ω at the bottom of the column containing p_{k}. The number at the top of the column containing q_{k} is then r_{k} +1 . By Theorem 2.8, the element F_{ω} ∈ ℂ ·S_{n} is divisible on the left by the product

∏_{s=p_{k},...,r_{k}}^{⟵}( ∏_{t=r_{k}+1,...,q_{k}}^{⟶} f_{st}(c_{s},c_{t})) = (3.6)

∏_{t=r_{k}+1,...,q_{k}}^{⟶}( ∏_{s=p_{k},...,r_{k}}^{⟵} f_{st}(c_{s},c_{t})) .

For distinct indices k the elements of ℂ · S_{n} defined by (3.6) pairwise commute. Let F be the element of ℂ ·S_{n} obtained by multiplying the elements (3.6) over all k. We can write F_{ω} = F G for some G ∈ ℂ ·S_{n}. We can also write F_{ω}^{∨} = F^{∨}G^{∨} in ℂ ·S_{2n}.

Put r = r_{d} for short. Using the relations (2.3) we can rewrite the product (3.5) as

F_{ω} · ∏_{q=1,...,n}^{⟶}( ∏_{p=1,...,r}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) × (3.7)

∏_{q=1,...,n}^{⟶}( ∏_{p=r+1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) · F_{ω}^{∨} .

Denote by Q the sequence obtained from the sequence 1, …,n by transposing its segments p_{k} , …,r_{k} and r_{k}+1, …,q_{k} for every possible index k. Using the relations (2.2),(2.3) and the decomposition F_{ω}^{∨} = F^{∨}G^{∨}, we can rewrite (3.7) as

F_{ω} · ∏_{q=1,...,n}^{⟶}( ∏_{p=1,...,r}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) × (3.8)

F^{∨} · ∏_{q ∈Q}^{⟶} ( ∏_{p=r+1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) · G^{∨} .

As well as in the product (3.5), here for every singular pair (p,q) of type (i) the factor f_{p,q+n}(c_{p} +u, c_{q}) is still followed by factor f_{p+1,q+n}(c_{p+1} +u, c_{q}) .

Let us now take any singular pair (n,q_{k}) of type (ii). In the second line of the ordered product (3.8), consider the sequence of factors

(3.9)  ∏_{q=q_{k},p_{k}}^{⟶} ( ∏_{p=r+1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) .

Note that c_{p} = c_{q} in (3.9) only if (p,q) = (n,q_{k}) . The rightmost factor in the product (3.6) is f_{p_{k}q_{k}}(c_{p_{k}},c_{q_{k}}) . Due to (2.2) and (2.3), the product of all factors in (3.8) preceding the sequence (3.9) is divisible by f_{p_{k}+n,q_{k}+n}(c_{p_{k}},c_{q_{k}}) on the right. But the cumulate product

f_{p_{k}+n,q_{k}+n}(c_{p_{k}}, c_{q_{k}}) · ∏_{q=q_{k},p_{k}}^{⟶} ( ∏_{p=r+1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) =

∏_{q=p_{k},q_{k}}^{⟶} ( ∏_{p=r+1,...,n−1}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) ×

f_{n,p_{k}+n}(c_{n} +u, c_{p_{k}}) f_{n,q_{k}+n}(c_{n} +u, c_{q_{k}}) f_{p_{k}+n,q_{k}+n}(c_{p_{k}}, c_{q_{k}})

has no pole at u = 0 due to Lemma 2.1. Thus when we estimate the order of pole of (3.8) at u = 0 from above, any singular pair (n,q_{k}) of type (ii) does not count.

III. By rewriting the product (3.7) in its second line as (3.8), we excluded from our count all singular pairs (p,q) of type (ii) with p = n. If the skew Young diagram ω consists of only one column, there is no singular pairs of type (ii) and the proof finishes on step I already. Suppose that the diagram ω has more than one column. Now let ψ be the skew Young diagram obtained from ω by removing the last non-empty column. Let m be the total number of boxes in ψ. Due to Proposition 2.7, the element F_{ω}^{∨} is divisible by F_{ψ}^{∨} on the left. But we also have

∏_{q=1,...,n}^{⟶}( ∏_{p=r+1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) · F_{ψ}^{∨} = F_{ψ}^{∨} ×

∏_{q =m,...,1,m+1,...,n}^{⟶}( ∏_{p=r+1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) ,

see Proposition 3.1. Hence the entire product in the second line of (3.7) is divisible on the left by F_{ψ}^{∨}. Now arguing similarly to step II, we exclude from our count all singular pairs (p,q) of type (ii) with p = m. Continuing this argument, we exclude from our count all singular pairs of type (ii). All singular pairs of type (i) can be still excluded from our count, as it was done on step I □

For the skew Young diagram ω, let us introduce the rational function of u

(3.10)  h_{ω}(u) = (\frac{−1}{u})^{n} · ∏_{1⩽p<q⩽n} (1 − \frac{1}{(u−c_{p}+c_{q})^{2}} ) .

Recall once again, that here the contents c_{1} , …,c_{n} of ω are ordered by using the column tableau Ω . Now consider the skew Young diagram ω^{∗} conjugate to ω, and the corresponding rational function h_{ω^{∗}}(u) . In general, we do not have the equality h_{ω^{∗}}(u) = (−1)^{n} · h_{ω}(−u) . But the following proposition is always true.

Proposition 3.3. The leading term of the Laurent expansion at u = 0 of the function h_{ω^{∗}}(u) coincides with that of the function (−1)^{n} · h_{ω}(−u) .

Proof. Consider the sequence of numbers obtained by reading the entries of the column tableau Ω in the natural way, that is by rows downwards, from left to right in every row. This sequence can be also obtained from the sequence 1, …,n by a certain permutation g ∈ S_{n}. In general, the permutation g is non-trivial. The sequence of contents corresponding to the diagram ω^{∗} is −c_{g(1)} , …,−c_{g(n)} . So

(−1)^{n} · h_{ω^{∗}}(−u) = (\frac{−1}{u})^{n} · ∏_{1⩽p<q⩽n} (1 − \frac{1}{(u−c_{g(p)}+c_{g(q)})^{2}} ) .

Dividing the right hand side of this equality by that of (3.10), we get the product of

(3.11)  (1 − \frac{1}{(u+c_{p} −c_{q})^{2}} ) · (1 − \frac{1}{(u−c_{p} +c_{q})^{2}} )^{−1}

over all pairs (p,q) such that p < q but g^{−1}(p) > g^{−1}(q) . For any such pair, the number q stands above and to the right of p in the tableau Ω . In particular, then c_{p} −c_{q} ≠ ±1 . But then the factor (3.11) is regular at u = 0 , and takes the value 1 at that point □

The next proposition provides another expression for the function h_{ω}(u) with the arbitrary skew Young diagram ω = λ/μ. Consider the infinite product

(3.12)  ∏_{i} \frac{u−λ_{i} +μ_{i}}{u} · ∏_{i<j} \frac{(u+λ_{i} −μ_{j} −i+j) (u+μ_{i} −λ_{j} −i+j)}{(u+λ_{i} −λ_{j} −i+j) (u+μ_{i} −μ_{j} −i+j)}

where i,j = 1,2, …. Here the first fraction equals 1 for all but a finite number of indices i. The second fraction equals 1 for all but a finite number of indices i and j. Hence the product (3.12) determines a rational function of the variable u.

Proposition 3.4. The rational function (3.12) is equal to h_{ω^{∗}}(u) · (1−u)^{n}.

Proof. For each i = 1, 2, … denote by h_{i}(u) the product of the factors

(3.13)  1 − \frac{1}{(u−c_{p} +c_{q})^{2}} = \frac{u−c_{p}+c_{q} −1}{u−c_{p}+c_{q}} · \frac{u−c_{p}+c_{q} +1}{u−c_{p}+c_{q}}

over p < q such that the numbers p and q stand in the i-th column of the tableau Ω . If this column is empty, we put h_{i}(u) = 1 . For these numbers p and q we have c_{p} − c_{q} = q − p. The total number of boxes in this column is λ_{i}^{∗} − μ_{i}^{∗}. So by direct calcultion

h_{i}(u) = \frac{(u−λ_{i}^{∗} +μ_{i}^{∗}) u^{λ_{i}^{∗}−μ_{i}^{∗}−1}}{(u−1)^{λ_{i}^{∗}−μ_{i}^{∗}}} .

Further, for each i put a_{i} = i−μ_{i}^{∗}−1 and b_{i} = i−λ_{i}^{∗}. The numbers a_{i} and b_{i} are the contents respectively of the top and the bottom boxes of the i-th column of the diagram ω, if this column is not empty. If it is empty, λ_{i}^{∗} = μ_{i}^{∗} and b_{i} = a_{i} +1 .

Now for i < j denote by h_{ij}(u) the product of all those factors (3.13), where the numbers p and q stand respectively in the i-th and the j-th columns of Ω . If at least one of these two columns is empty, we put h_{ij}(u) = 1 . By direct calculation,

h_{ij}(u) = \frac{u−a_{i}+b_{j} −1}{u−a_{i}+a_{j}} · \frac{u−b_{i}+a_{j} +1}{u−b_{i}+b_{j}} .

Note that we also have λ_{1}^{∗} −μ_{1}^{∗} +λ_{2}^{∗} −μ_{2}^{∗} + … = n. So by the definition (3.10)

h_{ω}(u) · (1−u)^{n} = (\frac{u−1}{u})^{n} · ∏_{i} h_{i}(u) · ∏_{i<j} h_{ij}(u) =

∏_{i} \frac{u−λ_{i}^{∗} +μ_{i}^{∗}}{u} · ∏_{i<j} \frac{(u+μ_{i}^{∗} −λ_{j}^{∗} −i+j) (u+λ_{i}^{∗} −μ_{j}^{∗} −i+j)}{(u+μ_{i}^{∗} −μ_{j}^{∗} −i+j) (u+λ_{i}^{∗} −λ_{j}^{∗} −i+j)} .

Replacing here the diagram ω by its conjugate ω^{∗}, we get Proposition 3.4 □

The total number of indices i such that λ_{i} ≠ μ_{i}, is the number ℓ(ω) of non-empty rows in the skew Young diagram ω. For any i < j we also have the inequalities

λ_{i} −i > μ_{j} −j , λ_{i} −i > λ_{j} −j , μ_{i} −i > μ_{j} −j .

Therefore, the order of the pole of the rational function (3.12) at u = 0 is

ℓ(ω) − #{ (i,j) | λ_{j} −j = μ_{i} −i, i < j } = d(ω) = d(ω^{∗}) ,

see Proposition 1.2. Now Proposition 3.4 implies that the order of the pole of the function h_{ω}(u) at u = 0 is also d(ω) . Denote by c(ω) the coefficient at u^{−d(ω)} in the Laurent expansion of h_{ω}(u) at u = 0 . By Proposition 3.3, we have

(3.14)  c(ω^{∗}) = c(ω) · (−1)^{n+d(ω)} .

Here is the main result of this section. We will employ this result in Section 4.

Theorem 3.5. The leading term of the Laurent expansion at u = 0 of F_{ωω}(u) is

(3.15)  c(ω) · (1,n+1) (2,n+2) … (n,2n) · F_{ω} F_{ω}^{∨} u^{−d(ω)} .

Proof. We will use induction on n, the total number of boxes in ω. If n = 1 , then

F_{ω} = 1, F_{ω}^{∨} = 1 and F_{ωω}(u) = 1− (12)/u .

But for n = 1 we have h_{ω}(u) = −1/u by the definition (3.10). So the statement of Theorem 3.5 is obvious in this case. From now on we will assume that n > 1 .

Suppose that the diagram ω consists of one column only. Then the diagram ω^{∗} consists of only one row. It follows from Proposition 2.3 and Theorems 2.5,2.7 that then

F_{ω} = ∑_{g∈S_{n}} sgn(g) · g and F_{ω^{∗}} = ∑_{g∈S_{n}} g

where sgn(g) = ±1 depending on whether the permutation g is even or odd. Now denote by β the involutive automorphism of the group ring ℂ·S_{2n} defined by β(g) = sgn(g) · g for every g ∈ S_{2n}. Then we have β(F_{ωω}(u)) = F_{ω^{∗}}_{ω^{∗}}(−u) by the definition (3.4). On the other hand, the image of (3.15) under β is

(−1)^{n} c(ω) · (1,n+1) (2,n+2) … (n,2n) · F_{ω^{∗}}F_{ω^{∗}}^{∨} u^{−d(ω)} .

Using the equality (3.14) we see, that the statements of Theorem 3.5 for ω and ω^{∗} are equivalent in the case, when ω consists of one column only. Note that then d(ω) = 1 , but we did not use the last equality here. We only used F_{ω^{∗}} = β(F_{ω}) .

From now on we will assume that there is more than one column in the skew Young diagram ω. Let δ be the skew Young diagram consisting of the first non-empty column of ω. Let φ the diagram obtained from ω by removing this column, the diagram φ is also non-empty. Each of the diagrams δ and φ has less than n boxes. We will assume that the statement of Theorem 3.5 is valid for each of the diagrams δ and φ instead of ω.

I. Let l be the number of boxes in the column δ. As well as in Section 2, denote by σ_{l} the embedding ℂ ·S_{n−l} → ℂ ·S_{n} defined by σ_{l} : (pq) ↦ (p+l,q +l) for all distinct p, q = 1, …,n − l. Using (3.4) with (2.3), we can write F_{ωω}(u) as

(3.16)  F_{ω} · A(u)B(u)C(u)D(u) · F_{ω}^{∨}

where A(u), B(u), C(u), D(u) are ordered products of the factors f_{p,q+n}(c_{p} + u, c_{q}) respectively over

p,q = 1, …,l;
p = 1, …,l and q = l +1, …,n;
p = l +1, …,n and q = 1, …,l;
p,q = l +1, …,n.

The pairs (p,q) in each of these four products are ordered lexicographically. By Propositions 2.6 and 2.7, the element F_{ω} ∈ ℂ ·S_{n} is divisible on the left and on the right by σ_{l}(F_{φ}) and F_{δ}. The element F_{δ}/l! ∈ ℂ·S_{l} is an idempotent. In general, there maybe no non-zero idempotents in the subset ℂ·F_{φ} ⊂ ℂ·S_{n−l}. But there is always an element I_{φ} ∈ ℂ · S_{n−l} divisible on the left by F_{φ}, such that I_{φ}F_{φ} = F_{φ}. By Proposition 2.4 applied to F_{φ}, this follows from the semisimplicity of ℂ · S_{n−l}. Due to Proposition 2.3, the element J_{φ} = α(I_{φ}) is divisible by F_{φ} on the right, and we have F_{φ} J_{φ} = F_{φ}. Using Proposition 3.1, we can now rewrite (3.16) as

(3.17)  F_{ω} · F_{δ} A(u)F_{δ}^{∨} · F_{δ} B(u)σ_{l}(I_{φ})^{∨} · σ_{l}(J_{φ})C(u)D(u)F_{ω}^{∨} · (l!)^{−3} .

II. By the inductive assumption for δ, the factor F_{δ} A(u)F_{δ}^{∨} in (3.17) has a simple pole at u = 0 , and the corresponding residue coincides with that of

(3.18)  h_{δ}(u) · (1,n+1) … (l,n+l) · F_{δ} F_{δ}^{∨} .

Further, F_{δ} B(u)σ_{l}(F_{φ})^{∨} is conjugate by the involution (l + 1,n + l + 1) … (n,2n) to

(3.19)  F (u) = F_{δ} · ∏_{p=1,...,l}^{⟶}( ∏_{q=l+1,...,n}^{⟶} f_{pq}(c_{p} +u, c_{q})) · σ_{l}(F_{φ}) .

The function F (u) is regular at u = 0 , and F (0) = F_{ω}. This follows by setting z_{1} = … = z_{l} = u in Proposition 2.2. In particular, the factor F_{δ} B(u)σ_{l}(I_{φ})^{∨} in (3.17) is regular at u = 0 . We will show that the order of the pole at u = 0 of the factor σ_{l}(J_{φ})C(u)D(u)F_{ω}^{∨} in (3.17) does not exceed d(ω)−1 . This will allow us to replace the factor F_{δ} A(u)F_{δ}^{∨} in (3.17) by the product (3.18), without affecting the coefficient at u^{−d(ω)} in the Laurent expansion at u = 0 . Independently of Proposition 3.2, this will also imply that the order of the pole at u = 0 of F_{ωω}(u) does not exceed d(ω) .

It is the argument of Proposition 3.2 that we will use here. In particular, for any numbers p and q standing on the same diagonal of Ω , let us keep calling the pair (p,q) singular. Any singular pair belongs to one of the types (i,ii,iii) as described in the beginning of the proof of Proposition 3.2. Note that the number l stands at the bottom of the first column in the tableau Ω . Consider the product

σ_{l}(J_{φ})C(u)D(u)F_{ω}^{∨} = σ_{l}(J_{φ}) · ∏_{p=l+1,...,n}^{⟶}( ∏_{q=1,...,n}^{⟶} f_{p,q+n}(c_{p} +u, c_{q})) · F_{ω}^{∨} .

Here the factor f_{p,q+n}(c_{p} + u, c_{q}) has a pole at u = 0 if and only if the pair (p,q) is singular. The factor σ_{l}(J_{φ}) is divisible on the right by σ_{l}(F_{φ}) . By the argument of Proposition 3.2, when we estimate the order of pole of the above product at u = 0 from above, the singular pairs of types (i) and (ii) do not count. The total number of the singular pairs (p,q) of type (iii) is d(ω) . For any singular pair (p,q) of type (iii) we have p > l, except for the pair (p,q) = (l,l) . Hence in the above product, the number of singular pairs (p,q) of type (iii) is exactly d(ω)−1 .

III. Thus the coefficient at u^{−d(ω)} in the Laurent expansion at u = 0 of the function F_{ωω}(u) coincides with that of the function

h_{δ}(u) F_{ω} · (1,n+1) … (l,n+l) · F_{δ} F_{δ}^{∨} × (3.20)

F_{δ} B(u)σ_{l}(I_{φ})^{∨} · σ_{l}(J_{φ})C(u)D(u)F_{ω}^{∨} · (l!)^{−3} =
h_{δ}(u) F_{ω} · (1,n+1) … (l,n+l) · F_{δ} B(u)σ_{l}(I_{φ})^{∨} · C(u)D(u)F_{ω}^{∨} · (l!)^{−1} .

Without affecting that coefficient, we can also replace in (3.20) the constant factor F_{ω} by the function F (−u) ; see (3.19). The function F (−u) can be also written as

σ_{l}(F_{φ}) · ∏_{p=1,...,l}^{⟵}( ∏_{q=l+1,...,n}^{⟵} f_{pq}(c_{p} , c_{q} +u)) · F_{δ} ,

see Proposition 3.1. By exchanging the indices p,q and using the commutation relations (2.3), the last product can be rewritten as

(3.21)  σ_{l}(F_{φ}) · ∏_{p=l+1,...,n}^{⟵}( ∏_{q=1,...,l}^{⟵} f_{qp}(c_{q} , c_{p} +u)) · F_{δ} .

Let us now denote by E(u) the ordered product of the factors f_{q+n,p}(c_{q} , c_{p} + u) over

p = l +1, …,n and q = 1, …,l;

the pairs (p,q) being taken in the reversed lexicographical order. Using the relations (2.4) repeatedly, we obtain the equality

(3.22)  E(u)C(u) = ∏_{p=l+1}^{n} ∏_{q=1}^{l} (1 − \frac{1}{(u+c_{p}−c_{q})^{2}} ) .

Denote by h(u) the rational function of u at the right hand side of this equality.

The product (3.21) is conjugate by the involution (1,n + 1) … (l,n + l) ∈ S_{2n} to σ_{l}(F_{φ})E(u)F_{δ}^{∨}. By the above argument, the coefficient at u^{−d(ω)} in the Laurent expansion at u = 0 of the function (3.20) coincides with that of the function

h_{δ}(u) F (−u) · (1,n+1) … (l,n+l) · F_{δ} B(u)σ_{l}(I_{φ})^{∨} · C(u)D(u)F_{ω}^{∨} · (l!)^{−1} =

h_{δ}(u) · (1,n+1) … (l,n+l) · F_{δ} B(u)σ_{l}(I_{φ})^{∨} · σ_{l}(F_{φ})E(u)F_{δ}^{∨} C(u)D(u)F_{ω}^{∨} · (l!)^{−1}

= h_{δ}(u)h(u) · (1,n+1) … (l,n+l) · F_{δ} B(u)σ_{l}(I_{φ})^{∨} · σ_{l}(F_{φ})D(u)F_{ω}^{∨};

here we also used divisibility of the product of F_{ω}^{∨} and of the product C(u)D(u)F_{ω}^{∨} by F_{δ}^{∨} on the left, along with the equality (3.22).

IV. The element F_{ω}^{∨} is also divisible on the left by σ_{l}(F_{φ})^{∨}. By the inductive assumption for the diagram φ, the leading term in the Laurent expansion at u = 0 of σ_{l}(F_{φ})D(u)σ_{l}(F_{φ})^{∨} coincides with that of the function

h_{φ}(u) · (l +1,n+l +1) … (n,2n) · σ_{l}(F_{φ}) σ_{l}(F_{φ})^{∨} .

Therefore, by the result of step III, the coefficient at u^{−d(ω)} in the Laurent expansion at u = 0 of the function F_{ωω}(u) coincides with that of the function

h_{δ}(u)h(u)h_{φ}(u) · (1,n+1) … (l,n+l) ×
F_{δ} B(u)σ_{l}(I_{φ})^{∨} · (l +1,n+l +1) … (n,2n) · σ_{l}(F_{φ})F_{ω}^{∨} =
h_{δ}(u)h(u)h_{φ}(u) · (1,n+1) … (l,n+l) · (l +1,n+l +1) … (n,2n) · F (u)F_{ω}^{∨};

see the definition (3.19). But here we have h_{δ}(u)h(u)h_{φ}(u) = h_{ω}(u) by (3.10). The statement of Theorem 3.5 for the skew Young diagram ω now follows from the equality F (0) = F_{ω}. This equality has been already obtained on step II □

We will complete this section with one remark on the ordered product

(3.23)  ∏_{p=1,...,n}^{⟵}( ∏_{q=1,...,n′}^{⟶} f_{p,q+n}(c_{p} +u, c′_{q}))

for two non-empty skew Young diagrams ω and ω′. This product appeared in (3.2). Denote the product (3.23) by G_{ωω′}(u) , this is a rational function of u which may have poles only at u ∈ ℤ. By Proposition 3.1, we have F_{ωω′}(u) = G_{ωω′}(u)F_{ω} F_{ω′}^{∨} . Let us now extend the notation F^{∨} used earlier in this section for the elements G ∈ ℂ · S_{n′} as follows. For any G ∈ ℂ ·S_{n+n′} denote by G^{∨} the element g Gg^{−1} conjugate to G by the permutation from S_{n+n′}

g : (1, …,n′, n′+1, …,n+n′) ↦ (n+1, …,n+n′, 1, …,n) .

Proposition 3.6. We have the equality

G_{ωω′}(u) G_{ω′ω}^{∨}(−u) = ∏_{p=1}^{n} ∏_{q=1}^{n′} (1 − \frac{1}{(u+c_{p}−c′_{q})^{2}} ) .

Proof. By the definition of the function G_{ω′ω}(u) , we have

G_{ω′ω}^{∨}(−u) = ∏_{p=1,...,n′}^{⟵}( ∏_{q=1,...,n}^{⟶} f_{p+n,q}(c′_{p} −u, c_{q}))

= ∏_{p=1,...,n}^{⟶}( ∏_{q=1,...,n′}^{⟵} f_{q+n,p}(c′_{q} , c_{p} +u));

here the second equality has been obtained by exchanging the indices p,q and using the relations (2.3). Now the required statement can be derived from the definition (3.23) of the function G_{ωω′}(u) by using the relations (2.4) □

4. Irreducibility of Yangian modules

We will begin this section with recalling the definition of the Yangian of the general linear Lie algebra 𝔤𝔩_{N} . This is the unital associative algebra Y(𝔤𝔩_{N}) over ℂ, with the infinite family of generators T_{ij}^{(s)} where s = 1, 2, … and i,j = 1, …,N . The defining relations can be written in terms of the generating series (0.1) as

(u − v) · [T_{ij}(u), T_{kl}(v)] = T_{kj}(u)T_{il}(v) − T_{kj}(v)T_{il}(u) .

The Yangian Y(𝔤𝔩_{N}) is a Hopf algebra. Coproduct Δ : Y(𝔤𝔩_{N}) → Y(𝔤𝔩_{N}) ⊗ Y(𝔤𝔩_{N}) is given by

Δ(T_{ij}(u)) = ∑_{k=1}^{N} T_{ik}(u) ⊗ T_{kj}(u) .

The opposite coproduct Δ̃ : Y(𝔤𝔩_{N}) → Y(𝔤𝔩_{N}) ⊗ Y(𝔤𝔩_{N}) is the composition of the coproduct Δ and subsequent transposition of tensor factors in Y(𝔤𝔩_{N}) ⊗ Y(𝔤𝔩_{N}) . For any two Y(𝔤𝔩_{N})-modules U and V we will denote by U ⊗ V and U ˜⊗ V the Yangian modules, where the action of Y(𝔤𝔩_{N}) on the tensor product of vector spaces is determined via by the coproduct Δ and the opposite coproduct Δ̃ respectively.

Our definition implies, that for any z ∈ ℂ the assignment T_{ij}(u) ↦ T_{ij}(u−z) defines an automorphism of the Hopf algebra Y(𝔤𝔩_{N}) . Here the formal series in (u − z)^{−1} should be re-expanded in u^{−1}. Denote this automorphism by τ_{z}.

We will also need a matrix form of the definition of the Hopf algebra Y(𝔤𝔩_{N}) . Let us introduce the following notation. Consider the tensor product of any unital associative algebras A_{1} ⊗ … ⊗ A_{n}. For p = 1, …,n let ι_{p} : A_{p} → A_{1} ⊗ … ⊗ A_{n} be the embedding as the p-th tensor factor:

ι_{p}(X) = 1^{⊗(p−1)} ⊗ X ⊗ 1^{⊗(n−p)}, X ∈ A_{p} .

For X ∈ A_{p} and Y ∈ A_{q} where p ≠ q, let us write (X ⊗ Y)^{(pq)} = ι_{p}(X)ι_{q}(Y) . For any Z ∈ A_{p} ⊗ A_{q}, determine the element Z^{(pq)} ∈ A_{1} ⊗ … ⊗A_{n} by linearity. For any vector spaces U_{1} , …,U_{n} we identify the algebras End(U_{1} ⊗ … ⊗ U_{n}) and End(U_{1})⊗ … ⊗ End(U_{n}) .

Now take the vector space ℂ^{N} = U . For i, j = 1, …,N let E_{ij} ∈ End(U) be the matrix units. Let P : x ⊗ y ↦ y ⊗ x be the flip map in U ⊗ U . Put R(u) = u − P . Identifying End(U ⊗ U) = End(U) ⊗ End(U) , we can write

(4.1)  R(u) = u − ∑_{i,j=1}^{N} E_{ij} ⊗ E_{ji} .

Combine all series (0.1) into a series T (u) with coefficients in Y(𝔤𝔩_{N}) ⊗ End(U) :

(4.2)  T (u) = ∑_{i,j=1}^{N} T_{ij}(u) ⊗ E_{ji} .

Then the defining relations of the algebra Y(𝔤𝔩_{N}) are equivalent to the relation

(4.3)  T^{(01)}(u) T^{(02)}(v) R^{(12)}(u − v) = R^{(12)}(u − v) T^{(02)}(v)T^{(01)}(u)

for the series with coefficients in Y(𝔤𝔩_{N})⊗End(U)⊗End(U) , where we label the tensor factors by the indices 0,1, 2 .

Now label the tensor factors in Y(𝔤𝔩_{N}) ⊗ Y(𝔤𝔩_{N}) ⊗ End(U) by 1,2,3 as usual. Put

(4.4)  T^{[k]}(u) = T^{(k3)}(u) ; k = 1,2.

Then we have the equality

(4.5)  (Δ ⊗ id)(T (u)) = T^{[2]}(u) T^{[1]}(u) .

We refer the reader to the survey [MNO] for more details on the definition of the Yangian Y(𝔤𝔩_{N}) , and for description of its basic properties. Note that the series denoted by T (u) in [MNO] differs from our series (4.2) by transposition E_{ij} ↦ E_{ji} in the second tensor factor.

Let ω = λ/μ be a non-empty skew Young diagram. Let n be the number of boxes in ω. Take the vector space U^{⊗n}. The symmetric group S_{n} acts naturally in this vector space, permuting the tensor factors. Denote by π_{n} the corresponding homomorphism ℂ · S_{n} → End(U^{⊗n}) . In Section 2 we defined a certain element F_{ω} ∈ ℂ · S_{n}. Let the subspace V_{ω} ⊂ U^{⊗n} be the image of the operator π_{n}(F_{ω}) .

Proposition 4.1 : V_{ω} ≠ {0} if and only if the length of every column of ω does not exceed N .

The results of the present article do not depend on this proposition. Hence we omit the proof, and refer to [C1,NT2] instead.

Now take two non-empty skew Young diagrams ω and ω′ with n and n′ boxes respectively. Let c_{1} , …,c_{n} and c′_{1} , …,c′_{n′} be the corresponding sequences of contents, ordered according to the column tableaux. Consider the rational function G_{ωω′}(u) with the values in ℂ · S_{n+n′} , defined by (3.23). For any value of u ∉ ℤ the operator π_{n+n′}(G_{ωω′}(u)) is well defined, and by Proposition 3.1 preserves the subspace

V_{ω} ⊗ V_{ω′} ⊂ U^{⊗n}⊗ U^{⊗n′}= U^{⊗(n+n′)}.

Let R_{ωω′}(u) be the value of the restriction of π_{n+n′}(G_{ωω′}(u)) to V_{ω} ⊗ V_{ω′} ; R_{ωω′}(u) is a rational function of u valued in End(V_{ω}) ⊗ End(V_{ω′}) . Suppose that each of the vector spaces V_{ω} and V_{ω′} is non-zero, see Proposition 4.1. Then by Proposition 3.6

(4.6)  R_{ωω′}(u) R_{ω′ω}^{(21)}(−u) = ∏_{p=1}^{n} ∏_{q=1}^{n′} (1 − \frac{1}{(u + c_{p} − c′_{q})^{2}} ) .

Take one more non-empty skew Young diagram ω′′. Let n′′ be its number of boxes.

Proposition 4.2. We have the equality of rational functions of u,v with values in the tensor product End(V_{ω}) ⊗ End(V_{ω′}) ⊗ End(V_{ω′′})

(4.7)  R_{ωω′}^{(12)}(u − v) R_{ωω′′}^{(13)}(u) R_{ω′ω′′}^{(23)}(v) = R_{ω′ω′′}^{(23)}(v) R_{ωω′′}^{(13)}(u) R_{ωω′}^{(12)}(u − v) .

Proof. The relation (4.7) for the functions R_{ωω′}(u) , R_{ωω′′}(u) , R_{ω′ω′′}(u) follows from the respective relation for the functions G_{ωω′}(u) , G_{ωω′′}(u) , G_{ω′ω′′}(u). The latter relation is an equality of rational functions of u,v with the values in the algebra ℂ ·S_{n+n′+n′′} . That equality follows from (2.2) and (2.3), we will not write it here. By applying the homomorphism π_{n+n′+n′′} to both sides of that equality and taking the restriction to V_{ω} ⊗ V_{ω′} ⊗ V_{ω′′} ⊂ U^{⊗(n+n′+n′′)} , we obtain (4.7) □

Denote by ε the Young diagram with one box (1,1) . Then V_{ε} = U . Note that R_{εε}(u) = R(u)/u. We will now define a family of modules over the algebra Y(𝔤𝔩_{N}) , we call these modules elementary. Take any z ∈ ℂ. Consider the rational function R_{ωε}(z −u) of the variable u. This function takes values in End(V_{ω}) ⊗ End(U) . This function is regular at u = ∞. Moreover, at u = ∞ it takes the value 1 . Here we keep to the assumption V_{ω} ≠ {0}. Consider the Laurent expansion at u = ∞

R_{ωε}(z − u) = 1 + ∑_{i,j=1}^{N} u^{−s} E_{ij}^{(s)}(ω,z) ⊗ E_{ji} .

Here each tensor factor E_{ij}^{(s)}(ω,z) is a certain element of the algebra End(V_{ω}) .

Proposition 4.3. The assignment for all s = 1, 2, … and i,j = 1, …,N

(4.8)  T_{ij}^{(s)} ↦ E_{ij}^{(s)}(ω,z)

defines an Y(𝔤𝔩_{N})-module structure on the vector space V_{ω}.

Proof. We have to verify that the operators E_{ij}^{(s)}(ω,z) satisfy the defining relations of Y(𝔤𝔩_{N}) . Let us rewrite the definition (4.8) in a matrix form. It will be

(4.9)  T (u) ↦ R_{ωε}(z −u) .

In view of the defining relation (4.3), we have to verify the equality

R_{ωε}^{(01)}(z −u) R_{ωε}^{(02)}(z −v) R^{(12)}(u−v) = R^{(12)}(u−v) R_{ωε}^{(02)}(z −v) R_{ωε}^{(01)}(z −u)

of rational functions in u,v with values in End(V_{ω}) ⊗ End(U) ⊗ End(U) , where the tensor factors are labelled by the indices 0,1, 2 . Changing these labels respectively to 1,2, 3 and taking into account that R(u) = uR_{εε}(u) , we come to verifying

R_{ωε}^{(12)}(z −u) R_{ωε}^{(13)}(z −v) R_{εε}^{(23)}(u−v) = R_{εε}^{(23)}(u−v) R_{ωε}^{(13)}(z −v) R_{ωε}^{(12)}(z −u) .

But the latter equality follows from Proposition 4.2 when ω′ = ω′′ = ε □

The Y(𝔤𝔩_{N})-module defined by Proposition 4.3 will be denoted by V_{ω}(z) , and called an elementary module. Note that for any z ∈ ℂ the Y(𝔤𝔩_{N})-module V_{ω}(z) is obtained from the module V_{ω}(0) by pulling back through the automorphism τ_{z} of the algebra Y(𝔤𝔩_{N}) .

Let us now give another description of the elementary Y(𝔤𝔩_{N})-module V_{ω}(z) . First consider the module V_{ε}(z) . The vector space of this module is U = ℂ^{N} , and the action of the algebra Y(𝔤𝔩_{N}) in this space is determined by the assignment T_{ij}^{(s)} ↦ z^{s}E_{ij} , see (4.1).

Proposition 4.4. The action of the algebra Y(𝔤𝔩_{N}) in the tensor product of the modules V_{ε}(c_{1} +z) ⊗ … ⊗ V_{ε}(c_{n} +z) preserves the subspace V_{ω} ⊂ U^{⊗n}. The restriction of this action to V_{ω} gives exactly the Y(𝔤𝔩_{N})-module V_{ω}(z) .

Proof. Let us write the definition of the action of the algebra Y(𝔤𝔩_{N}) in V_{ε}(z) in a matrix form. It will be T (u) ↦ R_{εε}(z−u) . Then due to (4.5) the action of Y(𝔤𝔩_{N}) in the tensor product V_{ε}(c_{1} +z) ⊗ … ⊗ V_{ε}(c_{n} +z) is determined by the assignment

T (u) ↦ ∏_{p=1,...,n}^{⟵} R_{εε}^{(p,n+1)}(c_{p} + z − u) .

The right hand side of this assignment can be rewritten as

∏_{p=1,...,n}^{⟵} π_{n+1}(f_{p,n+1}(c_{p} + z − u,0)) = π_{n+1}(G_{ωε}(z −u)) .

We have already shown that π_{n+1}(G_{ωε}(z − u)) preserves the subspace V_{ω} ⊗ V_{ε} in U^{⊗n} ⊗ U . This means that the action of Y(𝔤𝔩_{N}) in V_{ε}(c_{1} + z) ⊗ … ⊗ V_{ε}(c_{n} + z) preserves the subspace V_{ω} ⊂ U^{⊗n}. Restriction of this action to V_{ω} is determined exactly by the assignment

T (u) ↦ R_{ωε}(z −u)

because of the definition of R_{ωε}(z − u) . But the last assignment coincides with the matrix definition of the Y(𝔤𝔩_{N})-module V_{ω}(z) □

For any z ∈ ℂ let us expand the rational function R_{ωω′}(u) into the Laurent series at u = z. Let (u − z)^{−a_{ωω′} (z)} I_{ωω′}(z) be the leading term of this expansion. This defines an integer a_{ωω′}(z) and a non-zero linear operator I_{ωω′}(z) in V_{ω} ⊗V_{ω′} . Again, here we keep to the assumption that V_{ω} , V_{ω′} ≠ {0}. The rational function R_{ωω′}(u) is regular at u = z for any z ∉ ℤ. Then by (4.6), this function also does not vanish at u = z for any z ∉ ℤ. Thus for any z ∉ ℤ we have a_{ωω′}(z) = 0 and I_{ωω′}(z) = R_{ωω′}(z) . Furthermore, the equality (4.6) implies

Proposition 4.5. For any z ∈ ℂ the product I_{ωω′}(z) I_{ω′ω}^{(21)}(−z) ∈ End(V_{ω} ⊗ V_{ω′}) is proportional to the identity map. In particular, the maps I_{ωω′}(z) and I_{ω′ω}(−z) are invertible simultaneously.

The basic property of the linear operator I_{ωω′}(z) is given by the next proposition.

Proposition 4.6. For any z, z′ ∈ ℂ the element I_{ωω′}(z − z′) ∈ End(V_{ω} ⊗ V_{ω′}) is an intertwiner of the Yangian modules V_{ω}(z) ⊗ V_{ω′}(z′) ⟶ V_{ω}(z) ˜⊗ V_{ω′}(z′) .

Proof. The formula (4.7) implies the following equality of rational functions in u, v:

R_{ωω′}^{(12)}(z −v) R_{ωε}^{(13)}(z −u) R_{ω′ε}^{(23)}(v −u) = R_{ω′ε}^{(23)}(v −u) R_{ωε}^{(13)}(z −u) R_{ωω′}^{(12)}(z −v) .

Taking the leading terms of the Laurent expansions of both sides at v = z′ we get

I_{ωω′}^{(12)}(z −z′) R_{ωε}^{(13)}(z −u) R_{ω′ε}^{(23)}(z′ −u) = R_{ω′ε}^{(23)}(z′ −u) R_{ωε}^{(13)}(z −u) I_{ωω′}^{(12)}(z −z′) .

The latter equality implies Proposition 4.6, see (4.5) and (4.9) □

Recall that the positive integer d(ω) is the Durfee rank of the non-empty skew Young diagram ω, see Section 1. The non-zero rational number c(ω) is the value of the function u^{d(ω)} h_{ω}(u) at u = 0 , see (3.10). Denote by P_{ω} be the flip map in V_{ω} ⊗ V_{ω}, so that P_{ω}(x ⊗ y) = y ⊗ x for any x, y ∈ V_{ω}. The main result of the previous section, Theorem 3.5, can be now restated as follows.

Proposition 4.7. We have a_{ωω}(0) = d(ω) and I_{ωω}(0) = c(ω)P_{ω}.

Proof. The rational function F_{ωω}(u) with the values in ℂ · S_{2n} can be written as G_{ωω}(u)F_{ω} F_{ω}^{∨}. Hence the required statement follows from Theorem 3.5, by the definition of the operator I_{ωω}(0) in V_{ω} ⊗ V_{ω} □

Now take k ⩾ 1 non-empty skew Young diagrams ω_{1} , …,ω_{k}. We will assume that each of the vector spaces V_{ω_{1}} , …,V_{ω_{k}} is not zero, see Proposition 4.1. Take also k arbitrary complex numbers z_{1} , …,z_{k}. Consider the elementary Y(𝔤𝔩_{N})-modules V_{ω_{1}}(z_{1}), …,V_{ω_{k}}(z_{k}) . The next theorem is the main result of the present paper.

Theorem 4.8. The Y(𝔤𝔩_{N})-module V_{ω_{1}}(z_{1}) ⊗ … ⊗ V_{ω_{k}}(z_{k}) is irreducible if and only if all the intertwiners I_{ω_{i}ω_{j}}(z_{i} −z_{j}) with 1 ⩽ i < j ⩽ k are invertible.

There is an immediate corollary, we state it as a theorem because of its importance.

Theorem 4.9. The Y(𝔤𝔩_{N})-module V_{ω_{1}}(z_{1}) ⊗ … ⊗ V_{ω_{k}}(z_{k}) is irreducible if and only if all the modules V_{ω_{i}}(z_{i}) ⊗ V_{ω_{j}}(z_{j}) with 1 ⩽ i < j ⩽ k are irreducible.

We shall prove Theorem 4.8 in the remainder of this section. However, before giving the proof, let us make a few remarks. Applying Theorem 4.8 when k = 1, 2 we obtain, in particular, the following properties of the elementary Y(𝔤𝔩_{N})-modules:

(a) the Y(𝔤𝔩_{N})-module V_{ω}(z) is irreducible;

(b) the Y(𝔤𝔩_{N})-module V_{ω}(z) ⊗ V_{ω′}(z′) is irreducible provided z − z′ ∉ ℤ;

(c) if V_{ω}(z) ⊗ V_{ω′}(z′) is irreducible, then it is equivalent to V_{ω}(z) ˜⊗ V_{ω′}(z′) . The properties (a,b,c) are well known, see for instance [C2,CP,NT1]. But we would like to emphasize here, that Theorem 4.8 gives each of them a new independent proof. Using Theorem 4.8 together with Proposition 4.7, we obtain for any k = 2, 3, … a new remarkable property of the elementary Y(𝔤𝔩_{N})-modules:

(d) the k-th tensor power of the Y(𝔤𝔩_{N})-module V_{ω}(z) is irreducible. Conversely, the property (d) for k = 2 implies that the operator I_{ωω}(0) in the vector space V_{ω} ⊗ V_{ω} is proportional to the flip map P_{ω}; see Proposition 4.6.

The only if part of Theorem 4.8 is an easy and well known statement. For the sake of completeness, with will give the proof of this statement at the end of this section. Our proof of the if part is based on the following general observation. Let A be any unital associative algebra over ℂ, and let W be any finite-dimensional A-module. Denote by ρ the corresponding homomorphism A → End(W) . Let P_{W} ∈ End(W ⊗ W) be the flip map.

Lemma 4.10. If P_{W} ∈ ρ(A) ⊗ End(W) , then the A-module W is irreducible.

Proof. Let 1 ∈ End(W) be the unit element, we have 1 ∈ ρ(A) by our assumption. Furthermore, we have (1 ⊗ X) · P_{W} ∈ ρ(A) ⊗ End(W) for any X ∈ End(W) . Now let tr : End(W) → ℂ be the trace map. Then the subalgebra ρ(A) ⊂ End(W) contains the element

(id ⊗ tr)((1 ⊗ X) · P_{W} ) = X .

So we actually have ρ(A) = End(W) □

Now take the algebra A = Y(𝔤𝔩_{N}) and its module W = V_{ω_{1}}(z_{1}) ⊗ … ⊗ V_{ω_{k}}(z_{k}) . Under the assumption that all the intertwiners I_{ω_{i}ω_{j}}(z_{i} −z_{j}) with 1 ⩽ i < j ⩽ k are invertible, we will show that in the above notation,

(4.10)  P_{W} ∈ ρ (Y(𝔤𝔩_{N})) ⊗ End(W) .

The if part of Theorem 4.8 will then follow by Lemma 4.10. We will split the proof of (4.10) into four propositions below.

Let ω be a non-empty skew Young diagram with n boxes. We keep assuming that the subspace V_{ω} ⊂ U^{⊗n} is non-zero. Let End_{ω}(U^{⊗n}) ⊂ End(U^{⊗n}) be the stabilizer of V_{ω}. Let

χ : End_{ω}(U^{⊗n}) → End(V_{ω})

be the canonical homomorphism: χ(X) = X|_{V_{ω}} . Introduce the series T̂_{ω}(u) with coefficients in the tensor product Y(𝔤𝔩_{N}) ⊗ End(U^{⊗n}) = Y(𝔤𝔩_{N}) ⊗ (End(U))^{⊗n}

T̂_{ω}(u) = T^{(01)}(u + c_{1}) … T^{(0n)}(u + c_{n}) .

Here we labelled the factors in Y(𝔤𝔩_{N}) ⊗ (End(U))^{⊗n} by the indices 0,1, …,n.

Proposition 4.11. Coefficients of the series T̂_{ω}(u) are in Y(𝔤𝔩_{N}) ⊗ End_{ω}(U^{⊗n}) .

Proof. Consider the rational function F_{ω}(z_{1} , …,z_{n}) , defined in Section 2 as the product (2.5). The relation (4.3) implies that

T^{(01)}(u+c_{1} +z_{1}) … T^{(0n)}(u+c_{n} +z_{n}) · π_{n}(F_{ω}(z_{1} , …,z_{n})) =
π_{n}(F_{ω}(z_{1} , …,z_{n})) · T^{(0n)}(u+c_{n} +z_{n}) … T^{(01)}(u+c_{1} +z_{1}) .

Hence, by Proposition 2.2

T̂_{ω}(u) π_{n}(F_{ω}) = π_{n}(F_{ω}) T^{(0n)}(u+c_{n}) … T^{(01)}(u+c_{1}) □

Due to the Proposition 4.11, we can apply the homomorphism id ⊗ χ to the series T̂_{ω}(u) , and obtain in this way a series with the coefficients in Y(𝔤𝔩_{N}) ⊗ End(V_{ω}) . We will denote the latter series by T_{ω}(u) .

Henceforth, we will use the following generalization of notation (4.4). Given a vector space V and an element X ∈ Y(𝔤𝔩_{N}) ⊗ End(V) , we will write

X^{[k]} = X^{(k3)} ∈ Y(𝔤𝔩_{N}) ⊗ Y(𝔤𝔩_{N}) ⊗ End(V) , k = 1,2 .

Proposition 4.12 : Δ(T_{ω}(u)) = T_{ω}^{[2]}(u) T_{ω}^{[1]}(u) .

Proof. The formula (4.5) implies that Δ(T̂_{ω}(u)) = T̂_{ω}^{[2]}(u) T̂_{ω}^{[1]}(u) . By applying the homomorphism id ⊗ id ⊗ χ to the last equality, the proposition follows □

Let γ : Y(𝔤𝔩_{N}) → End(V_{ω}) be the defining homomorphism of the module V_{ω}(z) . Take one more skew Young diagram ω′, we will again assume that V_{ω′} ≠ {0}.

Proposition 4.13 : (γ ⊗ id)(T_{ω′}(u)) = R_{ωω′}(z − u) .

Proof. Let n′ be the numbers of boxes in the diagram ω′. Consider the canonical homomorphism

χ′ : End_{ω′}(U^{⊗n′}) → End(V_{ω′}) .

It is straighforward to verify that each of (γ ⊗ id)(T_{ω′}(u)) and R_{ωω′}(z − u) equals

(( χ ⊗ χ′) ^{∘} π_{n+n′})(G_{ωω′}(z − u)) □

Now consider the formal power series in u^{−1}

T_{W} (u) = T_{ω_{1}}^{(01)}(u+z_{1}) … T_{ω_{k}}^{(0k)}(u+z_{k}) ∈ Y(𝔤𝔩_{N}) ⊗ End(W) ,

here we identify Y(𝔤𝔩_{N}) ⊗ End(W) with Y(𝔤𝔩_{N}) ⊗ End(V_{ω_{1}}) ⊗ … ⊗ End(V_{ω_{k}}) , and label the tensor factors by 0, 1, …,k. The next proposition completes the proof of the if part of Theorem 4.8.

Proposition 4.14. The image (ρ⊗id)(T_{W} (u)) is a rational function in u with values in

ρ(Y(𝔤𝔩_{N})) ⊗ End(W) ⊂ End(W) ⊗ End(W) .

The leading coefficient in its Laurent expansion at u = 0 is proportional to P_{W} .

Proof. It follows from Proposition 4.12 that Δ(T_{W} (u)) = T_{W}^{[2]}(u) T_{W}^{[1]}(u) . Using this relation and Proposition 4.13, we get

(4.11)  (ρ ⊗ id)(T_{W} (u)) = ∏_{j=1,...,k}^{⟶}( ∏_{i=1,...,k}^{⟵} R_{ω_{i}ω_{j}}^{(i,j+k)}(z_{i} −z_{j} −u)) .

Here we regard End(W) ⊗ End(W) as the 2k-fold tensor product

(4.12)  End(V_{ω_{1}}) ⊗ … ⊗ End(V_{ω_{k}}) ⊗ End(V_{ω_{1}}) ⊗ … ⊗ End(V_{ω_{k}}) .

The right hand side of the equality (4.11) is a rational function in u by definition. Denote

a = ∑_{i,j=1}^{k} a_{ω_{i}ω_{j}}(z_{i} −z_{j}) .

Then the product of the leading terms of the Laurent expansions at u = 0 of all factors in the product (4.11) equals (−u)^{−a} multiplied by

(4.13)  ∏_{j=1,...,k}^{⟶}( ∏_{i=1,...,k}^{⟵} I_{ω_{i}ω_{j}}^{(i,j+k)}(z_{i} −z_{j})) .

By Proposition 4.7, any factor I_{ω_{i}ω_{j}}^{(i,i+k)}(0) corresponding to i = j, equals P_{ω_{i}}^{(i,i+k)} up to a non-zero multiplier. Using this result, we can rewrite (4.13) as

(4.14)  ∏_{i=1}^{k} P_{ω_{i}}^{(i,i+k)} · ∏_{1⩽i<j⩽k}(I_{ω_{j}ω_{i}}^{(j+k,i)}(z_{j} −z_{i}) I_{ω_{i}ω_{j}}^{(i,j+k)}(z_{i} −z_{j}))

up to a nonzero multiplier. Here all factors P_{ω_{i}}^{(i,i+k)} pairwise commute. By Proposition 4.5, each of the factors in (4.14) corresponding to 1 ⩽ i < j ⩽ k, is proportional to the identity map in (4.12). In particular, these factors also pairwise commute. Moreover, each of these factors is non-zero, because the operators I_{ω_{i}ω_{j}}(z_{i} − z_{j}) with 1 ⩽ i < j ⩽ k are invertible by our assumption. So the entire product (4.13) is non-zero and proportional to

∏_{i=1}^{k} P_{ω_{i}}^{(i,i+k)} = P_{W} .

Thus the leading term in the Laurent expansion of the function (ρ⊗ id)(T_{W} (u)) near the origin u = 0 equals (−u)^{−a} P_{W} , up to a non-zero factor from ℂ □

We will complete this section with the proof of the only if part of Theorem 4.8. Consider the Y(𝔤𝔩_{N})-module W = V_{ω_{1}}(z_{1}) ⊗ … ⊗ V_{ω_{k}}(z_{k}) . Suppose there is a non-invertible intertwiner I_{ω_{i}ω_{j}}(z_{i} − z_{j}) with 1 ⩽ i < j ⩽ k. Assume that the pair (i,j) is lexicographically minimal with this non-invertibility property. The operator

I_{ω_{i}ω_{j}}^{(ij)}(z_{i} −z_{j}) … I_{ω_{i}ω_{i+1}}^{(i,i+1)}(z_{i} − z_{i+1})

is non-zero and non-invertible. By Proposition 4.6, this is an intertwiner of Yangian modules

W ⟶ V_{ω_{1}}(z_{1}) ⊗ … ⊗ V_{ω_{i−1}}(z_{i−1}) ⊗
( V_{ω_{i}}(z_{i}) ˜⊗ ( V_{ω_{i+1}}(z_{i+1}) ⊗ … ⊗ V_{ω_{j}}(z_{j}))) ⊗ V_{ω_{j+1}}(z_{j+1}) ⊗ … ⊗ V_{ω_{k}}(z_{k}) .

So the Y(𝔤𝔩_{N})-module W is reducible. This proves the only if part of Theorem 4.8.

Acknowledgements

It is our pleasure to thank N. Kitanine for a stimulating discussion and for bringing the results of [KMT,MT] to our attention. This work was done when the authors stayed at the Mathematisches Forschungsinstitut Oberwolfach, with the support from the Volkswagen-Stiftung. While finishing this article, the second author stayed at the Max-Planck-Institut für Mathematik in Bonn. The first author has been supported by the EPSRC, and by the EC under the TMR grant FMRX-CT97-0100. The second author has been supported by the RFFI grant 99-01-00101 and by the INTAS grant 99-01705.

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