New Combinatorial Formulae for Nested Bethe Vectors
We give new combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for the evaluation modules over the Yangian Y(₄). The case of Y(ₙ) for an arbitrary n is considered in [Lett. Math. Phys. 115 (2025), 12, 20 pages, arXiv:2402.15717].
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2025 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/213516 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | New Combinatorial Formulae for Nested Bethe Vectors. Maksim Kosmakov and Vitaly Tarasov. SIGMA 21 (2025), 060, 28 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860277679406710784 |
|---|---|
| author | Kosmakov, Maksim Tarasov, Vitaly |
| author_facet | Kosmakov, Maksim Tarasov, Vitaly |
| citation_txt | New Combinatorial Formulae for Nested Bethe Vectors. Maksim Kosmakov and Vitaly Tarasov. SIGMA 21 (2025), 060, 28 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We give new combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for the evaluation modules over the Yangian Y(₄). The case of Y(ₙ) for an arbitrary n is considered in [Lett. Math. Phys. 115 (2025), 12, 20 pages, arXiv:2402.15717].
|
| first_indexed | 2026-03-15T03:36:23Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 060, 28 pages
New Combinatorial Formulae
for Nested Bethe Vectors
Maksim KOSMAKOV a and Vitaly TARASOV b
a) Department of Mathematical Sciences, University of Cincinnati,
P.O. Box 210025, Cincinnati, OH 45221, USA
E-mail: kosmakmm@ucmail.uc.edu
b) Department of Mathematical Sciences, Indiana University Indianapolis,
402 North Blackford St, Indianapolis, IN 46202-3216, USA
E-mail: vtarasov@iu.edu
Received January 08, 2025, in final form July 08, 2025; Published online July 22, 2025
https://doi.org/10.3842/SIGMA.2025.060
Abstract. We give new combinatorial formulae for vector-valued weight functions (off-
shell nested Bethe vectors) for the evaluation modules over the Yangian Y (gl4). The case
of Y (gln) for an arbitrary n is considered in [Lett. Math. Phys. 115 (2025), 12, 20 pages,
arXiv:2402.15717].
Key words: Bethe ansatz; Yangian; weight functions
2020 Mathematics Subject Classification: 17B37; 81R50; 82B23
1 Introduction
In this paper we will give new combinatorial formulae for vector-valued weight functions for
evaluation modules over the Yangian Y (gln). The weight functions, also known as (off-shell)
nested Bethe vectors, play an important role in the theory of quantum integrable models and
representation theory of Lie algebras and quantum groups. Initially, they appeared in the
framework of the nested algebraic Bethe ansatz as a tool to find eigenvectors and eigenvalues
of transfer matrices of lattice integrable models associated with higher rank Lie algebras [6, 7],
see [21, 22] for a review of the algebraic Bethe ansatz. The results of [6] has been extended to
higher transfer matrices in [11].
Furthermore, the vector-valued weight functions were used to construct hypergeometric so-
lutions of the quantized (difference) Knizhnik–Zamolodchikov equations [10, 28]. They also
showed up in several related problems [3, 12, 24, 26]. In a more recent development, the weight
functions were connected to the stable envelopes for particular Nakajima quiver varieties, the
cotangent bundles of partial flag varieties [15, 16, 17, 18, 27].
For various applications, it is important to have expressions for vector-valued weight functions
for tensor products of evaluation modules over Y (gln). Such expressions can be obtained in two
steps. The first step is to consider weight functions for a single evaluation module, and the
second step is to combine expressions for individual evaluation modules into an expression for
the whole tensor product. In this paper, we will focus on the first step. The second step is fairly
standard and is not specifically discussed here.
By definition, an evaluation Y (gln)-module is a gln-module equipped with the action of Y (gln)
via the evaluation homomorphism Y (gln) → U(gln), see Section 2. The goal is to expand
the vector-valued weight function for the evaluation Y (gln)-module in a basis coming from
the representation theory of gln and find expressions for the coordinates. For Verma modules
mailto:kosmakmm@ucmail.uc.edu
mailto:vtarasov@iu.edu
https://doi.org/10.3842/SIGMA.2025.060
2 M. Kosmakov and V. Tarasov
over gln, such kind of expressions are given in [25]. In this paper, we give a generalization of
formulae from [25].
Combinatorial formulae for the vector-valued weight functions associated with the differential
Knizhnik–Zamolodchikov equations were developed in [1, 8, 9, 14, 19, 20].
The expressions for weight functions in [25] are based on recursions induced by the standard
embeddings of Lie algebras, gl1 ⊕ gln−1 ⊂ gln and gln−1 ⊕ gl1 ⊂ gln. The recursions allow
one to write down weight functions for Y (gln) via weight functions for Y (gln−1). This results
in formulae for coordinates of weight functions in bases of Verma gln-modules of the form{∏
i>j
e
mij
ij v, mij ∈ Z≥0
}
, (1.1)
where eij are the standard generators of gln, see (2.7), v is the highest weight vector, and some
ordering of noncommuting factors is imposed. The ordering is determined by the in-between
part of the involved chain of embeddings gl1 ⊕ · · · ⊕ gl1 ⊂ · · · ⊂ gln. For instance, the chain
gl1 ⊕ · · · ⊕ gl1 ⊂ · · · ⊂ gln−2 ⊕ gl1 ⊕ gl1 ⊂ gln−1 ⊕ gl1 ⊂ gln
yields the ordering
e⊛ij is to the left of e⊛kl if i > k or i = k, j > l, (1.2)
while the chain
gl1 ⊕ · · · ⊕ gl1 ⊂ · · · ⊂ gl1 ⊕ gl1 ⊕ gln−2 ⊂ gl1 ⊕ gln−1 ⊂ gln
yields the ordering
e⊛ij is to the left of e⊛kl if j < l or j = l, i < k. (1.3)
For example for n = 4, the product e43e42e41e32e31e21 obeys ordering (1.2), while the product
e21e31e41e32e42e43 obeys ordering (1.3).
However, some natural orderings of noncommuting factors in (1.1) important for applications
do not show up in the formulae established in [25], see, for instance, [8]. The first nontrivial
example occurs at n = 4 and is given by the basis{
em32
32 em31
31 em42
42 em41
41 em21
21 em43
43 v, mij ∈ Z≥0
}
. (1.4)
To make the set of covered orderings wider, one can consider recursions based on more general
embeddings
glm ⊕ gln−m ⊂ gln with 1 < m < n− 1. (1.5)
For instance, the embedding gl2 ⊕ gl2 ⊂ gl4 yields example (1.4). In this paper, we will work
out example (1.4) in detail with the main result given by Theorem 5.7. We consider the general
case in [5].
We would like to present the gl4 case separately in order to explain calculations more clearly
without introducing too cumbersome notation and to make the exposition paper more accessible.
For the same purpose, we show explicitly intermediate steps in the proofs that commonly might
be tacit for the sake of making a paper shorter. In particular, we give in Appendix A a proof
of Proposition 5.1. Although this statement has a long history, going back to [4], numerous
applications, and is explained in several lecture courses, see [22], its straightforward proof is not
easily available in the literature.
New Combinatorial Formulae for Nested Bethe Vectors 3
At the same time we point out that the proof of Theorem 5.7 in this paper extends almost
in a straightforward way to the proof of [5, Theorem 5.5] in the general gln case. In particular,
the proof of the key Proposition 6.3 in [5] is literally the same as the proof of Proposition 5.6
in this paper.
Unlike [25], we will consider only the case of weight functions for Yangian modules (the
rational case). It turns out that dealing with weight functions for modules over the quantum
loop algebra Uq
(
g̃ln
)
, the trigonometric case, hits an obstacle of essential noncommutativity of
q-analogues of the generators eij , i > j. This obstacle does not show up for the embeddings
gl1 ⊕ gln−1 ⊂ gln and gln−1 ⊕ gl1 ⊂ gln explored in [25], but reveals itself for embeddings (1.5).
For instance, the obstacle in example (1.4) comes from the relation
e42e31 − e31e42 =
(
q − q−1
)
e32e41
that holds in the trigonometric case.
There is an alternative approach to get explicit expressions for the vector-valued weight
functions in the trigonometric case, see [2, 3, 13], based on considering composed currents and
half-currents in the quantum affine algebra and their projections on two Borel subalgebras
of different kind. This approach allows one to recover combinatorial expression for vector-
valued weight functions in evaluation modules in the trigonometric case obtained in [25]. It is
an interesting open question whether the composed currents approach can be helpful to obtain
trigonometric analogues of new combinatorial expressions for vector-valued weight functions
developed in this paper.
2 Notations
We will be using the standard superscript notation for embeddings of tensor factors into tensor
products. For a tensor product of vector spaces V1⊗V2⊗· · ·⊗Vk and an operator A ∈ End(Vi),
denote
A(i) = 1⊗(i−1) ⊗A⊗ 1⊗(k−i) ∈ End(V1 ⊗ V2 ⊗ · · · ⊗ Vk).
Also, if B ∈ End(Vj), i ̸= j, denote (A⊗B)(ij) = A(i)B(j), etc.
Fix a positive integer n. All over the paper we identify elements of End Cn with n×nmatrices
using the standard basis of Cn. That is, for L ∈ EndCn we have L =
(
Lab
)n
a,b=1
, where Lab are
the entries of L. Entries of matrices acting in the tensor products (Cn)⊗k are naturally labeled
by multiindices. For instance, if M ∈ End(Cn ⊗ Cn), then M =
(
Mab
cd
)n
a,b,c,d=1
.
The rational R-matrix is R(u) ∈ End(Cn ⊗ Cn),
R(u) = 1 +
1
u
n∑
a,b=1
Eab ⊗ Eba, (2.1)
where Eab ∈ End (Cn) is the matrix with the only nonzero entry equal to 1 at the intersection
of the a-th row and b-th column. The entries of R(u) are
Rabcd(u) = δacδbd +
1
u
δadδbc.
The R-matrix satisfies the Yang–Baxter equation
R(12)(u− v)R(13)(u)R(23)(v) = R(23)(v)R(13)(u)R(12)(u− v). (2.2)
4 M. Kosmakov and V. Tarasov
The Yangian Y (gln) is a unital associative algebra with generators
(
T ab
){s}
, a, b = 1, . . . , n,
and s = 1, 2, . . . . Organize them into generating series
T ab (u) = δab +
∞∑
s=1
(
T ab
){s}
u−s, a, b = 1, . . . , n. (2.3)
The defining relations in Y (gln) are
(u− v)
[
T ab (u), T
c
d (v)
]
= T ad (u)T
c
b (v)− T ad (v)T
c
b (u) (2.4)
for all a, b, c, d = 1, . . . , n.
Combine series (2.3) into a matrix T (u) =
∑n
a,b=1Eab ⊗ T ab (u) with entries in Y (gln). Then
relations (2.4) amount to the following equality:
R(12)(u− v)T (1)(u)T (2)(v) = T (2)(v)T (1)(u)R(12)(u− v),
where T (1)(u) =
∑n
a,b=1Eab ⊗ 1⊗ T ab (u) and T
(2)(v) =
∑n
a,b=1 1⊗ Eab ⊗ T ab (v).
The Yangian Y (gln) is a Hopf algebra. In terms of generating series (2.3), the coproduct
∆: Y (gln) → Y (gln)⊗ Y (gln) reads as follows:
∆
(
T ab (u)
)
=
n∑
c=1
T cb (u)⊗ T ac (u), a, b = 1, . . . , n. (2.5)
Denote by ∆̃: Y (gln) → Y (gln)⊗ Y (gln) the opposite coproduct
∆̃
(
T ab (u)
)
=
n∑
c=1
T ac (u)⊗ T cb (u), a, b = 1, . . . , n. (2.6)
There is a one-parameter family of automorphisms ρx : Y (gln) → Y (gln) defined in terms of
the series T (u) by the rule ρxT (u) = T (u − x), where in the right-hand side, each expression
(u− x)−s has to be expanded as a power series in u−1.
Denote by eab, a, b = 1, . . . , n, the standard generators of the Lie algebra gln,
[eab, ecd] = eadδbc − ecbδad. (2.7)
A vector v in a gln-module is called singular of weight
(
Λ1, . . . ,Λn
)
if eabv = 0 for all a < b
and eaav = Λav for all a = 1, . . . , n.
The Yangian Y (gln) contains the universal enveloping algebra U(gln) as a Hopf subalgebra.
The embedding is given by the rule eab 7→
(
T ba
){1}
for all a, b = 1, . . . , n. We identify U(gln)
with its image in Y (gln) under this embedding.
The evaluation homomorphism ϵ : Y (gln) → U(gln) is given by the rule ϵ : (T ab )(u) 7→ δab +
ebau
−1 for all a, b = 1, . . . , n. Both the automorphisms ρx and the homomorphism ϵ restricted
to the subalgebra U(gln) are the identity maps.
For a gln-module V , denote by V (x) the Y (gln)-module induced from V by the homomor-
phism ϵ ◦ ρx. The module V (x) is called an evaluation module over Y (gln).
A vector v in a Y (gln)-module is called singular with respect to the action of Y (gln) if
T ab (u)v = 0 for all 1 ≤ b < a ≤ n. A singular vector v that is an eigenvector for the action of
T 1
1 (u), . . . , T
n
n (u) is called a weight singular vector, and the respective eigenvalues are denoted
by
〈
T 1
1 (u)v
〉
, . . . , ⟨Tnn (u)v⟩.
Example. Let V be a gln-module and v ∈ V be a gln-singular vector of weight
(
Λ1, . . . ,Λn
)
.
Then v is a weight singular vector with respect to the action of Y (gln) in the evaluation mod-
ule V (x) and ⟨T aa (u)v⟩ = 1 + Λa(u− x)−1, a = 1, . . . , n.
New Combinatorial Formulae for Nested Bethe Vectors 5
For k < n, we consider two embeddings of the algebra Y (glk) into Y (gln), called ϕk and ψk:
ϕk
(
T ⟨k⟩(u)
)a
b
=
(
T ⟨n⟩(u)
)a
b
, ψk
(
T ⟨k⟩(u)
)a
b
=
(
T ⟨n⟩(u)
)a+n−k
b+n−k (u), (2.8)
with a, b = 1, . . . , k. Here
(
T ⟨k⟩(u)
)a
b
and
(
T ⟨n⟩(u)
)a
b
are series T ab (u) for the algebras Y (glk)
and Y (gln), respectively.
3 Combinatorial formulae for rational weight functions
Fix a collection of nonnegative integers ξ1, ξ2, . . . , ξn−1. Set ξ = (ξ1, ξ2, . . . , ξn−1) and ξa =
ξ1 + · · ·+ ξa, a = 1, . . . , n− 1. Consider the variables tai , a = 1, . . . , n− 1, i = 1, . . . , ξa. We will
also write
ta =
(
ta1, . . . , t
a
ξa
)
, t =
(
t1, . . . , tn−1
)
.
We will use the ordered product notation for any noncommuting factors X1, . . . , Xk,
→∏
1≤i≤k
Xi = X1X2 · · ·Xk,
←∏
1≤i≤k
Xi = XkXk−1 · · ·X1.
Consider the vector space (Cn)⊗ξn−1
and define
[j]
T
(
tj
)
=
→∏
1≤k≤ξj
T (ξj−1+k)
(
tjk
)
,
[k,j]
R
(
tk, tj
)
=
→∏
1≤i≤ξk
( ←∏
1≤l≤ξj
R(ξk−1+i,ξj−1+l)
(
tki − tjl
))
,
where we consider T (ξj−1+k)
(
tjk
)
as a matrix with noncommuting entries belonging to Y (gln).
For the expression
T̂ξ(t) =
[1]
T
(
t1
)
· · ·
[n−1]
T
(
tn−1
) ←∏
1≤i≤n−1
( ←∏
1≤j<i
[i,j]
R
(
ti, tj
))
, (3.1)
denote by Bξ(t) the following entry
Bξ(t) =
(
T̂ξ(t)
)1ξ1 ,2ξ2 ,...,n−1ξn−1
2ξ1 ,3ξ2 ,...,nξn−1
,
where
1ξ1 ,2ξ2 , . . . , (n − 1)ξn−1 = 1, 1, . . . , 1︸ ︷︷ ︸
ξ1
, 2, 2, . . . , 2︸ ︷︷ ︸
ξ2
, . . . , n− 1, n− 1, . . . , n− 1︸ ︷︷ ︸
ξn−1
,
2ξ1 ,3ξ2 , . . . ,nξn−1 = 2, 2, . . . , 2︸ ︷︷ ︸
ξ1
, 3, 3, . . . , 3︸ ︷︷ ︸
ξ2
, . . . , n, n, . . . , n︸ ︷︷ ︸
ξn−1
.
To indicate the dependence on n, if necessary, we will write B⟨n⟩ξ (t).
Example. Let n = 2 and ξ = (ξ1). Then B⟨2⟩ξ (t) = T 1
2
(
t11
)
. . . T 1
2
(
t1ξ1
)
. Abusing notation, we
will farther write B⟨2⟩ξ1 (t) instead of B⟨2⟩ξ (t).
6 M. Kosmakov and V. Tarasov
Example. Let n = 4 and ξ = (1, 1, 1). Then
B⟨4⟩ξ (t) = T 1
2
(
t11
)
T 2
3
(
t21
)
T 3
4
(
t31
)
+
1
t21 − t11
T 1
3
(
t11
)
T 2
2
(
t21
)
T 3
4
(
t31
)
+
1
t31 − t21
T 1
2
(
t11
)
T 2
4
(
t21
)
T 3
3
(
t31
)
+
1(
t21 − t11
)(
t31 − t21
)(T 1
4
(
t11
)
T 2
2
(
t21
)
T 3
3
(
t31
)
+ T 1
3
(
t11
)
T 2
4
(
t21
)
T 3
2
(
t31
))
+
(
t21 − t11
)(
t31 − t21
)
+ 1(
t21 − t11
)(
t31 − t11
)(
t31 − t21
)T 1
4
(
t11
)
T 2
3
(
t21
)
T 3
2
(
t31
)
.
For a weight singular vector v with respect to the action of Y (gln), we call the expression
Bξ(t)v the (rational) vector-valued weight function of weight (ξ1, ξ2−ξ1, . . . , ξn−1−ξn−2,−ξn−1)
associated with v.
From now on, we will consider only the case n = 4. We are interested in writing down the
following expansion for a weight function in a evaluation module over the Y (gl4):
Bξ(t)v =
∑
m⃗∈Z6
≥0
Fm⃗(t) · em32
32 em31
31 em42
42 em41
41 em21
21 em43
43 v (3.2)
with the functions Fm⃗(t) given by explicit formulae. Various similar expansions for Bξ(t)v were
obtained in [25], however, expansion (3.2) is not covered there.
4 Splitting property of the weight functions
Let T
⟨2⟩
ab (u) be series (2.3) for the algebra Y (gl2), and R⟨2⟩(u) be the corresponding rational
R-matrix, see (2.1). Consider two Y (gl2)-module structures on the vector space C2. The first
one, called L(x), is given by the rule
π(x) : T ⟨2⟩(u) 7→ R⟨2⟩(u− x),
and the second one, called L̄(x), is given by the rule
ϖ(x) : T ⟨2⟩(u) 7→
((
R⟨2⟩(x− u)
)(21))t2 ,
where the superscript t2 stands for the matrix transposition in the second tensor factor.
Let w1, w2 be the standard basis of the space C2. The module L(x) is a highest weight eval-
uation module with gl2 highest weight (1, 0) and highest weight vector w1. The module L̄(x) is
a highest weight evaluation module with gl2 highest weight (0,−1) and highest weight vector w2.
For any X ∈ End
(
C2
)
, set ν(X) = Xw1 and ν̄(X) = Xw2.
Recall the coproducts ∆ and ∆̃, see (2.5) and (2.6), and the embeddings ψ2 : Y (gl2) → Y (gl4)
and ϕ2 : Y (gl2) → Y (gl4) given by (2.8). For any k, denote by
∆(k) : Y (gl2) → (Y (gl2))
⊗(k+1) and ∆̃(k) : Y (gl2) → (Y (gl2))
⊗(k+1)
the iterated coproduct and opposite coproduct. Consider the maps
ψ2(x1, . . . , xk) : Y (gl2) →
(
C2
)⊗k ⊗ Y (gl4),
ψ2(x1, . . . , xk) =
(
ν⊗k ⊗ id
)
◦ (π(x1)⊗ · · · ⊗ π(xk)⊗ ψ2) ◦∆(k),
and
ϕ2(x1, . . . , xk) : Y (gl2) →
(
C2
)⊗k ⊗ Y (gl4),
ϕ2(x1, . . . , xk) =
(
ν̄⊗k ⊗ id
)
◦ (ϖ(x1)⊗ · · · ⊗ϖ2(xk)⊗ ϕ2) ◦ ∆̃(k).
New Combinatorial Formulae for Nested Bethe Vectors 7
For any element g ∈
(
C2
)⊗k ⊗ Y (gl4), we define its components ga, a = (a1, . . . , ak), by the rule
g =
2∑
a1,...,ak=1
wa1 ⊗ · · · ⊗wak ⊗ ga.
In the gl4 case, we have ξ = (ξ1, ξ2, ξ3), and formula (3.1) takes the form
Bξ(t) =
([1]
T
(
t1
)[2]
T
(
t2
)[3]
T
(
t3
)[32]
R
(
t3, t2
)[31]
R
(
t3, t1
)[21]
R
(
t2, t1
))1ξ1 ,2ξ2 ,3ξ3
2ξ1 ,3ξ2 ,4ξ3
.
Proposition 4.1. Let v be a Y (gl4)-singular vector, ξ1, ξ2, ξ3 be nonnegative integers, and
t =
(
t11, . . . , t
1
ξ1
; t21, . . . , t
2
ξ2
; t31, . . . , t
3
ξ3
)
. Then
Bξ(t)v =
∑
a,b
(
T
(
t2
))a
b
(
ϕ2
(
t2
)(
B⟨2⟩ξ1
(
t1
)))a(
ψ2
(
t2
)(
B⟨2⟩ξ3
(
t3
)))b−2
v, (4.1)
where the sum is taken over all sequences a = (a1, a2, . . . , aξ2), b = (b1, b2, . . . , bξ2), such that
ai ∈ {1, 2}, bi ∈ {3, 4} for all i = 1, . . . , ξ2, b − 2 = (b1 − 2, b2 − 2, . . . , bξ2 − 2), and(
T
(
t2
))a
b
= T
(
t21
)a1
b1
T
(
t22
)a2
b2
· · ·T
(
t2ξ2
)aξ2
bξ2
.
Proof. Formula (4.1) follows from the definition of the maps ψ2
(
t2
)
and ϕ2
(
t2
)
and Lemma 4.2
below. ■
Lemma 4.2. One has
Bξ(t)v =
∑
a,b
(
T
(
t2
))a
b
([21]
R
(
t2, t1
)[1]
T
(
t1
))1ξ1 ,2ξ2 ,3ξ3
2ξ1 ,a,3ξ3
([3]
T
(
t3
)[32]
R
(
t3, t2
))1ξ1 ,b,3ξ3
1ξ1 ,3ξ2 ,4ξ3
v,
where the sum over a, b is the same as in formula (4.1).
Proof. Using Yang–Baxter equation (2.2), we can write B(t) in the following form:
Bξ(t)v =
([21]
R
(
t2, t1
)[2]
T
(
t2
)[1]
T
(
t1
)[3]
T
(
t3
)[31]
R
(
t3, t1
)[32]
R
(
t3, t2
))1ξ1 ,2ξ2 ,3ξ3
2ξ1 ,3ξ2 ,4ξ3
v.
Therefore,
Bξ(t)v =
∑
p,q,r,s
([21]
R
[2]
T
[1]
T
)1ξ1 ,2ξ2 ,3ξ3
p,q,3ξ3
([3]
T
)p,q,3ξ3
p,q,r
([31]
R
)p,q,r
2ξ1 ,q,s
([32]
R
)2ξ1 ,q,s
2ξ1 ,3ξ2 ,4ξ3
v, (4.2)
where p = (p1, . . . , pξ1), q = (q1, . . . , qξ2), r = (r1, . . . , rξ3), s = (s1, . . . , sξ3). In (4.2), we omit-
ted the arguments t1, t2, t3 since they can be restored from the context.
We say that r ≥ 3ξ3 if ri ≥ 3 for all i = 1, . . . , ξ3. Observe that by the definition of a singular
vector and the commutation relations
T 3
b (w)T
3
d (u) =
w − u− 1
w − u
T 3
d (u)T
3
b (w) +
1
w − u
T 3
d (w)T
3
b (u),
we have
[3]
T
(
t3
)3ξ3
r
v = 0 unless r ≥ 3ξ3 .
Furthermore, for r ≥ 3ξ3 , we have by induction on ξ3 that([31]
R
)p,q,r
2ξ1 ,q,s
= δp,2ξ1 δr,s.
8 M. Kosmakov and V. Tarasov
Indeed, for ξ3 = 0, the statement is true. Assume that([31]
R
)p,q,r
2ξ1 ,q,s
= δp,2ξ1 δr,s
for r ≥ 3ξ3 if ξ3 = n − 1, and consider the case ξ3 = n. Let r = (r1, . . . , rn), s = (s1, . . . , sn),
r̃ = (r1, . . . , rn−1), s̃ = (s1, . . . , sn−1), then we have
([31]
R
)p,q,r
2ξ1 ,q,s
=
∑
x
( →∏
1≤i≤n−1
( ←∏
1≤j≤ξ1
R(ξ2+i,j)
))p,q,r̃,rn
x,q,s̃,rn
( ←∏
1≤k≤ξ1
R(ξ3,k)
)x,q,s̃,rn
2ξ1 ,q,s̃,sn
. (4.3)
Observe that the R-matrix entry Rjlik with i ̸= l is not zero if and only if i = j and k = l, and
Rikik = 1. Because of that and since rn ≥ 3, the last factor
(∏
R(ξ3,k)
)x,q,s̃,rn
2ξ1 ,q,s̃,sn
in (4.3) equals
δx,2ξ1 δrn,sn , and we get
([31]
R
)p,q,r
2ξ1 ,q,s
=
( →∏
1≤i≤n−1
( ←∏
1≤j≤ξ1
R(ξ2+i,j)
))p,q,r̃,rn
2ξ1 ,q,s̃,rn
δrn,sn = δp,2ξ1 δr̃,s̃δrn,sn = δp,2ξ1 δr,s,
by the induction assumption.
Since([3]
T
)p,q,3ξ3
p,q,r
v = 0
unless r ≥ 3ξ3 and([31]
R
)p,q,r
2ξ1 ,q,s
= δp,2ξ1 δr,s
for r ≥ 3, formula (4.2) becomes
Bξ(t)v =
∑
p,q,r
([21]
R
[2]
T
[1]
T
)1ξ1 ,2ξ2 ,3ξ3
2ξ1 ,q,3ξ3
([3]
T
)2ξ1 ,q,3ξ3
2ξ1 ,q,r
([32]
R
)2ξ1 ,q,r
2ξ1 ,3ξ2 ,4ξ3
v,
and can be further transformed as
Bξ(t)v =
∑
a,b,c,r
([2]
T
)c,a,3ξ3
c,b,3ξ3
([21]
R
)1ξ1 ,2ξ2 ,3ξ3
c,a,3ξ3
([1]
T
)c,b,3ξ3
2ξ1 ,b,3ξ1
([3]
T
)2ξ1 ,b,3ξ3
2ξ1 ,b,r
([32]
R
)2ξ1 ,b,r
2ξ1 ,3ξ2 ,4ξ3
v, (4.4)
where the sum is over all sequences a = (a1. . . . , aξ2), b = (b1. . . . , bξ2), c = (c1, . . . , cξ1),
r = (r1, . . . , rξ3) such that ai, bi, ci, ri ∈ {1, 2, 3, 4}. Since
([21]
R
)1ξ1 ,2ξ2 ,3ξ3
c,a,3ξ3
= 0
if ai ≥ 3 for some i, and([32]
R
)2ξ1 ,b,r
2ξ1 ,3ξ2 ,4ξ3
= 0
if bi ≤ 2 for some i, terms in the sum in the right-hand side of (4.4) equal zero unless ai ∈ {1, 2}
and bi ∈ {3, 4} for all i. Taking the sum over c and r in formula (4.4), we get the statement of
Lemma 4.2. ■
New Combinatorial Formulae for Nested Bethe Vectors 9
Example. Here we illustrate the proof of the relation([31]
R
)p,q,r
2ξ1 ,q,s
= δp,2ξ1 δr,s
if r ≥ 3ξ3 for ξ1 = ξ3 = 2. In this case, p = (p1, p2), r = (r1, r2), s = (s1, s2), and([31]
R
)p,q,r
2ξ1 ,q,s
=
∑
a,b,c,d
Rp2r1ab
(
t31 − t12
)
Rp1bcs1
(
t31 − t11
)
Rar22d
(
t32 − t12
)
Rcd2s2
(
t32 − t11
)
.
For r1 ≥ 3, r2 ≥ 3, we have Rar22d
(
t32 − t12
)
= δa,2δr2,d, thus([31]
R
)p,q,r
2ξ1 ,q,s
=
∑
b,c
Rp2r12b
(
t31 − t12
)
Rp1bcs1
(
t31 − t11
)
Rcr22s2
(
t32 − t11
)
.
Then Rcr22s2
(
t32 − t11
)
= δc,2δr2,s2 and Rp2r12b
(
t31 − t12
)
= δp2,2δr1,b, so that([31]
R
)p,q,r
2ξ1 ,q,s
= Rp1r12s1
(
t31 − t11
)
δp2,2δr2,s2 = δp2,2δp1,2δr1,s1δr2,s2 = δp,2ξ1 δr,s.
5 Main theorem for the gl4 case
The main result of this paper is Theorem 5.7 formulated at the end of this section. We will
approach it in several steps.
For a nonnegative integer m, set
Qm(t1, . . . , tm) =
∏
1⩽i<j⩽m
ti − tj − 1
ti − tj
. (5.1)
For an expression f(t1, . . . , tm), define
Symt f(t1, . . . , tm) =
∑
σ∈Sm
f
(
tσ(1), . . . , tσ(m)
)
,
and
Symt f(t1, . . . , tm) = Symt(f(t1, . . . , tm)Qm(t1, . . . , tm)). (5.2)
To simplify notation, we will write T
⟨2⟩
ij instead of
(
T ⟨2⟩
)i
j
.
Proposition 5.1. Let ξ be a nonnegative integer and t = (t1, . . . , tξ). Then
∆
(
B⟨2⟩ξ (t)
)
=
ξ∑
η=0
1
(ξ − η)!η!
×Symt
((
B⟨2⟩η (t1, . . . , tη)⊗ B⟨2⟩ξ−η(tη+1, . . . , tξ)
)( ξ∏
i=η+1
T
⟨2⟩
22 (ti)⊗
η∏
j=1
T
⟨2⟩
11 (tj)
))
.
This proposition goes back to [4, 28]. For convenience, we give its proof in Appendix A.
Given a subset I of {1, 2, . . . , k} denote by I∗ the complement of I in {1, 2, . . . , k}. Define
a vector wI ∈
(
C2
)⊗k
by the rule
wI = wa1 ⊗wa2 ⊗ · · · ⊗wak ,
where ai = 2 if i ∈ I, and ai = 1 if i ̸∈ I.
10 M. Kosmakov and V. Tarasov
Fix a Y (gl2)-module V and a weight singular vector v ∈ V with respect to the Y (gl2)-action,
T
⟨2⟩
21 (u)v = 0, T
⟨2⟩
11 (u)v =
〈
T
⟨2⟩
11 (u)v
〉
v, T
⟨2⟩
22 (u)v =
〈
T
⟨2⟩
22 (u)v
〉
v.
Here
〈
T
⟨2⟩
11 (u)v
〉
and
〈
T
⟨2⟩
22 (u)v
〉
are the corresponding eigenvalues. Given complex numbers
z1, . . . , zk, consider the Y (gl2)-module L(z1)⊗· · ·⊗L(zk)⊗V . Observe that w⊗k1 ⊗ v is a weight
singular vector with respect to the action of Y (gl2) in this module.
Proposition 5.2. For the Y (gl2)-module L(z1)⊗ · · · ⊗ L(zk)⊗ V , we have
B⟨2⟩ξ (t)
(
w⊗k1 ⊗ v
)
=
∑
I
1
(ξ − |I|)!
Symt
[
FI(t, z)
|I|∏
a=1
〈
T
⟨2⟩
11 (ta)v
〉(
wI ⊗ B⟨2⟩ξ−|I|
(
t|I|+1, . . . , tξ
)
v
)]
, (5.3)
where the sum is over all subsets I ⊂ {1, . . . , k} such that |I| ≤ ξ, and for a given I = {i1 <
i2 < · · · < i|I|},
FI(t, z) =
|I|∏
a=1
(
1
ta − zia
k∏
m=ia+1
ta − zm + 1
ta − zm
)
. (5.4)
Proof. Observe that for each Y (gl2)-module L(zi), i = 1, . . . , k, the corresponding vector
w1 ∈ L(zi) is a weight singular vector,
T
⟨2⟩
11 (u)w1 =
(
1 + (u− zi)
−1)w1, T
⟨2⟩
22 (u)w1 = w1, T
⟨2⟩
21 w1 = 0.
Moreover, B⟨2⟩1 (u)w1 = T
⟨2⟩
12 (u)w1 = (u− zi)
−1w2 and B⟨2⟩ζ (u1, . . . , uζ)w1 = 0 for ζ ≥ 2. Then
formula (5.3) follows from Proposition 5.1 and identity (A.10) by induction on k. ■
Given complex numbers z1, . . . , zk, consider the Y (gl2)-module V ⊗ L̄(zk) ⊗ · · · ⊗ L̄(z1).
Observe that v ⊗w⊗k2 is a weight singular vector with respect to the action of Y (gl2) in this
module.
Proposition 5.3. For the Y (gl2) module V ⊗ L̄(zk)⊗ · · · ⊗ L̄(z1), we have
B⟨2⟩ξ (t)
(
v ⊗w⊗k2
)
(5.5)
=
∑
I
1
(ξ − |I|)!
Symt
[
F̃I(t, z)
|I|∏
i=1
〈
T
⟨2⟩
22
(
tξ−|I|+i
)
v
〉(
B⟨2⟩ξ−|I|
(
t1, . . . , tξ−|I|
)
v ⊗wI∗
)]
,
where the sum is over all subsets I ⊂ {1, . . . , k} such that |I| ≤ ξ, and for a given I = {i1 <
i2 < · · · < i|I|},
F̃I(t, z) =
|I|∏
a=1
(
1
zia − tξ−a+1
k∏
m=ia+1
zm − tξ−a+1 + 1
zm − tξ−a+1
)
. (5.6)
Proof. Observe that for each Y (gl2)-module L̄(zi), i = 1, . . . , k, the corresponding vector
w2 ∈ L̄(zi) is a weight singular vector,
T
⟨2⟩
11 (u)w2 = w2, T
⟨2⟩
22 (u)w2 =
(
1 + (zi − u)−1
)
w2, T
⟨2⟩
21 (u)w2 = 0.
Moreover, B⟨2⟩1 (u)w2 = T
⟨2⟩
12 (u)w2 = (zi − u)−1w1, and B⟨2⟩ζ (u1, . . . , uζ)w2 = 0 for ζ ≥ 2. Then
formula (5.5) follows from Proposition 5.1 and identity (A.10) by induction on k. ■
New Combinatorial Formulae for Nested Bethe Vectors 11
For t = (t1, . . . , tξ), z = (z1, . . . .zk), y ∈ C, and a subset I = {i1 < i2 < · · · < i|I|} ⊂
{1, . . . , k}, define the functions
VI(t, z, y) =
1
(ξ − |I|)!
Symt
(
FI(t, z)
|I|∏
a=1
(ta − y)
)
(5.7)
and
ṼI(t, z, y) =
1
(ξ − |I|)!
Symt
(
F̃I(t, z)
|I|∏
a=1
(tξ−a+1 − y)
)
. (5.8)
Consider the collection Sp,q,r,k of pairs of subsets of {1, . . . , k} with given cardinalities of the
subsets and their intersection. Namely,
Sp,q,r,k = {(I, J) | I, J ⊂ {1, . . . , k}, |I| = p, |J | = q, |I ∩ J | = r}.
For I ⊂ {1, . . . , k}, set Ǐ = {k − i+ 1, i ∈ I}.
Theorem 5.4. Let V be a gl4-module and v ∈ V a gl4-singular vector of weight
(
Λ1,Λ2,Λ3,Λ4
)
.
Let ξ1, ξ2, ξ3 be nonnegative integers, t1 =
(
t11, . . . , t
1
ξ1
)
, t2 =
(
t21, . . . , t
2
ξ2
)
, t3 =
(
t31, . . . , t
3
ξ3
)
,
and t =
(
t1, t2, t3
)
. For every triple (p, q, r), p = 0, . . . ,min(ξ2, ξ3), q = 0, . . . ,min(ξ2, ξ1),
r = max(0, p+ q − ξ2), . . . ,min(p, q), fix a pair (Ip,q,r, Jp,q,r) ∈ Sp,q,r,ξ2. Then,
(a) In the evaluation Y (gl4)-module V (x), one has
Bξ(t)v =
3∏
a=1
ξa∏
i=1
1
tai − x
×
min(ξ2,ξ3)∑
p=0
min(ξ2,ξ1)∑
q=0
min(p,q)∑
r=max(0,p+q−ξ2)
(
Symt2
(
VǏp,q,r
(
t3, t2, x− Λ3
)
ṼJp,q,r
(
t1, ť
2
, x− Λ2
))
× eξ2−p−q+r32 eq−r31 ep−r42 er41e
ξ1−q
21 eξ3−p43 v
(p− r)!(q − r)!r!(ξ2 − p− q − r)!
)
, (5.9)
where ť
2
=
(
t2ξ2 , . . . , t
2
1
)
.
(b) The function Symt2
(
VǏp,q,r
(
t3, t2, x− Λ3
)
ṼJp,q,r
(
t1, ť
2
, x− Λ2
))
in (5.9) does not depend
on the choice of the pair (Ip,q,r, Jp,q,r).
Proof. Item (a) follows from Propositions 5.5 and 5.6 given below. Item (b) is an immediate
corollary of Proposition 5.6.
Propositions 5.5 and 5.6 are proved in Sections 6 and 7, respectively. ■
Proposition 5.5. In the notation of Theorem 5.4, we have
Bξ(t)v =
3∏
a=1
ξa∏
i=1
1
tai − x
×
min(ξ2,ξ3)∑
p=0
min(ξ2,ξ1)∑
q=0
min(p,q)∑
r=max(0,p+q−ξ2)
( ∑
(I,J)∈Sp,q,r,ξ2
ṼJ
(
t1, t2, x− Λ2
)
VI
(
t3, t2, x− Λ3
)
× eξ2−p−q+r32 eq−r31 ep−r42 er41e
ξ1−q
21 eξ3−p43 v
)
. (5.10)
12 M. Kosmakov and V. Tarasov
Proposition 5.6. In the notation of Theorem 5.4, we have∑
(I,J)∈Sp,q,r,ξ2
ṼJ
(
t1, t2, x− Λ2
)
VI
(
t3, t2, x− Λ3
)
=
Symt2
(
VǏ0
(
t3, t2, x− Λ3
)
ṼJ0
(
t1, ť
2
, x− Λ2
))
(p− r)!(q − r)!r!(ξ2 − p− q − r)!
, (5.11)
where (I0, J0) is any pair from Sp,q,r,ξ2.
Below we reformulate Theorem 5.4 in a more closed form.
Theorem 5.7. Let V be a gl4-module and v ∈ V a gl4-singular vector of weight
(
Λ1,Λ2,Λ3,Λ4
)
.
Let ξ1, ξ2, ξ3 be nonnegative integers, t1 =
(
t11, . . . , t
1
ξ1
)
, t2 =
(
t21, . . . , t
2
ξ2
)
, t3 =
(
t31, . . . , t
3
ξ3
)
,
and t =
(
t1, t2, t3
)
. For every triple (p, q, r), p = 0, . . . ,min(ξ2, ξ3), q = 0, . . . ,min(ξ2, ξ1), r =
max(0, p+ q − ξ2), . . . ,min(p, q), fix two sequences i = {i1 < · · · < ip} and j = {j1 < · · · < jq},
such that |{i1, . . . , ip} ∩ {j1, . . . , jq}| = r. Then,
(a) In the evaluation Y (gl4)-module V (x), one has
Bξ(t)v =
3∏
a=1
ξa∏
i=1
1
tai − x
×
min(ξ2,ξ3)∑
p=0
min(ξ2,ξ1)∑
q=0
min(p,q)∑
r=max(0,p+q−ξ2)
Symt1 Symt2 Symt3 Gi,j(t)
× eξ2−p−q+r32 eq−r31 ep−r42 er41e
ξ1−q
21 eξ3−p43 v
(ξ2 − p− q + r)!(q − r)!(p− r)!r!(ξ1 − q)!(ξ3 − p)!
, (5.12)
where
Gi,j(t) =
p∏
a=1
(
t3a − x+ Λ3
t3a − t2ia
ξ2∏
m=ia+1
t3a − t2m + 1
t3a − t2m
)
×
q∏
s=1
(
t1ξ1−q+s − x+ Λ2
t2js − t1ξ1−q+s
ξ2−js∏
l=1
t2l − t1ξ1−q+s + 1
t2l − t1ξ1−q+s
)
. (5.13)
(b) The function Symt1 Symt2 Symt3Gi,j(t) does not depend on the choice of the sequences i, j.
Proof. Given the pair (Ip,q,r, Jp,q,r) from the formulation of Theorem 5.4, define the sequences
i = {i1 < · · · < ip} and j = {j1 < · · · < jq} by the rule
Ip,q,r = {ξ2 − i1 + 1, ξ2 − i2 + 1, . . . , ξ2 − ip + 1},
Jp,q,r = {ξ2 − j1 + 1, ξ2 − j2 + 1, . . . , ξ2 − jq + 1}.
Notice that {i1, . . . , ip} = Ǐp,q,r, {j1, . . . , jq} = J̌p,q,r, and
|{i1, . . . , ip} ∩ {j1, . . . , jq}| = |Ip,q,r ∩ Jp,q,r| = r.
Then combining formulae (5.4) and (5.6)–(5.8), we obtain that
VǏp,q,r
(
t3, t2, x− Λ3
)
=
1
(ξ3 − p)!
Symt3
p∏
a=1
(
t3a − x+ Λ3
t3a − t2ia
ξ2∏
m=ia+1
t3a − t2m + 1
t3a − t2m
)
(5.14)
New Combinatorial Formulae for Nested Bethe Vectors 13
and
ṼJp,q,r
(
t1, ť
2
, x− Λ2
)
=
1
(ξ1 − q)!
Symt1
q∏
b=1
(
t1ξ1−b+1 − x+ Λ2
t2jq−b+1
− t1ξ1−b+1
ξ2−jq−b+1∏
l=1
t2l − t1ξ1−b+1 + 1
t2l − t1ξ1−b+1
)
.
After substituting b = q + 1− s, the last formula becomes
ṼJp,q,r
(
t1, ť
2
, x− Λ2
)
=
1
(ξ1 − q)!
Symt1
q∏
s=1
(
t1ξ1−q+s − x+ Λ2
t2js − t1ξ1−q+s
ξ2−js∏
l=1
t2l − t1ξ1−q+s + 1
t2l − t1ξ1−q+s
)
. (5.15)
Plugging (5.14) and (5.15) into formula (5.9), we obtain formulae (5.12) and (5.13).
Item (b) of Theorem 5.7 is a reformulation of item (b) of Theorem 5.4. ■
Example. Below we give two examples of natural choices of the sequences i, j in Theorem 5.7
and write down the corresponding expressions for the function Gi,j(t), see formula (5.13).
(a) i = i1 = {1 < · · · < p}, j = j1 = {p+ 1− r < · · · < p+ q − r}. Then
Gi1,j1(t) =
p∏
a=1
(
t3a − Λ3
t3a − t2a
ξ2∏
m=a+1
t3a − t2m + 1
t3a − t2m
)
×
q∏
c=1
(
t1ξ1−q+c − Λ2
t2p−r+c − t1ξ1−q+c
ξ2−p+r−c∏
l=1
t2l − t1ξ1−q+c + 1
t2l − t1ξ1−q+c
)
.
(b) i = i2 = {q + 1− r < · · · < q + p− r}, j = j2 = {1 < · · · < q}. Then
Gi2,j2(t) =
p∏
a=1
(
t3a − Λ3
t3a − t2q−r+a
ξ2∏
m=q−r+a+1
t3a − t2m + 1
t3a − t2m
)
×
q∏
b=1
(
t1ξ1−b+1 − Λ2
t2b − t1ξ1−b+1
ξ2−b∏
l=1
t2l − t1ξ1−b+1 + 1
t2l − t1ξ1−b+1
)
.
Notice that the equality
Symt1 Symt2 Symt3 Gi1,j1(t) = Symt1 Symt2 Symt3 Gi2,j2(t),
stated in item (b) of Theorem 5.7, is not obvious.
6 Proof of Proposition 5.5
Let V be a gl4-module and v ∈ V a gl4-singular vector of weight
(
Λ1,Λ2,Λ3,Λ4
)
. Let ξ1, ξ2, ξ3 be
nonnegative integers, t1 =
(
t11, . . . , t
1
ξ1
)
, t2 =
(
t21, . . . , t
2
ξ2
)
, t3 =
(
t31, . . . , t
3
ξ3
)
and t =
(
t1, t2, t3
)
.
Recall that in the evaluation Y (gl4)-module V (x), we have T ab (u) = δab + eba(u− x)−1, thus
T aa (u)v =
u− x+ Λa
u− x
v.
By Proposition 4.1,
Bξ(t)v =
∑
a,b
([2]
T
(
t2
))a
b
(
ϕ2
(
t2
)(
B⟨2⟩ξ1
(
t1
)))a(
ψ2
(
t2
)(
B⟨2⟩ξ3
(
t3
)))b−2
v, (6.1)
14 M. Kosmakov and V. Tarasov
where the sum is taken over all sequences a = (a1, a2, . . . , aξ2), b = (b1, b2, . . . , bξ2), such that
ai ∈ {1, 2}, and bi ∈ {3, 4} for all i = 1, . . . , ξ2.
Let ψV (x) be the Y (gl2)-module obtained by pulling back the module V (x) through the em-
bedding ψ2. To compute
(
ψ2
(
t2
)(
B⟨2⟩ξ3
(
t3
)))b
v, we take the weight function B⟨2⟩ξ3
(
t3
)(
w⊗ξ21 ⊗ v
)
in the Y (gl2)-module L
(
t21
)
⊗ · · · ⊗ L
(
t2ξ2
)
⊗ ψV (x) and apply Proposition 5.2 for k = ξ2.
Then we obtain(
ψ2
(
t2
)(
B⟨2⟩ξ3
(
t3
)))b−2
v
=
1
(ξ3 − |I|)!
Symt3
[
FI
(
t3, t2
) |I|∏
m=1
t3m − x+ Λ3
t3m − x
ξ3∏
r=|I|+1
1
t3r − x
]
e
ξ3−|I|
43 v, (6.2)
where the subset I ⊂ {1, . . . , ξ2} and the sequence b = (b1, . . . , bξ2) are related as follows: bj = 3
if j /∈ I and bj = 4 if j ∈ I. Therefore, by formula (5.8) we have
(
ψ2
(
t2
)(
B⟨2⟩ξ3
(
t3
)))b−2
v =
ξ3∏
r=1
1
t3r − x
VI
(
t3, t2, x− Λ3
)
e
ξ3−|I|
43 v.
The next step is to compute
(
ϕ2
(
t2
)(
B⟨2⟩ξ1
(
t1
)))a
e
ξ3−|I|
43 v. Notice that for any nonnegative
integer m, we have
T 2
1 (u)e
m
43v = 0, T 2
2 (u)e
m
43v =
u− x+ Λ2
u− x
em43v, T 1
1 (u)e
m
43v =
u− x+ Λ1
u− x
em43v.
Let ϕV (x) the Y (gl2)-module obtained by pulling back V (x) through the embedding ϕ2. To
compute the
(
ϕ2
(
t2
)(
B⟨2⟩ξ1
(
t1
)))a
e
ξ3−|I|
43 v, we take the weight function B⟨2⟩ξ1
(
t1
)(
e
ξ3−|I|
43 v ⊗w⊗ξ22
)
in the Y (gl2)-module ϕV (x)⊗ L
(
t2ξ2
)
⊗ · · · ⊗ L
(
t21
)
and apply Proposition 5.3 for k = ξ2. Then
we obtain(
ϕ2
(
t2
)(
B⟨2⟩ξ1
(
t1
)))a
e
ξ3−|I|
43 v
=
1
(ξ1 − |J |)!
Symt1
[
F̃J
(
t1, t2
) |J |∏
m=1
t1m − x+ Λ2
t1m − x
ξ1∏
r=|J |+1
1
t1r − x
]
e
ξ1−|J |
21 e
ξ3−|I|
43 v, (6.3)
where the subset J ⊂ {1, . . . , ξ2} and the sequence a = (a1, . . . , aξ2) are related as follows:
aj = 1 if j ∈ J and aj = 2 if j /∈ J . Therefore, by formula (5.7) we have
(
ϕ2
(
t2
)(
B⟨2⟩ξ1
(
t1
)))a
e
ξ3−|I|
43 v =
ξ1∏
r=1
1
t1r − x
ṼJ
(
t1, t2, x− Λ2
)
e
ξ1−|J |
21 e
ξ3−|I|
43 v.
Finally, for the sequences a, b that are related to the sets I, J as above, we have
([2]
T
(
t2
))a
b
=
ξ2∏
r=1
1
t2r − x
eξ2−p−q+s32 eq−s31 ep−s42 es41, (6.4)
where p = |I|, q = |J |, s = |I ∩ J |. Now formula (5.10) follows from formulae (6.1)–(6.4).
7 Proof of Proposition 5.6
Consider the algebra A generated by two commuting copies of the symmetric group Sk and
rational functions of z1, . . . , zk subject to relations (7.1) below. We denote the copies of Sk
New Combinatorial Formulae for Nested Bethe Vectors 15
in A by Ṡk and S̈k, and mark elements of Ṡk and S̈k by the corresponding dots, keeping the
notation Sk without dots for the abstract symmetric group.
Let z = (z1, . . . , zk) and zσ = (zσ(1), . . . , zσ(k)). The additional relations in A are
σ̇f(z) = f(zσ)σ̇, τ̈ f(z) = f(z)τ̈ . (7.1)
For a = 1, . . . , k − 1, let sa ∈ Sk be the transposition of a and a+ 1. Consider the elements
ŝ1, . . . , ŝk of A,
ŝa =
(
za − za+1
za − za+1 − 1
s̈a −
1
za − za+1 − 1
)
ṡa. (7.2)
It is straightforward to check that they satisfy the following relations:
ŝaŝa+1ŝa = ŝa+1ŝaŝa+1, ŝ2a = 1.
Therefore, the assignment sa 7→ ŝa defines an algebra homomorphism CSk → A. For any σ ∈ Sk,
we denote by σ̂ the corresponding element of A. Every element σ̂ can be written in the following
form:
σ̂ =
∑
τ∈Sk
Xσ,τ (z)τ̈ σ̇, (7.3)
where Xσ,τ (z) are functions of z1, . . . , zk.
Let |σ| denote the length of σ ∈ Sk.
Lemma 7.1. The functions Xσ,τ (z) have the following properties:
Xσ,τ (z) = 0 if |τ | > |σ|, (7.4)
Xσ,τ (z) = δσ,τXσ,σ(z) if |τ | = |σ|, (7.5)
Xσ,σ(z) =
∏
a<b,σ−1(a)>σ−1(b)
za − zb
za − zb − 1
. (7.6)
Proof. Formulae (7.4) and (7.5) follow from formula (7.2) by inspection. Formula (7.6) can be
shown by induction on |σ|. ■
Denote by σ0 the longest element of Sk, σ0(i) = k − i+ 1, i = 1, . . . , k. Let
Φ(z) =
∏
a<b
za − zb − 1
za − zb
.
Notice that
Φ(z) =
1
Xσ0,σ0(z)
. (7.7)
Lemma 7.2. One has∑
λ∈Sk
Xλ,ρ(z)Φ
(
zλσ0
)
Xσ0λ−1,σ0τ−1
(
zλσ0
)
= δρ,τ . (7.8)
Proof. Since σ̂τ = σ̂τ̂ , by formula (7.3) we have
Xστ,ρ(z) =
∑
π
Xσ,π(z)Xτ,π−1ρ(z
σ). (7.9)
16 M. Kosmakov and V. Tarasov
Taking here ρ = σ0, and using Lemma 7.1 and formula (7.7), we get∑
π
Xσ,π(z)Xτ,π−1σ0(z
σ) = δστ,σ0
1
Φ(z)
. (7.10)
Replacing now z by zσ
−1
in formula (7.10) and taking there τ = µ−1σ0, we get∑
π
Xσ,π
(
zσ
−1)
Xµ−1σ0,π−1σ0(z) = δσ,µ
1
Φ
(
zσ−1
) . (7.11)
Formula (7.11) can be understood as the matrix equality AB = C for k!× k! matrices A, B, C
with entries labeled by permutations:
Aσ,π = Xσ,π
(
zσ
−1)
, Bπ,µ = Xµ−1σ0,π−1σ0(z), Cσ,µ = δσ,µ
1
Φ
(
zσ−1
) .
Therefore, the product BC−1A equals the identity matrix, which can be written as follows:∑
µ
Xµ−1σ0,π−1σ0(z)Φ
(
zµ
−1)
Xµ,σ
(
zµ
−1)
= δπ,σ.
After the substitution λ = µ−1σ0, ρ = π−1σ0, τ = σ−1σ0, we get formula (7.8). ■
Lemma 7.3. One has
Xµ,σ(z
sa) =
za − za+1
za − za+1 + 1
Xsaµ,saσ(z) +
1
za − za+1 + 1
Xsaµ,σ(z). (7.12)
Proof. By formulae (7.2) and (7.3), we have
Xsa,sa(z) =
za − za+1
za − za+1 − 1
, Xsa,id(z) =
−1
za − za+1 − 1
,
and Xsa,τ (z) = 0, otherwise. Therefore, by formula (7.9) we obtain
Xsaπ,σ(z) =
za − za+1
za − za+1 − 1
Xπ,saσ(z
sa)− 1
za − za+1 − 1
Xπ,σ(z
sa).
Replacing here z by zsa and making the substitution π = saµ, we get formula (7.12). ■
Lemma 7.4. One has
Xµ,σ
(
zsaµ
−1)
=
za − za+1
za − za+1 − 1
Xµsa,σsa
(
zsaµ
−1)− 1
za − za+1 − 1
Xµsa,σ
(
zsaµ
−1)
. (7.13)
Proof. By formula (7.9), we have
Xµsa,σ(z) =
∑
π
Xµ,π(z)Xsa,π−1σ(z
µ).
Thus
Xµsa,σ(z) =
∑
π
Xµ,π(z)Xsa,π−1σ(z
µ) = Xµ,σ(z)Xsa,id(z
µ) +Xµ,σsa(z)Xsa,sa(z
µ).
Replacing here z by zµ
−1
, we get
Xµsa,σ
(
zµ
−1)
= Xµ,σ
(
zµ
−1)
Xsa,id(z) +Xµ,σsa
(
zµ
−1)
Xsa,sa(z).
Substituting now µ with µsa, we obtain (7.13). ■
New Combinatorial Formulae for Nested Bethe Vectors 17
For σ ∈ Sn and a subset I = {i1, . . . , im} ⊂ {1, . . . , n}, denote
σ(I) = {σ(i1), . . . , σ(im)}.
Recall the functions VI(t, z, y), ṼJ(t, z, y), see formulae (5.4), (5.7) and (5.6), (5.8).
Lemma 7.5. For each a = 1, . . . , k − 1, we have
VI(t, z
sa, y) =
za+1 − za
za+1 − za − 1
Vsa(I)(t, z, y)−
1
za+1 − za − 1
VI(t, z, y), (7.14)
ṼI(t, z
sa, y) =
za − za+1
za − za+1 − 1
Ṽsa(I)(t, z, y)−
1
za − za+1 − 1
ṼI(t, z, y). (7.15)
Proof. By the structure of formulae (5.4) and (5.7) for the function VI(t, z, y), it is enough to
prove formula (7.14) for k = 2. In this case, the statement follows from the identities
1 =
z′ − z
z′ − z − 1
− 1
z′ − z − 1
,
1
t− z′
· t− z + 1
t− z
=
z′ − z
z′ − z − 1
· 1
t− z′
− 1
z′ − z − 1
· 1
t− z
· t− z′ + 1
t− z′
,(
t′ − z
)(
t− z′ + 1
)(
t− t′ − 1
)
− (t− z)
(
t′ − z′ + 1
)(
t′ − t− 1
)
=
(
t′ − z′
)
(t− z + 1)
(
t− t′ − 1
)
−
(
t− z′
)(
t′ − z + 1
)(
t′ − t− 1
)
.
The proof of (7.15) is similar by using formulae (5.6) and (5.8) for functions ṼJ(t, z, y). ■
The statement of Proposition 5.6 is given by formula (5.11). It can be written as follows:∑
(I,J)∈Sp,q,r,k
VI
(
t3, z, x− Λ3
)
ṼJ
(
t1, z, x− Λ2
)
=
1
Cp,q,r,k
∑
σ∈Sk
Vσ0(I0)
(
t3, zσ, x− Λ3
)
ṼJ0
(
t1, zσσ0 , x− Λ2
)
Φ(zσ), (7.16)
where Sp,q,r,k in the left-hand side is the set of all pairs of subsets I, J of {1, . . . , k}, such that
|I| = p, |J | = q, |I ∩ J | = r, and we use k = ξ2, z = t2. In the right-hand side,
Cp,q,r,k = (p− r)!(q − r)!r!(k − p− q − r)!
and (I0, J0) is any fixed pair from Sp,q,r,k. We also expanded Symt2 according to formulae (5.1)
and (5.2), and observed that
Ǐ0 = σ0(I0), ť
2
= zσ0 .
In the rest of the proof, we will suppress the arguments t1, t3, x− Λ2, x− Λ3 because they
are the same in both sides of formula (7.16) and will never be changed in the reasoning.
Notice that every pair (I, J) ∈ Sp,q,r,k can be obtained from an arbitrary fixed pair (I0, J0) ∈
Sp,q,r,k by the action of the symmetric group Sk. Therefore, the left-hand side of the for-
mula (7.16) can be written in the following way:
∑
(I,J)∈Sp,q,r,k
ṼJ(z)VI(z) =
1
Cp,q,r,k
∑
σ∈Sk
Ṽσ(J0)(z)Vσ(I0)(z). (7.17)
18 M. Kosmakov and V. Tarasov
Using Lemma 7.2, we get∑
σ∈Sk
Ṽσ(J0)(z)Vσ(I0)(z)
=
∑
σ,π,τ∈Sk
Ṽσ(J0)(z)Xπ,σ(z)Φ(z
πσ0)Xσ0π−1,σ0τ−1(zπσ0)Vτ(I0)(z), (7.18)
since from formula (7.8)∑
π∈Sk
Xπ,σ(z)Φ(z
πσ0)Xσ0π−1,σ0τ−1(zπσ0) = δσ,τ .
Lemma 7.6. We have∑
σ∈Sk
Ṽσ(J0)(z)Xπ,σ(z) = ṼJ0(z
π). (7.19)
Proof. We will use induction on the length of the permutation π. For π = id, formula (7.19) is
clear, and for π = sa with some a = 1, . . . , k − 1, formula (7.19) coincides with formula (7.15).
For the induction step, we find a such that |saπ| = |π| − 1, and denote ρ = saπ. Then by the
induction assumption∑
σ
Ṽσ(J0)(z)Xρ,σ(z) = ṼJ0(z
ρ).
Replacing here z by zsa , we get∑
σ
Ṽσ(J0)(z
sa)Xρ,σ(z
sa) = ṼJ0(z
saρ) = ṼJ0(z
π). (7.20)
Using formulae (7.12) and (7.15), the left-hand side of (7.20) becomes
∑
σ
(za − za+1)
2
(za − za+1)2 − 1
Ṽsaσ(J0)(z)Xsaρ,saσ(z)
+
∑
σ
za − za+1
(za − za+1)2 − 1
Ṽsaσ(J0)(z)Xsaρ,σ(z)
−
∑
σ
za − za+1
(za − za+1)2 − 1
Ṽσ(J0)(z)Xsaρ,saσ(z)
−
∑
σ
1
(za − za+1)2 − 1
Ṽσ(J0)(z)Xsaρ,σ(z).
Changing the summation index in the first and second sums from σ to saσ, we observe that the
second and third sums cancel each other, while the first and forth sums combine together and
simplify to the expression∑
σ
Ṽσ(J0)(z)Xsaρ,σ(z) =
∑
σ
Ṽσ(J0)(z)Xπ,σ(z),
which appears in the left-hand side of formula (7.19). ■
Lemma 7.7. We have∑
τ
Vτ(I0)(z)Xσ0π−1,σ0τ−1(zπσ0) = Vσ0(I0)(z
πσ0). (7.21)
New Combinatorial Formulae for Nested Bethe Vectors 19
Proof. Recall the notation σ0(I0) = Ǐ0. Transform formula (7.21) by making the substitutions
µ = σ0π
−1, σ = σ0τ
−1,∑
σ
Vσ−1(Ǐ0)
(z)Xµ,σ
(
zµ
−1)
= VǏ0
(
zµ
−1)
. (7.22)
The rest of the proof is analogous to that of Lemma 7.6.
To prove formula (7.22), we will use induction on the length of µ. For µ = id, formula (7.22)
is clear, and for µ = sa with some a = 1, . . . , k− 1, formula (7.22) coincides with formula (7.14).
For the induction step, we find a such that |µsa| = |µ| − 1, and denote ρ = µsa. Then by the
induction assumption,∑
σ
Vσ−1(Ǐ0)
(z)Xρ,σ
(
zρ
−1)
= VǏ0
(
zρ
−1)
,
and replacing here z by zsa , we get∑
σ
Vσ−1(Ǐ0)
(zsa)Xρ,σ
(
zsaρ
−1)
= VǏ0
(
zµ
−1)
= VǏ0
(
zsaρ
−1)
. (7.23)
Using formulae (7.13), (7.14), the left-hand of (7.23) side becomes
∑
σ
(za+1 − za)
2
(za+1 − za)2 − 1
Vsaσ−1(Ǐ0)
(z)Xρsa,σsa
(
zsaρ
−1)
−
∑
σ
za − za+1
(za − za+1)2 − 1
Vsaσ−1(Ǐ0)
(z)Xρsa,σ
(
zsaρ
−1)
+
∑
σ
za − za+1
(za − za+1)2 − 1
Vσ−1(Ǐ0)
(z)Xρsa,σsa
(
zsaρ
−1)
−
∑
σ
1
(za+1 − za)2 − 1
Vσ−1(Ǐ0)
(z)Xρsa,σ
(
zsaρ
−1)
.
Changing the summation index in the first and second sums from σ to σsa, we observe that the
second and third sums cancel each other, while the first and the forth sums combine together
and simplify to the expression∑
σ
Vσ−1(Ǐ0)
(z)Xρsa,σ
(
zsaρ
−1)
=
∑
σ
Vσ−1(Ǐ0)
(z)Xµ,σ
(
zµ
−1)
,
which appears in the left-hand side of formula (7.21). ■
Using Lemmas 7.6 and 7.7, we evaluate the sums over σ and τ in the right-hand side of the
formula (7.18) and get the equality∑
σ∈Sk
Ṽσ(J0)(z)Vσ(I0)(z) =
∑
π
Vσ0(I0)(z
πσ0)ṼJ0(z
π)Φ(zπσ0). (7.24)
Using formula (7.17) in the left-hand side and making the substitution π = σσ0 in the right-hand
side, we obtain that (7.24) can be written as∑
(I,J)∈Sp,q,r,k
ṼJ(z)VI(z) =
1
Cp,q,r,k
∑
σ
Vσ0(I0)(z
σ)ṼJ0(z
σσ0)Φ(zσ),
which is formula (7.16). Proposition 5.6 is proved.
20 M. Kosmakov and V. Tarasov
A Proof of Proposition 5.1
In this appendix, we will consider only the algebra Y (gl2) and, for convenience, we will not write
the superscript ⟨2⟩. We will use the commutation relations
T11(u)T11(t) = T11(t)T11(u), T12(u)T12(t) = T12(t)T12(u),
T22(u)T22(u) = T22(t)T22(u), (A.1)
T11(u)T12(t) =
u− t− 1
u− t
T12(t)T11(u) +
1
u− t
T12(u)T11(t), (A.2)
T22(u)T12(t) =
u− t+ 1
u− t
T12(t)T22(u)−
1
u− t
T12(u)T22(t), (A.3)
following from the defining relations in Y (gl2), see (2.4). We will also use the next statement.
Proposition A.1. One has
T11(u)T12(t1) · · ·T12(tk) =
k∏
i=1
u− ti − 1
u− ti
T12(t1) · · ·T12(tk)T11(u) (A.4)
+
k∑
l=1
1
u− tl
k∏
m=1
m ̸=l
tl − tm − 1
tl − tm
T12(t1) · · ·T12(tl−1)T12(tl+1) · · ·T12(tk)T12(u)T11(tl),
T22(u)T12(t1) · · ·T12(tk) =
k∏
i=1
u− ti + 1
u− ti
T12(t1) · · ·T12(tk)T22(u) (A.5)
−
k∑
l=1
1
u− tl
k∏
m=1
m ̸=l
tl − tm + 1
tl − tm
T12(t1) · · ·T12(tl−1)T12(tl+1) · · ·T12(tk)T12(u)T22(tl).
Proof. The statement goes back to [23]. We will prove it by induction on k. Consider for-
mula (A.4). The statement for k = 1 is given by formula (A.2). We use the induction assumption
to move T11(u) through the product T12(t1) · · ·T12(tk−1):
T11(u)T12(t1) · · ·T12(tk−1)T12(tk) =
k−1∏
i=1
u− ti − 1
u− ti
T12(t1) · · ·T12(tk−1)T11(u)T12(tk) (A.6)
+
k−1∑
l=1
1
u− tl
k−1∏
m=1
m̸=l
tl − tm − 1
tl − tm
T12(t1) · · ·T12(tl−1)T12(tl+1) · · ·T12(tk−1)T12(u)T11(tl)T12(tk).
Then we apply (A.2) to the product T11(u)T12(tk) and T11(tl)T12(tk) and the right-hand side
of (A.6) becomes
k∏
i=1
u− ti − 1
u− ti
T12(t1) · · ·T12(tk−1)T12(tk)T11(u)
+
1
u− tk
k−1∏
i=1
u− ti − 1
u− ti
T12(t1) · · ·T12(tk−1)T12(u)T11(tk)
+
k−1∑
l=1
1
u− tl
k∏
m=1
m ̸=l
tl − tm − 1
tl − tm
× T12(t1) · · ·T12(tl−1)T12(tl+1) · · ·T12(tk−1)T12(u)T12(tk)T11(tl)
New Combinatorial Formulae for Nested Bethe Vectors 21
+
k−1∑
l=1
1
u− tl
1
tl − tk
k−1∏
m=1
m ̸=l
tl − tm − 1
tl − tm
× T12(t1) · · ·T12(tl−1)T12(tl+1) · · ·T12(tk−1)T12(u)T12(tl)T11(tk).
The first term here coincide with the first term in the right-hand side of formula (A.4). The
third term here is the second term of (A.4) without l = k summand. We also used that T12(u)
and T12(tk) commute, see (A.1). The second and forth summands combine into the product
1
u− tk
k−1∏
m=1
tk − tm − 1
tk − tm
T12(t1) · · ·T12(tl−1)T12(tl+1) · · ·T12(tk−1)T12(u)T11(tl), (A.7)
using the following identity:
1
u− tk
k−1∏
i=1
u− ti − 1
u− ti
+
k−1∑
l=1
1
(u− tl)(tl − tk)
k−1∏
m=1
m̸=l
tl − tm − 1
tl − tm
=
1
u− tk
k−1∏
m=1
tk − tm − 1
tk − tm
.
The product (A.7) is exactly the summand with l = k of the second term in (A.4). Formula (A.4)
is proved.
The proof of formula (A.5) is similar to that of formula (A.4) with relation (A.3) used instead
of (A.2). ■
Recall that for the gl2 case we have
Bξ(t) = T12(t1) · · ·T12(tξ),
and thus Proposition 5.1 can be rewritten as follows.
Proposition A.2. Let ξ be a nonnegative integer and t = (t1, . . . , tξ). Then
∆(T12(t1) · · ·T12(tξ)) (A.8)
=
ξ∑
η=0
1
(ξ − η)!η!
Symt
[(
η∏
i=1
T12(ti)⊗
ξ∏
j=η+1
T12(tj)
)(
ξ∏
k=η+1
T22(tk)⊗
η∏
l=1
T11(tl)
)]
.
Remark. Notice that according to (A.1), the factors in each of the large products commute
among themselves, so the order of the factors is irrelevant. (A.8).
Proof. Consider the summand from the right-hand side of (A.8) with a given η,
Fη,ξ−η(t) = Symt
[(
η∏
i=1
T12(ti)⊗
ξ∏
j=η+1
T12(tj)
)(
ξ∏
k=η+1
T22(tk)⊗
η∏
l=1
T11(tl)
)]
. (A.9)
Let
Pη,ξ−η(t) =
∏
1≤i≤η<j≤ξ
ti − tj − 1
ti − tj
×
(
η∏
i=1
T12(ti)⊗
ξ∏
j=η+1
T12(tj)
)(
ξ∏
k=η+1
T22(tk)⊗
η∏
l=1
T11(tl)
)
,
Uη,ξ−η(t) =
∏
1≤i<j≤η
ti − tj − 1
ti − tj
∏
η+1≤i<j≤ξ
ti − tj − 1
ti − tj
, tσ =
(
tσ(1), . . . , tσ(ξ)
)
.
22 M. Kosmakov and V. Tarasov
Using this notation, formula (A.9) can be written as
Fη,ξ−η(t) =
∑
σ∈Sξ
Uη,ξ−η(t
σ)Pη,ξ−η(t
σ).
Observe that Fη,ξ−η(t) is symmetric in t1, . . . , tξ. Denote by Sη × Sξ−η the subgroup of Sξ
stabilizing the subsets {1, . . . , η} and {η + 1, . . . , ξ}. We have
Fη,ξ−η(t) =
1
η!(ξ − η)!
∑
τ∈Sη×Sξ−η
Fη,ξ−η(t
τ )
=
1
η!(ξ − η)!
∑
τ∈Sη×Sξ−η
∑
σ∈Sξ
Uη,ξ−η(t
τσ)Pη,ξ−η(t
τσ).
Changing the summation variable in the inner sum, σ = τ−1ρτ , and using the fact that
Pη,ξ−η(t
ρτ ) = Pη,ξ−η(t
ρ) for all τ ∈ Sη × Sξ−η, we get
Fη,ξ−η(t) =
1
η!(ξ − η)!
∑
τ∈Sη×Sξ−η
∑
ρ∈Sξ
Uη,ξ−η(t
ρτ )Pη,ξ−η(t
ρτ )
=
1
η!(ξ − η)!
∑
ρ∈Sξ
Pη,ξ−η(t
ρ)
∑
τ∈Sη×Sξ−η
Uη,ξ−η(t
ρτ ).
Furthermore, using the identity∑
τ∈Sn
∏
1≤i<j≤n
xτ(i) − xτ(j) − 1
xτ(i) − xτ(j)
= n!, (A.10)
we obtain that
∑
τ∈Sη×Sξ−η
Uη,ξ−η(t
ρτ ) = η!(ξ − η)! and
Fη,ξ−η(t) =
∑
ρ∈Sξ
Pη,ξ−η(t
ρ). (A.11)
Using formula (A.11), the statement of Proposition A.2 can be formulated as follows:
∆(T12(t1) · · ·T12(tξ)) =
ξ∑
η=0
1
(ξ − η)!η!
∑
ρ∈Sξ
Pη,ξ−η(t
ρ). (A.12)
We will prove this formula using the induction on ξ. The base of induction at ξ = 1 is given by
formula (2.5):
∆(T12(t1)) = T12(t1)⊗ T11(t1) + T22(t1)⊗ T12(t1). (A.13)
To make the induction step, we use that
∆(Bξ(t)) = ∆
(
T12(t1)
)
∆(T12(t2) · · ·T12(tξ)), (A.14)
expand the first factor according to (A.13), and apply the induction assumption to expand the
second factor. Denote by S′ξ−1 ⊂ Sξ the subgroup of permutations ρ, such that ρ(1) = 1. Then
the right-hand side of formula (A.14) becomes
T12(t1)⊗ T11(t1)
ξ∑
η=1
1
(ξ − η)!(η − 1)!
∑
τ∈S′
ξ−1
Pη−1,ξ−η
(
tτ(2), . . . , tτ(ξ)
)
+ T22(t1)⊗ T12(t1)
ξ−1∑
η=0
1
(ξ − 1− η)!η!
∑
ρ∈S′
ξ−1
Pη,ξ−η−1
(
tρ(2), . . . , tρ(ξ)
)
, (A.15)
New Combinatorial Formulae for Nested Bethe Vectors 23
where in the first term we shifted the summation variable of the exterior sum. Using the
definition of Pη,ξ−1−η(t) and Pη−1,ξ−η(t), we further expand expression (A.15):
ξ∑
η=1
1
(ξ − η)!(η − 1)!
∑
τ∈S′
ξ−1
∏
1<i≤η<j≤ξ
tτ(i) − tτ(j) − 1
tτ(i) − tτ(j)
×
(
T12(t1)
η∏
i=2
T12
(
tτ(i)
)
⊗ T11(t1)
ξ∏
j=η+1
T12
(
tτ(j)
))( ξ∏
k=η+1
T22
(
tτ(k)
)
⊗
η∏
l=2
T11
(
tτ(l)
))
+
ξ−1∑
η=0
1
(ξ − η − 1)!η!
∑
ρ∈S′
ξ−1
∏
1<i≤η+1<j≤ξ
tρ(i) − tρ(j) − 1
tρ(i) − tρ(j)
×
(
T22(t1)
η+1∏
i=2
T12
(
tρ(i)
)
⊗ T12(t1)
ξ∏
j=η+2
T12
(
tρ(j)
))( ξ∏
k=η+2
T22
(
tρ(k)
)
⊗
η+1∏
l=2
T11
(
tρ(l)
))
.
In the first term, we move T11(t1) through the product
∏ξ
j=η+1 T12
(
tτ(j)
)
using formula (A.4):
T11(t1)
ξ∏
j=η+1
T12
(
tτ(j)
)
=
(
ξ∏
j=η+1
t1 − tτ(j) − 1
t1 − tj
T12
(
tτ(j)
))
T11(t1)
+
ξ∑
p=η+1
1
t1 − tτ(p)
(
ξ∏
j=η+1
j ̸=p
tτ(p) − tτ(j) − 1
tτ(p) − tτ(j)
T12
(
tτ(j)
))
T12(t1)T11
(
tτ(p)
)
.
Similarly, in the second term we move T22(t1) through the product
∏η
m=2 T12
(
tρ(m)
)
using for-
mula (A.5):
T22(t1)
η+1∏
i=2
T12
(
tρ(i)
)
=
(
η+1∏
i=2
t1 − tρ(i) + 1
t1 − tρ(i)
T12
(
tρ(m)
))
T22(t1)
−
η+1∑
s=2
1
t1 − ts
(
η+1∏
i=1
i ̸=s
tρ(s) − tρ(i) + 1
tρ(s) − tρ(i)
T12
(
tρ(i)
))
T12(t1)T22
(
tρ(s)
)
,
After all, the right-hand side of (A.14) becomes a sum of four terms:
∆(Bξ(t)) = Y1(t) + Y2(t) + Y3(t) + Y4(t),
where
Y1(t) =
ξ∑
η=1
1
(ξ − η)!(η − 1)!
∑
τ∈S′
ξ−1
[
ξ∏
l=η+1
t1 − tτ(l) − 1
t1 − tτ(l)
∏
1<i≤η<j≤ξ
tτ(i) − tτ(j) − 1
tτ(i) − tτ(j)
×
(
T12(t1)
η∏
i=2
T12
(
tτ(i)
)
⊗
ξ∏
j=η+1
T12
(
tτ(j)
))
×
(
ξ∏
k=η+1
T22
(
tτ(k)
)
⊗ T11(t1)
η∏
l=2
T11
(
tτ(l)
))]
,
24 M. Kosmakov and V. Tarasov
Y2(t) =
ξ−1∑
η=1
1
(ξ − η)!(η − 1)!
×
∑
τ∈S′
ξ−1
ξ∑
l=η+1
[
1
t1 − tτ(l)
∏
1<i≤η<j≤ξ
tτ(i) − tτ(j) − 1
tτ(i) − tτ(j)
ξ∏
m=η+1
m ̸=l
tτ(l) − tτ(m) − 1
tτ(l) − tτ(m)
×
(
T12(t1)
η∏
k=2
T12
(
tτ(k)
)
⊗ T12(t1)
ξ∏
m=η+1
m ̸=l
T12
(
tτ(m)
))
×
(
ξ∏
i=η+1
T22
(
tτ(i)
)
⊗ T11
(
tτ(l)
) η∏
j=2
T11
(
tτ(j)
))]
.
Y3(t) =
ξ−1∑
η=0
1
(ξ − η − 1)!η!
∑
ρ∈S′
ξ−1
[
η+1∏
m=2
tρ(m) − t1 − 1
tρ(m) − t1
∏
1<i≤η+1<j≤ξ
tρ(i) − tρ(j) − 1
tρ(i) − tρ(j)
×
(
η+1∏
m=2
T12
(
tρ(m)
)
⊗ T12(t1)
ξ∏
l=η+2
T12
(
tρ(l)
))
×
(
T22(t1)
ξ∏
i=η+2
T22
(
tρ(i)
)
⊗
η+1∏
j=2
T11
(
tρ(j)
))]
.
Y4(t) = −
ξ−1∑
η=1
1
(ξ − η − 1)!η!
×
∑
ρ∈S′
ξ−1
[
η+1∑
k=2
1
t1 − tρ(k)
∏
1<i≤η+1<j≤ξ
tρ(i) − tρ(j) − 1
tρ(i) − tρ(j)
η+1∏
m=2
m ̸=k
tρ(k) − tρ(m) + 1
tρ(k) − tρ(m)
×
(
T12(t1)
η+1∏
m=2
m ̸=k
T12
(
tρ(m)
)
⊗ T12(t1)
ξ∏
l=η+2
T12
(
tρ(l)
))
×
(
T22
(
tρ(k)
) ξ∏
i=η+2
T22
(
tρ(i)
)
⊗
η+1∏
j=2
T11
(
tρ(j)
))]
.
To complete the proof, we will show that
Y1(t) + Y3(t) =
ξ∑
η=0
1
(ξ − η)!η!
∑
ρ∈Sξ
Pη,ξ−η(t
ρ) and Y2(t) + Y4(t) = 0. (A.16)
We will start with the first equality in (A.16).
Observe that
Y1(t) =
ξ∑
η=1
1
(ξ − η)!(η − 1)!
∑
σ∈Sξ
σ(1)=1
Pη,ξ−η(t
σ),
and
Y3(t) =
ξ−1∑
η=0
1
(ξ − η − 1)!η!
∑
σ∈Sξ
σ(η+1)=1
Pη,ξ−η(t
σ).
New Combinatorial Formulae for Nested Bethe Vectors 25
On the other hand, we have
ξ∑
η=0
1
(ξ − η)!η!
∑
σ∈Sξ
Pη,ξ−η(t
σ) =
ξ∑
η=1
1
(ξ − η)!η!
∑
σ∈Sξ
σ−1(1)∈{1,...,η}
Pη,ξ−η(t
σ)
+
ξ−1∑
η=0
1
(ξ − η)!η!
∑
σ∈Sξ
σ−1(1)∈{η+1,...,ξ}
Pη,ξ−η(t
σ).
Denote by sa,b ∈ Sξ the transposition of a and b. Then we have
∑
σ∈Sξ
σ−1(1)∈{1,...,η}
Pη,ξ−η(t
σ) =
η∑
l=1
∑
τ∈Sξ,
τ(1)=1
Pη,ξ−η(t
τs1,l) = η
∑
τ∈Sξ,
τ(1)=1
Pη,ξ−η(t
τ ).
For the first step, we used l = σ−1(1) and τ = σs1,σ−1(1), so that τ(1) = 1. For the second step,
we used the equality Pη,ξ−η(t
τs1,l) = Pη,ξ−η(t
τ ). Similarly,∑
σ∈Sξ
σ−1(1)∈{η+1,...,ξ}
Pη,ξ−η(t
σ) = (ξ − η)
∑
ρ∈Sξ,
ρ(η+1)=1
Pη,ξ−η(t
ρ).
Therefore,
ξ∑
η=0
1
(ξ − η)!η!
∑
σ∈Sξ
Pη,ξ−η(t
σ) =
ξ∑
η=1
1
(ξ − η)!(η − 1)!
∑
σ∈Sξ
σ(1)=1
Pη,ξ−η(t
σ)
+
ξ−1∑
η=0
1
(ξ − 1− η)!η!
∑
σ∈Sξ
σ(η+1)=1
Pη,ξ−η(t
σ)
= Y1(t) + Y3(t).
Finally, we show that Y2(t) + Y4(t) = 0. Observe that Y2(t) can be written as
Y2(t) =
ξ−1∑
η=1
1
(ξ − η)!(η − 1)!
×
ξ∑
l=η+1
∑
τ∈S′
ξ−1
[
1
t1 − tτ(l)
∏
1<i≤η<j≤ξ
tτ(i) − tτ(j) − 1
tτ(i) − tτ(j)
ξ∏
m=η+1
m ̸=l
tτ(l) − tτ(m) − 1
tτ(l) − tτ(m)
×
(
T12(t1)
η∏
k=2
T12
(
tτ(k)
)
⊗ T12(t1)
ξ∏
m=η+1
m ̸=l
T12
(
tτ(m)
))
×
(
ξ∏
i=η+1
T22
(
tτ(i)
)
⊗ T11(tτ(l))
η∏
j=2
T11
(
tτ(j)
))]
.
26 M. Kosmakov and V. Tarasov
Changing the summation variable in the inner sum, τ = σsl,η+1, we obtain that
Y2(t) =
ξ−1∑
η=1
1
(ξ − η)!(η − 1)!
×
ξ∑
l=η+1
∑
σ∈S′
ξ−1
[
1
t1 − tσ(η+1)
ξ∏
m=η+2
tσ(η+1) − tσ(m) − 1
tσ(η+1) − tσ(m)
∏
1<i≤η<j≤ξ
tσ(i) − tσ(j) − 1
tσ(i) − tσ(j)
×
(
T12(t1)
η∏
k=2
T12
(
tσ(k)
)
⊗ T12(t1)
ξ∏
m=η+1
T12
(
tσ(m)
))
×
(
ξ∏
i=η+2
T22
(
tσ(i)
)
⊗
η+1∏
j=2
T11
(
tσ(j)
))]
.
The expression under the inner sum over σ does not depend on l and after a redistribution of
factors, we get
Y2(t) =
ξ−1∑
η=1
1
(ξ − η − 1)!(η − 1)!
×
∑
σ∈S′
ξ−1
∑
σ∈S′
ξ−1
[
1
t1 − tσ(η+1)
ξ∏
m=η+2
tσ(η+1) − tσ(m) − 1
tσ(η+1) − tσ(m)
η∏
i=2
tσ(i) − tσ(η+1) − 1
tσ(i) − tσ(η+1)
×
∏
1<i<η+1<j≤ξ
tσ(i) − tσ(j) − 1
tσ(i) − tσ(j)
×
(
T12(t1)
η∏
k=2
T12
(
tσ(k)
)
⊗ T12(t1)
ξ∏
m=η+1
T12
(
tσ(m)
))
×
(
ξ∏
i=η+2
T22
(
tσ(i)
)
⊗
η+1∏
j=2
T11
(
tσ(j)
))]
. (A.17)
Similarly, Y4(t) can be written as
Y4(t) =
ξ−1∑
η=1
1
(ξ − 1− η)!η!
×
η+1∑
k=2
∑
ρ∈S′
ξ−1
[
1
t1 − tρ(k)
∏
1<i≤η+1<j≤ξ
tρ(i) − tρ(j) − 1
tρ(i) − tρ(j)
η+1∏
m=2
m ̸=k
tρ(k) − tρ(m) + 1
tρ(k) − tρ(m)
×
(
T12(t1)
η+1∏
m=2
m ̸=k
T12
(
tρ(m)
)
⊗ T12(t1)
ξ∏
l=η+2
T12
(
tρ(l)
))
×
(
T22
(
tρ(k)
) ξ∏
i=η+2
T22
(
tρ(i)
)
⊗
η+1∏
j=2
T11
(
tρ(j)
))]
,
and changing the summation variable in the inner sum, ρ = σsk,η+1, we get
Y4(t) = −
ξ−1∑
η=1
1
(ξ − 1− η)!η!
New Combinatorial Formulae for Nested Bethe Vectors 27
×
η+1∑
k=2
∑
σ∈S′
ξ−1
[
1
t1 − tσ(η+1)
η∏
m=2
tσ(η+1) − tσ(m) + 1
tσ(η+1) − tσ(m)
∏
1<i≤η+1<j≤ξ
tσ(i) − tσ(j) − 1
tσ(i) − tσ(j)
×
(
T12(t1)
η∏
m=2
T12
(
tσ(m)
)
⊗ T12(t1)
ξ∏
l=η+2
T12
(
tσ(l)
))
×
(
ξ∏
i=η+1
T22
(
tσ(i)
)
⊗
η+1∏
j=2
T11
(
tσ(j)
))]
.
The expression under the inner sum over σ does not depend on k and after a redistribution of
factors, we get
Y4(t) = −
ξ−1∑
η=1
1
(ξ − η − 1)!(η − 1)!
×
∑
σ∈S′
ξ−1
[
1
t1 − tσ(η+1)
η∏
m=2
tσ(η+1) − tσ(m) + 1
tσ(η+1) − tσ(m)
ξ∏
q=η+2
tσ(η+1) − tσ(q) − 1
tσ(η+1) − tσ(q)
×
∏
1<i<η+1<j≤ξ
tσ(i) − tσ(j) − 1
tσ(i) − tσ(j)
×
(
T12(t1)
η∏
m=2
T12
(
tσ(m)
)
⊗ T12(t1)
ξ∏
l=η+2
T12
(
tσ(l)
))
×
(
ξ∏
i=η+1
T22
(
tσ(i)
)
⊗
η+1∏
j=2
T11
(
tσ(j)
))]
. (A.18)
Formulae (A.17) and (A.18) show that Y2(t) +Y4(t) = 0. This completes the proof of for-
mula (A.12). Proposition A.2 is proved. ■
Acknowledgements
The authors thank the referees for very careful reading of this paper and their valuable sugges-
tions. The second author is supported in part by Simons Foundation grants 430235, 852996.
References
[1] Felder G., Rimányi R., Varchenko A., Poincaré–Birkhoff–Witt expansions of the canonical elliptic differential
form, in Quantum Groups, Contemp. Math., Vol. 433, American Mathematical Society, Providence, RI, 2007,
191–208, arXiv:math.RT/0502296.
[2] Khoroshkin S., Pakuliak S., Generating series for nested Bethe vectors, SIGMA 4 (2008), 081, 23 pages,
arXiv:0810.3131.
[3] Khoroshkin S., Pakuliak S., Tarasov V., Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57
(2007), 1713–1732, arXiv:math.QA/0610517.
[4] Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391–418.
[5] Kosmakov M., Tarasov V., New combinatorial formulae for nested Bethe vectors II, Lett. Math. Phys. 115
(2025), 12, 20 pages, arXiv:2402.15717.
[6] Kulish P.P., Reshetikhin N.Yu., Diagonalisation of GL(N) invariant transfer matrices and quantum N -wave
system (Lee model), J. Phys. A 16 (1983), L591–L596.
[7] Kulish P.P., Reshetikhin N.Yu., GL3-invariant solutions of the Yang–Baxter equation and associated quan-
tum systems, J. Sov. Math. 34 (1986), 1948–1971.
https://doi.org/10.1090/conm/433/08327
http://arxiv.org/abs/math.RT/0502296
https://doi.org/10.3842/SIGMA.2008.081
http://arxiv.org/abs/0810.3131
https://doi.org/10.1016/j.geomphys.2007.02.005
http://arxiv.org/abs/math.QA/0610517
https://doi.org/10.1007/BF01212176
https://doi.org/10.1007/s11005-025-01896-2
http://arxiv.org/abs/2402.15717
https://doi.org/10.1088/0305-4470/16/16/001
https://doi.org/10.1007/BF01095104
28 M. Kosmakov and V. Tarasov
[8] Markov Y., Varchenko A., Hypergeometric solutions of trigonometric Knizhnik–Zamolodchikov equations
satisfy dynamical difference equations, Adv. Math. 166 (2002), 100–147.
[9] Matsuo A., An application of Aomoto–Gel’fand hypergeometric functions to the SU(n) Knizhnik–
Zamolodchikov equation, Comm. Math. Phys. 134 (1990), 65–77.
[10] Miwa T., Takeyama Y., Tarasov V., Determinant formula for solutions of the quantum Knizhnik–
Zamolodchikov equation associated with Uq(sln) at |q| = 1, Publ. Res. Inst. Math. Sci. 35 (1999), 871–892,
arXiv:math.QA/9812096.
[11] Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory
Exp. 2006 (2006), P08002, 44 pages, arXiv:math.QA/0605015.
[12] Mukhin E., Tarasov V., Varchenko A., Spaces of quasi-exponentials and representations of the Yangian
Y (glN ), Transform. Groups 19 (2014), 861–885, arXiv:1303.1578.
[13] Oskin A., Pakuliak S., Silantyev A., On the universal weight function for the quantum affine algebra Uq(ĝlN ),
St. Petersburg Math. J. 21 (2009), 651–680, arXiv:0711.2821.
[14] Rimányi R., Stevens L., Varchenko A., Combinatorics of rational functions and Poincaré–Birchoff–
Witt expansions of the canonical U(n−)-valued differential form, Ann. Comb. 9 (2005), 57–74,
arXiv:math.CO/0407101.
[15] Rimányi R., Tarasov V., Varchenko A., Cohomology classes of conormal bundles of Schubert varieties and
Yangian weight functions, Math. Z. 277 (2014), 1085–1104, arXiv:1204.4961.
[16] Rimányi R., Tarasov V., Varchenko A., Partial flag varieties, stable envelopes, and weight functions, Quan-
tum Topol. 6 (2015), 333–364, arXiv:1212.6240.
[17] Rimányi R., Tarasov V., Varchenko A., Trigonometric weight functions as K-theoretic stable envelope maps
for the cotangent bundle of a flag variety, J. Geom. Phys. 94 (2015), 81–119, arXiv:1411.0478.
[18] Rimányi R., Tarasov V., Varchenko A., Elliptic and K-theoretic stable envelopes and Newton polytopes,
Selecta Math. (N.S.) 25 (2019), 16, 43 pages, arXiv:1705.09344.
[19] Schechtman V., Varchenko A., Hypergeometric solutions of Knizhnik–Zamolodchikov equations, Lett. Math.
Phys. 20 (1990), 279–283.
[20] Schechtman V., Varchenko A., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106
(1991), 139–194.
[21] Slavnov N., Algebraic Bethe ansatz, arXiv:1804.07350.
[22] Slavnov N., Algebraic Bethe ansatz and correlation functions—an advanced course, World Scientific Pub-
lishing, Hackensack, NJ, 2022.
[23] Takhtadzhan L.A., Faddeev L.D., The quantum method for the inverse problem and the XY Z Heisenberg
model, Russ. Math. Surv. 34 (1979), 11–68.
[24] Tarasov V., Varchenko A., Selberg-type integrals associated with sl3, Lett. Math. Phys. 65 (2003), 173–185,
arXiv:math.QA/0302148.
[25] Tarasov V., Varchenko A., Combinatorial formulae for nested Bethe vectors, SIGMA 9 (2013), 048, 28 pages,
arXiv:math.QA/0702277.
[26] Tarasov V., Varchenko A., Hypergeometric solutions of the quantum differential equation of the cotangent
bundle of a partial flag variety, Cent. Eur. J. Math. 12 (2014), 694–710, arXiv:1301.2705.
[27] Tarasov V., Varchenko A., q-hypergeometric solutions of quantum differential equations, quantum Pieri
rules, and gamma theorem, J. Geom. Phys. 142 (2019), 179–212, arXiv:1710.03177.
[28] Varchenko A., Tarasov V., Jackson integral representations for solutions of the Knizhnik–Zamolodchikov
quantum equation, St. Petersburg Math. J. 6 (1994), 275–313, arXiv:hep-th/9311040.
https://doi.org/10.1006/aima.2001.2027
https://doi.org/10.1007/BF02102089
https://doi.org/10.2977/prims/1195143360
http://arxiv.org/abs/math.QA/9812096
https://doi.org/10.1088/1742-5468/2006/08/p08002
https://doi.org/10.1088/1742-5468/2006/08/p08002
http://arxiv.org/abs/math.QA/0605015
https://doi.org/10.1007/s00031-014-9275-8
http://arxiv.org/abs/1303.1578
https://doi.org/10.1090/S1061-0022-2010-01110-5
http://arxiv.org/abs/0711.2821
https://doi.org/10.1007/s00026-005-0241-3
http://arxiv.org/abs/math.CO/0407101
https://doi.org/10.1007/s00209-014-1295-5
http://arxiv.org/abs/1204.4961
https://doi.org/10.4171/QT/65
https://doi.org/10.4171/QT/65
http://arxiv.org/abs/1212.6240
https://doi.org/10.1016/j.geomphys.2015.04.002
http://arxiv.org/abs/1411.0478
https://doi.org/10.1007/s00029-019-0451-5
http://arxiv.org/abs/1705.09344
https://doi.org/10.1007/BF00626523
https://doi.org/10.1007/BF00626523
https://doi.org/10.1007/BF01243909
http://arxiv.org/abs/1804.07350
https://doi.org/10.1142/12776
https://doi.org/10.1142/12776
https://doi.org/10.1070/RM1979v034n05ABEH003909
https://doi.org/10.1023/B:MATH.0000010712.67685.9d
http://arxiv.org/abs/math.QA/0302148
https://doi.org/10.3842/SIGMA.2013.048
http://arxiv.org/abs/math.QA/0702277
https://doi.org/10.2478/s11533-013-0376-8
http://arxiv.org/abs/1301.2705
https://doi.org/10.1016/j.geomphys.2019.04.005
http://arxiv.org/abs/1710.03177
http://arxiv.org/abs/hep-th/9311040
1 Introduction
2 Notations
3 Combinatorial formulae for rational weight functions
4 Splitting property of the weight functions
5 Main theorem for the gl_4 case
6 Proof of Proposition 5.5
7 Proof of Proposition 5.6
A Proof of Proposition 5.1
References
|
| id | nasplib_isofts_kiev_ua-123456789-213516 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T03:36:23Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kosmakov, Maksim Tarasov, Vitaly 2026-02-18T11:22:56Z 2025 New Combinatorial Formulae for Nested Bethe Vectors. Maksim Kosmakov and Vitaly Tarasov. SIGMA 21 (2025), 060, 28 pages 1815-0659 2020 Mathematics Subject Classification: 17B37; 81R50; 82B23 arXiv:2312.00980 https://nasplib.isofts.kiev.ua/handle/123456789/213516 https://doi.org/10.3842/SIGMA.2025.060 We give new combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for the evaluation modules over the Yangian Y(₄). The case of Y(ₙ) for an arbitrary n is considered in [Lett. Math. Phys. 115 (2025), 12, 20 pages, arXiv:2402.15717]. The authors thank the referees for their very careful reading of this paper and their valuable suggestions. The second author is supported in part by Simons Foundation grants 430235 and 852996. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications New Combinatorial Formulae for Nested Bethe Vectors Article published earlier |
| spellingShingle | New Combinatorial Formulae for Nested Bethe Vectors Kosmakov, Maksim Tarasov, Vitaly |
| title | New Combinatorial Formulae for Nested Bethe Vectors |
| title_full | New Combinatorial Formulae for Nested Bethe Vectors |
| title_fullStr | New Combinatorial Formulae for Nested Bethe Vectors |
| title_full_unstemmed | New Combinatorial Formulae for Nested Bethe Vectors |
| title_short | New Combinatorial Formulae for Nested Bethe Vectors |
| title_sort | new combinatorial formulae for nested bethe vectors |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213516 |
| work_keys_str_mv | AT kosmakovmaksim newcombinatorialformulaefornestedbethevectors AT tarasovvitaly newcombinatorialformulaefornestedbethevectors |