Duality for Knizhnik–Zamolodchikov and Dynamical Operators

Duality for Knizhnik–Zamolodchikov
and Dynamical Operators††This paper is a contribution to the Special Issue on
Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the
60th birthday and Alexander Varchenko on the 70th birthday. The full
collection is available at https://www.emis.de/journals/SIGMA/Tarasov-
Varchenko.html

Vitaly TARASOV †‡ and Filipp UVAROV †

V. Tarasov and F. Uvarov

† Department of Mathematical Sciences, Indiana University – Purdue University
Indianapolis,
† 402 North Blackford St, Indianapolis, IN 46202-3216, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ St. Petersburg Branch of Steklov Mathematical Institute,
‡ Fontanka 27, St. Petersburg, 191023, Russia<EMAIL_ADDRESS>
Received February 25, 2020, in final form April 10, 2020; Published online
April 25, 2020

We consider the Knizhnik–Zamolodchikov and dynamical operators, both
differential and difference, in the context of the
$(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality for the space of polynomials
in $kn$ anticommuting variables. We show that the Knizhnik–Zamolodchikov and
dynamical operators naturally exchange under the duality.

Knizhnik–Zamolodchikov operators; dynamical operators; the
$(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality

17B37; 81R10; 81R50; 39A12

##  1 Introduction

The Knizhnik–Zamolodchikov (KZ) operators is a family of pairwise commuting
differential operators acting on $U(\mathfrak{gl}_{k})^{\otimes n}$-valued
functions. They play an important role in conformal field theory,
representation theory, and they are closely related to the famous Gaudin
Hamiltonians. The difference analogue of the KZ operators is the quantum
Knizhnik–Zamolodchikov (qKZ) operators. There are rational, trigonometric, and
elliptic versions of the KZ and qKZ operators. For a review and references
see, for example, [2].

There exist other families of commuting differential or difference operators
called the dynamical differential (DD) or dynamical difference (qDD)
operators, respectively. There are rational and trigonometric versions of the
DD and qDD operators as well. It is known that the rational DD operators
commute with the rational KZ operators, the trigonometric DD operators commute
with the rational qKZ operators, and the rational qDD operators commute with
the trigonometric KZ operators, see [4, 5, 7]. The qDD operators appear as the
action of the dynamical Weyl groups [3]. The DD operators are also known as
the Casimir connection, see [8, 9].

Together with the KZ, DD, qKZ, and qDD operators associated with
$U(\mathfrak{gl}_{k})^{\otimes n}$, we will simultaneously consider similar
operators associated with $U(\mathfrak{gl}_{n})^{\otimes k}$ interchanging $k$
and $n$. Let $P_{kn}$ be the space of polynomials in $kn$ commuting variables.
One can define actions of $U(\mathfrak{gl}_{k})^{\otimes n}$ and
$U(\mathfrak{gl}_{n})^{\otimes k}$ on the space $P_{kn}$. It is well known
that the $U(\mathfrak{gl}_{k})^{\otimes n}$\- and
$U(\mathfrak{gl}_{n})^{\otimes k}$-actions on $P_{kn}$ commute, which
manifests the $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality, see, for
example, [1]. Consider the images of the KZ, DD, qKZ, qDD operators associated
with $U(\mathfrak{gl}_{k})^{\otimes n}$ and $U(\mathfrak{gl}_{n})^{\otimes k}$
under the corresponding actions. It was proved in [6] that the images of the
rational (trigonometric) KZ operators associated with
$U(\mathfrak{gl}_{n})^{\otimes k}$ coincide with the images of the rational
(trigonometric) DD operators associated with $U(\mathfrak{gl}_{k})^{\otimes
n}$. Similarly, the images of the rational qKZ operators associated with
$U(\mathfrak{gl}_{n})^{\otimes k}$, up to an action of a central element in
$\mathfrak{gl}_{n}$, coincide with the images of the rational qDD operators
associated with $U(\mathfrak{gl}_{k})^{\otimes n}$. In this paper we obtain a
similar duality for the case of $U(\mathfrak{gl}_{k})^{\otimes n}$\- and
$U(\mathfrak{gl}_{n})^{\otimes k}$-actions on the space of polynomials in $kn$
anticommuting variables, see Theorem 4.4.

The duality for the rational and trigonometric KZ and DD operators is proved
in a straightforward way. To prove the duality for the rational qKZ and qDD
operators, we study the eigenvalues of the rational $R$-matrix and compare
them to the eigenvalues of the operator $B^{\langle 2\rangle}_{12}(t)$, which
is used in the construction of the qDD operators.

The $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality for classical integrable
models related to Gaudin Hamiltonians and the actions of $\mathfrak{gl}_{k}$
and $\mathfrak{gl}_{n}$ on the space of polynomials in anticommuting variables
was studied in [10, Section 3.3]. The result of [10] resembles what one can
expect for Bethe algebras of Gaudin models discussed in our work. We will
study those Bethe algebras in an upcoming paper.

The paper is organized as follows. In Section 2, we introduce necessary
notations. In Section 3, we define the KZ, DD, qKZ, and qDD operators. In
Section 4, we formulate and prove the main result.

##  2 Basic notation

Let $e_{ab}$, $a,b=1,\dots,k$, be the standard basis of the Lie algebra
$\mathfrak{gl}_{k}$: $[e_{ab},e_{cd}]=\delta_{bc}e_{ad}-\delta_{ad}e_{cb}$. We
take the Cartan subalgebra $\mathfrak{h}\subset\mathfrak{gl}_{k}$ spanned by
$e_{11},\dots,e_{kk}$, and the nilpotent subalgebras $\mathfrak{n}_{+}$ and
$\mathfrak{n}_{-}$ spanned by the elements $e_{ab}$ for $a<b$ and $a>b$
respectively. We have standard Gauss decomposition
$\mathfrak{gl}_{k}=\mathfrak{n}_{+}\oplus\mathfrak{h}\oplus\mathfrak{n}_{-}$.

Let $\varepsilon_{1},\dots,\varepsilon_{k}$ be the basis of $\mathfrak{h}^{*}$
dual to $e_{11},\dots,e_{kk}$:
$\langle\varepsilon_{a},e_{bb}\rangle=\delta_{ab}$. We identify
$\mathfrak{h}^{*}$ with $\mathbb{C}^{k}$ mapping
$l_{1}\varepsilon_{1}+\dots+l_{k}\varepsilon_{k}$ to $(l_{1},\dots,l_{k})$.
The root vectors of $\mathfrak{gl}_{k}$ are $e_{ab}$ for $a\neq b$, the
corresponding root being equal to
$\alpha_{ab}=\varepsilon_{a}-\varepsilon_{b}$. The roots $\alpha_{ab}$ for
$a<b$ are positive. The simple roots are $\alpha_{1},\dots,\alpha_{k-1}$:
$\alpha_{a}=\varepsilon_{a}-\varepsilon_{a+1}$. Denote by $\rho$ the half-sum
of positive roots.

We choose the standard invariant bilinear form $(\,,\,)$ on
$\mathfrak{gl}_{k}$: $(e_{ab},e_{cd})=\delta_{ad}\delta_{bc}$. It defines an
isomorphism $\mathfrak{h}\rightarrow\mathfrak{h}^{*}$ The induced bilinear
form on $\mathfrak{h}^{*}$ is $(\varepsilon_{a},\varepsilon_{b})=\delta_{ab}$.

For a $\mathfrak{gl}_{k}$-module $W$ and a weight
$\boldsymbol{l}\in\mathfrak{h}^{*}$, let $W[\boldsymbol{l}]$ be the weight
subspace of $W$ of weight $\boldsymbol{l}$.

For any $\boldsymbol{l}=(l_{1},\dots,l_{k})$ with $l_{a}\geq l_{a+1}$, we
denote by $V_{\boldsymbol{l}}$ the irreducible $\mathfrak{gl}_{k}$-module with
highest weight $\boldsymbol{l}$. Also, for any $m\in\mathbb{Z}_{\geq 0}$ we
write $V_{m}$ instead of $V_{(1^{m})}$, where
$(1^{m})=(\underbrace{1,\dots,1}_{k},0,\dots,0)$. Thus, $V_{0}=\mathbb{C}$ is
the trivial $\mathfrak{gl}_{k}$-module, $V_{1}=\mathbb{C}^{k}$ with the
natural action of $\mathfrak{gl}_{k}$, and $V_{m}$ is the $m$-th antisymmetric
power of $V_{1}$.

The element $I=\underset{a,b=1}{\overset{k}{\sum}}e_{ab}e_{ba}$ is central in
$U(\mathfrak{gl}_{k})$. It acts as multiplication by
$(\boldsymbol{l},\boldsymbol{l}+2\rho)$ in the irreducible
$\mathfrak{gl}_{k}$-module $V_{\boldsymbol{l}}$.

Consider the algebra $\mathfrak{X}_{k}=\bigwedge^{\bullet}\mathbb{C}^{k}$,
which will be identified with the ring of all polynomials in anticommuting
variables $x_{1},\dots,x_{k}$. In parrticular, $x_{i}^{2}=0$ for all
$i=1,\dots,k$.

We define left derivation $\partial_{a}$ as follows: if $g(x)=x_{b_{1}}\dots
x_{b_{m}}$ for some $m$, where $b_{s}\neq a$ for any $s$, then

$\displaystyle\partial_{a}g(x)=0,\qquad\partial_{a}(x_{a}g(x))=g(x).$

The operators of left multiplication by $x_{1},\dots,x_{k}$ and left
derivations $\partial_{1},\dots,\partial_{k}$ make the space
$\mathfrak{X}_{k}$ into the irreducible representation of the Clifford algebra
$\operatorname{Cliff}_{k}$.

The Lie algebra $\mathfrak{gl}_{k}$ acts on the space $\mathfrak{X}_{k}$ by
the rule: $e_{ab}\cdot p=x_{a}\partial_{b}p$ for any $p\in\mathfrak{X}_{k}$.
Denote the obtained $\mathfrak{gl}_{k}$-module by $V_{\bullet}$, then

$\displaystyle V_{\bullet}=\bigoplus_{l=0}^{k}V_{l},$ (2.1)

the submodule $V_{l}$ being spanned by homogeneous polynomials of degree $l$.
A highest weight vector of the submodule $V_{l}$ is $x_{1}x_{2}\cdots x_{l}$.

The $\mathfrak{gl}_{k}$-action on $\mathfrak{X}_{k}$ naturaly extends to a
$U(\mathfrak{gl}_{k})^{\otimes n}$-action on $(\mathfrak{X}_{k})^{\otimes n}$.
For any $g\in U(\mathfrak{gl}_{k})$, set
$g_{(i)}=1\otimes\cdots\otimes\underset{i\text{-th}}{g}\otimes\dots\otimes
1\in U(\mathfrak{gl}_{k})^{\otimes n}$. We consider $U(\mathfrak{gl}_{k})$ as
the diagonal subalgebra of $U(\mathfrak{gl}_{k})^{\otimes n}$, that is, the
embedding $U(\mathfrak{gl}_{k})\hookrightarrow U(\mathfrak{gl}_{k})^{\otimes
n}$ is given by the n-fold coproduct: $x\mapsto x_{(1)}+\dots+x_{(n)}$ for any
$x\in\mathfrak{gl}_{k}$. This corresponds to the standard
$\mathfrak{gl}_{k}$-module structure on $(\mathfrak{X}_{k})^{\otimes n}$ as
the tensor product of $\mathfrak{gl}_{k}$-modules.

Let $\Omega=\underset{a,b=1}{\overset{k}{\sum}}e_{ab}\otimes e_{ba}$ be the
Casimir tensor, and let

$\displaystyle\Omega^{+}=\frac{1}{2}\underset{a=1}{\overset{k}{\sum}}e_{aa}\otimes
e_{aa}+\underset{1\leq a<b\leq k}{\sum}e_{ab}\otimes e_{ba},$
$\displaystyle\Omega^{-}=\frac{1}{2}\,\underset{a=1}{\overset{k}{\sum}}e_{aa}\otimes
e_{aa}+\underset{1\leq a<b\leq k}{\sum}e_{ba}\otimes e_{ab},$

so that $\Omega=\Omega^{+}+\Omega^{-}$.

##  3 The KZ, qKZ, DD and qDD operators

Fix a nonzero complex number $\kappa$. Consider differential operators
$\nabla_{z_{1}},\dots,\nabla_{z_{n}}$, and
$\widehat{\nabla}_{z_{1}},\dots,\widehat{\nabla}_{z_{n}}$ with coefficients in
$U(\mathfrak{gl}_{k})^{\otimes n}$ depending on complex variables
$z_{1},\dots,z_{n}$, and $\lambda_{1},\dots,\lambda_{k}$:

$\displaystyle\nabla_{z_{i}}(z;\lambda)=\kappa\frac{\partial}{\partial
z_{i}}-\sum_{a=1}^{k}\lambda_{a}(e_{aa})_{(i)}-\sum_{j=1,\,j\neq
i}^{n}\frac{\Omega_{(ij)}}{z_{i}-z_{j}},$
$\displaystyle\widehat{\nabla}_{z_{i}}(z;\lambda)=\kappa
z_{i}\frac{\partial}{\partial
z_{i}}-\sum_{a=1}^{k}\left(\lambda_{a}-\frac{e_{aa}}{2}\right)(e_{aa})_{(i)}-\sum_{j=1,\,j\neq
i}^{n}\frac{z_{i}\Omega^{+}_{(ij)}+z_{j}\Omega^{-}_{(ij)}}{z_{i}-z_{j}}.$

The differential operators $\nabla_{z_{1}},\dots,\nabla_{z_{n}}$ (resp.,
$\widehat{\nabla}_{z_{1}},\dots,\widehat{\nabla}_{z_{n}}$) are called the
rational (resp., trigonometric) Knizhnik–Zamolodchikov (KZ) operators.

Introduce differential operators $D_{\lambda_{1}},\dots,D_{\lambda_{k}}$, and
$\widehat{D}_{\lambda_{1}},\dots,\widehat{D}_{\lambda_{k}}$ with coefficients
in $U(\mathfrak{gl}_{k})^{\otimes n}$ depending on complex variables
$z_{1},\dots,z_{n}$, and $\lambda_{1},\dots,\lambda_{k}$:

$\displaystyle
D_{\lambda_{a}}(z;\lambda)=\kappa\frac{\partial}{\partial\lambda_{a}}-\sum_{i=1}^{n}z_{i}(e_{aa})_{(i)}-\sum_{b=1,\,b\neq
a}^{k}\frac{e_{ab}e_{ba}-e_{aa}}{\lambda_{a}-\lambda_{b}},$
$\displaystyle\widehat{D}_{\lambda_{a}}(z;\lambda)=\kappa\lambda_{a}\frac{\partial}{\partial\lambda_{a}}+\frac{e_{aa}^{2}}{2}-\sum_{i=1}^{n}z_{i}(e_{aa})_{(i)}$
$\displaystyle\hphantom{\widehat{D}_{\lambda_{a}}(z;\lambda)=}{}-\sum_{b=1}^{k}\sum_{1\leq
i<j\leq n}(e_{ab})_{(i)}(e_{ba})_{(j)}-\sum_{b=1,\,b\neq
a}^{k}\frac{\lambda_{b}}{\lambda_{a}-\lambda_{b}}(e_{ab}e_{ba}-e_{aa}).$

The differential operators $D_{\lambda_{1}},\dots,D_{\lambda_{k}}$ (resp.
$\widehat{D}_{\lambda_{1}},\dots,\widehat{D}_{\lambda_{k}}$) are called the
rational (resp., trigonometric) differential dynamical (DD) operators, see [4,
7].

For any $a,b=1,\dots,k$, $a\neq b$, introduce the series $B_{ab}(t)$ depending
on a complex variable $t$:

$\displaystyle
B_{ab}=1+\sum_{s=1}^{\infty}\frac{e_{ba}^{s}e_{ab}^{s}}{s!}\prod_{j=1}^{s}(t-e_{aa}+e_{bb}-j)^{-1}.$

The action of this series is well defined in any finite-dimensional
$\mathfrak{gl}_{k}$-module $W$, giving an End($W$)-valued rational function of
$t$.

Denote $\lambda_{bc}=\lambda_{b}-\lambda_{c}$. Consider the products
$X_{1},\dots,X_{k}$ depending on the complex variables $z_{1},\dots,z_{n}$,
and $\lambda_{1},\dots,\lambda_{k}$:

$\displaystyle X_{a}(z;\lambda)=(B_{ak}(\lambda_{ak})\cdots
B_{a,a+1}(\lambda_{a,a+1}))^{-1}$
$\displaystyle\hphantom{X_{a}(z;\lambda)=}{}\times\prod_{i=1}^{n}\big{(}z_{i}^{-e_{aa}}\big{)}_{(i)}B_{1a}(\lambda_{1a}-\kappa)\cdots
B_{a-1,a}(\lambda_{a-1,a}-\kappa).$

The products $X_{1},\dots,X_{k}$ act in any $n$-fold tensor product
$W_{1}\otimes\dots\otimes W_{n}$ of finite-dimensional
$\mathfrak{gl}_{k}$-modules.

Denote by $T_{u}$ a difference operator acting on a function $f(u)$ by

$(T_{u}f)(u)=f(u+\kappa).$

Introduce difference operators $Q_{\lambda_{1}},\dots,Q_{\lambda_{k}}$:

$\displaystyle Q_{\lambda_{a}}(z;\lambda)=X_{a}(z;\lambda)T_{\lambda_{a}}.$

The operators $Q_{\lambda_{1}},\dots,Q_{\lambda_{k}}$ are called the
(rational) difference dynamical (qDD) operators [5].

For any finite-dimensional irreducible $\mathfrak{gl}_{k}$-modules $V$ and
$W$, there is a distinguished rational function $R_{VW}(t)$ of $t$ with values
in $\operatorname{End}(V\otimes W)$ called the rational $R$-matrix. It is
uniquely determined by the $\mathfrak{gl}_{k}$-invariance

$\displaystyle[R_{VW}(t),g\otimes 1+1\otimes g]=0\qquad\text{for any}\quad
g\in\mathfrak{gl}_{k},$ (3.1)

the commutation relations

$\displaystyle R_{VW}(t)\left(te_{ab}\otimes 1+\sum_{c=1}^{k}e_{ac}\otimes
e_{cb}\right)=\left(te_{ab}\otimes 1+\sum_{c=1}^{k}e_{cb}\otimes
e_{ac}\right)R_{VW}(t),$ (3.2)

and the normalization condition

$\displaystyle R_{VW}(t)v\otimes w=v\otimes w,$ (3.3)

where $v$ and $w$ are the highest weight vectors of $V$ and $W$, respectively.

Denote $z_{ij}=z_{i}-z_{j}$ and $R_{ij}(t)=(R_{W_{i}W_{j}}(t))_{(ij)}$.
Consider the products $K_{1},\dots,K_{n}$ depending on the complex variables
$z_{1},\dots,z_{n}$, and $\lambda_{1},\dots,\lambda_{k}$:

$\displaystyle K_{i}(z;\lambda)=(R_{in}(z_{in})\cdots
R_{i,i+1}(z_{i,i+1}))^{-1}\prod_{a=1}^{k}\big{(}\lambda_{a}^{-e_{aa}}\big{)}_{(i)}R_{1i}(z_{1i}-\kappa)\cdots
R_{i-1,i}(z_{i-1,i}-\kappa).$

The products $K_{1},\dots,K_{n}$ act in any $n$-fold tensor product
$W_{1}\otimes\cdots\otimes W_{n}$ of $\mathfrak{gl}_{k}$-modules.

Introduce difference operators $Z_{z_{1}},\dots,Z_{z_{n}}$:

$\displaystyle Z_{z_{i}}(z;\lambda)=K_{i}(z;\lambda)T_{z_{i}}.$

The operators $Z_{z_{1}},\dots,Z_{z_{n}}$ are called (rational) quantized
Knizhnik–Zamolodchikov (qKZ) operators.

It is known that the introduced operators combine into three commutative
families, see [4, 5, 7] for more references.

######  Theorem 3.1.

The operators $\nabla_{z_{1}},\dots,\nabla_{z_{k}}$,
$D_{\lambda_{1}},\dots,D_{\lambda_{k}}$ pairwise commute.

######  Theorem 3.2.

The operators $\widehat{\nabla}_{z_{1}},\dots,\widehat{\nabla}_{z_{n}}$,
$Q_{\lambda_{1}},\dots,Q_{\lambda_{k}}$ pairwise commute.

######  Theorem 3.3.

The operators $\widehat{D}_{\lambda_{1}},\dots,\widehat{D}_{\lambda_{k}}$,
$Z_{z_{1}},\dots,Z_{z_{n}}$ pairwise commute.

##  4 The $\boldsymbol{(\mathfrak{gl}_{k},\mathfrak{gl}_{n})}$-duality

Consider the ring $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting
variables $x_{ai}$, $a=1,\dots,k$, $i=1,\dots,n$. As a vector space,
$\mathfrak{P}_{kn}$ is isomorphic to $(\mathfrak{X}_{k})^{\otimes n}$, the
isomorphism $\phi_{1}\colon(\mathfrak{X}_{k})^{\otimes
n}\rightarrow\mathfrak{P}_{kn}$ being given by

$\displaystyle\phi_{1}(p_{1}\\!\otimes\dots\otimes
p_{n})(x_{11},\dots,x_{kn})=p_{1}(x_{11},\dots,x_{k1})p_{2}(x_{12},\dots,x_{k2})\cdots
p_{n}(x_{1n},\dots,x_{kn}),\\!\\!\\!$ (4.1)

and to $(\mathfrak{X}_{n})^{\otimes k}$, the isomorphism
$\phi_{2}:(\mathfrak{X}_{n})^{\otimes k}\rightarrow\mathfrak{P}_{kn}$ being
given by

$\displaystyle\phi_{2}(p_{1}\\!\otimes\dots\otimes
p_{k})(x_{11},\dots,x_{kn})=p_{1}(x_{11},\dots,x_{1n})p_{2}(x_{21},\dots,x_{2n})\cdots
p_{k}(x_{k1},\dots,x_{kn}).\\!\\!\\!$ (4.2)

We transfer the $\mathfrak{gl}_{k}$-action on $(\mathfrak{X}_{k})^{\otimes n}$
to $\mathfrak{P}_{kn}$ using isomorphism $\phi_{1}$. Similarly, we transfer
the $\mathfrak{gl}_{n}$-action on $(\mathfrak{X}_{n})^{\otimes k}$ to
$\mathfrak{P}_{kn}$ using isomorphism $\phi_{2}$.

We will write superscripts $\langle k\rangle$ and $\langle n\rangle$ to
distinguish objects associated with Lie algebras $\mathfrak{gl}_{k}$ and
$\mathfrak{gl}_{n}$, respectively. For example, $e_{ab}^{\langle k\rangle}$,
$a,b=1,\dots,k$ denote the generators of $\mathfrak{gl}_{k}$, and
$e_{ij}^{\langle n\rangle}$, $i,j=1,\dots,n$ denote the generators of
$\mathfrak{gl}_{n}$. Then $\mathfrak{P}_{kn}$ is isomorphic to
$\big{(}V_{\bullet}^{\langle k\rangle}\big{)}^{\otimes n}$ as a
$\mathfrak{gl}_{k}$-module by (4.1), and it is isomorphic to
$\big{(}V_{\bullet}^{\langle n\rangle}\big{)}^{\otimes k}$ as a
$\mathfrak{gl}_{n}$-module by (4.2).

It is easy to check that $\mathfrak{gl}_{k}$\- and $\mathfrak{gl}_{n}$-actions
on $\mathfrak{P}_{kn}$ commute, therefore $\mathfrak{P}_{kn}$ is a
$\mathfrak{gl}_{k}\oplus\mathfrak{gl}_{n}$-module. We have the following
theorem, see for example [1]:

######  Theorem 4.1.

For any partition $\boldsymbol{l}$, denote its transpose by
$\boldsymbol{l}^{\prime}$. The
$\mathfrak{gl}_{k}\oplus\mathfrak{gl}_{n}$-module $\mathfrak{P}_{kn}$ has the
decomposition:

$\displaystyle\mathfrak{P}_{kn}=\bigoplus_{\begin{subarray}{c}\boldsymbol{l}=(l_{1},\dots,l_{k}),\\\
l_{1}\leq n.\end{subarray}}V^{\langle k\rangle}_{\boldsymbol{l}}\otimes
V^{\langle n\rangle}_{\boldsymbol{l}^{\prime}}.$

Fix vectors $\boldsymbol{l}=(l_{1},\dots,l_{n})\in\mathbb{Z}_{\geq 0}^{n}$ and
$\boldsymbol{m}=(m_{1},\dots,m_{k})\in\mathbb{Z}_{\geq 0}^{k}$ such that
$\sum\limits_{i=1}^{n}l_{i}=\sum\limits_{a=1}^{k}m_{a}$. Let

$\displaystyle\mathcal{Z}_{kn}[\boldsymbol{l},\boldsymbol{m}]=\left\\{(d_{ai})_{\begin{subarray}{1}a=1,\dots,k\\\
i=1,\dots,n\end{subarray}}\in\\{0,1\\}^{kn}\,\bigg{|}\,\sum_{a=1}^{k}d_{ai}=l_{i},\,\sum_{i=1}^{n}d_{ai}=m_{a}\right\\}.$

Denote by
$\mathfrak{P}_{kn}[\boldsymbol{l},\boldsymbol{m}]\subset\mathfrak{P}_{kn}$ the
span of all monomials $x^{\boldsymbol{d}}=x_{11}^{d_{11}}\cdots
x_{k1}^{d_{k1}}\cdots x_{1n}^{d_{1n}}\cdots x_{kn}^{d_{kn}}$ such that
$\boldsymbol{d}=(d_{ai})\in\mathcal{Z}_{kn}[\boldsymbol{l},\boldsymbol{m}]$.
Then by (2.1), the maps $\phi_{1}$ and $\phi_{2}$ induce the isomorpfisms of
the respective weight subspaces $\big{(}V_{l_{1}}^{\langle
k\rangle}\otimes\cdots\otimes V_{l_{n}}^{\langle
k\rangle}\big{)}[m_{1},\dots,m_{k}]$ and $\big{(}V_{m_{1}}^{\langle
n\rangle}\otimes\cdots\otimes V_{m_{k}}^{\langle
n\rangle}\big{)}[l_{1},\dots,l_{n}]$ with the space
$\mathfrak{P}_{kn}[\boldsymbol{l},\boldsymbol{m}]$.

There is another description of the isomorphisms $\phi_{1}$ and $\phi_{2}$.

For any $\boldsymbol{a}=(a_{1},\dots,a_{r})$,
$\boldsymbol{i}=(i_{1},\dots,i_{s})$, such that $1\leq a_{1}<\dots<a_{r}\leq
k$, $1\leq i_{1}<\dots<i_{s}\leq n$, define: $e_{\boldsymbol{a}}^{\langle
k\rangle}=e^{\langle k\rangle}_{a_{1}1}e^{\langle k\rangle}_{a_{2}2}\cdots
e^{\langle k\rangle}_{a_{r}r}$, $e_{\boldsymbol{i}}^{\langle
n\rangle}=e^{\langle n\rangle}_{i_{1}1}e^{\langle n\rangle}_{i_{2}2}\cdots
e^{\langle n\rangle}_{i_{s}s}$.

Fix
$\boldsymbol{d}=(d_{ai})\in\mathcal{Z}_{kn}[\boldsymbol{l},\boldsymbol{m}]$.
Let $\boldsymbol{a}_{j}=\big{(}a_{1}^{j},\dots,a_{l_{j}}^{j}\big{)}$ be such
that $a_{1}^{j}<a_{2}^{j}<\dots<a_{l_{j}}^{j}$ and $d_{a_{p}^{j},j}=1$ for all
$j=1,\dots,n$, $p=1,\dots,l_{j}$. Similarly, let
$\boldsymbol{i}_{s}=\big{(}i_{1}^{s},\dots,i_{m_{l}}^{s}\big{)}$ be such that
$i_{1}^{s}<i_{2}^{s}<\dots<i_{m_{s}}^{s}$ and $d_{s,i_{p}^{s}}=1$ for all
$s=1,\dots,k$, $p=1,\dots,m_{s}$. Introduce the following vectors:

$\displaystyle v_{\boldsymbol{d}}^{\langle k\rangle}=e^{\langle
k\rangle}_{\boldsymbol{a}_{1}}v_{l_{1}}^{\langle k\rangle}\otimes\dots\otimes
e^{\langle k\rangle}_{\boldsymbol{a}_{n}}v_{l_{n}}^{\langle k\rangle},\qquad
v_{\boldsymbol{d}}^{\langle n\rangle}=e^{\langle
n\rangle}_{\boldsymbol{i}_{1}}v_{m_{1}}^{\langle n\rangle}\otimes\dots\otimes
e^{\langle n\rangle}_{\boldsymbol{i}_{k}}v_{m_{k}}^{\langle n\rangle},$

where $v_{l_{j}}^{\langle k\rangle}=x_{1}x_{2}\cdots x_{l_{j}}$ and
$v_{m_{s}}^{\langle n\rangle}=x_{1}x_{2}\cdots x_{m_{s}}$ are highest weight
vectors for the modules $V_{l_{j}}^{\langle k\rangle}$ and $V_{m_{l}}^{\langle
n\rangle}$, respectively.

######  Lemma 4.2.

The vectors $v_{\boldsymbol{d}}^{\langle k\rangle}$,
$\boldsymbol{d}\in\mathcal{Z}_{kn}[\boldsymbol{l},\boldsymbol{m}]$ form a
basis of the weight subspace $\big{(}V_{l_{1}}^{\langle
k\rangle}\otimes\dots\otimes V_{l_{n}}^{\langle
k\rangle}\big{)}[m_{1},\dots,m_{k}]$. Similarly, the vectors
$v_{\boldsymbol{d}}^{\langle n\rangle}$,
$\boldsymbol{d}\in\mathcal{Z}_{kn}[\boldsymbol{l},\boldsymbol{m}]$ form a
basis of the weight subspace $\big{(}V_{m_{1}}^{\langle
n\rangle}\otimes\cdots\otimes V_{m_{k}}^{\langle
n\rangle}\big{)}[l_{1},\dots,l_{n}]$.

Let $\varepsilon(\boldsymbol{d})$ be a sign function such that
$x_{11}^{d_{11}}\cdots x_{1n}^{d_{1n}}\cdots x_{k1}^{d_{k1}}\cdots
x_{kn}^{d_{kn}}=\varepsilon(\boldsymbol{d})x^{\boldsymbol{d}}$.

######  Lemma 4.3.

We have $\phi_{1}\big{(}v^{\langle
k\rangle}_{\boldsymbol{d}}\big{)}=x^{\boldsymbol{d}}$ and
$\phi_{2}\big{(}v^{\langle
n\rangle}_{\boldsymbol{d}}\big{)}=\varepsilon(\boldsymbol{d})x^{\boldsymbol{d}}$.

Consider the action of KZ, qKZ, DD and qDD operators for the Lie algebras
$\mathfrak{gl}_{k}$ and $\mathfrak{gl}_{n}$ on $\mathfrak{P}_{kn}$-valued
functions of $z_{1},\dots,z_{n}$, and $\lambda_{1},\dots,\lambda_{k}$,
treating the space $\mathfrak{P}_{kn}$ as a tensor product
$\big{(}V_{\bullet}^{\langle k\rangle}\big{)}^{\otimes n}$ of
$\mathfrak{gl}_{k}$-modules and as a tensor product
$\big{(}V_{\bullet}^{\langle n\rangle}\big{)}^{\otimes k}$ of
$\mathfrak{gl}_{n}$-modules. We will write $F\backsimeq G$ if the operators
$F$ and $G$ act on the $\mathfrak{P}_{kn}$-valued functions in the same way.

Denote $\boldsymbol{1}^{\langle
k\rangle}=(\underset{k}{\underbrace{1,1,\dots,1}})$ and
$\boldsymbol{1}^{\langle n\rangle}=(\underset{n}{\underbrace{1,1,\dots,1}})$

######  Theorem 4.4.

For any $i=1,\dots,n$ and $a=1,\dots,k$ the following relations hold

$\displaystyle\nabla_{z_{i}}^{\langle k\rangle}(z,\lambda,\kappa)\backsimeq
D_{z_{i}}^{\langle n\rangle}(\lambda,-z,-\kappa),$ (4.3)
$\displaystyle\nabla_{\lambda_{a}}^{\langle
n\rangle}(\lambda,z,\kappa)\backsimeq D_{\lambda_{a}}^{\langle
k\rangle}(z,-\lambda,-\kappa),$ (4.4)
$\displaystyle\widehat{\nabla}_{z_{i}}^{\langle
k\rangle}(z,\lambda,\kappa)\backsimeq-\widehat{D}_{z_{i}}^{\langle
n\rangle}\big{(}{-}\lambda+\boldsymbol{1}^{\langle
k\rangle},z,-\kappa\big{)},$ (4.5)
$\displaystyle\widehat{\nabla}_{\lambda_{a}}^{\langle
n\rangle}(\lambda,z,\kappa)\backsimeq-\widehat{D}_{\lambda_{a}}^{\langle
k\rangle}\big{(}{-}z+\boldsymbol{1}^{\langle
n\rangle},\lambda,-\kappa\big{)},$ (4.6) $\displaystyle Z_{z_{i}}^{\langle
k\rangle}(z,\lambda,\kappa)\backsimeq N_{i}^{\langle
n\rangle}(z)Q_{z_{i}}^{\langle n\rangle}(\lambda,-z,-\kappa),$ (4.7)
$\displaystyle Z_{\lambda_{a}}^{\langle n\rangle}(\lambda,z,\kappa)\backsimeq
N_{a}^{\langle k\rangle}(\lambda)Q_{\lambda_{a}}^{\langle
k\rangle}(z,-\lambda,-\kappa),$ (4.8)

where

$\displaystyle N_{i}^{\langle n\rangle}(z)=\frac{\prod\limits_{1\leq
j<i}C_{ji}^{n}(z_{ji}-\kappa)}{\prod\limits_{i<j\leq
n}C_{ij}^{n}(z_{ij})},\qquad N_{a}^{\langle
k\rangle}(\lambda)=\frac{\prod\limits_{1\leq
b<a}C_{ba}^{k}(\lambda_{ba}-\kappa)}{\prod\limits_{a<b\leq
k}C_{ab}^{k}(\lambda_{ab})}.$

and

$\displaystyle C_{ab}^{\langle
k\rangle}(t)=\frac{\Gamma\big{(}t+e_{aa}^{\langle
k\rangle}+1\big{)}\Gamma\big{(}t-e_{bb}^{\langle
k\rangle}\big{)}}{\Gamma\big{(}t+e_{aa}^{\langle k\rangle}-e_{bb}^{\langle
k\rangle}+1\big{)}\Gamma(t)},\qquad C_{ij}^{\langle
n\rangle}(t)=\frac{\Gamma\big{(}t+e_{ii}^{\langle
n\rangle}+1\big{)}\Gamma\big{(}t-e_{jj}^{\langle
n\rangle}\big{)}}{\Gamma\big{(}t+e_{ii}^{\langle n\rangle}-e_{jj}^{\langle
n\rangle}+1\big{)}\Gamma(t)}.$ (4.9)

###### Proof.

Verification of relations (4.3), (4.4), (4.5), (4.6) is straightforward. For
(4.7) and (4.8), we have to show that

$\displaystyle R_{ij}^{\langle k\rangle}(t)\backsimeq C_{ij}^{\langle
n\rangle}(t)B_{ij}^{\langle n\rangle}(-t),$ (4.10) $\displaystyle
R_{ab}^{\langle n\rangle}(t)\backsimeq C_{ab}^{\langle
k\rangle}(t)B_{ab}^{\langle k\rangle}(-t).$ (4.11)

We will prove relation (4.11). Relation (4.10) can be proved similarly.

Note, that both action of $R_{ab}^{\langle n\rangle}(t)$ on
$\mathfrak{P}_{kn}$ and action of $C_{ab}^{\langle k\rangle}(t)B_{ab}^{\langle
k\rangle}(-t)$ on $\mathfrak{P}_{kn}$ involve only the variables
$x_{a1},\dots,x_{an}$, $x_{b1},\dots,x_{bn}$. Therefore, it is sufficient to
prove (4.11) for the case of $k=2$, $a=1$, $b=2$.

The $\mathfrak{gl}_{n}$\- module $\mathfrak{P}_{2,n}$ is isomorphic to
$V_{\bullet}^{\langle n\rangle}\otimes V_{\bullet}^{\langle n\rangle}$.
Consider the submodule $V_{m_{1}}^{\langle n\rangle}\otimes V_{m_{2}}^{\langle
n\rangle}\subset\mathfrak{P}_{2,n}$. We have the following decomposition of
the $\mathfrak{gl}_{n}$-module:

$\displaystyle V_{m_{1}}^{\langle n\rangle}\otimes V_{m_{2}}^{\langle
n\rangle}=\bigoplus_{m=\max(0,m_{1}+m_{2}-n)}^{\min(m_{1},m_{2})}V_{\boldsymbol{l}(m)}^{\langle
n\rangle}.$ (4.12)

Here $\boldsymbol{l}(m)=(2,2,\dots,2,1,\dots,1,0,\dots,0)$, where $2$ repeats
$m$ times and $1$ repeats $m_{1}+m_{2}-2m$ times. Denote by $v_{m}$ a highest
weight vector of the summand $V_{\boldsymbol{l}(m)}^{\langle n\rangle}$ given
by formula (A.1).

Define the scalar product on $\mathfrak{P}_{2,n}$ by the rule: $\langle
f,f\rangle=1$, if $f\in\mathfrak{P}_{2,n}$ is a nonzero monomial, and $\langle
f,h\rangle=0$, if $f,h\in\mathfrak{P}_{2,n}$ are two non-proprosional
monomials.

######  Lemma 4.5.

We have $\langle v_{m},v_{m}\rangle\neq 0$ for every $m$.

The proof is straightforward by formula (A.1).

######  Lemma 4.6.

$\big{\langle}w_{1},e_{ij}^{\langle
n\rangle}w_{2}\big{\rangle}=\big{\langle}e_{ji}^{\langle
n\rangle}w_{1},w_{2}\big{\rangle}$ for any $w_{1},w_{2}\in\mathfrak{P}_{2,n}$,
and $i,j=1,\dots,n$.

The proof is straightforward.

######  Corollary 4.7.

If vectors $w$ and $\tilde{w}$ belong to distinct summands of decomposition
(4.12), then $\langle w,\tilde{w}\rangle=0$.

###### Proof.

The summands of decomposition (4.12) are eigenspaces of the operator
$I^{\langle n\rangle}$, and the corresponding eigenvalues are distinct. Lemma
4.6 implies that the operator $I^{\langle n\rangle}$ is symmetric with respect
to the scalar product $\langle\cdot,\cdot\rangle$, which implies the
statement. ∎

Denote

$\displaystyle L_{ij}(t)=t\big{(}e_{ij}^{\langle
n\rangle}\big{)}_{(1)}+\sum_{k=1}^{n}\big{(}e_{ik}^{\langle
n\rangle}\big{)}_{(1)}\big{(}e_{kj}^{\langle n\rangle}\big{)}_{(2)},\qquad$
$\displaystyle M_{ij}(t)=t\big{(}e_{ij}^{\langle
n\rangle}\big{)}_{(1)}+\sum_{k=1}^{n}\big{(}e_{kj}^{\langle
n\rangle}\big{)}_{(1)}\big{(}e_{ik}^{\langle n\rangle}\big{)}_{(2)},$
$\displaystyle\alpha_{m}(t)=\langle L_{m_{1}+m_{2}-m+1,m}(t)\cdot
v_{m},v_{m-1}\rangle,\qquad$ $\displaystyle\beta_{m}(t)=\langle
M_{m_{1}+m_{2}-m+1,m}(t)\cdot v_{m},v_{m-1}\rangle.$

######  Lemma 4.8.

The functions $\alpha_{m}(t)$ and $\beta_{m}(t)$ are nonzero, and

$\displaystyle\frac{\alpha_{m}(t)}{\beta_{m}(t)}=\frac{t+1+m_{1}-m}{t-1+m-m_{2}}.$
(4.13)

The proof is given in Appendix A.

Due to relation (3.1), for any $m$, there exists a scalar function
$\rho_{m}(t)$ such that $R_{12}^{\langle n\rangle}(t)w=\rho_{m}(t)w$ for any
$w\in V_{\boldsymbol{l}(m)}^{\langle n\rangle}$.

######  Lemma 4.9.

It holds that

$\frac{\rho_{m}(t)}{\rho_{m-1}(t)}=\frac{\alpha_{m}(t)}{\beta_{m}(t)}.$ (4.14)

###### Proof.

Let us single out the term $V_{\boldsymbol{l}(m-1)}$ in the decomposition
(4.12):

$V_{m_{1}}^{\langle n\rangle}\otimes V_{m_{2}}^{\langle
n\rangle}=V_{\boldsymbol{l}(m-1)}\bigoplus\tilde{V}.$

Then we can write $L_{m_{1}+m_{2}-m+1,m}(t)\cdot v_{m}=w+\tilde{w}$, where
$w\in V_{\boldsymbol{l}(m-1)}$ and $\tilde{w}\in\tilde{V}$. By the definition
of $L_{m_{1}+m_{2}-m+1,m}(t)$, the vector $w$ has weight
$\boldsymbol{l}(m-1)$. Therefore, $w=av_{m-1}$ for some scalar $a$. By
Corollary 4.7, we have

$\alpha_{m}(t)=\langle L_{m_{1}+m_{2}-m+1,m}(t)\cdot
v_{m},v_{m-1}\rangle=a\langle v_{m-1},v_{m-1}\rangle.$

Notice that $R_{12}^{\langle n\rangle}(t)\tilde{w}\in\tilde{V}$, because
$R$-matrix $R_{12}^{\langle n\rangle}(t)$ acts as a multiplication by a scalar
function on each summand of the decomposition (4.12). Then, by Corollary 4.7,
$\big{\langle}R_{12}^{\langle
n\rangle}(t)\tilde{w},v_{m-1}\big{\rangle}\allowbreak=0$, and

$\displaystyle\begin{split}&\big{\langle}R_{12}^{\langle
n\rangle}(t)L_{m_{1}+m_{2}-m+1,m}(t)\cdot v_{m},v_{m-1}\big{\rangle}\\\
&\qquad{}=\big{\langle}R_{12}^{\langle
n\rangle}(t)w,v_{m-1}\big{\rangle}=\rho_{m-1}(t)a\langle
v_{m-1},v_{m-1}\rangle=\rho_{m-1}(t)\alpha_{m}(t).\end{split}$

On the other hand, relation (3.2) gives

$\displaystyle\big{\langle}R_{12}^{\langle
n\rangle}(t)L_{m_{1}+m_{2}-m+1,m}(t)\cdot v_{m},v_{m-1}\big{\rangle}$
$\displaystyle\qquad{}=\big{\langle}M_{m_{1}+m_{2}-m+1,m}(t)R_{12}^{\langle
n\rangle}(t)\cdot v_{m},v_{m-1}\big{\rangle}=\rho_{m}(t)\beta_{m}(t).$

Thus we get $\alpha_{m}(t)\rho_{m-1}(t)=\rho_{m}(t)\beta_{m}(t)$, which is
relation (4.14). ∎

By formulae (4.14), (4.13),

$\displaystyle\rho_{m}(t)=\prod_{s=1}^{m}\frac{\rho_{s}(t)}{\rho_{s-1}(t)}=\prod_{s=0}^{m-1}\frac{t+m_{1}-s}{t-m_{2}+s},$
(4.15)

where we used that $\rho_{0}=1$ by the normalization condition (3.3).

Consider a decomposition of the $\mathfrak{gl}_{2}$-module:

$\displaystyle V_{l_{1}}^{\langle 2\rangle}\otimes\dots\otimes
V_{l_{n}}^{\langle 2\rangle}=\bigoplus_{0\leq
m\leq|\boldsymbol{l}|/2}V_{(|\boldsymbol{l}|-m,m)}^{\langle 2\rangle}\otimes
W_{m}^{\langle 2\rangle},$

where $|\boldsymbol{l}|=\sum_{i}l_{i}$ and $W_{m}^{\langle 2\rangle}$ are
multiplicity spaces. Let

$\big{(}V_{l_{1}}^{\langle 2\rangle}\otimes\dots\otimes V_{l_{n}}^{\langle
2\rangle}\big{)}[m_{1},m_{2}]_{m}=\big{(}V_{l_{1}}^{\langle
2\rangle}\otimes\dots\otimes V_{l_{n}}^{\langle
2\rangle}\big{)}[m_{1},m_{2}]\cap\big{(}V_{(|\boldsymbol{l}|-m,m)}^{\langle
2\rangle}\otimes W_{m}^{\langle 2\rangle}\big{)}.$

######  Lemma 4.10.

It holds that

$\displaystyle B_{12}^{\langle 2\rangle}(t)\big{|}_{\big{(}V_{l_{1}}^{\langle
2\rangle}\otimes\dots\otimes V_{l_{n}}^{\langle
2\rangle}\big{)}[m_{1},m_{2}]_{m}}=\prod_{s=m}^{m_{2}-1}\frac{t+m_{2}-s}{t-m_{1}+s}.$
(4.16)

###### Proof.

The modules $V_{l_{i}}^{\langle 2\rangle}$ can have only the following highest
weights: $(1,1)$, $(1,0)$, $(0,0)$. Moreover, the modules $V_{(0,0)}^{\langle
2\rangle}$ and $V_{(1,1)}^{\langle 2\rangle}$ are one-dimensional and the
elements $e_{12}$, $e_{21}$, and $e_{11}-e_{22}$ act there by zero. Thus, it
is enough to consider the case when $V_{l_{i}}^{\langle
2\rangle}=V_{(1,0)}^{\langle 2\rangle}$ for all $i$, so formula (4.16) follows
from [6, formula (5.13)]. ∎

Comparing formulas (4.15), (4.16), and (4.9), we conclude

$\displaystyle\rho_{m}(t)=\big{(}B_{12}^{\langle 2\rangle}(-t)C_{12}^{\langle
2\rangle}(t)\big{)}\big{|}_{\big{(}V_{l_{1}}^{\langle
2\rangle}\otimes\dots\otimes V_{l_{n}}^{\langle
2\rangle}\big{)}[m_{1},m_{2}]_{m}}.$ (4.17)

######  Lemma 4.11.

For the Casimir elements $I^{\langle 2\rangle}$ and $I^{\langle n\rangle}$, we
have

$I^{\langle 2\rangle}-2\sum_{a=1}^{2}e_{aa}^{\langle
2\rangle}\backsimeq-I^{\langle n\rangle}+n\sum_{i=1}^{n}e_{ii}^{\langle
n\rangle}.$ (4.18)

The proof is straightforward.

Recall that in the irreducible $\mathfrak{gl}_{k}$-module $V_{\boldsymbol{l}}$
the element $I^{\langle k\rangle}$ acts as a multiplication by
$(\boldsymbol{l},\boldsymbol{l}+2\rho)$. Then it is easy to verify that

$\displaystyle\left(I^{\langle 2\rangle}-2\sum_{a=1}^{2}e_{aa}^{\langle
2\rangle}\right)\bigg{|}_{V_{(|\boldsymbol{l}|-m,m)}^{\langle 2\rangle}\otimes
W_{m}^{\langle 2\rangle}}=\left(-I^{\langle
n\rangle}+n\sum_{i=1}^{n}e_{ii}^{\langle
n\rangle}\right)\bigg{|}_{V_{\boldsymbol{l}(m)}^{\langle n\rangle}}.$ (4.19)

Comparing formulae (4.18) and (4.19), and using the fact that the Casimir
elements act on distinct irreducible modules as a multiplication by distinct
scalar functions, we get that under isomorphisms (4.1) and (4.2) the
respective images of $V_{(|\boldsymbol{l}|-m,m)}^{\langle 2\rangle}\otimes
W_{m}^{\langle 2\rangle}$ and $V_{\boldsymbol{l}(m)}^{\langle n\rangle}$ in
$\mathfrak{P}_{2,n}$ coincide. To indicate that, we will write
$V_{(|\boldsymbol{l}|-m,m)}^{\langle 2\rangle}\otimes W_{m}^{\langle
2\rangle}\backsimeq V_{\boldsymbol{l}(m)}^{\langle n\rangle}$.

We also have $\big{(}V_{l_{1}}^{\langle 2\rangle}\otimes\dots\otimes
V_{l_{n}}^{\langle
2\rangle}\big{)}[m_{1},m_{2}]\backsimeq\big{(}V_{m_{1}}^{\langle
n\rangle}\otimes V_{m_{2}}^{\langle n\rangle}\big{)}[l_{1},\dots,l_{n}]$.
Therefore $\big{(}V_{m_{1}}^{\langle n\rangle}\otimes V_{m_{2}}^{\langle
n\rangle}\big{)}[l_{1},\dots,l_{n}]\cap V_{\boldsymbol{l}(m)}^{\langle
n\rangle}\backsimeq\big{(}V_{l_{1}}^{\langle 2\rangle}\otimes\dots\otimes
V_{l_{n}}^{\langle 2\rangle}\big{)}[m_{1},m_{2}]_{m}$. Now we see that (4.17)
gives us relation between actions of operators $B_{12}^{\langle 2\rangle}(t)$,
$C_{12}^{\langle 2\rangle}(t)$, and $R_{12}^{\langle n\rangle}(t)$ on one
particular submodule of $\mathfrak{P}_{2,n}$.

Theorem 4.4 is proved. ∎

##  Appendix A Proof of Lemma 4.8

Let

$v_{m}=\underset{i=1}{\overset{m}{\prod}}x_{1i}x_{2i}\underset{\varepsilon}{\sum}x_{\varepsilon_{1},m+1}x_{\varepsilon_{2},m+2}..x_{\varepsilon_{m_{1}+m_{2}-2m},m_{1}+m_{2}-m}$
(A.1)

with
$\\{\varepsilon\\}=\big{\\{}(\varepsilon_{1,}\varepsilon_{2},\dots,\varepsilon_{m_{1}+m_{2}-2m})\colon\varepsilon_{i}=1\text{
or }2\text{, }\sum_{i}\varepsilon_{i}=m_{1}-m+2(m_{2}-m)\big{\\}}$. One can
easily prove that $v_{m}$ is a highest weight vector of weight
$\boldsymbol{l}(m)$ .

It follows from the construction of the scalar product
$\langle\cdot,\cdot\rangle$ that $\alpha_{m}(t)$ (resp. $\beta_{m}(t)$) equals
the sum of the coefficients of those monomials presented in

$L_{m_{1}+m_{2}-m+1,m}(t)\cdot v_{m}\qquad\text{(resp.}\
M_{m_{1}+m_{2}-m+1,m}(t)\cdot v_{m})$

that also appear in $v_{m-1}$. In fact, all monomials either in

$L_{m_{1}+m_{2}-m+1,m}(t)\cdot v_{m}\qquad\text{or in}\
M_{m_{1}+m_{2}-m+1,m}(t)\cdot v_{m}$

appear in $v_{m-1}$ as well.

We will start with $\alpha_{m}$. Denote $s=m_{1}+m_{2}-m+1$. Let us inspect
what happens when we apply various terms of $L_{sm}(t)$ to $v_{m}$. For the
sum $\underset{k=1}{\overset{n}{\sum}}(e_{sk})_{(1)}(e_{km})_{(2)}$, we can
assume that $m\leq k<s$. If $k>m$, then the operator
$(e_{sk})_{(1)}(e_{km})_{(2)}$ will send a monomial in $v_{m}$ to zero if and
only if this monomial does not depend on $x_{1k}$. That is, we look at all
terms in $v_{m}$ corresponding to $\varepsilon_{k-m}=1$. There are
$C_{m_{1}+m_{2}-2m-1}^{m_{2}-m}$ such terms with the same contribution
$(-1)^{m_{1}+m_{2}+m}$. We leave the details of this calculation to a reader.
Under the assumption $m<k<s$, there are $m_{1}+m_{2}-2m$ different values of
$k$, which yield the overall contribution
$(-1)^{m_{1}+m_{2}+m}(m_{1}+m_{2}-2m)C_{m_{1}+m_{2}-2m-1}^{m_{2}-m}$ to
$\alpha_{m}(t)$.

If $k=m$, then we have $(e_{sk})_{(1)}(e_{km})_{(2)}\cdot
v_{m}=(e_{sm})_{(1)}\cdot v_{m}$. Therefore all $C_{m_{1}+m_{2}-2m}^{m_{1}-m}$
terms in $v_{m}$ equally contribute $(-1)^{m_{1}+m_{2}+m}(m_{1}+m_{2}-2m)$.

Finally, the term $t(e_{ij})_{(1)}$ in $L_{sm}(t)$ generates the contribution
$t(-1)^{m_{1}+m_{2}+m}C_{m_{1}+m_{2}-2m}^{m_{1}-m}$ to $\alpha_{m}(t)$, which
can be seen similarly to the case $k=m$ considered above.

Thus we obtained

$\displaystyle(-1)^{m_{1}+m_{2}+m}\alpha_{m}=(t+1)C_{m_{1}+m_{2}-2m}^{m_{1}-m}+(m_{1}+m_{2}-2m)C_{m_{1}+m_{2}-2m-1}^{m_{2}-m}.$

The similar arguments give us

$\displaystyle(-1)^{m_{1}+m_{2}+m}\beta_{m}=(t-1)C_{m_{1}+m_{2}-2m}^{m_{1}-m}-(m_{1}+m_{2}-2m)C_{m_{1}+m_{2}-2m-1}^{m_{1}-m}.$

Since

$(m_{1}+m_{2}-2m)C_{m_{1}+m_{2}-2m-1}^{m_{1}-m}=(m_{1}-m)C_{m_{1}+m_{2}-2m}^{m_{1}-m},$

and

$(m_{1}+m_{2}-2m)C_{m_{1}+m_{2}-2m-1}^{m_{2}-m}=(m_{2}-m)C_{m_{1}+m_{2}-2m}^{m_{1}-m}.$

Lemma 4.8 is proved.

### Acknowledgements

V. Tarasov was supported in part by Simons Foundation grant 430235.

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# Simplify-then-Translate: Automatic Preprocessing for Black-Box Translation

Sneha Mehta1, Bahareh Azarnoush2, Boris Chen2, Avneesh Saluja2,
Vinith Misra2, Ballav Bihani2, Ritwik Kumar2
1Department of Computer Science, Virginia Tech, VA
2Netflix Inc., CA
<EMAIL_ADDRESS>{bazarnoush, bchen, asaluja,
vmisra, bbihani<EMAIL_ADDRESS>
###### Abstract

Black-box machine translation systems have proven incredibly useful for a
variety of applications yet by design are hard to adapt, tune to a specific
domain, or build on top of. In this work, we introduce a method to improve
such systems via automatic pre-processing (APP) using sentence simplification.
We first propose a method to automatically generate a large in-domain
paraphrase corpus through back-translation with a black-box MT system, which
is used to train a paraphrase model that “simplifies” the original sentence to
be more conducive for translation. The model is used to preprocess source
sentences of multiple low-resource language pairs. We show that this
preprocessing leads to better translation performance as compared to non-
preprocessed source sentences. We further perform side-by-side human
evaluation to verify that translations of the simplified sentences are better
than the original ones. Finally, we provide some guidance on recommended
language pairs for generating the simplification model corpora by
investigating the relationship between ease of translation of a language pair
(as measured by BLEU) and quality of the resulting simplification model from
back-translations of this language pair (as measured by SARI), and tie this
into the downstream task of low-resource translation.

##  1 Introduction

Modern translation systems built on top of a sequence transduction approach
(Sutskever, Vinyals, and Le, 2014; Bahdanau, Cho, and Bengio, 2014) have
greatly advanced the state and quality of machine translation (MT). These
systems generally rely on the availability of large-scale parallel corpora,
and while unsupervised (Lample et al., 2017) or semi-supervised (Saluja et
al., 2014) approaches are a popular area of research, production-grade
translation systems still primarily leverage bitexts when training. Efforts
such as WMT111http://www.statmt.org/wmt18/ provide such corpora for select
language pairs, which has enabled neural MT systems to achieve state-of-the-
art performance on those pairs. However, low resource MT (language pairs for
which parallel data is scarce) remains a challenge.

In this work, we focus on improving translation quality for low-resource
translation (i.e., from English _into_ a low-resource language) in the _black-
box MT_ (BBMT) setting - namely a system which has been trained and tuned a
priori and for which we cannot access the model parameters or training data
for fine-tuning or improvements. Examples of such systems include those
provided by commercial vendors e.g., Google
Translate222https://translate.google.com/ or Microsoft
Translator333https://www.bing.com/translator. While some provide the option of
fine-tuning on domain-specific data under certain conditions, how to improve
the performance of such black-box systems on domain-specific translation tasks
remains an open question. We investigate methods to leverage the BBMT system
to preprocess input source sentences in a way that preserves meaning and
improves translation in the target language. Specifically, a large-scale
parallel corpus for English simplification is obtained by back-translating
(Sennrich, Haddow, and Birch, 2016a) the reference translations of several
high-resource target languages. The resulting parallel corpus is used to train
an Automatic Preprocessing model (APP) (§ 2), which transforms source sentence
into a form that preserves meaning while also being easier to translate into a
low-resource language. In effect, the APP model attacks the longstanding
problem of handling complex idiomatic and non-compositional phrases (Lin,
1999; Sag et al., 2002), and simplifies these expressions to more literal,
compositional ones that we hypothesize are easier to translate.

We use the APP model to simplify the source sentences of a variety of low-
resource language pairs and compare the performance of the black-box MT system
on the original and simplified sentences. Note that only one APP model needs
to be trained per source language and this model can be applied to a variety
of low-resource language pairs as long as the source language is the same. In
our study we focus on the domain of conversational language as used in
dialogues of TV shows. We picked this domain since here language tends to be
colloquial and idiomatic. We empirically show improvement in translation
quality in this domain across a variety of low resource target languages (§3).
This improvement is further corroborated with side-by-side human evaluations
(§4.2) and evaluating on post-editing efficiency (§4.1). Lastly, we perform an
empirical analysis to probe further into which high-resource language pairs
should be selected to obtain a good quality simplification corpus for a given
language, before discussing connections with related work (§5) and concluding.

Figure 1: Machine vs. Human Translation

##  2 APP with Back Translations

Consider the example in Fig. 1. The source sentence “The vice president should
feel free to jump in” has been translated by Google Translate 444This specific
translation is observed in translate.google.com as of September 5, 2019.
incorrectly to Hindi as “Vice President should feel free to jump inside”. The
system was unable to correctly translate the idiomatic and non-compositional
phrase “jump in”, which in this context means “intervene” or “get involved”,
and instead translated it literally. An expert human Hindi translator would
take these idiomatic expressions into account when generating the Hindi
translation. Indeed, when back-translating the reference translation into
English we obtain “The Vice President should feel free to take part” (in the
conversation). Such instances where MT systems incorrectly translate sentences
containing phrases, idioms or complex words for low-resources (i.e. with
smaller training sets) languages pairs are fairly common.

In other words, the back-translation is different in meaning than the natural
source sentence. This problem is prevalent even when these BBMT systems are
fine-tuned on domain-specific data and is exacerbated when dealing with low-
resource language pairs, simply because the paucity of data does not allow the
MT models to infer the translations of the myriad of phrases and complex
words. For instance, in the example above ‘jump in’ was interpreted
compositionally as ‘jump’ $+$ ‘in’. To generate better quality or even
acceptable translations, it is imperative to simplify such complex sentences
into simpler forms while still preserving meaning.

Using automated models to simplify such sentences is a well-studied problem.
Though, when it comes to the ultimate task of domain-specific translation, it
is not entirely clear what data is best suited to train such simplification
models. Open source datasets like WikiLarge (Zhang and Lapata, 2017) or Simple
PPDB (Zhao et al., 2018) are good options to explore, but the domain mismatch
and dataset size may pose a challenge. In particular, WikiLarge dataset
contains 296K sentence pairs of descriptive text while our domain of interest
in this study is conversational dialogues from TV Shows and movies. Collecting
a large amount of domain-specific simplification data could be prohibitive,
forcing one to consider alternatives when constructing their simplification
models.

To address this problem we make use of the observation that translating back-
translations is easier than translating naturally occurring source sentences,
which has been corroborated by numerous studies (Graham, Haddow, and Koehn,
2019; Zhang and Toral, 2019). Consider a set of around 30K uniformly-sampled
sentence pairs from the English-Bulgarian (En-Bg) subtitles corpus appearing
on TV shows and movies from a subscription video-on-demand
provider555www.netflix.com. The BLEU score in the natural or direct direction
(En $\rightarrow$ Bg) is only 10.20, whereas when following the reverse
direction (Bg $\rightarrow$ En $\rightarrow$ Bg) and translating back-
translations instead of original source sentences, the BLEU score dramatically
improves to 33.39. Probing a little deeper, Fig. 2 shows the distributions of
sentence-level GLEU scores (Wu and others, 2016) for two language pairs:
English-Bulgarian (right) and English-Hindi (left). We observe the trend that
GLEU scores have generally improved when using back-translations. The area
where the blue curve dominates the red curve can be considered as the ‘scope-
of-simplification’.

(a) English-Hindi

(b) English-Bulgarian

Figure 2: Sentence-level GLEU for direct translations and back-translations.
The red density curve represents the distribution of GLEU scores obtained by
the direction translation direction and the blue density curve represents the
distribution of GLEU scores obtained by using back-translations.

Thus it seems that human reference translations when back-translated to the
original language (in this case English) are a rich source of simplifications
(e.g. “jump in” is simplified to “take part”). This observation leads to two
immediate corollaries - 1) by back-translating the ground truth human
translations to the source language we obtain a (perhaps noisy) simplified
version of the original source, and 2) we can learn a function to map the
source sentences to their simplified forms by training a sequence-to-sequence
(S2S) model from the aforementioned generated parallel corpus. We term the
resulting simplification model an Automated Preprocessing (or APP) model.

We formalize our APP model as a S2S model from source sentences in one
language to back-translations of ground truth sentences i.e., translationese
targets in the same language. The translationese targets can be obtained from
multiple high-resource language pairs using black-box MT systems and the
trained APP model can be applied to a variety of low resource language pairs.
Let $(X^{i},Y^{i})$ be the $i^{\textrm{th}}$ training bitext corpus ($X$ is
source, $Y$ is target) with source language $s_{i}$ and target language
$t_{i}$ for $i\in\\{1,\dots,M\\}$, where $M$ is the number of training
language pairs, and let $j\in\\{M+1,\dots,M+N\\}$ refer to the $N$ test bitext
corpora. Note that for the experiments in this paper, $s_{i}$ is fixed to
English $\forall i\in\\{1,\dots,M+N\\}$ so we simply refer to it as $s$. The
APP procedure is as follows:

  * •

Obtain back-translations of target train sets $Y^{i}$ for $i=1$ to $M$ to
language $s_{i}$ given by $T^{1},T^{2},\dots,T^{M}$ using BBMT models
$MT_{t_{i}\rightarrow s}\forall i$.

  * •

Train an APP simplification model $f_{APP}$ on the combined parallel corpus
$\bigcup_{i=1}^{M}\\{(X^{i},T^{i})\\}$.

  * •

At test time, preprocess the source $X_{s}^{j}$ for each test language pair
$j$ using the trained APP model to obtain the simplified source $X^{j*}$,
where

$X^{j*}=f_{APP}(X^{j})$ (1)

  * •

Translate the simplified source using the BBMT model for the $j^{th}$ test
language pair.

$\hat{Y}^{j*}=MT_{s\rightarrow t_{j}}(X^{j*})$ (2)

APP provides in-domain simplification bitext at scale and from the same BBMT
system that we eventually use to translate into low-resource languages, thus
providing a more flexible solution than using precompiled simplification
corpora. In the next sections, we compare the performance of BBMT system
outputs with and without APP simplifications i.e. $\hat{Y}^{j}*$ &
$\hat{Y}^{j}$ respectively. Further, we compare the APP models trained on in-
domain vs out-of-domain corpora.

##  3 Evaluation

We first compare the in-domain APP model to a S2S model trained on the
WikiLarge corpus, by evaluating downstream translation performance on low-
resource languages, followed by an evaluation based on human judgements and
post-editing efficiency. These experiments are conducted on subtitles dataset
from a subscription video-on-demand provider. We also validate the approach on
another subtitles dataset and verify that the improvements we see in the first
set of experiments generalize to other corpora in the same domain. In all of
our experiments we used the Google Translate BBMT system.

###  3.1 Datasets and Metrics

#### FIGS

This dataset comes from subtitles appearing on 12,301 TV shows and movies from
a subscription video-on-demand provider. Titles include but are not limited
to: “How to Get Away with Murder”, “Star Trek: Deep Space Nine”, and “Full
Metal Alchemist”. We take four high-resource language pairs namely: English-
French (1.3 million parallel subtitle events i.e., sentence pairs), English-
Italian (1.0M), English-German (1.2M) and English-Spanish (6.5M). We
collectively refer to this dataset as FIGS. We use the APP simplification
procedure to obtain English simplification parallel corpora resulting in 9.5M
subtitle events i.e., sentence pairs. This dataset contains short sentences
with an average length of 7. For evaluation, we pick 7 low-resource language
pairs namely: English-Hungarian (En-Hu), English-Ukrainian (En-Uk), English-
Czech (En-Cs), English-Romanian (En-Ro), English-Bulgarian (En-Bg), English-
Hindi (En-Hi), and English-Malay (En-Ms). The test set statistics for each
dataset is given in Table 1. We refer to the APP model trained on this dataset
as ‘FigsAPP’.

Table 1: FIGS test set statistics  Language | #sentences
---|---
En-Hu | 27,393
En-Uk | 30,761
En-Cs | 35,505
En-Ro | 47,054
En-Bg | 30,714
En-Hi | 23,496
En-Ms | 11,713

#### Wikilarge

The WikiLarge dataset (Zhang and Lapata, 2017) was compiled by using sentence
alignments from other Wikipedia-based datasets (Zhu, Bernhard, and Gurevych,
2010; Woodsend and Lapata, 2011; Kauchak, 2013), and contains 296K instances
of 1-to-1 and 1-to-many alignments. This is a widely used benchmark for test
simplification tasks. The train split contains 296K sentence pairs and the
validation split contains 992 sentence pairs. We call the simplification model
trained on this dataset as ‘WikiAPP’.

#### Open Subtitles

The Open Subtitles dataset (Lison and Tiedemann, 2016) is a collection of
translated movie subtitles obtained from
opensubtitles.org666http://www.opensubtitles.org/ . It contains 62 languages
and 1,782 bitexts. We first train two MT models using two high resource
language pairs (Es-En and Ru-En) using the Transformer architecture (Vaswani
et al., 2017). Then using the MT models above we train two APP models obtained
from the same language pairs English-Spanish (En-Es) and English-Russian (En-
Ru). We sample 5M sentence pairs each for training MT and APP models from the
corresponding Open Subtitles corpora and filter out short ($length<3$) and
long ($length>50$) sentence pairs. Note that training sets for both MT and APP
models are disjoint. For evaluation, we pick the following six language pairs:
three randomly picked pairs English-Armenian (En-Hy), English-Ukrainian(En-Uk)
and English-Bulgarian (En-Bg) and three pairs in which the target language is
similar to Spanish including English-Catalan (En-Ca), English-Portuguese (En-
Pt) and English-Romanian (En-Ro). We sample 50,000 pairs from the low-resource
test bitexts and filter out pairs with length less than 3 and greater than 50.
We call the APP model obtained from En-Es dataset as ‘OSEsAPP’ and from En-Ru
dataset as ‘OSRuAPP’.

#### Turk and PWKP

To test the performance of APP models obtained from different language pairs (
§ 4.3) we pick two open source simplification datasets. The first dataset is
the Turk (Xu et al., 2016) dataset which contains 1-to-1 alignments focused on
paraphrasing transformations, and multiple (8) simplification references per
original sentence (collected through Amazon Mechanical Turk). To evaluate the
performance on this dataset we use the SARI metric they introduced. The next
dataset we use is the test set of the PWKP dataset (Zhu, Bernhard, and
Gurevych, 2010). This dataset contains only 1-to-1 mapping between source and
reference and hence we use the BLEU metric to evaluate the simplification
performance.

#### Metrics

We evaluate the translation performance of the BBMT system using the commonly
used BLEU metric (Papineni et al., 2002). For subtitle generation, expert
linguists post-edit subtitles at the event or dialog level, hence it is useful
to look at the impact that simplification brings at the sentence-level,
motivating the choice of sentence-level GLEU (Wu and others, 2016), which has
been shown to be better correlated with human judgements at the sentence-level
as compared to sentence-level BLEU. Furthermore, we compute the normalized
edit (Levenshtein) distance between a translation output and human reference
translation also known as translation error rate (TER), which has been shown
to correlate well with the amount of post-editing effort required by a human
(Snover et al., 2006). This metric provides yet another way to evaluate the
quality of translation and completes our comprehensive suite of metrics.

###  3.2 Implementation

For training the APP simplification model we use the Transformer architecture
(Vaswani et al., 2017) through the
tensor2tensor777https://github.com/tensorflow/tensor2tensor library. We also
evaluate BLEU using the implementation in that library and report the uncased
version of the metric. For computing the TER score we use the implementation
provided by (Snover et al., 2006) 888http://www.cs.umd.edu/ snover/tercom/.

All experiments are based on the transformer base architecture with 6 blocks
in the encoder and decoder. We use the same hyper-parameters for all
experiments, i.e., word representations of size 512 and feed-forward layers
with inner dimension 4096. Dropout is set to 0.2 and we use 8 attention heads.
Models are optimized with Adam (Kingma and Ba, 2014) using $\beta_{1}=0.9$,
$\beta_{2}=0.98$, and $\epsilon=1e-9$, with the same learning rate schedule as
Vaswani et al. (2017). We use 50,000 warmup steps. All models use label
smoothing of 0.1 with a uniform prior distribution over the vocabulary. We run
all experiments using machines with 4 Nvidia V100 GPUs. We use a sub-word
vocabulary of size 32K implemented using the word-piece algorithm (Sennrich,
Haddow, and Birch, 2016a) to deal with out-of-vocabulary words and the open
vocabulary problem in S2S language models.

##  4 Results

Table 2 compares the performance of APP models trained on FIGS (in-domain)
dataset and the WikiLarge (out-of-domain) datasets on the FIGS test set using
the BBMT system. The values in columns 1-3 indicate the BLEU score after
translating the original sentence and after simplifying using the FigsAPP and
WikiAPP models respectively. There is uniform improvement across all languages
when using the FigsAPP model, ranging from 3.5% (relative) for En-Uk to 11.6%
for En-Bg. On the other hand, performance degrades significantly on all
languages when simplified using a model trained on the WikiLarge dataset. Fig.
3 shows the distribution of sentence-level GLEU for each target language in
the FIGS test set. Mean GLEU increases for En-Hu, En-Ro, En-Ms and En-Bg.

Table 3 shows the results of APP on the Open Subtitles dataset. It can be
noted that the performance improves for Catalan (ca) and Portuguese (pt) which
are languages similar to the language used for training the simplification
corpus. Additionally for Bulgarian an improvement of 4.1% can be observed.
Moreover, the performance of OSEsAPP is better than the performance of
OSRuAPP. This can be attributed to the fact that En-Ru is a harder language
pair to translate than En-Es and hence the simplification model obtained from
En-Ru is of worse quality than En-Es. We further elaborate this point in
section § 4.3.

Table 2: In-domain vs out-of-domain simplification performance. It is evident that APP models trained on an out-of-domain simplification corpus (WikiLarge) degrades performance, whereas in-domain simplification corpora (FIGS) boosts performance. language pair | original | FigsAPP | WikiApp
---|---|---|---
En-Hu | 17.69 | 18.92 | 11.86
En-Uk | 17.57 | 18.18 | 12.51
En-Cs | 21.62 | 22.35 | 16.71
En-Ro | 26.08 | 27.98 | 21.56
En-Bg | 14.9 | 16.63 | 12.39
En-Hi | 14.45 | 15.53 | 11.39
En-Ms | 19.14 | 20.37 | 13.00
Table 3: Translation performance (BLEU) before and after of OSEsAPP and OSRuAPP models on six low resource target language pairs from the Open Subtitles corpus. language pair | original | OSEsAPP | OSRuAPP
---|---|---|---
En-Ca | 27.25 | 27.84 | 23.36
En-Hu | 7.05 | 6.28 | 5.79
En-Pt | 25.11 | 25.5 | 22.28
En-Ro | 26.18 | 25.03 | 22.40
En-Uk | 11.73 | 11.77 | 11.61
En-Bg | 23.71 | 24.68 | 23.90

###  4.1 Post-Editing Efficiency

Figure 3: Sentence-level GLEU scores  Table 4: Translation Edit Rate (TER) score of translations before and after applying the APP simplification for test language-pairs from the FIGS dataset. ‘original’ and ‘simple’ columns show the TER for translations before and after APP where the last column indicates the percentage decrease in TER. language | original | simple | %$\Delta$
---|---|---|---
En-Hu | 0.86 | 0.80 | -7%
En-Uk | 0.74 | 0.70 | -5.4%
En-Cs | 0.78 | 0.77 | -1.3%
En-Ro | 0.72 | 0.67 | -6.9%
En-Bg | 0.83 | 0.77 | -7.3%
En-Hi | 0.62 | 0.60 | -3.3%
En-Ms | 0.76 | 0.72 | -5.3%

We also observe that TER decreases for all languages, which is intuitive to
understand because the APP simplification brings the sentences closer to their
literal human translation. Table 4 shows TER score for the FIGS test corpora
for translating into seven low resource languages, before and after applying
the APP simplification. As can be seen, TER decreases for all languages after
simplification with a reduction of 6.9%, 6.9% and 7.2% for target languages
Hungarian, Romanian and Bulgarian respectively. The reduction in TER
correspondingly translates to a reduction in post-editing effort required by
translators using the BBMT system as an assistive tool.

###  4.2 Human Evaluation

Simplified sentences with worse GLEU than their baseline non-simplified
counterparts might not necessarily be of worse quality; rather, they may just
be phrased differently than the reference sentence. We thus perform a

Figure 4: Human Evaluation Results

side-by-side human evaluation to verify if APP-simplified translations improve
MT quality, which allows us to assess via human evaluation these supposedly
worse translations and validate translations with GLEU improvements at the
same time. For this purpose, we restricted evaluation to five languages from
the FIGS test set (hu, uk, cs, ro, and bg) and sampled 100 sentences from the
fraction of sentences for which $\Delta GLEU>0.4$ and 100 sentences from the
sentences for which $\Delta GLEU<0$, where $\Delta GLEU$ for one sentence pair
(x, y) is defined as;

$\Delta GLEU=GLEU(MT(x^{*}),y)-GLEU(MT(x),y)$ (3)

and $x^{*}$ is the simplified source sentence. For each language we show the
source sentence, the BBMT output of the original sentence, the BBMT output of
the simplified sentence, and the ground truth human translation per language
side-by-side to expert linguists. We ask them to rate the quality of the three
translations (original BBMT, simplified BBMT and human translation) according
to the scale used by Wu and others (2016) described in Table 5. Since
translation is generally easier for shorter sentences, in order to get a
representative sample of challenging sentences we only selected sentences with
more than 4 tokens for this study.

Table 5: Human evaluation ratings and their description Rating | Description
---|---
0 | completely nonsense translation
2 |  | the sentence preserves some of the meaning of the
---
source sentence but misses significant parts
4 |  | the sentence retains most of the meaning of the source
---
sentence, but may have some grammar mistakes
6 |  | perfect translation: the meaning of the translation is
---
completely consistent with the source, and the grammar
is correct

Fig 4 displays results from the human evaluation, with corresponding values
stated in Table 6. The green box represents scores from translations of
original sentences, the orange box represents scores from translations from
simplified sentences and the blue box represents the ground truth human
translation. As expected the human translation has the highest score. Also
worth noting is the jump in score intervals between original and simplified
translations. For Hungarian, Czech, and Bulgarian median scores jump from
$2\rightarrow 4$, with improvements for Ukrainian ($3\rightarrow 4$) and
Romanian ($2\rightarrow 3$) as well. Thus at least for these languages we can
conclude that APP simplification results in improved translation output, as
determined by expert human translators. This concurs with the initial
observation e.g., for Bulgarian in Fig. 2, where the number of sentences with
score of 0.0 are squashed and their scores shifted higher. This means
simplification can improve upon erroneous or bad-quality translations. Table 6
shows the mean human scores for original, simplified and reference
translations as well as percentage of sentences for which human score
improved, worsened and remained the same after simplification for all 200
sentences per language.

Table 6: Human Evaluation statistics for the FIGS test language pairs. ‘Original Mean’, ‘Simple Mean’ and ‘Human Mean’ represent the mean human scores before, after APP and ground truth human scores respectively. The next three columns indicates the percentage of samples with improved (% +ve), worse (% -ve), and same (% same) performance after simplification.  lang |  | Original
---
Mean
| Simple
---
Mean
| Human
---
Mean
% +ve | % -ve | % same
Hu | 2.52 | 3.11 | 4.45 | 38.5% | 18.5% | 43%
Uk | 3.02 | 3.61 | 5.8 | 43% | 15% | 42%
Cs | 2.43 | 3.12 | 4.77 | 49% | 16.5% | 34.5%
Ro | 2.56 | 3.0 | 3.91 | 41.5% | 25.5% | 33%
Bg | 2.34 | 3.33 | 5.365 | 51% | 22% | 27%

###  4.3 Ease of Translation vs Simplification Quality

We further seek to investigate the relationship between the translation
language pair used to generate the simplification corpus and the quality of
the resulting simplification model. To this end, we train translation systems
on two language pairs: English-Spanish (En-Es) and English-Russian (En-Ru).
The En-Es pair is an easier language pair to translate than En-Ru as reflected
by the BLEU scores of the SOTA MT systems on these pairs. We pick an En-Es
system trained to reach a BLEU of 42.7 and an En-Ru system trained to a BLEU
of 34.23. We hypothesize that the APP models resulting from an easier language
pair will be of better quality because it is easier to generate a good quality
parallel simplification corpus. To test this hypothesis, we train
simplification models obtained from the En-Es and En-Ru translation pairs and
test the simplification performance on two standard simplification test sets.

Table 7: Simplification performance of APP models trained on corpora generated by En-Es and En-Ru datasets on Turk and PWKP open source datasets. (Metric) Dataset | Wiki | En-Es | En-Ru
---|---|---|---
(SARI) Turk | 31.9 | 26.7 | 23.2
(BLEU) PWKP | 53.5 | 32.7 | 23.9

Table 7 presents the results of this experiment, specifically the performance
of the simplification models trained via automatically-generated
simplification corpora obtained from the En-Es and En-Ru OpenSubtitles models.
We also test the performance of an in-domain simplification model trained on
the WikiLarge dataset. As expected, the performance of the in-domain model
exceeds the performance of the models trained on En-Es and En-Ru datasets. It
is also interesting to note that the En-Es model outperforms the En-Ru models
on both datasets, underlining our hypothesis that easier language pairs result
in better APP models. This idea can be used to inform the high-resource
language pair(s) to pick for training an APP model for a target language pair.
For instance, to simplify English it would be better to pick a high-resource
pair which is easier to translate from English, e.g. Spanish. While
simplifying Catalan(ca), it would be good to pick a translation pair like
Catalan-Spanish which would be easier to translate than Catalan-English.

###  4.4 Qualitative Analysis

Table 8: Positive and negative qualitative examples of simplifications brought about by APP.  | Original:
---
Simple:
| I still, think you’re nuts, but not as nuts as I thought
---
I still think you’re crazy, but not as crazy as I thought
| Original:
---
Simple:
| in a town only five miles from Kabul.
---
in a city eight kilometers from Kabul.
| Original:
---
Simple:
| This case is so far over your head, it’d make your nose
---
bleed.
This case is so complicated that it would bleed your nose.
| Original:
---
Simple:
| When I was marooned here, my first meal was a pheasant.
---
When I was stranded here, my first meal was a pheasant.
| Original:
---
Simple:
|  She jumped from the window of Room 180.
---
He jumped out of the window of Room 180.

Here we provide qualitative examples of simplifications generated by the APP
approach and how it helps in improving BBMT performance. Table 8 shows some
example simplifications from the FIGS test datasets. Phrases highlighted in
bold were converted to ‘simpler’ phrases. In the first example “you’re nuts”
was replaced by a non-idiomatic/simpler phrase “you’re crazy”. In the second
example, distance of five miles was almost accurately converted from imperial
to the metric system whereas in the third example non-compositional phrase “so
far over your head” was translated into the compositional phrase “so
complicated”. In the next example, an infrequent word “marooned” was replaced
by its more frequent counterpart “stranded”. Finally, the last example shows
the kinds of errors introduced by the APP model where it occasionally replaces
pronouns like ‘it’, ‘she’ by ‘he’.

Table 9 gives examples of how APP simplifications can help the BBMT systems
make fewer errors. The first example shows a sample from English-Romanian
translation pair. Direct translation of the source makes the BBMT system
incorrectly translate “fixating” as “fixing” (as observed in the
backtranslation of the BBMT output) where as simplifying “fixating on” as
“thinking about” produces a more meaningful translation. Similarly in the
second example from the English-Catalan pair of the Open Subtitles corpus, APP
replaces the colloquial word ‘swell’ by its meaning ‘great’ and hence results
in a translation that is identical to the reference.

Table 9: Qualitative examples of how APP simplification can help mitigate BBMT errors on the FIGS and Open Subtitles datasets. Here x is the input source.  | x:
---
BBMT(x):
APP(x):
BBMT(APP(x)):
Reference:
| If only we can stop fixating on the days
---
Dacă numai putem opri fixarea pe zile
If only we could stop thinking about the days
Dacă am putea să nu ne mai gândim la zile
Trebuie să nu ne mai gândim la zile
| x:
---
BBMT(x):
APP(x):
BBMT(APP(x)):
Reference:
| Another swell party, Jay.
---
Una altra festa de l’onatge, Jay.
Another great party, Jay.
Una altra gran festa, Jay.
Una altra gran festa, Jay.

##  5 Related Work

Automatic text simplification (ATS) systems aim to transform original texts
into their lexically and syntactically simpler variants. In theory, they could
also simplify texts at the discourse level, but most systems still operate
only on the sentence level. The motivation for building the first ATS systems
was to improve the performance of machine translation systems and other text
processing tasks, e.g. parsing, information retrieval, and summarization
(Chandrasekar, Doran, and Srinivas, 1996). It was argued that simplified
sentences (which have simpler sentential structures and reduced ambiguity)
could lead to improvements in the quality of machine translation
(Chandrasekar, Doran, and Srinivas, 1996) A large body of work, since then has
investigated text simplification for machine translation and found that this
approach can improve fluency of the translation output and reduce technical
post-editing effort (Štajner and Popovic, 2016).

Researchers have attempted to build simplification systems for different
languages such as English, Spanish, (Saggion et al., 2015), and Portuguese
(Aluísio et al., 2008). Wubben, van den Bosch, and Krahmer (2012) use phrase-
based machine translation for sentence simplification based on the PWKP
dataset (Zhu, Bernhard, and Gurevych, 2010). However, these systems are
modular, rule-based (Poornima, Dhanalakshmi, and Soman, 2011), limited by data
or language specific.

End-to-end simplification which is more similar to our work also has been
studied by applying RNN-based sequence-to-sequence models to the PWKP dataset
or transformer based models that are integrated with paraphrase rules (Zhao et
al., 2018) and trained on English to English parallel simplification corpora
and are current state-of-the-art. These methods are limited by the
availability of parallel simplification corpora and especially the ones that
can adapt to new domains. We propose a general framework that can be used to
collect large-scale data for any language to train in-domain end-to-end data-
driven lexical simplification systems.

Our work capitalizes on the observation that synthetically-generated source
sentences resulting from reversing the translation direction on a parallel
corpus yield better translations. These “back-translations” (Sennrich, Haddow,
and Birch, 2016b; Poncelas et al., 2018) can augment relatively scarce
parallel data with by translating the plentiful target monolingual data to the
source language.

Various methods have been explored to improve low-resource translation. (Zoph
et al., 2016) transfer parameters from an MT model trained on a high-resource
language pair to low-resource language pairs and observe performance
improvement. To improve performance on spoken language domain, researchers
have finetuned state-of-the-art models trained on domains in which data is
more abundant (Luong and Manning, ) whereas others have used data augmentation
techniques (Fadaee, Bisazza, and Monz, 2017) to bring improvements in low-
resource translation. Above approaches assume access to the underlying MT
system whereas we assume a black-box scenario.

##  6 Conclusion

In this work we introduced a framework for generating a large-scale parallel
corpus for sentence simplification, and demonstrated how the corpus can be
used to improve the performance of black-box MT systems (especially on low-
resource language pairs) and increase the post-editing efficiency at the
subtitle-event i.e., sentence level. Moreover, we perform thorough empirical
analysis to give insights into language pairs to select for simplifying a
given language. Our results suggest that easier a language pair to translate,
the better the simplification model that will result.

It should be noted that even though this work mainly focuses on simplification
of English, our method is general and can be used to automatically generate
simplification parallel corpora and thus data-driven simplification models
using state-of-the-art architectures for any given language. Moreover, it
accommodates collecting multiple reference simplifications for a given source
sentence by leveraging open-source multilingual corpora. Using the insight
that translating multiword expressions and non-compositional phrases is hard
and simplifying these expressions before translating helps, our work merges
two important sub-fields of NLP (machine translation and sentence
simplification) and paves the path for future research in both of these
fields.

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