# Howe duality and dynamical Weyl group

Rea Dalipi  Department of Mathematics, University of Geneva, Rue du Conseil-
Général 7-9, 1205 Geneva, Switzerland<EMAIL_ADDRESS>and  Giovani
Felder  Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
<EMAIL_ADDRESS>
###### Abstract.

We give a fermionic formula for $R$-matrices of exterior powers of the vector
representations of $U_{q}(\widehat{\mathfrak{gl}}_{N})$ and relate it to the
dynamical Weyl group of Tarasov–Varchenko and Etingof–Varchenko, via a Howe
($\mathfrak{gl}_{N},\mathfrak{gl}_{M})$-duality. As a consequence we obtain a
dynamical action of the Weyl group on integrable
$U_{q}\mathfrak{gl}_{M}$-modules, extending the known action on zero weight
spaces.

##  1\. Introduction

The symmetric groups play two different roles in the representation theory of
general linear groups. On one side, let us call it the Weyl side, the
symmetric group $S_{N}$ of permutations of $N$ letters is the subgroup of
$\mathit{GL}_{N}$ of permutation matrices and thus acts on any representation
of $\mathit{GL}_{N}$. On the other side, say the Schur side, $S_{M}$ is the
group of permutation of factors of the $M$th tensor power $U^{\otimes
M}=U\otimes\cdots\otimes U$ of a representation $U$ of $\mathit{GL}_{N}$ and
the action of $S_{M}$ commutes with the natural action of $\mathit{GL}_{N}$ on
$U^{\otimes M}$. We will focus on the case where $U$ is an exterior power of
the vector representation, but let us be general for the time being.

Both roles admit a $q$-deformed version but the symmetric group needs to be
replaced by the braid group. On the Weyl side one has an action of the braid
group $B_{N}$ on finite dimensional representations of the quantum enveloping
algebras $U_{q}\mathfrak{gl}_{N}$ deforming the above action of the symmetric
group. This is a special case of the action on integrable representations of
the Artin braid group corresponding to the Weyl group of any semisimple Lie
algebra [14, 13]. On the Schur side, the universal $R$-matrix of
$U_{q}\mathfrak{g}l_{N}$ defines a representation of the braid group $B_{M}$
on the tensor power $U^{\otimes M}$ of a finite dimensional representation $U$
of $U_{q}{\mathfrak{gl}}_{N}$. The action of the generator $s_{i}$, braiding
the $i$th with the $i+1$st strand is given by the action of the $R$-matrix on
the $i$th and $i+1$st factor.

Both sides are degenerate cases of versions with parameters, called spectral
(or equivariant) parameters on the Schur side and dynamical (or Kähler)
parameters on the Weyl side. In terms of representation theory, this
generalization amounts to replacing $U_{q}\mathfrak{gl}_{N}$ by its affine
version $U_{q}\widehat{\mathfrak{gl}}_{N}$. A feature of the versions with
parameters, whose clarification is a goal of this paper, is that the symmetric
group plays the leading role, rather than the braid group that is relevant in
the degenerate limit. The $R$-matrices of quantum groups first appeared in
statistical mechanics and quantum field theory as solutions
$R(z)\in\operatorname{End}(U\otimes U)$ of the Yang–Baxter equation, depending
meromorphically on a complex spectral parameter $z$. The Yang–Baxter equation
is equivalent to the braiding relation (2.3) for the braiding matrix
$\check{R}(z)=P\circ R(z)$ obtained by composing the $R$-matrix with the
permutation $P(u\otimes v)=v\otimes u$ of factors. Moreover one has the
inversion (or unitarity) relation $\check{R}(z^{-1})\check{R}(z)=\mathrm{id}$.
One obtains, on the Schur side, a representation of the symmetric group
$S_{M}$ on $U^{\otimes M}$-valued meromorphic functions $f(z_{1},\dots,z_{M})$
of $M$ variables, such that the generator $s_{i}=(i,i+1)$ is mapped to

$s_{i}f(z_{1},\dots,z_{N})=\mathrm{id}^{\otimes(i-1)}\otimes\check{R}(z_{i}/z_{i+1})\otimes\mathrm{id}^{\otimes(N-i-1)}f(z_{1},\dots,z_{i+1},z_{i},\dots,z_{N}).$

Representations of the braid group on $U^{\otimes M}$ appear in the limits
$z_{i}/z_{i+1}\to 0$, $i=1,\dots,M-1$ or $\infty$. Indeed $R(z)$ converges in
the limit $z\to\infty$ to the solution of the Yang–Baxter equation without
spectral parameter corresponding to $U_{q}{\mathfrak{gl}}_{N}$. The relation
$s_{i}^{2}=1$ is lost in this limit and one is left with a representation of
the braid group.

On the Weyl side, the discovery of a version of the quantum Weyl group action
with parameters is more recent [17, 6] and is called dynamical Weyl group. Let
$T\subset SL_{M}(\mathbb{C})$ be the Cartan torus of diagonal matrices,
$P_{M}=\operatorname{Hom}(T,\mathbb{C}^{\times})$ the weight lattice. Then
finite dimensional representations $U$ of $U_{q}\mathfrak{sl}_{M}$ (of type I
with $q$ not a root of unity) have a weight decomposition $U=\oplus_{\mu\in
P_{M}}U[\mu]$. The Weyl group $S_{M}$ acts on $T$ and $P_{M}$, defining a
action of the braid group via the canonical homomorphism $B_{M}\to S_{M}$. A
dynamical action of the braid group $B_{M}$ on an $\mathfrak{sl}_{M}$-module
$U$ is a representation $\rho$ of the braid group on rational $U$-valued
function on $T$ of the form

$(\rho(g)f)(t)=A_{g}(g^{-1}t)f(g^{-1}t),$

for some rational functions $t\mapsto A_{g}(t)\in\operatorname{End}(U)$,
compatible with the $T$-action on $U$ in the sense that
$A_{g}(t)(U[\mu])\subset U[g\mu]$. The theory of intertwining operators
developed in [17, 6], which works for arbitrary simple Lie algebras, gives
rise to a dynamical action of the braid group with many interesting
properties, such as universal formulas for $A_{g}(t)|_{U[\mu]}$ as the action
of an element of $U_{q}\mathfrak{sl}_{M}$ independent of $U$ and a
compatibility with inclusions of Lie subalgebras, implying that $A_{s_{i}}(t)$
for generators $s_{i}$ is obtained as the image of the corresponding element
of $U_{q}(\mathfrak{sl}_{2})$ by the inclusion associated with the $i$th
simple root.

In this paper we take $U=\bigwedge V$, the direct sum of exterior powers of
the vector representation $V=\mathbb{C}^{N}$ of $U_{q}{\mathfrak{gl}}_{N}$,
and relate the two actions of the symmetric group via quantum (skew) Howe
duality. The classical Howe duality [9] is based on the observation that on
$\bigwedge(\mathbb{C}^{N}\otimes\mathbb{C}^{M})$ we have natural commuting
actions of $\mathit{GL}_{N}$ and $\mathit{GL}_{M}$ whose images span each
other’s commutant in the endomorphism ring. This implies that endomorphisms of
$(\bigwedge V)^{\otimes M}=\bigwedge V\otimes\cdots\otimes\bigwedge V$
commuting with the $\mathit{GL_{N}}$ action on $V=\mathbb{C}^{N}$ are spanned
by a commuting action of $\mathit{GL}_{M}$. There is a also a symmetric
version of Howe duality, where exterior powers are replaced by symmetric
powers. Quantum version of Howe dualities are known. The skew case was first
considered in [2] and [12]. See [16] for recent developments of this story. In
Section 3 we present an approach based on the action of the Clifford algebra
which is more elementary than the categorification approach of [3], see
Proposition 3.1.

In particular one obtains a $U_{q}{\mathfrak{gl}}_{M}$-action on $(\bigwedge
V)^{\otimes M}$ commuting with the diagonal $U_{q}{\mathfrak{gl}}_{N}$-action.
The upshot is that the action of the symmetric group on $(\bigwedge
V)^{\otimes M}$-valued functions by braiding matrices with spectral parameter
is realized by a dynamical Weyl group action for the
$U_{q}{\mathfrak{gl}}_{M}$-action, after a suitable identification of spectral
with dynamical parameters. Thus we connect the Schur with the Weyl side. In
the limit of infinite spectral/dynamical parameter we recover the result of
[2] relating braid group action with quantum Weyl group action.

For $M=2$ we obtain an explicit formula for the $R$-matrix for pairs of
exterior powers of the vector representation in terms of
$U_{q}{\mathfrak{gl}}_{2}$, see Theorem 2.6. These $R$-matrices were
calculated in [5] by the fusion procedure and are expressed in terms of
projections onto irreducible subrepresentations of the tensor product. Our
alternative formula does not rely on the decomposition into irreducible
subrepresentations. It is a sum of terms, each with a simple pole in the
spectral parameters, and gives an interpretation of the residues at the poles.
When written in terms of Clifford algebra generators, our formula for the
$R$-matrix of $\bigwedge^{k}V\otimes\bigwedge^{k}V$ converges in the limit
$q\to 1,z\to 1$ to the formula proposed by Smirnov [15] in the case of the
Yangian. As in [15], it has the property that it also applies to the limit
$N\to\infty$, see Theorem 4.5. Finding a conceptual proof of Smirnov’s formula
was an initial motivation of this work.

As an application of our result, we obtain in Theorem 4.1 and Corollary 4.2 a
dynamical action of the symmetric group on integrable representations of
$U_{q}{\mathfrak{gl}}_{M}$, rather than of the braid group as in [6]. In fact
our dynamical action differs from the dynamical action of the braid group
defined in [6] by a transformation $A_{g}(t)|_{U[\mu]}\mapsto
A_{g}(t)|_{U[\mu]}f_{g,\mu}(t)$ for some scalar meromorphic functions
$f_{g,\mu}(t)$ obeying the cocycle condition
$f_{gh,\mu}(t)=f_{g,h\mu}(ht)f_{h,\mu}(t)$ for any braids $g,h$, see
Proposition 4.3. While in [6] the dynamical action of the braid group factors
through the symmetric group only when restricted to the zero weight space
$U[0]$, our modified action defines a representation of the symmetric group on
the whole space of $U$-valued functions. We expect that the same
transformation gives rise to a dynamical action of the Weyl group on
integrable representations of arbitrary reductive Lie algebras.

A summary of the results of this paper was announced in the Oberwolfach report
[7].

###  1.1. Acknowledgments

This research was supported in part by the National Centre of Competence in
Research SwissMAP (grant numbers 182902, 205607) of the Swiss National Science
Foundation. The second author is also supported by the grant 196892 of the
Swiss National Science Foundation. This article was written in part while the
second author was at IHES, which he thanks for hospitality.

We thank Tommaso Botta for various useful discussions and suggestions. The
second author thanks the organizers of the mini-workshop on “Recent
developments in representation theory and mathematical physics” at the
Mathematisches Forschungsinstitut Oberwolfach, where these results were
presented, and the participants, expecially Evgeny Mukhin and Catharina
Stroppel for discussions. We thank Andrey Smirnov and Huafeng Zhang for useful
correspondence.

##  2\. Braiding matrices for exterior powers of the vector representation

###  2.1. $q$-number notation

We adopt the standard $q$-number notation: for $n\in\mathbb{Z}_{\geq 0}$,
$[n]_{q}=\frac{q^{n}-q^{-n}}{q-q^{-1}}$. The $q$-factorial is defined by
$[0]_{q}!=1$, $[n]_{q}!=\prod_{j=1}^{n}[j]_{q}$, $n\geq 1$ and the
$q$-binomial coefficient (or Gauss polynomial) is

$\genfrac{[}{]}{0.0pt}{}{n}{j}_{q}=\frac{[n]_{q}!}{[j]_{q}![n-j]_{q}!}.$

All these objects belong to $\mathbb{Z}[q,q^{-1}]$.

###  2.2. Quantum enveloping algebras of type $A_{N-1}$ and $A_{N-1}^{(1)}$

Drinfeld–Jimbo quantum enveloping algebras are the subject of several
textbooks, for example [14, 4]. Here we collect our conventions. Let $q$ be a
non-zero complex number which is not a root of unity. Let
$(a_{ij})_{i,j=0}^{N-1}$ be the Cartan matrix of $A_{N-1}^{(1)}$: $a_{ij}=2$
if $i=j$, $a_{ij}=-1$ if $j=i\pm 1\mod N$, and $a_{ij}=0$ otherwise. It is
convenient to identify the index set $\\{0,\dots,N-1\\}$ with
$\mathbb{Z}/N\mathbb{Z}$ in view of the symmetry $a_{i+1,j+1}=a_{ij}$. The
Hopf algebra $U^{\prime}_{q}(\widehat{\mathfrak{sl}}_{N})$ is the unital
algebra over $\mathbb{C}$ generated by $e_{i},f_{i},k_{i}^{\pm 1}$,
$(i=0,\dots,N-1)$ and relations

$\displaystyle k_{i}k_{j}$ $\displaystyle=k_{j}k_{i},\quad
k_{i}k_{i}^{-1}=k_{i}^{-1}k_{i}=1$ $\displaystyle k_{i}e_{j}k_{i}^{-1}$
$\displaystyle=q^{a_{ij}}e_{j},\quad k_{i}f_{j}k_{i}^{-1}=q^{-a_{ij}}f_{j},$
$\displaystyle e_{i}f_{j}-f_{j}e_{i}$
$\displaystyle=\delta_{ij}\frac{k_{i}-k_{i}^{-1}}{q-q^{-1}}$
$\displaystyle\sum_{r=0}^{1-a_{ij}}$
$\displaystyle\genfrac{[}{]}{0.0pt}{}{1-a_{ij}}{r}_{q}e_{i}^{r}e_{j}e_{i}^{1-a_{ij}-r}=0,\quad\text{if
$i\neq j$,}$ $\displaystyle\sum_{r=0}^{1-a_{ij}}$
$\displaystyle\genfrac{[}{]}{0.0pt}{}{1-a_{ij}}{r}_{q}f_{i}^{r}f_{j}f_{i}^{1-a_{ij}-r}=0,\quad\text{if
$i\neq j$.}$

The coproduct is defined on generators as $\Delta(e_{i})=e_{i}\otimes
1+k_{i}\otimes e_{i}$, $\Delta(f_{i})=f_{i}\otimes k_{i}^{-1}+1\otimes f_{i}$
and $\Delta(k^{\pm 1}_{i})=k^{\pm 1}_{i}\otimes k^{\pm 1}_{i}$. The element
$\prod_{i=0}^{N-1}k_{i}$ is central and group-like. For finite dimensional
representations of $U_{q}^{\prime}(\widehat{\mathfrak{sl}}_{N})$ this central
element acts by 1. The quantum loop algebra $U_{q}L{\mathfrak{sl}}_{N}$ is the
quotient of $U^{\prime}_{q}(\widehat{\mathfrak{sl}}_{N})$ by the relation
$\prod_{i=0}^{N-1}k_{i}=1$. The Hopf algebra structure descends to this
quotient. It is a deformation of the universal enveloping algebra of the Lie
algebra $L\mathfrak{sl}_{N}={\mathfrak{sl}}_{N}[t,t^{-1}]$ of loops in
${\mathfrak{sl}}_{N}$. The Hopf algebra $U_{q}L{\mathfrak{sl}}_{N}$ contains
the Hopf subalgebra $U_{q}{\mathfrak{sl}}_{N}$ generated by
$e_{i},f_{i},k_{i}$ with $1\leq i\leq N-1$.

For $k\in\mathbb{Z}_{\geq 0}$ the $k$th divided power of generators is defined
as

$e_{i}^{(k)}=\frac{e_{i}^{k}}{[k]_{q}!},\quad
f_{i}^{(k)}=\frac{f_{i}^{k}}{[k]_{q}!},$

with the convention that $e_{i}^{(0)}=f_{i}^{(0)}=1$.

Let $\mathbb{Z}^{N}=\oplus_{i\in\mathbb{Z}/N\mathbb{Z}}\mathbb{Z}\epsilon_{i}$
be the (co)weight lattice of ${\mathfrak{gl}}_{N}$ with standard inner product
$(\ |\ )$, which identifies the Cartan subalgebra of diagonal matrix with its
dual. The root lattice of vectors of zero coordinate sum is spanned by the
simple roots $\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}$, $(i=1,\dots,N-1)$. We
also set $\alpha_{0}=\epsilon_{N}-\epsilon_{1}$, the highest root. The Hopf
algebra $U_{q}L{\mathfrak{gl}}_{N}$ is obtained by adjoining generators
$t_{\lambda}$ for $\lambda\in\mathbb{Z}^{N}$ such that $t_{\alpha_{i}}=k_{i}$
($i=1,\dots,N-1$) with the relations

$t_{\lambda}t_{\mu}=t_{\lambda+\mu},\quad t_{0}=1,\quad
t_{\lambda}e_{i}t_{-\lambda}=q^{(\lambda|\alpha_{i})}e_{i},\quad
t_{\lambda}f_{i}t_{-\lambda}=q^{-(\lambda|\alpha_{i})}e_{i}.$

It is a Hopf algebra with
$\epsilon(t_{\lambda})=1,S(t_{\lambda})=t_{-\lambda},\Delta(t_{\lambda})=t_{\lambda}\otimes
t_{\lambda}$. We set $t_{i}=t_{\epsilon_{i}}$ so that
$k_{i}=t_{i}t_{i+1}^{-1}$ for $i\in\mathbb{Z}/N\mathbb{Z}$. The Hopf
subalgebra $U_{q}{\mathfrak{gl}}_{N}$ is generated by $e_{i},f_{i}$ with
$1\leq i\leq N-1$ and all $t_{i}^{\pm 1}$.

The multiplicative group $\mathbb{C}^{\times}$ acts on
$U_{q}L{\mathfrak{gl}}_{N}$ by algebra automorphisms, deforming the rescaling
action of the loop parameter: the automorphism $\varphi_{z}$ associated with
$z\in\mathbb{C}^{\times}$ acts as the identity on all generators except
$e_{0}$, $f_{0}$, for which

$\varphi_{z}(e_{0})=ze_{0},\quad\varphi_{z}(f_{0})=z^{-1}f_{0}.$

The weight space of a $U_{q}{\mathfrak{gl}}_{N}$-module of weight
$\mu\in\mathbb{Z}^{N}$ is the simultaneous eigenspace of $t_{\lambda}$ with
eigenvalue $q^{(\mu|\lambda)}$. A $U_{q}{\mathfrak{gl}}_{N}$-module is of type
I if it is the direct sum of its finite dimensional weight spaces. All modules
considered in this paper are of type I.

###  2.3. Exterior powers of the vector representation

Let $V=\mathbb{C}^{N}$ with basis $v_{1},\dots,v_{N}$ and $C_{N}$ be the
Clifford algebra of $V\oplus V^{*}$ with quadratic form
$Q(v\oplus\alpha)=\alpha(v)$. It is the unital algebra generated by the images
of the standard basis and dual basis vectors $\psi^{*}_{i},\psi_{i}$, labeled
by $i\in\\{1,\dots,N\\}$, which we often identify with
$\mathbb{Z}/N\mathbb{Z}$, subject to the relations

$\psi_{i}\psi_{j}=-\psi_{j}\psi_{i},\quad\psi_{i}^{*}\psi_{j}^{*}=-\psi_{j}^{*}\psi_{i}^{*},\quad\psi_{i}\psi_{j}^{*}+\psi_{j}^{*}\psi_{i}=\delta_{ij}1.$
(1)

The Fock representation of $C_{N}$ is $\bigwedge V$ with $\psi^{*}_{i}$ acting
as $v\mapsto v_{i}\wedge v$ and $\psi_{i}$ by contraction. It is irreducible
and generated by $|0\rangle=1$ such that $\psi_{i}|0\rangle=0$ for all $i$.
The commuting “fermion number” operators $n_{i}=\psi_{i}^{*}\psi_{i}$ have a
common eigenbasis $v_{i_{1}}\wedge\cdots\wedge v_{i_{k}}$
($i_{1}<\dots<i_{k}$) and eigenvalue $1$ if $i=i_{j}$ for some $j$ and $0$
otherwise. Thus $t^{\pm 1}_{i}:=q^{\pm n_{i}}=q^{\pm
1}\psi_{i}^{*}\psi_{i}+\psi_{i}\psi_{i}^{*}$ obeys the relations

$t_{i}t_{i}^{-1}=1,\quad t_{i}t_{j}=t_{j}t_{i},\quad
t_{i}\psi_{j}^{*}t_{i}^{-1}=q^{\delta_{ij}}\psi_{j}^{*},\quad
t_{i}\psi_{j}t_{i}^{-1}=q^{-\delta_{ij}}\psi_{j}.$ (2)

for all $i,j$. Let $C_{N}(q)$ be the algebra with generators
$t_{i},\psi_{i},\psi_{i}^{*}$, ($i=1,\dots,N$) and relations (1), (2). It is a
$\mathbb{Z}/2\mathbb{Z}$-graded algebra with $\psi,\psi_{i}^{*}$ odd and
$t_{i}$ even, and $\bigwedge V$ is a $\mathbb{Z}/2\mathbb{Z}$-graded module,
with the convention that $|0\rangle$ is even.

######  Proposition 2.1 (Hayashi [8]).

There is a homomorphism $U_{q}L\mathfrak{gl}_{N}\to C_{N}(q)$ such that

$e_{i}\mapsto\psi_{i}^{*}\psi_{i+1},\quad
f_{i}\mapsto\psi_{i+1}^{*}\psi_{i},\quad t_{i}^{\pm 1}\mapsto t_{i}^{\pm 1},$

for $i\in\mathbb{Z}/N\mathbb{Z}$.

Thus we have a representation of $U_{q}L{\mathfrak{gl}}_{N}$ on $\bigwedge V$.
It is the direct sum of the irreducible eigenspaces $\bigwedge^{k}V$ of the
central element $\prod_{i=1}^{N}t_{i}$. These irreducible representations are
generated by the highest weight vectors

$v^{k}=\psi_{1}^{*}\cdots\psi_{k}^{*}|0\rangle$ (3)

of weight $\sum_{i=1}^{k}\epsilon_{i}$. In fact they are obtained from the
representation of the classical Lie algebra $L{\mathfrak{gl}}_{N}$ on the same
representation space and with the same action of the generators $e_{i},f_{i}$
by setting $t_{\lambda}=q^{\lambda}$ for Cartan elements $\lambda$. The point
is that, since the eigenvalues of $\epsilon_{i}$ are $1$ or $0$, the right-
hand side $(q^{h_{i}}-q^{-h_{i}})/(q-q^{-1})$ of the commutation relation of
$e_{i},f_{i}$ coincides with the action of its classical counterpart
$h_{i}=\epsilon_{i}-\epsilon_{i+1}$.

######  Definition 2.2.

The $j$-th exterior power of the vector representation of
$U_{q}L{\mathfrak{gl}}_{N}$ with spectral parameter $z$ is the twist
$\bigwedge^{j}V(z)$ of $\bigwedge^{j}V$ by the automorphism $\varphi_{z}$. The
vector representation is $V(z)=\bigwedge^{1}V(z)$.

Thus the action of the generators is given by the formulae of Proposition 2.1
except that $e_{0},f_{0}$ are mapped to $z\psi_{N}^{*}\psi_{1}$ and
$z^{-1}\psi_{1}^{*}\psi_{N}$, respectively.

######  Remark 2.3.

Our definition of the exterior power representation has a slightly non-
standard choice of signs. A more common convention is to let $e_{0}$ and
$f_{0}$ act as $(-1)^{j-1}z\psi_{N}^{*}\psi_{1}$,
$(-1)^{j-1}z^{-1}\psi_{1}^{*}\psi_{N}$, respectively, which in the limit $q\to
1$ correspond to the action of $zE_{N,1},z^{-1}E_{1,N}$, in terms of the
standard basis $(E_{i,j})$ of ${\mathfrak{gl}}_{N}$, so that $z$ has the
interpretation of an “evaluation point.” Our convention is better suited to
the fermionic representation and somewhat reduces the number of signs in the
formulas. The relation of our definition of the exterior power
$\bigwedge^{j}V(z)$ to the definition $V^{(j)}(z)$ in [5] is
$\bigwedge^{j}V(z)=V^{(j)}((-1)^{j-1}z)$,

Another interpretation of the exterior powers, useful to compute $R$-matrices
by the fusion procedure, is in terms of $q$-deformed exterior products:

######  Proposition 2.4 (Jimbo [10]).

Let $k\geq 2$ and $S$ be the subspace of $V^{\otimes k}$ spanned by
$\alpha\otimes v_{i}\otimes v_{i}\otimes\beta$, $\alpha\otimes(v_{i}\otimes
v_{j}+qv_{j}\otimes v_{i})\otimes\beta$ for $i>j$, $\alpha\in V^{\otimes
r},\beta\in V^{\otimes k-r-2}$, $r=0,\dots,k-2$. Then $S$ is a
$U_{q}L{\mathfrak{gl}}_{N}$ submodule of

$V^{k}(z):=V(q^{-k+1}z)\otimes\cdots\otimes V(q^{k-3}z)\otimes V(q^{k-1}z)$

and we have the isomorphism of $U_{q}L{\mathfrak{gl}}_{N}$-modules
$V^{k}(z)/S\to\bigwedge^{k}V(z)$ such that $v_{i_{1}}\otimes\cdots\otimes
v_{i_{k}}+S\mapsto v_{i_{1}}\wedge\cdots\wedge v_{i_{k}}$ for $1\leq
i_{1}<\cdots<i_{k}\leq N$.

For generic $z_{1},z_{2}$ the tensor products module
$\bigwedge^{k}V(z_{1})\otimes\bigwedge^{k^{\prime}}V(z_{2})$ with action of
$U_{q}{\mathfrak{gl}}_{N}$ given by the coproduct are irreducible and do not
depend on the order of factors up to isomorphism. Thus there is an isomorphism
of $U_{q}\widehat{{\mathfrak{gl}}}_{N}$-modules

$\check{R}_{k,k^{\prime}}(z_{1}/z_{2})\colon\textstyle{\bigwedge^{k}V(z_{1})\otimes\bigwedge^{k^{\prime}}V(z_{2})\to\bigwedge^{k^{\prime}}V(z_{2})\otimes\bigwedge^{k}V(z_{1})},$

which, viewed as a linear map between the underlying vector spaces, depends on
the ratio of spectral parameters. We find it convenient to normalize it in
such a way that $v^{k}\otimes v^{k^{\prime}}$ is mapped to
$(-1)^{kk^{\prime}}v^{k^{\prime}}\otimes v^{k}$. Then the $R$-matrix

$R_{k,k^{\prime}}(z_{1}/z_{2})=P\check{R}_{k,k^{\prime}}(z_{1}/z_{2})\in\operatorname{End}_{\mathbb{C}}(\textstyle{\bigwedge^{k}V(z_{1})\otimes\bigwedge^{k^{\prime}}V(z_{2})}),$

obtained by the composition with the graded flip $P\colon v\otimes
w\to(-1)^{kk^{\prime}}w\otimes v$, restricts to the identity on the product of
highest weight spaces.

Then by the irreducibility of tensor products at generic values of the
spectral parameter we have the braiding relation

$\displaystyle(\check{R}_{k_{2}k_{3}}(z_{2}/z_{3})$
$\displaystyle\otimes\mathrm{id})(\mathrm{id}\otimes\check{R}_{k_{1}k_{3}}(z_{1}/z_{3}))(\check{R}_{k_{1}k_{2}}(z_{1}/z_{2})\otimes\mathrm{id})$
$\displaystyle=(\mathrm{id}\otimes\check{R}_{k_{1}k_{2}}(z_{1}/z_{2}))(\check{R}_{k_{1}k_{3}}(z_{1}/z_{3})\otimes\mathrm{id})(\mathrm{id}\otimes\check{R}_{k_{2}k_{3}}(z_{2}/z_{3})),$
(4)

on
$\bigwedge^{k_{1}}V(z_{1})\otimes\bigwedge^{k_{2}}V(z_{2})\otimes\bigwedge^{k_{3}}V(z_{3})$,
equivalent to the Yang–Baxter equation for the $R$-matrices. Moreover we have
the inversion (or unitarity) relation

$\check{R}_{k^{\prime}k}(z^{-1})\check{R}_{kk^{\prime}}(z)=\mathrm{id}.$ (5)

These $R$-matrices were computed by Date and Okado [5] in terms of projections
onto irreducible submodules for the action of $U_{q}{\mathfrak{gl}}_{N}$. The
coefficients were computed using the fusion procedure [11]. Here we give an
alternative fermionic formula.

Let $\bigwedge V\otimes\bigwedge V$ be the tensor product of
$\mathbb{Z}/2\mathbb{Z}$-graded vector spaces. The action of tensor products
of linear maps is given by the sign rule: $(f\otimes g)(v\otimes
w)=(-1)^{|g||v|}fv\otimes gw$ for $g$ of degree $|g|$ and $v$ of degree $|v|$.
We introduce the following even endomorphisms of $\bigwedge V\otimes\bigwedge
V$:

$E=\sum_{i=1}^{N}(\psi^{*}_{i}\otimes{\psi}_{i})\prod_{j>i}{K}_{j}\qquad
F=-\sum_{i=1}^{N}\prod_{j<i}{K}^{{-1}}_{j}({\psi}_{i}\otimes\psi^{*}_{i})\qquad{K}=\prod_{i=1}^{N}{{K}}_{i},$
(6)

with $K_{i}=t_{i}\otimes t_{i}^{-1}$ ($i=1,\dots,N$).

######  Lemma 2.5.

The endomorphisms $E,F,K$ commute with the action of
$U_{q}{\mathfrak{gl}}_{N}$ and obey the commutation relations of
$U_{q}\mathfrak{sl}_{2}$:

$[E,F]=\frac{K-K^{-1}}{q-q^{-1}},\quad KEK^{-1}=q^{2}E,\quad
KFK^{-1}=q^{-2}F.$

This is an instance of Howe duality and is discussed in the next section. It
is convenient to extend this action to an action of $U_{q}{\mathfrak{gl}}_{2}$
by setting $T_{1}=\prod_{i=1}^{N}t_{i}\otimes 1$,
$T_{2}=\prod_{i=1}^{N}1\otimes t_{i}$ so that
$\bigwedge^{k}V\otimes\bigwedge^{k^{\prime}}V$ is the weight space of weight
$(k,k^{\prime})$.

######  Theorem 2.6.

The braiding matrix

$\check{R}_{k,k^{\prime}}(z_{1}/z_{2})\colon\textstyle{\bigwedge^{k}V(z_{1})\otimes\bigwedge^{k^{\prime}}V(z_{2})\to\bigwedge^{k^{\prime}}V(z_{1})\otimes\bigwedge^{k}V(z_{1})},$

normalized so that $\check{R}_{k,k^{\prime}}(z_{1}/z_{2})\,v^{k}\otimes
v^{k^{\prime}}=(-1)^{kk^{\prime}}v^{k^{\prime}}\otimes v^{k}$, is given by

$\check{R}_{k,k^{\prime}}(z_{1}/z_{2})=(-1)^{\min(k,k^{\prime})}A_{k-k^{\prime}}(z_{1}/z_{2})$

where $A_{k-k^{\prime}}(z_{1}/z_{2})\in U_{q}\mathfrak{gl}_{2}$ is defined by

$A_{m}(z)=\sum_{j=0}^{n}(-q)^{j}\frac{1-q^{|m|}z}{1-q^{2j+|m|}z}\begin{cases}E^{(j)}F^{(j+m)},&\text{if
$m\geq 0$,}\\\ E^{(j-m)}F^{(j)},&\text{if $m\leq 0$,}\end{cases}$

with the divided power notation $E^{(l)}=E^{l}/[l]_{q}!$,
$F^{(l)}=F^{l}/[l]_{q}!$ and

$n=\min(k,k^{\prime},N-k,N-k^{\prime}).$

The proof relies on Howe duality, which we proceed to explain, and the
computation [5] of the $R$-matrices by the fusion procedure.

##  3\. Quantum skew Howe duality

The classical skew-symmetric version of Howe duality [9, Section 4] states
that the natural actions of $\mathit{GL}_{N}$ and $\mathit{GL}_{M}$ on
$\bigwedge(\mathbb{C}^{N}\otimes\mathbb{C}^{M})$ generate each other’s
commutant and one has a decomposition into irreducibles of
$\mathit{GL}_{N}\times\mathit{GL}_{M}$

$\bigwedge(\mathbb{C}^{N}\otimes\mathbb{C}^{M})\cong\bigoplus_{\lambda}V^{N}_{\lambda}\otimes
V^{M}_{\lambda^{t}}.$

The sum is over all Young diagrams $\lambda$ with at most $N$ rows such that
the transposed diagrams $\lambda^{t}$ has at most $M$ rows (equivalently, the
length of the first row of $\lambda$ is at most $M$) and $V^{N}_{\lambda}$
denotes the irreducible representation of $\mathit{GL}_{N}$ of highest weight
$\lambda$. The following nice pictorial description of the corresponding
highest weight vectors is due to Howe in loc. cit.: write the tensor products
of basis vectors $v_{i,r}=v_{i}\otimes w_{r}$ $(i=1,\dots,N,r=1,\dots,M)$ in
the boxes of an $N\times M$ grid (the tensors $v_{1,r}$ go in the first row,
$v_{2,r}$ in the second row, etc.). The Young diagrams occuring in the
decomposition fit in this rectangle. The highest weight vector corresponding
to $\lambda$ is obtained by taking the wedge product of the basis vectors in
the boxes of the Young diagram $\lambda$. For example, if $\lambda=(3,1)$, we
take $v_{1,1}\wedge v_{1,2}\wedge v_{1,2}\wedge v_{2,1}$. It is indeed of
weight $\lambda$ for $\mathit{GL}_{N}$ and of weight $\lambda^{t}$ for
$\mathit{GL}_{M}$.

The quantum group version of Howe duality is known: in the analogous case of
the symmetric algebra it is discussed in [1], [18] and [19]. The skew-
symmetric case we use here is explained in [2] (for $M=2$), [12] and [3]. Here
we present an equivalent variant based on the Clifford algebra action.

Let $V=\mathbb{C}^{N}$ with basis $v_{1},\dots,v_{N}$ be the vector
representation of $U_{q}\mathfrak{gl}_{N}$ and $W=\mathbb{C}^{M}$ with basis
$w_{1},\dots,w_{M}$ be the vector representation of $U_{q}\mathfrak{gl}_{M}$.
Then we have an isomorphism of graded commutative algebras

$\bigwedge(V\otimes W)\cong\bigwedge(V\oplus\cdots\oplus V)\to\left(\bigwedge
V\right)^{\otimes M}=\bigwedge V\otimes\cdots\otimes\bigwedge V$

sending $v_{i}\otimes w_{j}$ to $1\otimes\cdots\otimes 1\otimes v_{i}\otimes
1\otimes\cdots\otimes 1$ with $v_{i}$ placed in the $j$-th factor. The right-
hand side is a tensor product of $U_{q}\mathfrak{gl}_{N}$-modules and thus
$U_{q}\mathfrak{gl}_{N}$ acts on it by the iterated coproduct. Similarly we
have an isomorphism $\bigwedge(V\otimes W)\cong\bigwedge
W\otimes\cdots\otimes\bigwedge W$ sending $v_{i}\otimes w_{j}$ to
$1\otimes\cdots\otimes 1\otimes w_{j}\otimes 1\otimes\cdots\otimes 1$ with
$i$-th factor $w_{j}$. Here we let $U_{q}\mathfrak{gl}_{M}$ act via the
opposite coproduct $\Delta^{\prime}=\sigma\circ\Delta$, with $\sigma(a\otimes
b)=b\otimes a$.

######  Proposition 3.1.

The actions of $(U_{q}\mathfrak{gl}_{N},\Delta)$ and
$(U_{q}\mathfrak{gl}_{M},\Delta^{\prime})$ on $\bigwedge(V\otimes W)$ pulled
back by the isomorphisms

$\left(\bigwedge W\right)^{\otimes N}\leftarrow\bigwedge(V\otimes
W)\rightarrow\left(\bigwedge V\right)^{\otimes M}$

commute. If $q$ is not a root of unity the images of $U_{q}\mathfrak{gl}_{N}$
and $U_{q}\mathfrak{gl}_{M}$ in $\operatorname{End}\bigwedge(V\otimes W)$ are
commutants of each other and we have a decomposition

$\bigwedge(V\otimes W)\cong\bigoplus_{\lambda}V^{N}_{\lambda}\otimes
V^{M}_{\lambda^{t}}$ (7)

into simple $U_{q}\mathfrak{gl}_{N}\otimes U_{q}\mathfrak{gl}_{M}$-modules,
where $\lambda$ runs over Young diagrams with at most $N$ rows and at most $M$
columns.

For the proof we view $\bigwedge(V\otimes W)$ as the Fock module of the
Clifford algebra of $V\otimes W\oplus(V\otimes W)^{*}$. It has generators
$\psi^{*}_{i,r},\psi_{i,r}$ corresponding to the tensor basis $v_{i}\otimes
w_{r}$ and its dual basis. As in Section 2.3 we introduce the operators
$t_{i,r}^{\pm 1}=q^{\pm 1}\psi_{i,r}^{*}\psi_{i,r}+\psi_{i,r}\psi_{i,r}^{*}$.
Via the isomorphism to $(\bigwedge V)^{\otimes M}$, $\psi_{i,r}^{*}$ acts as
$\mathrm{id}^{\otimes(r-1)}\otimes\psi_{i}^{*}\otimes\mathrm{id}^{\otimes(N-r)}$
and $\psi_{i,r}$ acts as
$\mathrm{id}^{\otimes(r-1)}\otimes\psi_{i}\otimes\mathrm{id}^{\otimes(N-r)}$.
Thus the pull-back of the action of $U_{q}\mathfrak{gl}_{N}$ on
$\bigwedge(V\otimes W)$ is given on generators by

$\displaystyle e_{i}$
$\displaystyle\mapsto\sum_{r=1}^{M}\prod_{s=1}^{r-1}k_{i,s}\psi^{*}_{i,r}\psi_{i+1,r},\quad
k_{i,s}=t_{i,s}t_{i+1,s}^{-1}$ $\displaystyle f_{i}$
$\displaystyle\mapsto\sum_{r=1}^{M}\psi^{*}_{i+1,r}\psi_{i,r}\prod_{s={r+1}}^{M}k_{i,s}^{-1}$
$\displaystyle t_{i}^{\pm 1}$ $\displaystyle\mapsto\prod_{r=1}^{M}t_{i,r}^{\pm
1}.$

Similary we can pull-back the action of $U_{q}\mathfrak{gl}_{M}$ on
$(\bigwedge W)^{\otimes N}$ with the opposite coproduct. The action of
generators (denoted by capital letters to distinguish them)

$\displaystyle E_{r}$
$\displaystyle\mapsto\sum_{i=1}^{N}\psi^{*}_{i,r}\psi_{i,r+1}\prod_{j=i+1}^{N}K_{j,r},\quad
K_{j,r}=t_{j,r}t_{j,r+1}^{-1},$ $\displaystyle F_{r}$
$\displaystyle\mapsto\sum_{i=1}^{N}\prod_{j=1}^{i-1}K_{j,r}^{-1}\psi^{*}_{i,r+1}\psi_{i,r},$
$\displaystyle T_{r}^{\pm 1}$ $\displaystyle\mapsto\prod_{i=1}^{N}t_{i,r}^{\pm
1}.$

In the case $M=2$ this reduces to the formulae of the previous section, by
remembering the sign rule in the definition of $F$. By construction, we obtain
actions of $U_{q}{\mathfrak{gl}}_{N}$ and $U_{q}{\mathfrak{gl}}_{M}$ and we
need to check that they commute:

######  Lemma 3.2.

The actions of $U_{q}{\mathfrak{gl}}_{N}$ and $U_{q}{\mathfrak{gl}}_{M}$ on
$\bigwedge(V\otimes W)$ commute.

###### Proof.

It is clear that $t_{i}^{\pm 1}$ commutes with the action of
$U_{q}{\mathfrak{gl}}_{M}$ and that $T_{r}^{\pm 1}$ commutes with the action
of $U_{q}{\mathfrak{gl}}_{N}$. Let us check that $[e_{i},E_{r}]=0$. The cases
of the other generators are dealt with in the same way. The only terms in the
sum defining $e_{i}$ and $E_{r}$ that contribute non-trivially to the
commutator are

$\displaystyle[e_{i},E_{r}]$
$\displaystyle=\prod_{s<r}k_{i,s}[\psi_{i,r}^{*}\psi_{i+1,r}+k_{i,r}\psi_{i,r+1}^{*}\psi_{i+1,r+1},$
$\displaystyle\quad\psi_{i,r}^{*}\psi_{i,r+1}K_{i+1,r}+\psi_{i+1,r}^{*}\psi_{i+1,r+1}]\prod_{j>i+1}K_{j,r}.$

The bracket on the right-hand side is

$\displaystyle[\psi_{i,r}^{*}\psi_{i+1,r},\psi_{i+1,r}^{*}\psi_{i+1,r+1}]+[k_{i,r}\psi_{i,r+1}^{*}\psi_{i+1,r+1},\psi_{i,r}^{*}\psi_{i,r+1}K_{i+1,r}]$
$\displaystyle=\psi_{i,r}^{*}\psi_{i+1,r+1}+k_{i,r}\psi_{i+1,r+1}\psi_{i,r}^{*}K_{i+1,r}$
$\displaystyle=\psi_{i,r}^{*}\psi_{i+1,r+1}+t_{i,r}\psi_{i+1,r+1}\psi_{i,r}^{*}t_{i+1,r+1}^{-1}$

We have the relations $t_{i,r}\psi_{i,r}^{*}=q\psi_{i,r}^{*}$ and
$\psi_{i+1,r+1}t_{i+1,r+1}^{-1}=q^{-1}\psi_{i+1,r+1}$ and we are left with
$\psi_{i,r}^{*}\psi_{i+1,r+1}+\psi_{i+1,r+1}\psi_{i,r}^{*}=0$. ∎

To complete the proof of Proposition 3.1, we observe that for $q$ not a root
of unity the decomposition into irreducible representations in the classical
case deforms to the quantum group case. In fact the highest weight vectors
generating irreducible $U_{q}{\mathfrak{gl}}_{N}\otimes
U_{q}{\mathfrak{gl}}_{M}$-modules are still given by Howe’s construction.

###### Proof of Theorem 2.6.

We use the formula of Date and Okado [5] for the action of the $R$-matrix on
highest weight vectors of irreducible
$U_{q}{\mathfrak{gl}}_{N}$-subrepresentations in
$\bigwedge^{k}V\otimes\bigwedge^{k^{\prime}}V$. The decomposition into
irreducibles is best understood in terms of Howe duality: we consider the case
$M=2$ of the decomposition (7) of Proposition 3.1. The left-hand side is

$\textstyle{\bigwedge(V\otimes\mathbb{C}^{2})=\bigwedge(V\oplus V)=\bigwedge
V\otimes\bigwedge
V=\oplus_{k,k^{\prime}=0}^{N}\bigwedge^{k}V\otimes\bigwedge^{k^{\prime}}V}$

and the summand labeled by $(k,k^{\prime})\in\\{0,\dots,N\\}^{2}$ is the
weight space of weight $(k,k^{\prime})$ for the
$U_{q}{\mathfrak{gl}}_{2}$-action. The right-hand side is a sum over Young
diagram $\lambda$ with at most two columns and at most $N$ rows. Thus
$\lambda=(2,\dots,2,1,\dots,1)=2^{\ell^{\prime}}1^{\ell-\ell^{\prime}}$
($\ell^{\prime}$ rows of length 2, $\ell-\ell^{\prime}$ rows of length 1) and
$\lambda^{t}=(\ell,\ell^{\prime})$ with $N\geq\ell\geq\ell^{\prime}\geq 0$.
The irreducible $U_{q}{\mathfrak{gl}}_{2}$-module
$V^{2}_{(\ell,\ell^{\prime})}$ has a non-trivial weight space of weight
$(k,k^{\prime})$ if and only if $k+k^{\prime}=\ell+\ell^{\prime}$ for some
$s\geq 0$ and $\ell^{\prime}-\ell\leq k-k^{\prime}\leq\ell-\ell^{\prime}$. In
this case the weight space is one-dimensional. If $k\geq k^{\prime}$ the
corresponding highest weights are

$(\ell,\ell^{\prime})=(k,k^{\prime}),(k+1,k^{\prime}-1),\dots,(k+n,k^{\prime}-n),$

where $n=\min(k^{\prime},N-k)$, whereas if $k\leq k^{\prime}$ they are

$(\ell,\ell^{\prime})=(k^{\prime},k),(k^{\prime}+1,k-1),\ldots,(k^{\prime}+n,k-n),$

where $n=\min(k,N-k^{\prime})$. In both cases we can write

$n=\min(k,k^{\prime},N-k,N-k^{\prime}).$

With this notation, the weight $(k,k^{\prime})$ subspace in the decomposition
(7) gives the decomposition

${\textstyle\bigwedge^{k}V\otimes\bigwedge^{k^{\prime}}V}\cong\oplus_{s=0}^{n}V^{N}_{(\overline{k}+s,\underline{k}-s)^{t}},\quad\text{with
$\overline{k}=\max(k,k^{\prime})$ and $\underline{k}=\min(k,k^{\prime})$},$

as an $U_{q}{\mathfrak{gl}}_{N}$-module. The $s=0$ component is generated by
the highest weight vector
$v^{(\overline{k},\underline{k})}_{0}=v^{\overline{k}}\otimes
v^{\underline{k}}$, see (3), and the highest weight vector
$v^{(k,k^{\prime})}_{s}$ of $V^{N}_{(\overline{k}+s,\underline{k}-s)^{t}}$ is
proportional to $F^{(s)}v_{0}^{(k+s,k^{\prime}-s)}$ if $k\geq k^{\prime}$ and
to $F^{(k^{\prime}-k+s)}v_{0}^{(k^{\prime}+s,k-s)}$ if $k\leq k^{\prime}$. We
normalize it as in [5] by fixing the coefficient of the basis vector
$\psi^{*}_{k}\cdots\psi^{*}_{1}|0\rangle$ in the first factor. Namely,

$v_{s}^{(k,k^{\prime})}=\psi_{k}^{*}\cdots\psi_{1}^{*}|0\rangle\otimes\psi^{*}_{\overline{k}+s}\cdots\psi^{*}_{\overline{k}+1}\psi^{*}_{\underline{k}-s}\cdots\psi_{1}^{*}|0\rangle+\cdots.$

Let

$\textstyle
Q_{s}\colon\bigwedge^{k}V\otimes\bigwedge^{k^{\prime}}V\to\bigwedge^{k^{\prime}}V\otimes\bigwedge^{k}V$

be the unique $U_{q}{\mathfrak{gl}}_{N}$-linear map sending
$v^{(k,k^{\prime})}_{s}$ to the vector of the same weight
$v^{(k^{\prime},k)}_{s^{\prime}}$, $s^{\prime}=k-k^{\prime}+s$ and vanishing
on the other irreducible components. In our notation and with the
normalization of Theorem 2.6, the formula of [5], obtained by the fusion
procedure, is

$\check{R}_{k,k^{\prime}}(z)=\sum_{s=0}^{n}(-1)^{kk^{\prime}+(k-k^{\prime})s}\prod_{j=1}^{s}\frac{z-q^{|k-k^{\prime}|+2j}}{1-zq^{|k-k^{\prime}|+2j}}Q_{s}.$

To deduce a formula in terms of the $U_{q}{\mathfrak{gl}}_{2}$-action we need
to compare the normalizations of the highest weight vectors. If $k\geq
k^{\prime}$ a straightforward computation using the formula for $F$ in (6)
shows that

$v^{(k,k^{\prime})}_{s}=(-1)^{sk}q^{ss^{\prime}}F^{(s)}v^{(k+s,k^{\prime}-s)}_{0},\quad
v^{(k^{\prime},k)}_{s^{\prime}}=(-1)^{s^{\prime}k^{\prime}}q^{ss^{\prime}}F^{(s^{\prime})}v^{(k+s,k^{\prime}-s)}_{0}.$

It follows that for $k\geq k^{\prime}$

$\check{R}_{k,k^{\prime}}(z)F^{(s)}v_{0}^{(k+s,k^{\prime}-s)}=(-1)^{k^{\prime}}\prod_{j=1}^{s}\frac{z-q^{|k-k^{\prime}|+2j}}{1-zq^{|k-k^{\prime}|+2j}}F^{(s^{\prime})}v_{0}^{(k+s,k^{\prime}-s)}$

To prove the formula for $\check{R}$ given in the Theorem we need to compute
the action of $A_{k-k^{\prime}}(z)$ on the irreducible representation
generated by the highest weight vector $v^{(k+s,k^{\prime}-s)}_{0}$ of
$\mathfrak{sl}_{2}$-weight $\ell=k-k^{\prime}+2s\in\mathbb{Z}_{\geq 0}$.
Comparing with the formula for $A_{m}(z)$ from the Appendix (see Lemma A.1) we
see that $\check{R}_{k,k^{\prime}}(z)$ acts by the same factor on
$F^{(s)}v_{0}^{(k+s,k^{\prime}-s)}$ up to the sign
$(-1)^{k^{\prime}}=(-1)^{\min(k,k^{\prime})}$.

If $k\leq k^{\prime}$ the reasoning is similar: in this case the highest
weight vector $v^{(k,k^{\prime})}_{s}$ is proportional to
$F^{(s^{\prime})}v_{0}^{(k^{\prime}+s,k-s)}$ with $s^{\prime}=k^{\prime}-k+s$
and

$v^{(k,k^{\prime})}_{s}=(-1)^{sk^{\prime}}q^{ss^{\prime}}F^{(s^{\prime})}v^{(k^{\prime}+s,k-s)}_{0},\quad
v^{(k^{\prime},k)}_{s^{\prime}}=(-1)^{s^{\prime}k}q^{ss^{\prime}}F^{(s)}v^{(k^{\prime}+s,k-s)}_{0}.$

From this we obtain

$\check{R}_{k,k^{\prime}}(z)F^{(s^{\prime})}v_{0}^{(k^{\prime}+s,k-s)}=(-1)^{k}\prod_{j=1}^{s}\frac{z-q^{|k-k^{\prime}|+2j}}{1-zq^{|k-k^{\prime}|+2j}}F^{(s)}v_{0}^{(k^{\prime}+s,k-s)},$

which is the same as the action of $A_{k-k^{\prime}}(z)$ up to the sign
$(-1)^{k}=(-1)^{\min(k,k^{\prime})}$. ∎

##  4\. Dynamical Weyl group

In Theorem 2.6 the braiding matrix for the tensor product of two
representations of $U_{q}L\mathfrak{gl}_{N}$ obtained from exterior powers of
the vector representation of $U_{q}\mathfrak{gl}_{N}$ are expressed in terms
of the action of $U_{q}\mathfrak{gl}_{2}$ via Howe duality. In this section we
consider braiding matrices on the tensor product of $M$ exterior powers and
notice that they are correspondingly expressed in terms of the action of
$U_{q}\mathfrak{gl}_{M}$ via the simple root embeddings of
$U_{q}{\mathfrak{gl}}_{2}$. The braid and inversion relations of the
$\check{R}$-matrices acting on a $M$-fold tensor product is then translated by
Howe duality to braid and inversion relations in $U_{q}{\mathfrak{gl}}_{M}$.
The result is then that these are relations of the dynamical Weyl group.

Recall that the (ordinary) Weyl group of $\mathit{GL}_{M}$ is the symmetric
group $S_{M}$. It acts on $\mathbb{C}^{M}$ and the weight lattice
$\mathbb{Z}^{M}$ by permutations. It is generated by the transpositions
$s_{i}\in S_{M}$ of $i$ and $i+1$ ($i=1,\dots,M-1$) with the relations
$s_{i}^{2}=1$, $s_{i}s_{j}=s_{j}s_{i}$ for $|i-j|\geq 2$ and
$s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}$ for $1\leq i\leq M-2$; the dynamical
Weyl group relations are a version with parameters.

Let us define formal series $A_{i,\mu}(z)$ of rational functions of
$M$-variables $z=(z_{1},\dots,z_{M})$ with values in
$U_{q}{\mathfrak{sl}}_{M}\subset U_{q}{\mathfrak{gl}}_{M}$ for $1\leq i\leq
M-1$ and $\mu\in P_{M}=\mathbb{Z}^{M}$:

$A_{i,\mu}(z)=\sum_{j=0}^{\infty}(-q)^{j}\frac{z_{i+1}-q^{|\mu_{i}-\mu_{i+1}|}z_{i}}{z_{i+1}-q^{2j+|\mu_{i}-\mu_{i+1}|}z_{i}}E_{i}^{(j+(\mu_{i+1}-\mu_{i})_{+})}F_{i}^{(j+(\mu_{i}-\mu_{i+1})_{+})}.$

Here we set $(x)_{+}=\max(x,0)$ and, as before, we denote the generators by
capital letters to distinguish them from the generators of
$U_{q}{\mathfrak{gl}}_{N}$. For any finite dimensional
$U_{q}{\mathfrak{gl}}_{N}$-module $U$ of type I only finitely many terms
contribute non-trivially to the sum and $A_{i,\mu}(z)$ maps the weight space
$U[\mu]$ of weight $\mu$ to $U[s_{i}\mu]$. Notice that $A_{i,\mu}(z)$ depends
on $\mu$ through its class in the weight lattice
$P_{M}=\mathbb{Z}^{M}/\mathbb{Z}(1,\dots,1)$ of $\mathfrak{sl}_{M}$. For
$M=2$, the map $\mathbb{Z}^{2}\to\mathbb{Z}$, $\mu\mapsto\mu_{1}-\mu_{2}$
induces a bijection $P_{2}\cong\mathbb{Z}$. With this identification,
$A_{1,m}(z_{1},z_{2})=A_{m}(z_{1}/z_{2})$ where

$A_{m}(z)=\sum_{j\geq
0}(-q)^{j}\frac{1-q^{|m|}z}{1-q^{2j+|m|}z}\begin{cases}E^{(j)}F^{(j+m)},&\text{if
$m\geq 0$,}\\\ E^{(j+|m|)}F^{(j)},&\text{if $m\leq 0$.}\end{cases}$ (8)

The general series $A_{i,\mu}(z)$ reduces to this case via the embedding
$j_{i}\colon U_{q}\mathfrak{sl}_{2}\hookrightarrow U_{q}\mathfrak{sl}_{M}$
sending $E,F,K^{\pm 1}$ to $E_{i},F_{i},K_{i}^{\pm 1}$:

$A_{i,\mu}(z)=j_{i}\left(A_{\mu(h_{i})}(z_{i}/z_{i+1})\right).$

By construction, see Theorem 2.6, the action of $A_{i,\mu}(z)$ on the tensor
product of exterior powers $\bigwedge^{\mu_{i}}V(z_{i})$ via Howe duality is
the action of the braiding matrix
$\check{R}_{\mu_{i},\mu_{i+1}}(z_{i}/z_{i+1})$ on the $i$th and $i+1$st
factors (up to an irrelevant sign). Note that this tensor product is the
weight space of weight $\mu$ for the action of $U_{q}{\mathfrak{gl}}_{M}$ on
the tensor product of $M$ copies of $\bigwedge V$.

######  Theorem 4.1.

Let $U$ be any finite dimensional $U_{q}{\mathfrak{gl}}_{M}$-module of type I.
Then the restriction of $A_{i,\mu}(z)$ to the weight space $U[\mu]$ of weight
$\mu$ obeys the following relations.

  1. (i)

For $1\leq i,j\leq M-1$ such that $|i-j|\geq 2$,

$A_{i,s_{i}\mu}(s_{j}z)A_{j,\mu}(z)=A_{j,s_{j}\mu}(s_{i}z)A_{i,\mu}(z),$

in $\operatorname{Hom}(U[\mu],U[s_{i}s_{j}\mu])$.

  2. (ii)

For all $1\leq i\leq M-1$,

$A_{i,s_{i}\mu}(s_{i}z)A_{i,\mu}(z)=\mathrm{id}$

in $\operatorname{End}(U[\mu])$.

  3. (iii)

For $1\leq i\leq M-2$,

$\displaystyle A_{i,s_{i+1}s_{i}\mu}$
$\displaystyle(s_{i+1}s_{i}z)A_{i+1,s_{i}\mu}(s_{i}z)A_{i,\mu}(z)$
$\displaystyle=A_{i+1,s_{i}s_{i+1}\mu}(s_{i}s_{i+1}z)A_{i,s_{i+1}\mu}(s_{i+1}z)A_{i+1,\mu}(z),$

in $\operatorname{Hom}(U[\mu],U[s_{i}s_{i+1}s_{i}\mu])$.

###### Proof.

By complete reducibility we may assume that $U$ is irreducible. The relations
hold for the action of $A_{i,\mu}$ on
$\oplus_{i=1}^{M}\bigwedge^{\mu_{i}}V(z_{i})$ via Howe duality: (i) follows
from the fact that the corresponding $\check{R}$-matrices act on different
factors of the tensor product, (ii) is the inversion relation (5) and (iii) is
the braiding relation (2.3). Thus the relations hold for any irreducible
$U_{q}{\mathfrak{gl}}_{M}$-module occurring in the decomposition of the tensor
product. But from Proposition 3.1 we see that any partition does occurs as a
highest weight if we take $N$ sufficiently large. This covers all type I
irreducible representations of $U_{q}\mathfrak{sl}_{M}$. For
$U_{q}{\mathfrak{gl}}_{M}$ we can obtain all representations by tensoring
these with a power of the one dimensional representations on which
$U_{q}\mathfrak{sl}_{M}$ acts trivially (i.e. by the counit) and $t_{i}$ act
by $q^{-1}$ for all $i$. This has the effect of shifting the $\mu_{i}$s by a
common amount and does not affect the validity of the claim, since
$A_{i,\mu}(z)$ depends on differences of $\mu_{i}$s. ∎

This result can be reformulated as the construction of a representation of the
Weyl group on $U$-valued functions. Let $U$ be a finite-dimensional type I
$U_{q}{\mathfrak{gl}}_{M}$-module with weight decomposition
$U=\oplus_{\mu}U[\mu]$. Let $A_{i,U}(z)$ be the $\operatorname{End}U$-valued
rational function such that

$A_{i,U}(z)|_{U[\mu]}=A_{i,\mu}(z)\colon U[\mu]\to U[s_{i}\mu]$ (9)

for all weights $\mu$.

######  Corollary 4.2.

Let $U$ be a finite-dimensional $U_{q}{\mathfrak{gl}}_{M}$-module of type I.
The endomorphisms $s_{1},\dots,s_{M-1}$ of the vector space $U(z)$ of
$U$-valued rational functions in $M$ variables $z$ given by

$(s_{i}f)(z)=A_{i,U}(s_{i}z)f(s_{i}z),\quad i=1,\dots,M-1,$

define a representation of the symmetric group $S_{M}$.

###### Proof.

The proof is a straightforward verification that the relations of the
symmetric group are equivalent to the identities (i)–(iii) of Theorem 4.1. For
example, to check the relation $s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}$ we
write

$\displaystyle(s_{i}s_{i+1}s_{i}f)(z)$
$\displaystyle=A_{i,U}(s_{i}z)(s_{i+1}s_{i}f)(s_{i}z)$
$\displaystyle=A_{i,U}(s_{i}z)A_{i+1,U}(s_{i+1}s_{i}z)(s_{i}f)(s_{i+1}s_{i}z)$
$\displaystyle=A_{i,U}(s_{i}z)A_{i+1,U}(s_{i+1}s_{i}z)A_{i,U}(s_{i}s_{i+1}s_{i}z)f(s_{i}s_{i+1}s_{i}z).$

We get the left-hand side of (iii) if we replace $z$ by $s_{i}s_{i+1}s_{i}z$
and restrict to the weight space $U[\mu]$. ∎

We notice that the operators $A_{i,U}(z)$ have similar properties as the
operators of the dynamical Weyl group of $U$ introduced and studied in [17, 6]
in the framework of the theory of intertwiners. As in the case of the
dynamical Weyl group, their restriction to weight spaces are the image of
elements of $U_{q}\mathfrak{sl}_{2}$ by embeddings
$U_{q}\mathfrak{sl}_{2}\hookrightarrow U_{q}\mathfrak{sl}_{M}$ corresponding
to simple roots 111In [6] the dynamical Weyl group is defined for any
semisimple Lie algebra.. They obey relations (i) and (iii) of Theorem 4.1 but
not (ii). In other words they define a representation of the braid group
$B_{M}$ on rational $U$-valued functions. In this representation the squares
$s_{i}^{2}$ of the generators act by multiplication by a rational function on
each weight space.

Let us compare our operators $A_{i,U}(z)$ with the operators
$\mathscr{A}_{s_{i},U}(\lambda)$ of [6, Section 4.2] defining the dynamical
action of generators, in the case of $U_{q}\mathfrak{sl}_{M}$. The latter
operators are initially defined as functions of sufficiently large
antidominant weights $\lambda$. They extend (upon choosing a logarithm of $q$)
to meromorphic functions of $\lambda\in\mathfrak{h}^{*}$ in the dual of the
Cartan subalgebra $\mathfrak{h}=\\{x\in\mathbb{C}^{M}\,|\,\sum_{i}x_{i}=0\\}$
of $\mathfrak{sl}_{M}$. Let $h_{i}=(0,\dots,0,1,-1,0,\dots,0)$, with $1$ in
the position $i=1,\dots,M-1$, be the basis of coroots. To relate $z$ to
$\lambda$ we identify $\mathfrak{h}^{*}$ with
$\mathbb{C}^{M}/\mathbb{C}(1,\dots,1)$ and write
$z=q^{2\lambda}=(q^{2\lambda_{1}},\dots,q^{2\lambda_{M}})$. This is defined up
to simultaneous scaling of the variables $z_{i}$. Since $A_{i,U}(z)$ is a
function of the ratios of $z_{j}$ $A_{i,U}(q^{2\lambda})$ is well-defined.

######  Proposition 4.3.

$A_{i,U}(q^{2\lambda})|_{U[\mu]}=\begin{cases}q^{-\mu(h_{i})}\mathscr{A}_{s_{i},U}(\lambda),&\text{if
$\mu(h_{i})\geq 0$,}\\\
\displaystyle(-1)^{\mu(h_{i})}q^{-\mu(h_{i})}\frac{[\lambda(h_{i})-\mu(h_{i})/2]_{q}}{[\lambda(h_{i})+\mu(h_{i})/2]_{q}}\mathscr{A}_{s_{i},U}(\lambda),&\text{if
$\mu(h_{i})\leq 0$}.\end{cases}$

###### Proof.

Since $A_{i,\mu}$ are images of elements of $U_{q}\mathfrak{sl}_{2}$ by
embeddings into $U_{q}\mathfrak{sl}_{M}$, it is sufficient to consider the
case $M=2$ and we set $\lambda=\lambda_{1}-\lambda_{2}$ and
$z=q^{2\lambda}=z_{1}/z_{2}$. By complete reducibility we may also assume that
$U=L_{\ell}$ is the irreducible representation of $U_{q}\mathfrak{sl}_{2}$ of
highest weight $\ell$ and dimension $\ell+1$. It has one-dimensional weight
spaces $L_{\ell}[\ell-2k]$ with basis $v_{\ell-2k}=F^{(\ell)}v_{\ell}$,
($k=0,\dots,\ell$), in terms of the highest weight vector $v_{\ell}$. In this
case we have a unique generator $s_{1}$ sending $x\in\mathfrak{h}^{*}$ to
$-x$. The following formula for $\mathscr{A}_{s_{1},L_{\ell}}(\lambda)$ can be
extracted from [6], see Corollary 8 (iii), Proposition 12 and the definition
of the dynamical action of the braid group at the end of Section 4.2:

$\mathscr{A}_{s_{1},L_{\ell}}(\lambda)v_{m}=(-1)^{k}q^{m}\prod_{j=1}^{k}\frac{[\lambda+k-j+\ell/2]_{q}}{[\lambda-k+j-\ell/2]_{q}}v_{-m},\quad
m=\ell-2k.$

By setting $z=q^{2\lambda}$, this can be rewritten as

$\mathscr{A}_{s_{1},L_{\ell}}(\lambda)v_{\ell-2k}=(-1)^{k}q^{m+k(\ell-k+1)}\prod_{j=0}^{k-1}\frac{1-zq^{-\ell+2j}}{1-zq^{\ell-2j}}v_{2k-\ell}.$

On the other hand, $A_{1,L_{\lambda}}(z)$ acts on the weight $m$ subspace of
any representation as $A_{m}(z)$, see (8).

It follows from Lemma A.1 that
$A_{m}(z)=\varphi_{m}(z)\mathscr{A}_{s,L_{\ell}}(q^{2\lambda})$ with
$\varphi_{m}(z)=q^{-m}$ for $m\geq 0$. If $m\leq 0$,

$\varphi_{m}(z)=(-1)^{m}\prod_{j=\frac{\ell-|m|}{2}}^{\frac{\ell+|m|}{2}-1}\frac{1-zq^{\ell-2j}}{1-zq^{-\ell+2j}}=(-1)^{m}\frac{1-zq^{-m}}{1-zq^{m}},\quad
m\leq 0.$

so that
$\varphi_{m}(q^{2\lambda})=(-1)^{m}q^{-m}\frac{[\lambda-m/2]_{q}}{[\lambda+m/2]_{q}}$.
∎

###  4.1. $B$-operators, quantum Weyl group, Yangian limit

The limiting values of $R$-matrices as the spectral parameter tends to $0$ or
$\infty$ are solutions of the Yang–Baxter equations without spectral
parameter. They are $R$-matrices for quantum enveloping algebras of finite
dimensional Lie algebras. By dividing the $A$-operators by their limiting
values we obtain weight zero “$B$-operators” [6]. Define a formal series
$B_{m}(z)$ by the formula

$A_{m}(z)=A_{m}(0)B_{m}(z).$ (10)

The series $B_{m}(z)$ is well-defined as an endomorphism of the $m$-weight
space of any finite dimensional representation and one has a universal
formula:

######  Proposition 4.4.

  1. (i)

For $m\geq 0$,

$B_{m}(z)=\sum_{j=0}^{\infty}q^{\frac{j(j-3)}{2}}\frac{(-z)^{j}}{[j]_{q}!}F^{j}E^{j}\prod_{i=1}^{j}\frac{1-q^{2}}{1-zq^{m+2i}}$

  2. (ii)

For $m\leq 0$,

$B_{m}(z)=\sum_{j=0}^{\infty}q^{\frac{j(j-3)}{2}}\frac{(-z)^{j}}{[j]_{q}!}E^{j}F^{j}\prod_{i=1}^{j}\frac{1-q^{2}}{1-zq^{-m+2i}}$

###### Proof.

The first formula can be checked directly or deduced from the remark after
Corollary 40 in [6], by using Proposition 4.3. There is a similar formula in
loc. cit., Proposition 14, but probably the powers of $q$ need to be adjusted
there. The second formula follows by applying the Cartan involution, see
Corollary A.2 ∎

For $m=k-k^{\prime}$, these operators define via Howe duality
$U_{q}{\mathfrak{gl}}_{N}$-linear endomorphisms of
$\bigwedge^{k}V(z_{1})\otimes\bigwedge^{k^{\prime}}V(z_{2})$. Their
representation theoretical meaning in terms of $U_{q}L{\mathfrak{gl}}_{N}$ is
not clear but in the limit $q\to 1$, $A_{m}(0)$ tends up to signs to the
permutation $P$ and in a suitable $z\to 1$ limit, $B_{m}(z)$ converges to a
rational $R$ matrix, intertwining the coproduct and the opposite coproduct of
the Yangian. More precisely we let $q=\exp(\epsilon\hbar)$, $z=\exp(2\epsilon
u)$ and consider the limit $\epsilon\to 0$:

######  Theorem 4.5.

Let $\check{R}_{k,k^{\prime}}(z,q)$ be the family of braiding matrices of
Theorem 2.6, viewed as a linear map on the underlying vector space
$\bigwedge^{k}V\otimes\bigwedge^{k^{\prime}}V$. Let

$R_{k,k^{\prime}}(z,q)=P\check{R}_{k,k^{\prime}}(z,q)\in\textstyle{\operatorname{End}_{\mathbb{C}}(\bigwedge^{k}V\otimes\bigwedge^{k^{\prime}}V)},\text{
with $Pv^{\prime}\otimes v=(-1)^{kk^{\prime}}v\otimes v^{\prime}$,}$

be the corresponding $R$-matrix, normalized to be the identity on the product
of highest weight spaces. Then the limit

$R^{\mathrm{rat}}_{k,k^{\prime}}(u,\hbar)=\lim_{\epsilon\to
0}R_{k,k^{\prime}}(e^{\hbar\epsilon},e^{2\epsilon u})$

exists and is a ${\mathfrak{gl}}_{N}$-invariant solution of the Yang–Baxter
equation. Moreover

$R^{\mathrm{rat}}_{k,k^{\prime}}(u,\hbar)=\sum_{j\geq
0}\frac{(-\hbar)^{j}}{j!}\prod_{i=1}^{j}\frac{1}{u+\hbar(i+|k-k^{\prime}|/2)}\begin{cases}F^{j}E^{j}&\text{if
$k\geq k^{\prime}$,}\\\ E^{j}F^{j}&\text{if $k\leq k^{\prime}$.}\end{cases}$

Here $E=\sum_{i=1}^{N}\psi^{*}_{i}\otimes\psi_{i}$,
$F=-\sum_{i=1}^{N}\psi_{i}\otimes\psi_{i}^{*}$ are the $q\to 1$ limits of the
$U_{q}\mathfrak{sl}_{2}$-generators of Section 2.

###### Proof.

The limit exists since both $E,F$ and the coefficients in the formula for
$\check{R}_{k,k^{\prime}}$ have a limit as $\epsilon\to 0$. The Yang–Baxter
equation then follows from the braiding relation (2.3) of
$\check{R}_{k,k^{\prime}}$. The action of the subalgebra
$U_{q}{\mathfrak{gl}}_{N}$ does not involve the spectral parameter and becomes
an action of $U{\mathfrak{gl}}_{N}$, so that the limit is
${\mathfrak{gl}}_{N}$-invariant.

In a neighbourhood of $q=1$, $\check{R}_{k,k^{\prime}}(z,q)$ is holomorphic in
the disk $|z|<1/2$. Its value at $q=1$ is independent of $z$ and is a
homomorphism of $L{\mathfrak{gl}}_{N}$-modules, which are irreducible for
$z=z_{1}/z_{2}\neq 1$. By Schur’s lemma
$\check{R}_{k,k^{\prime}}(z,1)=\check{R}_{k,k^{\prime}}(0,1)$ must be
proportional to the homomorphism $P\colon u\otimes
v\mapsto(-1)^{kk^{\prime}}v\otimes u$. With the normalization of Theorem 2.6,
the factor of proportionality is 1. It follows that

$R^{\mathrm{rat}}_{k,k^{\prime}}(u,\hbar)=\lim_{\epsilon\to
0}\check{R}_{k,k^{\prime}}(0,e^{\epsilon\hbar})^{-1}\check{R}_{k,k^{\prime}}(e^{2\epsilon
u},e^{\epsilon\hbar}).$

By Theorem 2.6 this can be expressed in terms of $A_{m}(z)=A_{m}(z,q)$ for
$m=k-k^{\prime}$:

$R^{\mathrm{rat}}_{k,k^{\prime}}(u,\hbar)=\lim_{\epsilon\to
0}A_{m}(0,e^{\epsilon\hbar})^{-1}A_{m}(e^{2\epsilon
u},e^{\epsilon\hbar})=\lim_{\epsilon\to 0}B_{m}(e^{2\epsilon
u},e^{\epsilon\hbar}).$

Then, in the limit $\epsilon\to 0$, $E=\sum\psi_{i}\otimes\psi_{i}^{*}$ and
$F=-\sum\psi_{i}^{*}\otimes\psi_{i}$ obey $\mathfrak{sl}_{2}$ commutation
relations and, by Proposition 4.4, $B_{m}$ tends to

$\sum_{j=0}^{\infty}\frac{(-\hbar)^{j}}{j!}F^{j}E^{j}\prod_{i=1}^{j}\frac{1}{u+\hbar(i+m/2)}$

for $m\geq 0$ and

$\sum_{j=0}^{\infty}\frac{(-\hbar)^{j}}{j!}E^{j}F^{j}\prod_{i=1}^{j}\frac{1}{u+\hbar(i-m/2)}$

for $m\leq 0$. ∎

######  Remark 4.6.

If $k=k^{\prime}$ we recover the formula proposed by Smirnov, see [15], eq.
(115). Understanding this formula was part of the motivation of the present
work. For general $k,k^{\prime}$ our formulas seem to be related to the
formula in [15] by a shift of spectral parameter which could be partly
explained by a redefinition of exterior powers.

##  Appendix A $U_{q}\mathfrak{sl}_{2}$ yoga

We recall some basic facts about $U_{q}\mathfrak{sl}_{2}$, see e.g. [4], and
apply them to matrix elements of $A_{m}(z)$ and $B_{m}(z)$. The irreducible
representations of $U_{q}\mathfrak{sl}_{2}$ for $q$ not a root of unity are
deformations of those of $\mathfrak{sl}_{2}$. The irreducible representation
$L_{\ell}$ of highest weight $\ell\in\mathbb{Z}_{\geq 0}$ has a basis
$v^{\ell}_{\ell},v^{\ell}_{\ell-2},\cdots,v^{\ell}_{-\ell}$ of weight vectors
with $v^{\ell}_{\ell-2k}=F^{(k)}v^{\ell}_{\ell}$.

The Cartan involution is the algebra automorphism of $U_{q}\mathfrak{sl}_{2}$
such that $\theta(E)=-F,\theta(F)=-E,\theta(K)=K^{-1}$. It is a coalgebra
antihomomorphism and for $v\in L_{\ell}$, $X\in U_{q}\mathfrak{sl}_{2}$,
$\theta(X)v=s_{\ell}Xs_{\ell}^{-1}v$, where

$s_{\ell}v^{\ell}_{m}=(-1)^{(\ell-m)/2}v^{\ell}_{-m}.$

Let $C\in Z(U_{q}\mathfrak{sl}_{2})$ be the Casimir element

$C=EF+\frac{q^{-1}K+qK^{-1}}{(q-q^{-1})^{2}}=FE+\frac{qK+q^{-1}K^{-1}}{(q-q^{-1})^{2}},$

belonging to the centre of $U_{q}\mathfrak{sl}_{2}$. Its value in $L_{\ell}$
is

$c_{\ell}=\frac{q^{\ell+1}+q^{-\ell-1}}{(q-q^{-1})^{2}}.$

One has the well-known identity, easily checked by induction,

$E^{j}F^{j}=\prod_{i=0}^{j-1}\left(C-\frac{q^{2i+1}K^{-1}+q^{-2i-1}K}{(q-q^{-1})^{2}}\right).$
(11)

######  Lemma A.1.

Let $A_{m}(z)$ be the formal power series (8) and $B_{m}(z)$ the formal power
series defined by (10). Then

$\displaystyle A_{m}(z)v_{m}^{\ell}$
$\displaystyle=\prod_{j=0}^{\frac{\ell-|m|}{2}-1}\frac{z-q^{\ell-2j}}{1-zq^{\ell-2j}}v_{-m}^{\ell}$
$\displaystyle B_{m}(z)v_{m}^{\ell}$
$\displaystyle=\prod_{j=0}^{\frac{\ell-|m|}{2}-1}\frac{1-zq^{-\ell+2j}}{1-zq^{\ell-2j}}v_{m}^{\ell}$

###### Proof.

Let us first consider the case $m\geq 0$.

The identity (11) implies that on the subspace of $L_{\ell}$ of weight
$-\ell+2k$, we have

$\displaystyle E^{j}F^{j}|_{L_{\ell}[-\ell+2k]}$
$\displaystyle=\prod_{i=0}^{j-1}\frac{q^{\ell+1}+q^{-\ell-1}-q^{2i+1+\ell-2k}-q^{-2i-1-\ell+2k}}{(q-q^{-1})^{2}}$
$\displaystyle=\prod_{i=0}^{j-1}\frac{(q^{\ell+1-k+i}-q^{-\ell-1+k-i})(q^{k-i}-q^{-k+i})}{(q-q^{-1})^{2}}$
$\displaystyle=\prod_{i=0}^{j-1}[\ell+1-k+i]_{q}[k-i]_{q}.$

It follows that for $m=\ell-2k\geq 0$

$A_{m}(z)=\sum_{j=0}^{k}(-q)^{j}\frac{1-q^{m}z}{1-q^{2j+m}z}\frac{1}{[j]_{q}![j+m]_{q}!}\prod_{i=0}^{j-1}[\ell+1-k-i]_{q}[k-i]_{q}F^{m}$

Therefore,

$\displaystyle A_{m}(z)v^{\ell}_{m}$
$\displaystyle=\sum_{j=0}^{k}(-q)^{j}\frac{1-q^{m}z}{1-q^{2j+m}z}\frac{[\ell-k]_{q}!}{[j]_{q}![j+m]_{q}![k]_{q}!}\prod_{i=0}^{j-1}[\ell+1-k-i]_{q}[k-i]_{q}v^{\ell}_{-m}$
$\displaystyle=\sum_{j=0}^{k}(-q)^{j}\frac{1-q^{\ell-2k}z}{1-q^{2j+\ell-2k}z}\frac{\prod_{i=0}^{k-1}[\ell-k+j-i]_{q}}{[j]_{q}![k-j]_{q}!}v^{\ell}_{-m}.$

We thus have a rational function of $z$ with simple poles at
$z=q^{-\ell+2k-2},$ $q^{-\ell+2k-4},$ $\dots,q^{-\ell}$ which is bounded at
infinity. By comparing the residues at the poles and the value
$\genfrac{[}{]}{0.0pt}{1}{\ell-k}{k}_{q}$ at $z=q^{-\ell+2k}$, we see that

$A_{m}(z)v^{\ell}_{m}=\prod_{j=0}^{k-1}\frac{z-q^{\ell-2j}}{1-zq^{\ell-2j}}v^{\ell}_{-m},\quad\text{if
$m=\ell-2k\geq 0$.}$

For $m\leq 0$, the inversion relation implies that
$A_{m}(z)v^{\ell}_{m}=A_{|m|}(z^{-1})^{-1}v^{\ell}_{m}$. With $-m=\ell-2k\geq
0$ we find that

$\displaystyle A_{m}(z)v^{\ell}_{m}$
$\displaystyle=\prod_{j=0}^{k-1}\frac{1-z^{-1}q^{\ell-2j}}{z^{-1}-q^{\ell-2j}}v^{\ell}_{-m}$
$\displaystyle=\prod_{j=0}^{k-1}\frac{z-q^{\ell-2j}}{1-zq^{\ell-2j}}v^{\ell}_{-m},\quad\text{if
$-m=\ell-2k\geq 0$.}$

This proves the formula for $A_{m}(z)$ since $k=\frac{\ell-|m|}{2}$. In
particular,
$A_{m}(0)v^{\ell}_{m}=\prod_{j=0}^{k-1}(-q^{\ell-2j})v^{\ell}_{-m}$, and the
formula for $B_{m}(z)=A_{m}(0)^{-1}A_{m}(z)$ follows. ∎

######  Corollary A.2.

Let $U$ be any finite dimensional representation of $U_{q}\mathfrak{sl}_{2}$.
Then for any $m\in\mathbb{Z}$

$\theta(B_{m}(z))|_{U[-m]}=B_{-m}(z)|_{U[-m]}$

###### Proof.

It is sufficient to prove this for $U$ irreducible. In this case the statement
follows from Lemma A.1 and the identity $\theta(X)=s_{\ell}Xs_{\ell}^{-1}$ for
the action of $X\in U_{q}\mathfrak{sl}_{2}$ on $L_{\ell}$. ∎

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