%Paper: hep-th/9406180
%From: Jsuzuki <jsuzuki@tansei.cc.u-tokyo.ac.jp>
%Date: Tue, 28 Jun 94 14:05:39 JST

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%   version 28 June 1994
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%--------------------------------------------------------title
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\vskip2.5cm
\centerline{{\bf Analytic Bethe Ansatz }}\par
\centerline{{\bf for }}\par
\centerline{{\bf Fundamental Representations of
Yangians}\footnote*{Short title:  Analytic Bethe ansatz}}
\vskip1.0cm \centerline{by}
\vskip1.0cm
\centerline{Atsuo Kuniba\footnote\dag{
E-mail: atsuo@hep1.c.u-tokyo.ac.jp}
and Junji Suzuki\footnote\ddag{
E-mail: jsuzuki@tansei.cc.u-tokyo.ac.jp}}
\vskip0.3cm
\centerline{Institute of Physics, University of Tokyo}
\centerline{Komaba 3-8-1, Meguro-ku, Tokyo 153 Japan}
\vskip7.0cm
\centerline{\bf Abstract}
\vskip0.2cm
\par
%
We study the analytic Bethe ansatz in
solvable vertex models associated with
the Yangian $Y(X_r)$ or its quantum affine analogue
$U_q(X^{(1)}_r)$ for $X_r = B_r, C_r$ and $D_r$.
%
Eigenvalue formulas are
proposed for the transfer matrices
related to all the fundamental
representations of $Y(X_r)$.
%
Under the Bethe ansatz equation, we explicitly prove that
they are pole-free, a crucial
property in the ansatz.
%
Conjectures are also given
on higher representation cases by
applying the $T$-system,
the transfer matrix functional relations proposed recently.
%
The eigenvalues are neatly described in terms of
Yangian analogues of the semi-standard Young tableaux.
%
\vfill
\eject
%
%------------------------------------------------------------------main
%
\beginsection 1. Introduction

\noindent
{\bf 1.1 General remarks.}\hskip0.3cm
%
Among many studies on solvable lattice models,
the Bethe ansatz is one of the most successful and
widely applied machinery.
%
It was invented at very dawn of the field [1] and
is still providing rich insights.
%
Meanwhile, original Bethe's idea has evolved into
several versions of Bethe ans{\"a}tze called with the adjectives as
`thermodynamic' [2], `algebraic' [3], `analytic' [4,5],
`functional' [6] and so forth.
%
These are all powerful techniques that involve some
profound aspects.
%
We are yet to understand their full contents,
a challenge raised on Feynman's `last blackboard' [7].
%
\par
In this paper we step forward to it by developing
our recent works [8,9,10,11] further.
%
We shall propose eigenvalue formulas for
several transfer matrices in the models
with the Yangian symmetry [12]
or its quantum affine analogue [13,14,15].
%
An interesting interplay will thereby
be exposed between
the representation theory of these algebras
and the analytic Bethe ansatz.
%
Let us explain our basic setting of the problem.
%
\par
%
\noindent
{\bf 1.2 Yang-Baxter equation and transfer matrices.}\hskip0.3cm
%
Consider the quantum affine algebra
$U_q(X^{(1)}_r)$ [13,14] associated with
any classical simple Lie algebra $X_r$ of rank $r$.
%
Throughout the paper we assume that $q$ is generic.
%
Let $W^{(a)}_m\, (1 \le a \le r, m \in {\bf Z}_{\ge 1})$
be the irreducible finite dimensional $U_q(X^{(1)}_r)$-module
as sketched in section 2.1.
%
See also [16] and [8].
%
For $W, W^\prime \in
\{W^{(a)}_m \mid 1 \le a \le r, m \in {\bf Z}_{\ge 1}\}$,
let $R_{W, W^\prime}(u) \in \hbox{End}(W \otimes W^\prime)$
denote the quantum $R$-matrix satisfying the Yang-Baxter equation [17]
%
$$
R_{W, W^\prime}(u)R_{W, W^{\prime\prime}}(u+v)
R_{W^\prime, W^{\prime\prime}}(v) =
R_{W^\prime, W^{\prime\prime}}(v)
R_{W, W^{\prime\prime}}(u+v)
R_{W, W^\prime}(u).\eqno(1.1)
$$
%
Here, $u,v \in {\bf C}$ denote the spectral parameters and
$R_{W,W^\prime}(u)$ is supposed to act as identity on
$W^{\prime\prime}$, etc.
%
As is well known, one has a solvable vertex model on
planar square lattice by regarding the matrix elements
of the $R$-matrix as local Boltzmann weights.
%
For $R_{W,W^\prime}(u)$,
the vertices take ${\rm dim }W$-states
(resp. ${\rm dim }W^\prime$-states)
on, say, horizontal (resp. vertical) edges.
%
The row-to-row transfer matrix
under the periodic boundary condition is defined by
%
$$
T^{(a)}_m(u) = \hbox{Tr}_{W^{(a)}_m}\bigl(
R_{W^{(a)}_m,W^{(p)}_s}(u-w_1) \cdots
R_{W^{(a)}_m,W^{(p)}_s}(u-w_N)\bigr)
\eqno(1.2)
$$
up to an overall scalar multiple.
%
Here $N$ is the system size, $w_1,\ldots,w_N$ are complex
parameters representing the inhomogeneity,
$1 \le a, p \le r$ and $m, s \in {\bf Z}_{\ge 1}$.
%
Following the QISM terminology [3], we say that
(1.2) is the row-to-row transfer matrix with the
{\it auxiliary space} $W^{(a)}_m$ that acts on the
{\it quantum space} $\bigl(W^{(p)}_s\bigr)^{\otimes N}$.
%
(More precisely, $W^{(a)}_m(u)$ and
$\otimes_{j=1}^N W^{(p)}_s(w_j)$,
respectively.
%
See section 2.1.)
%
Note that in (1.2) we have suppressed the quantum space
dependence on the lhs.
%
Thanks to the Yang-Baxter equation (1.1), the transfer matrices
form a commuting family
%
$$[T^{(a)}_m(u), T^{(a^\prime)}_{m^\prime}(u^\prime)] = 0.\eqno(1.3)$$
%
They can be simultaneously diagonalized
and we shall write their eigenvalues as $\Lambda^{(a)}_m(u)$,
which is also dependent on $p$ and $s$.
%
Our aim is to find an explicit formula for them.
%
So far, the full answer is known only for
$X_r = A_r$ [18,19] and $X_r = C_2$ [10].
%
In this paper we extend the results in [20,21] for
$X_r = B_r, C_r$ and $D_r$ further
by combining the two basic ingredients,
the analytic Bethe ansatz [5]
and the transfer matrix functional relations
($T$-system) [8,9].
%
Our approach renders a new insight
into the base structure of
the module $W^{(a)}_m$ and leads to
several conjectures on $\Lambda^{(a)}_m(u)$.
%
Below we shall illustrate our idea
along an exposition of the analytic Bethe ansatz (section 1.3) and
the $T$-system (section 1.4) for the simplest example $X_r = sl(2)$.
%
\par\noindent
%
{\bf 1.3 Analytic Bethe ansatz.}\hskip0.3cm
%
Let us concentrate on the $X_r = sl(2)$ case
in this and the next subsections.
%
We write $T_m(u)$ for $T^{(1)}_m(u)$, etc. since
the rank of $sl(2)$ is 1.
%
Then $W_m$ denotes the $(m+1)$-dimensional
irreducible representation of $U_q(\hat{sl}(2))$.
%
For simplicity, we assume that $s=1$ in (1.2).
%
Then $T_1(u)$ is just the 6-vertex model transfer matrix acting on
the vectors labeled by length $N$ sequences of $+$ or $-$ states.
%
We take the local vertex Boltzmann weights as
$R_u(\pm,\pm,\pm,\pm) = [2+u]$,
$R_u(\pm,\mp,\pm,\mp) = [u]$ and
$R_u(\pm,\mp,\mp,\pm) = [2]$, where
the local states $+$ or $-$ are ordered anti-clockwise from
the left edge of the vertex.
%
The function $[u]$ is defined by
%
$$[u] = {q^u - q^{-u} \over q-q^{-1}}.$$
%
The eigenvalue $\Lambda_1(u)$
is well known and given by
%
$$\eqalignno{
\Lambda_1(u) &= {Q(u-1)\over Q(u+1)}\phi(u+2) +
{Q(u+3)\over Q(u+1)}\phi(u),&(1.4{\rm a})\cr
%
Q(u) &= \prod_{j=1}^n[u-iu_j],\quad
\phi(u) = \prod_{j=1}^N[u-w_j].&(1.4{\rm b})\cr
}$$
%
Here, $0 \le n \le N/2$ is the number of the
$-$ states in the eigenvector, which is
preserved under the action of $T_1(u)$.
%
$u_j \in {\bf C}$ are
any solution of the Bethe ansatz equation (BAE)
%
$$ -{\phi(iu_k + 1)\over \phi(iu_k - 1)} =
{Q(iu_k + 2)\over Q(iu_k - 2)}.\eqno(1.5)
$$
%
On the result (1.4-5), one makes a few observations.
\par\noindent (i)
The eigenvalue has the ``dressed vacuum form (DVF)", which
means the following.
%
The ``vacuum vector" $+,+,\ldots,+$ is the obvious
eigenvector with the vacuum eigenvalue
%
$$
\prod_{j=1}^NR_{u-w_j}(+,+,+,+) +
\prod_{j=1}^NR_{u-w_j}(-,+,-,+) = \phi(u+2) + \phi(u).
\eqno(1.6)
$$
%
Eq.(1.4) tells that general eigenvalues have a modified form
of this with the ``dress" factors $Q/Q$ which is certainly 1
when $n = 0$.
%
In particular, the number of the terms in
$\Lambda_1(u)$ is the dimension
of the auxiliary space $\hbox{dim} W_1 = 2$.
%
\par\noindent (ii)
The BAE (1.5) ensures that the eigenvalues are free of poles
for finite $u$.
%
The apparent pole at $u = iu_k - 1$ in (1.4a) is spurious
as the residues from the two terms cancel due to (1.5).
%
The eigenvalues must actually be pole-free because the
local Boltzmann weight, hence the matrix elements
of $T_1(u)$ are so.
%
\par\noindent
(iii) Properties inherited from
the asymptotic behavior in $\vert u \vert \rightarrow \infty$ and
the first/second inversion relations of
the $R$-matrix (vertex Boltzmann weights).
%
For example, one has
$\Lambda_1(u) = (-)^N\Lambda_1(-2-u)\vert_{w_j \rightarrow -w_j,
u_j \rightarrow -u_j}$ form the last property.
%
See also the remark after (2.12).
%
\par
%
The analytic Bethe ansatz is
the hypothesis that the postulates (i)-(iii)
essentially determine a function of $u$ uniquely and that
the so obtained is the actual transfer matrix eigenvalue.
%
As the input data, it only uses the BAE and the $R$-matrix
(or the vacuum eigenvalue (1.6)) which should be
normalized to be an entire function of $u$.
%
It was formulated in [5] by extracting the idea from
Baxter's solution of the 8-vertex model [4].
%
See [10,11,20,21] for other applications.
%
In section 2.4, we will introduce a few more conditions
than (i)-(iii) above.
%
\par\noindent
{\bf 1.4 Transfer matrix functional relations.}\hskip0.3cm
%
The transfer matrix (1.2) obeys various
functional relations.
%
For $X_r = sl(2)$ and $s=1$ in (1.2), it is known that [18,22]
%
$$\eqalign{
T_m(u+1)T_m(u-1) &= T_{m+1}(u)T_{m-1}(u) +
g_m(u)\hbox{Id},\cr
%
g_m(u) &= \prod_{k=0}^{m-1}
\phi(u+2k-m)\phi(u+4+2k-m),\cr}
\eqno(1.7)
$$
%
where $m \ge 0$ and $T_0(u) = \hbox{Id}$.
%
Since $T_m(u)$'s can be simultaneously diagonalized,
(1.7) may be regarded as an equation for the eigenvalues
$\Lambda_m(u)$.
%
By using (1.4a) and $\Lambda_0(u) = 1$ as the initial condition,
it is easy to solve the recursion (1.7) to find
%
$$
\Lambda_m(u) =
\bigl(\prod_{k=1}^{m-1}\phi(u+m+1-2k)\bigr)
\sum_{j=0}^m
{Q(u-m)Q(u+m+2)\phi(u+m+1-2j)\over Q(u+m-2j)Q(u+m+2-2j)},
\eqno(1.8)
$$
%
in agreement with [18].
%
To observe a representation theoretical content,
we now set
%
$$
\Fsquare(0.4cm,1) =
{Q(u-1)\over Q(u+1)}\phi(u+2),\quad
%
\Fsquare(0.4cm,2) =
{Q(u+3)\over Q(u+1)}\phi(u),\eqno(1.9)
$$
%
where we assume on the lhs that
the spectral parameter $u$ is
implicitly attached to the single box as well.
%
In this notation (1.4a) reads as
$\Lambda_1(u)
= \Fsquare(0.4cm,1) + \Fsquare(0.4cm,2)$.
%
Moreover, the result (1.8) for general $m$ can be expressed as follows.
%
$$\Lambda_m(u) = \sum_{j=0}^m \bazres . \eqno(1.10)$$
%
Here we interpret the tableau as the product of the $m$
functions (1.9) with the spectral parameter $u$ shifted to
$u-m+1, u-m+3, \ldots, u+m-1$ from the left to the right.
%
Notice that the tableaux appearing in (1.10) are exactly
the semi-standard ones that label the weight vectors
in the $(m+1)$-dimensional irreducible representation
$W_m$ of $U_q(\hat{sl}(2))$
(plainly, the spin $m\over 2$ representation of $sl(2)$).
%
In this sense
the eigenvalues $\Lambda_m(u)$
are analogues (``Yang-Baxterizations") of the characters of
the auxiliary space $W_m$, which may be natural from (1.2).
%
The functional relation (1.7) for $\Lambda_m(u)$ thereby
plays the role of a character identity.
%
\par\noindent
{\bf 1.5 General $X_r$ case.}\hskip0.3cm
%
Having seen the $sl(2)$ example,
an immediate question then would be, how
the ``tableau construction" of the eigenvalues as (1.10)
can be generalized to the other algebra cases.
%
For $X_r = A_r$, the answer has been given in [19]
for the RSOS models [23],
which essentially includes (1.10) for $r=1$.
%
In this case, the $U_q(A^{(1)}_r)$-module $W^{(a)}_m$
(the auxiliary space) is a $q$-analogue of the $sl(r+1)$-module
corresponding to the $a \times m$ rectangular
Young diagram representation.
%
The eigenvalue $\Lambda^{(a)}_m(u)$ is constructed as in (1.10)
from the set of the usual semi-standard tableaux labeling the
weight vectors.
%
\par
%
An interesting feature emerges for $X_r \neq A_r$ where
$U_q(X^{(1)}_r)$-module $W^{(a)}_m$ is a $q$-analogue of
a {\it reducible} $X_r$-module in general.
%
Evaluation of $\Lambda^{(a)}_m(u)$ amounts to
finding the tableau-like objects
that label the base of such $W^{(a)}_m$.
%
This can actually be done by
postulating the {\it T-system},
the transfer matrix functional relations, proposed in [8].
%
It is a generalization of (1.7) into arbitrary $X_r$ case
and can be solved for
$\Lambda^{(a)}_m(u)$ in terms of
$\Lambda^{(a)}_1(u+\hbox{shift})\, (1 \le a \le r)$
(and $\Lambda^{(a)}_0(u)$ = 1).
%
Thus one can play the following game.
\par\noindent
%
{\it Step 1.} Find $\Lambda^{(1)}_1(u), \ldots, \Lambda^{(r)}_1(u)$
by the analytic Bethe ansatz.
\par\noindent
%
{\it Step 2.} Solve the $T$-system for $\Lambda^{(a)}_m(u)$
recursively by taking the step 1 result as the initial condition.
%
\par\noindent
{\it Step 3.} Find such ``tableaux" that the step 2 result is
expressed, in an analogous sense to (1.10), as
%
$$\Lambda^{(a)}_m(u) = \sum \hbox{tableau}(u).\eqno(1.11)$$
%
We shall execute the above program in a number of cases for
$X_r = B_r, C_r$ and $D_r$.
%
The resulting tableau label for the base of $W^{(a)}_m$
exhibits an interesting contrast with
those for the crystal base [24,25] concerning
the irreducible $X_r$-modules.
%
We find in several cases that the DVF (1.11), hence
the base of $W^{(a)}_m$,
can also be labeled by semi-standard-like tableaux
obeying remarkably simple rules.

\par\noindent
{\bf 1.6 Plan of the paper.}\hskip0.3cm
%
In the next section, we begin by fixing our notations
and recall the family of the modules $W^{(a)}_m$,
the $T$-system [8] and the BAE [26,21] for models
with $U_q(X^{(1)}_r)$ symmetry.
%
The Yangian case $Y(X_r)$ corresponds to a smooth
rational limit $q \rightarrow 1$ of them.
%
Then we discuss the analytic Bethe ansatz
and propose a few more
hypotheses, ``dress universality", ``top term"
and ``coupling rule".
%
They supplement
(i)-(iii) in section 1.3
and work efficiently for models with general
$U_q(X^{(1)}_r)$ symmetry.
%
Sections 3, 4 and 5 are devoted to the cases
$X_r = C_r, B_r$ and $D_r$, respectively.
%
A peculiarity for the latter two algebras
is the presence of the spin representations,
whose $U_q(X^{(1)}_r)$-analogues are certainly the members
of the family
$\{W^{(a)}_m \mid 1 \le a \le r, m \in {\bf Z}_{\ge 1} \}$.
%
($W^{(r)}_1$ for $B_r$ and
$W^{(r-1)}_1, W^{(r)}_1$ for $D_r$.)
%
For these algebras, we introduce two kinds of
elementary boxes corresponding to the bases of the vector and the spin
representations.
%
We clarify their relation reflecting
the fact that the former representation is contained
in a tensor product of the latter.
%
These features are quite similar for $B_r$ and $D_r$ cases,
hence we shall omit many details for the latter.
%
Section 6 gives the summary and discussion.
%
\par
\beginsection 2. $T$-system, BAE and analytic Bethe ansatz

\noindent
{\bf 2.1 Modules $W^{(a)}_m$.}\hskip0.3cm
Let us fix our notations for the data from the simple Lie algebras $X_r$.
%
Let $\alpha_a, \omega_a (1 \le a \le r)$ and $(\cdot\mid\cdot )$
denote the simple roots, the fundamental weights and the invariant
bilinear form on $X_r$.
%
We identify the Cartan subalgebra and its dual via $(\cdot\mid\cdot)$
and normalize it as $(\alpha\mid\alpha) = 2$
for $\alpha$ = long root.
%
Put
$$\eqalign{
t_a & = {2 \over (\alpha_a\mid\alpha_a)} \quad 1 \le a \le r,\cr
%
g & = \hbox{ dual Coxeter number of } X_r.\cr}
\eqno(2.1)$$
%
By the definition $t_a = 1, 2$ or $3$ and
$(\omega_a \mid \alpha_b) = \delta_{a b}/t_a$.
%
Enumeration of the nodes $1 \le a \le r$
on the Dynkin diagram is same with Table I in [8].
%
For $X_r = B_r (r \ge 2), C_r (r \ge 2)$ and $D_r (r \ge 4)$,
(2.1) reads explicitly as
%
$$\eqalign{
&t_1 = \cdots = t_{r-1} = 1, t_r = 2\, \hbox{ for } B_r,\cr
%
&t_1 = \cdots = t_{r-1} = 2, t_r = 1\, \hbox{ for } C_r,\cr
%
&\forall t_a = 1\qquad\qquad\qquad\quad\quad \,\,\hbox{ for } D_r,\cr
%
&g = \cases{2r-1& for $B_r$\cr
            r+1&  for $C_r$\cr
            2r-2& for $D_r$\cr}.\cr}\eqno(2.2)
$$
%
\par
%
Now we recall
the family of modules
$\{W^{(a)}_m \mid 1 \le a \le r, m \in {\bf Z}_{\ge 1}\}$
first introduced in [16] for the Yangian $Y(X_r)$
extending the earlier examples [26].
%
Precisely speaking, Yangian modules carry
a spectral parameter hence the auxiliary and the quantum
spaces in (1.2) are to be understood as
$W^{(a)}_m(u)$ and $\otimes_{j=1}^N W^{(p)}_s(w_j)$, respectively.
%
See [27,28] and section 3.2 in [8].
%
Then $W^{(a)}_m(u)$ has a characterization by the Drinfel'd
polynomials [27,28]
$\{P_a(v) \mid 1 \le a \le r \}$ as
%
$$P_b(v) = \cases{
(v-u+{m-1\over t_a})(v-u+{m-3\over t_a})\cdots
(v-u-{m-1\over t_a})& for $b = a$\cr
1& otherwise \cr}.\eqno(2.3)
$$
%
In [28], $W^{(a)}_1(u) \, (1 \le a \le r)$ is called
the {\it fundamental representation} of $Y(X_r)$.
%
Viewed as a module over $X_r \subset Y(X_r)$,
$W^{(a)}_m(u)$ is reducible in general but the
contained irreducible components are independent of $u$.
%
Thus we let simply $W^{(a)}_m$ denote the $X_r$-module so obtained.
%
Then it is known that [16]
%
$$\eqalignno{
C_r;&&\cr
&W^{(a)}_m \simeq \cases{
\oplus V(k_1\omega_1 + \cdots + k_a\omega_a) \quad &$1 \le a \le r-1$\cr
%
V(m\omega_r) \quad &$a = r$\cr},&(2.4{\rm a})\cr
%
B_r \hbox{ and } D_r;&&\cr
&W^{(a)}_m \simeq \oplus
V(k_{a_0}\omega_{a_0} + k_{a_0 + 2}\omega_{a_0 + 2} + \cdots +
k_a\omega_a) \quad 1 \le a \le r^\prime,&(2.4{\rm b})\cr
%
&r^\prime = \cases{r & for $B_r$\cr
                   r-2 & for $D_r$\cr},\quad
a_0 \equiv a \hbox{ mod } 2, \quad a_0 = 0 \hbox{ or } 1,
&(2.4{\rm c})\cr
%
&W^{(a)}_m \simeq V(m\omega_a) \quad a = r-1, r
\quad \hbox{ only for } D_r.&(2.4{\rm d})\cr}
$$
%
Here $\omega_0 = 0$ and
$V(\lambda)$ denotes the irreducible $X_r$-module with
highest weight $\lambda$.
%
The sum in (2.4a) is taken over non-negative integers
$k_1, \ldots, k_a$ that satisfy
$k_1 + \cdots + k_a \le m, k_j \equiv m\delta_{j a}$
mod 2 for all $ 1 \le j \le a$.
%
The sum in (2.4b) extends over non-negative integers
$k_{a_0}, k_{a_0 + 2}, \ldots, k_a$ obeying the constraint
$t_a(k_{a_0} + k_{a_0+2} + \cdots + k_{a-2}) + k_a = m$.
%
If one depicts the highest weights in
the sum (2.4a) and (resp. (2.4b))
by Young diagrams as usual, they correspond to those
obtained from the $a \times m$ rectangular one by successively
removing $1\times 2$ and (resp. $2 \times 1$) pieces.
%
\par
%
As mentioned in section 3.2 of [8], we assume in this paper
that there exists
a natural $q$-analogue of these modules over the quantum affine algebra
$U_q(X^{(1)}_r)$, which will also be denoted by $W^{(a)}_m$.
%
When referring it
as an $X_r$-module, it means that the corresponding
$Y(X_r)$-module in the $q \rightarrow 1$ limit has been regarded so.
%
\par\noindent
%
{\bf 2.2 $T$-system.}\hskip0.3cm
Consider the transfer matrix (1.2)
acting on the quantum space
$\otimes_{j=1}^N W^{(p)}_s(w_j)$.
%
We shall reserve the letters $p$ and $s$ for this meaning
throughout the paper.
%
(See also the end of section 2.4.)
%
In [8], a set of functional relations, the $T$-system,
was conjectured for $U_q(X^{(1)}_r)$ symmetry models
for any $X_r$.
%
For $X_r = B_r, C_r$ and $D_r$ they read as follows.
%
$$\eqalign{
B_r:\qquad\qquad\qquad\qquad\qquad &\cr
%
T^{(a)}_m(u-1)T^{(a)}_m(u+1) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) +
g^{(a)}_m(u)T^{(a-1)}_m(u)T^{(a+1)}_m(u)\cr
%
&\qquad\qquad\qquad\qquad\qquad 1 \le a \le r-2,\cr
%
T^{(r-1)}_m(u-1)T^{(r-1)}_m(u+1) &=
T^{(r-1)}_{m+1}(u)T^{(r-1)}_{m-1}(u) +
g^{(r-1)}_m(u)T^{(r-2)}_m(u)T^{(r)}_{2m}(u),\cr
%
T^{(r)}_{2m}(u-{1\over 2})T^{(r)}_{2m}(u+{1\over 2}) &=
T^{(r)}_{2m+1}(u)T^{(r)}_{2m-1}(u) \cr
%
&+ g^{(r)}_{2m}(u)T^{(r-1)}_m(u-{1\over 2})T^{(r-1)}_m(u+{1\over 2}),\cr
%
T^{(r)}_{2m+1}(u-{1\over 2})T^{(r)}_{2m+1}(u+{1\over 2}) &=
T^{(r)}_{2m+2}(u)T^{(r)}_{2m}(u) +
g^{(r)}_{2m+1}(u)T^{(r-1)}_m(u)T^{(r-1)}_{m+1}(u).\cr}
\eqno(2.5{\rm a})
$$
%
$$\eqalign{
C_r:\qquad\qquad\qquad\qquad\qquad \quad&\cr
%
T^{(a)}_m(u-{1\over 2})T^{(a)}_m(u+{1\over 2}) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) +
g^{(a)}_m(u)T^{(a-1)}_m(u)T^{(a+1)}_m(u)\cr
%
&\qquad\qquad\qquad\qquad\qquad 1 \le a \le r-2,\cr
%
T^{(r-1)}_{2m}(u-{1\over 2})T^{(r-1)}_{2m}(u+{1\over 2}) &=
T^{(r-1)}_{2m+1}(u)T^{(r-1)}_{2m-1}(u) \cr
%
&+ g^{(r-1)}_{2m}(u)T^{(r-2)}_{2m}(u)
T^{(r)}_m(u-{1\over 2})T^{(r)}_m(u+{1\over 2}),\cr
%
T^{(r-1)}_{2m+1}(u-{1\over 2})T^{(r-1)}_{2m+1}(u+{1\over 2}) &=
T^{(r-1)}_{2m+2}(u)T^{(r-1)}_{2m}(u) \cr
%
&+ g^{(r-1)}_{2m+1}(u)T^{(r-2)}_{2m+1}(u)
T^{(r)}_m(u)T^{(r)}_{m+1}(u),\cr
%
T^{(r)}_m(u-1)T^{(r)}_m(u+1) &=
T^{(r)}_{m+1}(u)T^{(r)}_{m-1}(u) +
g^{(r)}_m(u)T^{(r-1)}_{2m}(u).\cr}
\eqno(2.5{\rm b})
$$
%
$$\eqalign{
D_r:\qquad\qquad\qquad\qquad\qquad &\cr
%
T^{(a)}_m(u-1)T^{(a)}_m(u+1) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) +
g^{(a)}_m(u)T^{(a-1)}_m(u)T^{(a+1)}_m(u)\cr
%
&\qquad\qquad\qquad\qquad\qquad 1 \le a \le r-3,\cr
%
T^{(r-2)}_m(u-1)T^{(r-2)}_m(u+1) &=
T^{(r-2)}_{m+1}(u)T^{(r-2)}_{m-1}(u) \cr
&+ g^{(r-2)}_m(u)T^{(r-3)}_m(u)T^{(r-1)}_m(u)T^{(r)}_m(u),\cr
%
T^{(a)}_{m}(u-1)T^{(a)}_{m}(u+1) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u)
+ g^{(a)}_{m}(u)T^{(r-2)}_m(u)\quad a = r-1, r.\cr}
\eqno(2.5{\rm c})
$$
%
Here the subscripts of the transfer matrices in the lhs
are taken to be positive and we assume that
$T^{(0)}_m(u) = T^{(a)}_0(u) \equiv \hbox{Id}$.
%
$g^{(a)}_m(u)$ is a scalar function
that depends on $W^{(p)}_s$ and satisfies
%
$$g^{(a)}_m(u-{1\over t_a})g^{(a)}_m(u+{1\over t_a})
= g^{(a)}_{m+1}(u)g^{(a)}_{m-1}(u).\eqno(2.6)$$
%
See eq.(3.18) in [8].
%
We have slightly changed the convention from [8]
so that $T^{(a)}_m(u+j)$ there
corresponds to $T^{(a)}_m(u+2j)$ here, etc.
%
A wealth of consistency for the $T$-system
have been observed in [8,9,10,11]
for any $X_r$ and we shall assume (2.5) henceforth.
%
Owing to the commutativity (1.3),
one can regard (2.5) as the functional relations
on the eigenvalues $\Lambda^{(a)}_m(u)$.
%
($\Lambda^{(0)}_m(u) = \Lambda^{(a)}_0(u) = 1$.)
%
Then it can be recursively solved for
$\Lambda^{(a)}_m(u)$ in terms of
$\Lambda^{(1)}_1(u+\hbox{shift}),\ldots,
\Lambda^{(r)}_1(u+\hbox{shift})$.
%
In fact, $\Lambda^{(a)}_m(u)$ will be obtainable
within a {\it polynomial}
in these functions as argued in [8].
%
This process corresponds to the {\it Step 2}
mentioned in section 1.5.
%
\par\noindent
{\bf 2.3 Bethe ansatz equation.}\hskip0.3cm
%
As in (1.4), the eigenvalues $\Lambda^{(a)}_m(u)$ will be
expressed by the solutions to the BAE [26,21]:
%
$$-{\phi(iu^{(a)}_k + {s\over t_p}\delta_{a p})\over
    \phi(iu^{(a)}_k - {s\over t_p}\delta_{a p})} =
\prod_{b=1}^r
{Q_b(iu^{(a)}_k + (\alpha_a\mid \alpha_b))\over
 Q_b(iu^{(a)}_k - (\alpha_a\mid \alpha_b))},\eqno(2.7)
$$
%
where $s$ and $p$ are the labels of the quantum space
$\otimes_{j=1}^N W^{(p)}_s(w_j)$,
$\phi(u)$ is given in (1.4b) and $Q_a(u)$ is defined by
%
$$
Q_a(u) = \prod_{j=1}^{N_a}[u - iu^{(a)}_j]\quad
1 \le a \le r.\eqno(2.8)
$$
%
Here $N_a$ is a non-negative integer analogous to $n$ in (1.4b).
%
The system size $N$ in $\phi(u)$ and $N_a$
are to be taken so that
$\omega^{(p)}_s \buildrel \rm def \over =
Ns\omega_p - \sum_{a=1}^r N_a \alpha_a \in \sum_{a=1}^r
{\bf R}_{\ge 0}\omega_a$.
%
The BAE (2.7) is imposed on the numbers
$\{u^{(a)}_k \mid 1 \le a \le r, 1 \le k \le N_a \}$.
%
In section 5, we will consider a slightly modified version of
(2.7) that suits the diagram automorphism symmetry in $X_r = D_r$.
%
\par\noindent
%
{\bf 2.4 Empirical rules in
Analytic Bethe ansatz.}\hskip0.3cm
%
As in (1.4a), the functions $Q_a(u)$ and $\phi(u)$ are the constituents
of the dress and the vacuum parts in
the analytic Bethe ansatz, respectively.
%
In handling the formulas like (1.10) or (1.11),
we find it convenient to specify these parts as
$dr(\hbox{tableau})$ and $vac(\hbox{tableau})$.
%
For example, from the first equation in (1.9) one has
%
$$
\Fsquare(0.4cm,1) = dr \Fsquare(0.4cm,1)\, vac \Fsquare(0.4cm,1),\quad
dr \Fsquare(0.4cm,1) = {Q(u-1)\over Q(u+1)},\quad
vac \Fsquare(0.4cm,1) = \phi(u+2).\eqno(2.9{\rm a})
$$
%
In general the DVF reads
%
$$
\Lambda^{(a)}_m(u) = \sum
{Q_{a_1}(u+x_1) \cdots Q_{a_n}(u+x_n) \over
Q_{a_1}(u+y_1) \cdots Q_{a_n}(u+y_n)}
\phi(u+z_1) \cdots \phi(u+z_k),\eqno(2.9{\rm b})
$$
in which ratios of $Q_a$'s are the dress parts and
products of $\phi$'s are the vacuum parts.
%
Using these notations we now introduce
three hypotheses, ``dress universality", ``top term" and
``coupling rule" in the analytic Bethe ansatz.
%
They are the properties of mathematical interest
rendering valuable insights into the
auxiliary space $W^{(a)}_m$ as the
$U_q(X^{(1)}_r)$ or the Yangian modules.
%
Roughly speaking, the latter two are the information on the
``highest weight vector" and the ``action" of the
Chevalley-like generators.
%
The hypotheses have been confirmed in several examples
and we believe they should rightly be added to
the postulates (i)-(iii) explained in section 1.3.
\par
%
{\it Dress universality.}\hskip0.3cm
%
Let $T^{(a)}_m(u)$ and
$T^{\prime (a)}_m(u)$ be the
transfer matrices with the same auxiliary space
$W^{(a)}_m(u)$ but acting on the
different quantum spaces
\par
$\otimes_{k=1}^N W^{(p)}_s(w_k)$ and
$\otimes_{k=1}^{N^\prime} W^{(p^\prime)}_{s^\prime}(w^\prime_k)$,
respectively.
%
Denote by $Q_a(u)$ and $Q^\prime_a(u)$ the functions (2.8)
specified from the solutions to the BAE (2.7) for these
quantum space choices.
%
Suppose one got their eigenvalues in the DVFs
%
$$
\Lambda^{(a)}_m(u) = \sum_{j=1}^{\hbox{dim } W^{(a)}_m}
\hbox{tab}_j,\qquad
%
\Lambda^{\prime (a)}_m(u) = \sum_{j=1}^{\hbox{dim } W^{(a)}_m}
\hbox{tab}^\prime_j,\eqno(2.10)
$$
%
where $\hbox{tab}_j$ and $\hbox{tab}^\prime_j$ denote
the terms whose vacuum parts correspond
to the same (i.e., ``$j$-th") vector
from $W^{(a)}_m$ in the trace (1.2).
%
Then the dress universality is stated as
%
$$dr (\hbox{tab}_j) = dr (\hbox{tab}^\prime_j)
\vert_{Q^\prime_a(u) \rightarrow Q_a(u)}
\quad \hbox{ for all }\, j.
\eqno(2.11)$$
%
Namely, the dress part is independent of the
quantum space choice if it is expressed in terms of $Q_a(u)$.
%
On the contrary, one has
$vac(\hbox{tab}_j) \neq vac(\hbox{tab}^\prime_j)
\vert_{N^\prime \rightarrow N, w^\prime_k \rightarrow w_k}$
in general if $(p^\prime, s^\prime) \neq (p,s)$.
%
\par
%
{\it Top term.}\hskip0.3cm
%
Among the $\hbox{dim } W^{(a)}_m$ terms in (2.10),
let $\hbox{tab}_1$ denote
the one corresponding to the
``highest weight vector" in $W^{(a)}_m$.
%
By this we mean more precisely the unique vector
of weight $m\omega_a$ when $W^{(a)}_m$ is regarded as an
$X_r$-module in the sense of section 2.1.
%
Plainly, $\hbox{tab}_1$ is the analogue of the first term on the
rhs of (1.4a).
%
Then the top term hypothesis reads
%
$$dr(\hbox{tab}_1) = {Q_a(u-{m\over t_a})\over
                      Q_a(u+{m\over t_a})}\eqno(2.12)
$$
%
in (2.10), which is certainly consistent to the dress universality.
%
It follows from (2.12) that
%
$$\Lambda^{(a)}_m(u)
\bigl(\Lambda^{(a)}_m(-u)\vert_{w_j \rightarrow -w_j,
u^{(b)}_k \rightarrow -u^{(b)}_k} \bigr)
= \Phi(u)\bigl(\Phi(-u)\vert_{w_j \rightarrow -w_j}\bigr) + \cdots,
$$
%
where $\Phi(u) = vac(\hbox{tab}_1)$ is a product of
$\phi$'s.
%
This is essentially eq.(5) in [21], which is
a consequence of the first inversion relation of the
relevant $R$-matrix.
\par
{\it Coupling rule.}\hskip0.3cm
Regard the auxiliary space $W^{(a)}_m$ as an $X_r$-module
in the sense of section 2.1
and let $\lambda$ be a weight
without multiplicity
%
$$\hbox{mult}_\lambda W^{(a)}_m = 1.\eqno(2.13)
$$
%
Then it makes sense to denote by
$\Fsquare(0.4cm,\lambda)$ the term in (2.10)
corresponding to the $\lambda$-weight vector from $W^{(a)}_m$.
%
Thus
$\Lambda^{(a)}_m(u) = \cdots + \Fsquare(0.4cm,\lambda) + \cdots$.
%
Now the coupling rule is stated as follows.
%
\par
If $\lambda$ and $\mu$ are multiplicity-free weights such that
$\lambda - \mu = \alpha_a$, then
%
$$\eqalignno{
({\rm a}) &\, \Fsquare(0.4cm,\lambda) \hbox{ and }
\Fsquare(0.4cm,\mu) \hbox{ share common poles of the form }
1/ Q_a(u+\xi)\cr
%
&\, \hbox{ for a certain } \xi \hbox{ depending on }
\lambda \hbox{ and } a.&(2.14{\rm a})\cr
%
({\rm b}) &\, \hbox{The BAE (2.7)
guarantees }
Res_{u=-\xi+iu^{(a)}_k}( \Fsquare(0.4cm,\lambda) +
\Fsquare(0.4cm,\mu) ) = 0 \hbox{ in such a way that}\cr
%
&\, {dr \Fsquare(0.4cm,\mu)\over
  dr \Fsquare(0.4cm,\lambda)} =
\prod_{b=1}^r{Q_b(u+\xi+(\alpha_a\vert \alpha_b))\over
Q_b(u+\xi-(\alpha_a\vert \alpha_b))}.&(2.14{\rm b})\cr}
$$
%
The hypothesis tells that for $\lambda - \mu = \alpha_a$,
spurious ``poles of color $a$" in
$\Fsquare(0.4cm, \lambda)$ and
$\Fsquare(0.4cm,\mu)$
couple into a pair yielding zero residue in total.
%
This is a more detailed information than just saying that
the BAE assures pole-freeness as in (ii) in section 1.3.
%
To determine $\xi$ is a non-trivial task in general.
%
{}From (2.14b), (2.7) and
$\Fsquare(0.4cm,\lambda) =
dr \Fsquare(0.4cm,\lambda) vac \Fsquare(0.4cm,\lambda)$ etc,
one deduces
%
$$
{vac \Fsquare(0.4cm,\lambda) \over
 vac \Fsquare(0.4cm,\mu) }
 = {\phi(u+ \xi + {s\over t_p}\delta_{a p})\over
    \phi(u+ \xi - {s\over t_p}\delta_{a p})}\eqno(2.15)
$$
%
for the vacuum parts.
%
The last equation in (2.14b) excludes the possibility to
exchange $\lambda$ and $\mu$ in (2.14b) and (2.15) simultaneously,
in which case
the BAE could also have ensured the pole-freeness.
%
The coupling rule is certainly valid in (1.8) and (1.9) since
$\Fsquare(0.4cm,1)$ corresponds to the weight
$\omega_1$ and $\Fsquare(0.4cm,2)$ to
$-\omega_1 = \omega_1 - \alpha_1$ in $sl(2)$.
%
We will visualize (2.14) and (2.15) as
%
$$
\Fsquare(0.4cm,\lambda)
{\buildrel a \over \longrightarrow} \Fsquare(0.4cm,\nu),
$$
%
where the number over the arrow signifies
the color of the pole shared by the two boxes.
\par
%
There are two more postulates
that embody the asymptotics and
the second inversion properties mentioned in (iii) in section 1.3.
%
The first one is stated as
\par
%
{\it Character limit.}\hskip0.3cm
%
As said in the end of section 1.4,
the eigenvalue $\Lambda^{(a)}_m(u)$ is a Yang-Baxterization
($u$-dependent version) of the
character of the auxiliary space $W^{(a)}_m(u)$ viewed as
an $X_r$-module in the sense of section 2.1.
%
Indeed, the latter can be recovered from the former as
%
$$\lim_{u \rightarrow \sigma_1\infty,
(\vert q \vert^{\sigma_2} > 1)}
q^{\tau(\sigma_1,\sigma_2)} \Lambda^{(a)}_m(u) =
\sum_\lambda \bigl(\hbox{mult}_\lambda W^{(a)}_m\bigr)
q^{2\sigma_1\sigma_2(\omega^{(p)}_s \vert \lambda )}
\quad \sigma_1, \sigma_2 = \pm 1,
\eqno(2.16)
$$
%
where the sum extends over
all the weights in $W^{(a)}_m$,
$q^{\tau(\sigma_1,\sigma_2)}$ is some convergence factor and
$\omega^{(p)}_s$ has been specified after (2.8).
%
One readily sees that (2.16) is
consistent with (2.14b) and (2.15)
by computing the asymptotics of
$\Fsquare(0.4cm,\lambda)/\Fsquare(0.4cm,\mu)$.
%
Eq. (2.16) is also asserting that DVFs always contain
$Q_a$ via the combination
$Q_a(u+\cdots)/Q_a(u+\cdots)$ as in (2.9b)
and that they are homogeneous polynomials w.r.t
$\phi(u+\cdots)$.
%
Thus $k$ is common in all the terms in (2.9b).
%
In [8,9,29], the rhs of (2.16) was denoted by
$Q^{(a)}_m(\omega^{(p)}_s)$.
%
It obeys the $Q$-system,
the character identity in [16], which
was extensively
used to formulate the conjectures on
dilogarithm identity [29,30,8,9],
$q$-series formula for an $X^{(1)}_r$ string function [31]
and to find the $T$-system [8].
%
The limit (2.16) is essentially eq.(12) in [21].
%
Now we state the second postulate.
%
\par
%
{\it Crossing symmetry.}\hskip0.3cm
%
Most $R$-matrices enjoy the so-called
crossing symmetry, eq.(4) in [21], from which
the second inversion relation follows.
%
The eigenvalue $\Lambda^{(a)}_m(u)$ inherits
the following property form it.
%
$$\Lambda^{(a)}_m(u) = (-)^{kN}\Lambda^{(a)}_m(-g-u)
\vert_{w_j \rightarrow -w_j, u^{(a)}_i \rightarrow -u^{(a)}_i}.
\eqno(2.17)
$$
%
Here $g$ is defined in (2.1), $k$ is the order of the DVF
w.r.t $\phi$ as in (2.9b) and $N$ is the number of lattice sites
entering $\phi$ via (1.4b).
%
This is essentially eq.(6) in [21], which we call
the crossing symmetry as well.
%
Note that the BAE (2.7) remains unchanged under the
simultaneous replacement
$w_j \rightarrow -w_j$ and $u^{(a)}_k \rightarrow -u^{(a)}_k$.
%
In particular, if $\pm \lambda$ are multiplicity-free weights
of $W^{(a)}_m$,
the combination
$\Fsquare(0.4cm, \lambda) + \Flect(0.4cm,0.6cm,-\lambda)$
in $\Lambda^{(a)}_m(u)$ becomes same on both sides of
(2.17) as
%%
$$\Flect(0.4cm,0.6cm,-\lambda) = (-)^{kN}
\Fsquare(0.4cm,\lambda)
\vert_{u \rightarrow -g-u,
w_j \rightarrow -w_j, u^{(a)}_k \rightarrow -u^{(a)}_k}.
\eqno(2.18)
$$
%
{}From the definitions of $\phi(u)$ (1.4b) and $Q_a(u)$ (2.8),
the rhs of (2.17) is then obtained from (2.9b)
by the simultaneous replacements
%
$$x_i \rightarrow g-x_i,\,
  y_i \rightarrow g-y_i,\,
  z_i \rightarrow g-z_i.\eqno(2.19)
$$
%
\par
The dress universality, top term, coupling rule,
character limit and
crossing symmetry severely limit the possible
form of the DVF in the
analytic Bethe ansatz.
%
In particular if all the weights in $W^{(a)}_m$ are
multiplicity-free, (2.12), (2.14) and (2.15)
determine the DVF for $\Lambda^{(a)}_m(u)$
completely up to an overall scalar multiple.
%
In such cases, one even does not need the vacuum parts
a priori hence can avoid a tedious computation of the
$R$-matrices.
%
The DVFs given in the subsequent sections have actually been derived
in that manner for such cases.
%
Except a few cases,
it is yet to be verified if
those DVFs with $\forall Q_a(u) = 1$ yield the actual
vacuum eigenvalues obtainable from the relevant $R$-matrix
as in (1.6).
%
In a sense we have partially absorbed the postulate (i) of
section 1.3 into (2.11)-(2.15) here, which may be viewed as a
modification of the analytic Bethe ansatz itself.
%
\par
%
Let us include a remark before closing this section.
%
Suppose one has found the DVF
when the the quantum space is
$\otimes_{j=1}^N W^{(p)}_1(w_j)$.
%
Then, the one for
$\otimes_{j=1}^N W^{(p)}_s(w_j)$
can be deduced from it by the
replacement
%
$$
\phi(u) \rightarrow \phi_s(u) \buildrel \rm def \over =
\prod_{k=1}^s\phi(u+{s+1-2k\over t_p}).\eqno(2.20)
$$
%
To see this one just notes that
the lhs of (2.7) is equal to
$-{\phi_s(iu^{(a)}_k+\delta_{a p}/t_p)
\over \phi_s(iu^{(a)}_k-\delta_{a p}/t_p)}$.
%
See also (2.15).
%
In view of this we shall
hereafter consider the $s=1$ case only with no loss of generality.
%
\beginsection 3. Eigenvalues for $C_r$

\noindent
{\bf 3.1 Eigenvalue $\Lambda^{(1)}_1(u)$.}\hskip0.3cm
%
The family of $U_q(C^{(1)}_r)$-modules
$\{W^{(a)}_m \mid 1 \le a \le r, m \in {\bf Z}_{\ge 1} \}$
is generated by decomposing tensor products of
$W^{(1)}_1$ as suggested in [8].
%
In terms of the eigenvalues, it implies that
all the $\Lambda^{(a)}_m(u)$ are contained in a suitable
product $\prod_k\Lambda_1^{(1)}(u+c_k)$.
%
Thus we first do the analytic Bethe ansatz for the fundamental
eigenvalue $\Lambda^{(1)}_1(u)$.
%
The relevant auxiliary space is $W^{(1)}_1 \simeq V(\omega_1)$
as an $C_r$-module from (2.4a), which is the vector
representation.
%
Then all the weights are multiplicity-free and
one can apply the coupling rule (2.14).
%
To be concrete, we introduce the orthogonal vectors
$\epsilon_a, 1 \le a \le r$ normalized as
$(\epsilon_a \mid \epsilon_b) = \delta_{a b}/2$
and realize the root system as follows.
%
$$\eqalign{
\alpha_a &= \cases{\epsilon_a - \epsilon_{a+1} &
for $1 \le a \le r-1$\cr
2\epsilon_r & for $a = r$\cr},\cr
%
\omega_a &= \epsilon_1 + \cdots + \epsilon_a.\cr}
\eqno(3.1)$$
%
Then the weights in $V(\omega_1)$ are
$\epsilon_a$ and $-\epsilon_a (1 \le a \le r)$,
which we will abbreviate to $a$ and ${\bar a}$, respectively.
%
In this notation the set of weights reads
%
$$J = \{1, 2, \ldots, r, {\bar r},\ldots, {\bar 2}, {\bar 1}\}.
\eqno(3.2)
$$
%
Starting from the top term (2.12), one successively applies the
coupling rule (2.14) to find the DVF
%
$$\Lambda^{(1)}_1(u) = \sum_{a \in J}\Fsquare(0.4cm,a),
\eqno(3.3)$$
%
with the elementary boxes defined by
%
$$\eqalign{
\Fsquare(0.4cm,a) &= \psi_a(u)
{Q_{a-1}(u+{a+1 \over 2})Q_a(u+{a-2 \over 2})\over
Q_{a-1}(u+{a-1 \over 2})Q_a(u+{a \over 2})}\quad 1 \le a \le r-1,\cr
%
\Fsquare(0.4cm,r) &= \psi_r(u)
{Q_{r-1}(u+{r+1 \over 2})Q_r(u+{r-3 \over 2})\over
Q_{r-1}(u+{r-1 \over 2})Q_r(u+{r+1 \over 2})},\cr
%
\Fsquare(0.4cm,{\bar r}) &= \psi_{\bar r}(u)
{Q_{r-1}(u+{r+1 \over 2})Q_r(u+{r+5 \over 2})\over
Q_{r-1}(u+{r+3 \over 2})Q_r(u+{r+1 \over 2})},\cr
%
\Fsquare(0.4cm,{\bar a}) &= \psi_{\bar a}(u)
{Q_{a-1}(u+{2r-a+1 \over 2})Q_a(u+{2r-a+4 \over 2})\over
Q_{a-1}(u+{2r-a+3 \over 2})Q_a(u+{2r-a+2 \over 2})}
\quad 1 \le a \le r-1,\cr}\eqno(3.4{\rm a})
$$
%
where we have set $Q_0(u) = 1$.
%
In the above, the vacuum part $\psi_a(u) = vac \Fsquare(0.4cm,a)$
is given by
%
$$
\psi_a(u) = \cases{
\phi(u+{p+1 \over 2})\phi(u+{2r-p+3 \over 2}) &$1 \preceq a \preceq p$\cr
\phi(u+{p-1 \over 2})\phi(u+{2r-p+3 \over 2})
&$p+1 \preceq a \preceq \overline{p+1}$\cr
\phi(u+{p-1 \over 2})\phi(u+{2r-p+1 \over 2})
&${\bar p} \preceq a \preceq {\bar 1}$\cr}
\eqno(3.4{\rm b})
$$
%
depending on the quantum space
$\otimes_{j=1}^N W^{(p)}_1(w_j)$.
%
The symbol $\prec$ here stands for
a total order in the set $J$ specified as
%
$$
1 \prec 2 \prec \cdots \prec r \prec {\bar r} \prec \cdots \prec
{\bar 2} \prec {\bar 1}.\eqno(3.5)
$$
%
When $p=r$, the second possibility in (3.4b) is absent.
%
The case $p=1$ was obtained in [21].
%
Note that $\Fsquare(0.4cm, 1)$ is the top term (2.12).
%
By the construction,
$p$ enters only the vacuum parts (3.4b)
hence the dress universality (2.11) is valid.
%
The crossing symmetry (2.18) holds
between $\Fsquare(0.4cm,{\bar a})$ and $\Fsquare(0.4cm,a)$.
%
Under the BAE (2.7), (3.3) is pole-free because
the coupling rule (2.14) and (2.15) as follows.
%
$$\eqalignno{
&Res_{u=-{b\over 2} + iu^{(b)}_k}
(\Fsquare(0.4cm,b) + \Flect(0.4cm,1.0cm,b+1)) = 0\quad 1 \le b \le r-1,
&(3.6{\rm a})\cr
%
&Res_{u=-{r+1\over 2} + iu^{(r)}_k}
(\Fsquare(0.4cm,r) + \Fsquare(0.4cm,{\bar r})) = 0,
&(3.6{\rm b})\cr
%
&Res_{u=-{2r-b+2\over 2} + iu^{(b)}_k}
(\Flect(0.4cm,1.0cm,\overline{b+1}) + \Fsquare(0.4cm,{\bar b})) = 0
\quad 1 \le b \le r-1.
&(3.6{\rm c})\cr}
$$
%
Following section 2.4, this can be summarized in the diagram
%
$$
\Fsquare(0.4cm,1) {\buildrel 1 \over \longrightarrow} \Fsquare(0.4cm,2)
{\buildrel 2 \over \longrightarrow} \cdots
{\buildrel r-1 \over \longrightarrow} \Fsquare(0.4cm,r)
{\buildrel r \over \longrightarrow} \Fsquare(0.4cm,\bar{r})
{\buildrel {r-1} \over \longrightarrow}\cdots
{\buildrel 2 \over \longrightarrow}
\Fsquare(0.4cm,\bar{2})
{\buildrel 1 \over \longrightarrow}
\Fsquare(0.4cm,\bar{1})
$$
%
This turns out to be identical with the crystal graph [24,25].
%
\par\noindent
{\bf 3.2 Eigenvalue $\Lambda^{(a)}_1(u)$.}\hskip0.3cm
%
Let us proceed to $\Lambda^{(a)}_1(u)$, which can be constructed
from the elementary boxes (3.4).
%
For $1 \le a \le r$, let ${\cal T}^{(a)}_1$ be the set
of the tableaux of the form
%
$$
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\fsquare(0.5cm,\hbox{$i_1$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\fsquare(0.5cm,\hbox{$i_a$})$}\vskip-0.4pt
        }
      }
\eqno(3.7{\rm a})
$$
%
with entries $i_k \in J$ obeying the following conditions
%
$$\eqalignno{
&1 \preceq i_1 \prec i_2 \prec \cdots \prec i_a
\preceq {\bar 1},&(3.7{\rm b})\cr
%
&\hbox{If } i_k = c \hbox{ and } i_l = {\bar c},
\hbox{ then } r + k - l \ge c.&(3.7{\rm c})\cr}
$$
%
We remark that these constraints are very similar but
different from the crystal base [24,25], where
(3.7c) is replaced by $a+1+k-l \le c$.
%
We identify each element (3.7a) of ${\cal T}^{(a)}_1$ with
the product of (3.4) with the spectral parameters
$u+{a-1\over 2}, u+{a-3\over 2}, \ldots, u-{a-1\over 2}$
from the top to the bottom, namely,
%
$$\prod_{k=1}^a \Fsquare(0.4cm,i_k)
\vert_{u \rightarrow u + {a+1-2k\over 2}}. \eqno(3.8)
$$
%
Then the analytic Bethe ansatz yields the following DVF.
%
$$
\Lambda^{(a)}_1(u) = \sum_{T \in {\cal T}^{(a)}_1} T\quad
\qquad 1 \le a \le r,
\eqno(3.9)
$$
%
which reduces to (3.3) when $a=1$.
%
Let us observe consistency of this result before
proving that it is pole-free in section 3.3.
%
Firstly, the dress part of
%
$$
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\fsquare(0.5cm,\hbox{$1$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\fsquare(0.5cm,\hbox{$a$})$}\vskip-0.4pt
        }
      }
$$
%
is $Q_a(u-{1\over t_a})/Q_a(u+{1\over t_a})$,
telling that the above tableau indeed gives
the top term (2.12).
%
Secondly, the set ${\cal T}^{(a)}_1$ is invariant
under the interchange of the two tableaux
%
$$
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\fsquare(0.5cm,\hbox{$i_1$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\fsquare(0.5cm,\hbox{$i_a$})$}\vskip-0.4pt
        }
      }
%
\qquad
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\fsquare(0.5cm,\hbox{$\overline{i_a}$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\fsquare(0.5cm,\hbox{$\overline{i_1}$})$}\vskip-0.4pt
        }
      }
$$
%
and the crossing symmetry (2.18) is
valid among them.
%
Thirdly, the character limit (2.16) can be proved.
%
This is essentially done by showing
$$
\sharp {\cal T}^{(a)}_1 = \hbox{dim }V(\omega_a)
= {2r\choose a} - {2r\choose a-2},\eqno(3.10)
$$
%
which corresponds to the $q \rightarrow 1$ limit of (2.16)
since $W^{(a)}_1 \simeq V(\omega_a)$ as a $C_r$-module by (2.4a).
%
We have verified (3.10) by building injections in both directions
between the sets of depth $a$ tableaux (3.7a) breaking (3.7c)
and the depth $a-2$ ones only obeying the
constraint as (3.7b).
%
Once (3.10) is established, the weight counting in (2.16) for $q \neq 1$
is shown consistent with
eq.(2.2.2) of [25] by noting that the injections are
weight preserving and
%
$\lim_{u \rightarrow \infty, \vert q \vert > 1}
q^* \Fsquare(0.4cm,a) = q^{2(\omega^{(p)}_1\vert \epsilon_a)}$
for some $*$.
%
\par\noindent
{\bf 3.3 Pole-freeness of $\Lambda^{(a)}_1(u)$.}\hskip0.3cm
%
The quantity (3.9) passes the crucial
condition in the analytic Bethe ansatz, namely,
%
\proclaim Theorem 3.3.1.
$\Lambda^{(a)}_1(u) \, (1 \le a \le r)$
(3.9) is free of poles provided that
the BAE (2.7) (for $s=1$) is valid.
\par\noindent
%
For the proof we prepare a few Lemmas.
%
\proclaim Lemma 3.3.2.
For $1 \le b \le r-1$, the products
%
$$
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b$})$}\vskip-0.4pt
	     \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b+1$})$}\vskip-0.4pt
        }
      }
%
\qquad
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$\overline{b+1}$})$}\vskip-0.4pt
	     \hbox{$\Flect(0.4cm,1.0cm,\hbox{${\bar b}$})$}\vskip-0.4pt
        }
      }\eqno(3.11)
$$
%
with the spectral parameter $v$ ($v-1$) for the upper (lower)
box do not involve $Q_b$ function.
\par
The proof is straightforward by using the explicit form (3.4).
%
It is also elementary to check
%
\proclaim Lemma 3.3.3.
%
For $1 \le b \le r-1$, put
%
$$\eqalignno{
\Fsquare(0.4cm,b)_v \Flect(0.4cm,1.0cm,\overline{b+1})_{v-r+b}
&= {Q_b(v+{b\over 2}-1)\over Q_b(v+{b\over 2}+1)}X_1,
&(3.12{\rm a})\cr
%
\Fsquare(0.4cm,b)_v \Fsquare(0.4cm,\overline{b})_{v-r+b}
&= {Q_b(v+{b\over 2}-1)Q_b(v+{b\over 2}+2)
\over Q_b(v+{b\over 2})Q_b(v+{b\over 2}+1)}X_2,
&(3.12{\rm b})\cr
%
\Flect(0.4cm,1.0cm,b+1)_v \Fsquare(0.4cm,\overline{b})_{v-r+b}
&= {Q_b(v+{b\over 2}+2)\over Q_b(v+{b\over 2})}X_3,
&(3.12{\rm c})\cr}
$$
%
where the indices specify the spectral parameters attached
to the boxes (3.4).
%
Then $X_i$'s do not involve $Q_b$ function.
\par
%
The point is that (3.12a) and (3.12c)
have only one $Q_b$ function in their denominators after some
cancellations owing to the spectral parameter choice
$v, v-r+b$.
%
\proclaim Lemma 3.3.4.
For $1 \le b \le r-1$, let the tableaux
%
$$
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\eta$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$\overline{b+1}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\zeta$})$}\vskip-0.4pt
        }
      }
\quad\hbox{ or }\quad
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b+1$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\eta$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$\overline{b}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\zeta$})$}\vskip-0.4pt
        }
      }
\eqno(3.13)
$$
%
be the elements in ${\cal T}^{(a)}_1$ such that the columns
$\Fsquare(0.4cm,\xi), \Fsquare(0.4cm,\eta)$ and
$\Fsquare(0.4cm,\zeta)$ do not contain the boxes with entries
$b,b+1,\overline{b+1}$ and $\overline{b}$.
%
Then the length of $\Fsquare(0.4cm,\eta)$ is less than $r-b$.
\par
%
\noindent
{\it Proof.}\hskip0.3cm
%
We shall show this with respect to the first tableau in (3.13).
%
The proof for the second one is similar.
%
Suppose on the contrary that
the length $L$ of $\Fsquare(0.4cm,\eta)$ satisfies
$$L \ge r-b. \eqno(3.14)$$
%
Due to (3.7b) and (3.5), $\Fsquare(0.4cm,\eta)$
then consists of the elementary boxes with entries from
$\{b+2,b+3,\ldots,r,\overline{r},\overline{r-1},\ldots,\overline{b+2}\}$.
%
By (3.14), there must be at least one letter
$c \in \{b+2,b+3,\ldots,r\}$ such that
both $\Fsquare(0.4cm,c)$ and $\Fsquare(0.4cm,\overline{c})$
occur in $\Fsquare(0.4cm,\eta)$.
%
Let $c_0$ be the smallest among such $c$'s.
%
Then $\Fsquare(0.4cm,\eta)$ has the structure
%
$$
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,0.6cm,\hbox{$\eta_1$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,0.6cm,\hbox{$c_0$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,0.6cm,\hbox{$\eta_2$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,0.6cm,\hbox{$\overline{c_0}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,0.6cm,\hbox{$\eta_3$})$}\vskip-0.4pt
        }
      }\eqno(3.15)
$$
%
By the definition, the columns
$\Fsquare(0.4cm,\eta_1)$ and $\Fsquare(0.4cm,\eta_3)$
contain only boxes
$\Fsquare(0.4cm,q)$ and $\Fsquare(0.4cm,\overline{q})$
for $b+2 \le q \le c_0-1$, respectively.
%
Moreover,
$\Fsquare(0.4cm,q)$ and $\Fsquare(0.4cm,\overline{q})$
must not be present simultaneously.
%
Thus the total length $L_{13}$ of
$\Fsquare(0.4cm,\eta_1)$ and $\Fsquare(0.4cm,\eta_3)$
should satisfy
%
$$L_{13}\le (c_0-1)-(b+2)+1 = c_0-b-2.\eqno(3.16)$$
%
Since (3.15) is a part of the first tableau
in (3.13) belonging to ${\cal T}^{(a)}_1$,
the condition (3.7c) must be valid for the
$c_0, \overline{c_0}$ pair.
%
In terms of the length $L_2$ of the column
$\Fsquare(0.4cm,\eta_2)$, (3.7c) reads
%
$$r - L_2 - 1 \ge c_0.\eqno(3.17)$$
%
Combining (3.14), (3.16) and (3.17), we have the contradiction
%
$$r - b \le L = L_{13} + L_2 + 2 \le (c_0 - b - 2) +
(r - c_0 - 1) + 2 = r - b - 1,
$$
owing to the onset assumption
(3.14), and thus finish the proof.
%
\par
With these Lemmas, we proceed to\par\noindent
%
{\it Proof of Theorem 3.3.1.}\hskip0.3cm
We shall show that color $b$ singularity
is spurious, i.e.,
\par\noindent
$Res_{u = iu^{(b)}_k + \cdots}\Lambda^{(a)}_1(u) = 0$
for each $2 \le b \le r-1$.
%
The remaining cases $b=1$ and $r$ can be verified similarly and
more easily.
%
Among the elementary boxes (3.4a),
the factor $1/Q_b(u+\cdots)$ enters only
$\Fsquare(0.4cm,b), \Flect(0.4cm,1.0cm,b+1),
\Flect(0.4cm,1.0cm,\overline{b+1})$ and
$\Fsquare(0.4cm,{\bar b})$.
%
Thus one has to keep track of only these four
boxes appearing in (3.7a).
%
Accordingly, let us write (3.9) as
%
$\Lambda^{(a)}_1(u) = S_0 + S_1 + \cdots + S_4$, where
%
$S_k$ denotes the partial sum over the tableaux (3.7a)
containing precisely $k$ boxes among the above four.
%
Obviously $S_0$ is free of $1/Q_b(u+\cdots)$.
%
So is $S_4$ because the relevant tableaux
involve
both of the $2\times 1$ patterns in (3.11) and therefore
do not contain $Q_b$ by Lemma 3.3.2.
%
Next consider $S_1$ which is the sum over the tableaux of the form
%
$$
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,0.4cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,0.4cm,\hbox{$b$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,0.4cm,\hbox{$\eta$})$}\vskip-0.4pt
        }
      }
\quad
%
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b+1$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\eta$})$}\vskip-0.4pt
        }
      }
\quad
%
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$\overline{b+1}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\eta$})$}\vskip-0.4pt
        }
      }
\quad
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,0.4cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,0.4cm,\hbox{${\bar b}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,0.4cm,\hbox{$\eta$})$}\vskip-0.4pt
        }
      }
$$
%
Here $\Fsquare(0.4cm,\xi)$ and $\Fsquare(0.4cm,\eta)$
stand for columns with total length $a-1$ and they do not contain
$\Fsquare(0.4cm,b),
\Flect(0.4cm,1.0cm,b+1),
\Flect(0.4cm,1.0cm,\overline{b+1})$ and
$\Fsquare(0.4cm,\overline{b})$.
%
{}From (3.6),
color $b$ residues in the first and second (third and fourth) tableaux
sum up to zero.
%
By the same reason $S_3$ is free of color $b$ singularities since
the relevant tableaux must contain one of (3.11).
%
Thus we are left with $S_2$, whose summands are classified into
the following four types:
%
$$
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\eta$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$\overline{b+1}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\zeta$})$}\vskip-0.4pt
        }
      }
\quad
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\eta$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$\overline{b}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\zeta$})$}\vskip-0.4pt
        }
      }
\quad
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b+1$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\eta$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$\overline{b+1}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\zeta$})$}\vskip-0.4pt
        }
      }
\quad
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$b+1$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\eta$})$}\vskip-0.4pt
      \hbox{$\Flect(0.4cm,1.0cm,\hbox{$\overline{b}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.6cm,1.0cm,\hbox{$\zeta$})$}\vskip-0.4pt
        }
      }
\eqno(3.18)
$$
%
Here, $\Fsquare(0.4cm,\xi), \Fsquare(0.4cm,\eta)$ and
$\Fsquare(0.4cm,\zeta)$ are columns without
$\Fsquare(0.4cm,b),
\Flect(0.4cm,1.0cm,b+1),
\Flect(0.4cm,1.0cm,\overline{b+1})$ and
$\Fsquare(0.4cm,\overline{b})$.
%
We are going to show that the sum of the four tableaux (3.18)
is free of color $b$ singularity for any fixed
$\Fsquare(0.4cm,\xi), \Fsquare(0.4cm,\eta)$ and
$\Fsquare(0.4cm,\zeta)$.
%
Denoting their lengths by $k-1, l-k-1$ and $a-l$, respectively,
we consider the cases
$r+k-l\ge b+1, r+k-l=b$ and $r+k-l\le b-1$ separately.
%
If $r+k-l \ge b+1$, all the four tableaux (3.18) actually belong
to ${\cal T}^{(a)}_1$ and
the pole-freeness of their sum follows straightforwardly
from (3.6).
%
If $r+k-l = b$, the third tableau in (3.18) is absent
since it breaks (3.7c).
%
Up to an overall factor not containing $Q_b$,
the remaining three terms
are proportional to those in (3.12) for some $v$.
%
{}From Lemma 3.3.3,
their sum
has zero residue both at
$v=-{b\over 2}+iu^{(b)}_k$ by (3.6a) and at
$v=-{b\over 2}-1+iu^{(b)}_k$ by (3.6c).
%
Finally, we consider the case $r+k-l\le b-1$, when
the second and third tableaux in (3.18) do not exist
because they both break (3.7c).
%
In fact, the first and the fourth ones are also absent.
%
This is because $r+k-l \le b-1$ is equivalent to
saying that the length of
$\Fsquare(0.4cm,\eta)$ is not less than $r-b$
against Lemma 3.3.4.
%
Thus $S_2$ is free of color $b$ poles, which completes the
proof of the Theorem.
\par
\noindent
{\bf 3.4 Eigenvalue $\Lambda^{(1)}_m(u)$.}\hskip0.3cm
%
The result (3.9) accomplishes {\it Step} 1 in section 1.5
already with the tableau language sought in {\it Step} 3.
%
The remaining task is {\it Step} 2, i.e., to find the
eigenvalues $\Lambda^{(a)}_m(u)$ for higher $m$
by solving the $T$-system (2.5b) with
%
$$\eqalign{
g^{(a)}_m(u) &= 1 \qquad\hbox{ for } 1 \le a \le r-1,\cr
%
g^{(r)}_m(u) &= \prod_{k=1}^m g^{(r)}_1(u+m+1-2k),\cr
%
g^{(r)}_1(u) &= \psi_1(u+{r+1\over 2})
\psi_{\overline{1}}(u-{r+1\over 2}),\cr}
$$
%
under the
initial conditions $\Lambda^{(a)}_0(u) = 1$ and (3.9).
%
So far we have done this only partially to get a
conjecture on $\Lambda^{(1)}_m(u)$.
%
To present it we introduce
a set ${\cal T}^{(1)}_m$ ($m \in {\bf Z}_{\ge 1}$)
of the tableaux having the form
%
$$
\fsquare(0.4cm,i_1)\naga\fsquare(0.4cm,i_k)\hskip-0.4pt
\overbrace{
\addsquare(0.4cm,\overline{r})\addsquare(0.4cm,r)\naga
\fsquare(0.4cm,\overline{r})\addsquare(0.4cm,r)
}^{2n} \hskip-0.4pt
\addsquare(0.4cm,\overline{j_l})\naga
\fsquare(0.4cm,\overline{j_1})
\eqno(3.19{\rm a})
$$
with the conditions
%
$$\eqalignno{
&k, n, l \ge 0,\qquad k + 2n + l = m,&(3.19{\rm b})\cr
%
&1 \le i_1 \le i_2 \le \cdots \le i_k \le r,
\quad
1 \le j_1 \le j_2 \le \cdots \le j_l \le r.&(3.19{\rm c})\cr}
$$
%
Writing (3.19a) simply as
$\Fsquare(0.4cm,i_1)\naga\Fsquare(0.4cm,i_m)$ with
$i_k \in J$ (3.2),
we identify it with the product of (3.4) with
the spectral parameters
$u-{m-1\over 2}, u-{m-3\over 2}, \ldots,
u+{m-1\over 2}$ from the left to the right, namely,
%
$$\prod_{k=1}^m \Fsquare(0.4cm,i_k)
\vert_{u \rightarrow u - {m+1-2k\over 2}}.\eqno(3.20)
$$
%
Then we conjecture that the
$T$-system (2.5b) with the
initial condition (3.9) leads to
%
$$
\Lambda^{(1)}_m(u) = \sum_{T \in {\cal T}^{(1)}_m} T
\quad\qquad m \in {\bf Z}_{\ge 1}.\eqno(3.21)
$$
%
This is just (3.3) when $m=1$.
%
We remark that (3.21) consists of the correct number of terms,
%
$$\sharp {\cal T}^{(1)}_m = \hbox{dim } W^{(1)}_m.\eqno(3.22)
$$
%
To see this, note from (2.4a) that the rhs is equal to
%
$$\hbox{dim } V(m\omega_1) + \hbox{dim } V((m-2)\omega_1) + \cdots
+ \cases{
\hbox{dim } V(0)& $m$ even\cr
\hbox{dim } V(\omega_1) & $m$ odd\cr}.
\eqno(3.23)
$$
%
On the other hand, the set ${\cal T}^{(1)}_m$ is the disjoint
union of those tableaux (3.19a) with
$n=0, 1, 2, \ldots$.
%
Thus it suffices to check
%
$$\hbox{dim } V(m\omega_1) =
\sharp \{ (3.19{\rm a}) \in {\cal T}^{(1)}_m \mid n = 0 \}.
\eqno(3.24)
$$
%
Obviously the rhs is ${m+2r-1 \choose m}$, which
agrees with the lhs calculated from Weyl's dimension formula.
%
\par\noindent
{\bf 3.5 $C_2$ case.}\hskip0.3cm
For $C_2$ it is possible to provide the full solution
$\Lambda^{(1)}_m(u), \Lambda^{(2)}_m(u)$ to the
$T$-system [10].
%
In terms of the tableaux,
$\Lambda^{(1)}_m(u)$ in [10] is certainly given by (3.21)
up to an inessential overall scalar reflecting a
different convention on $\Lambda^{(a)}_0(u)$.
%
To present the other eigenvalue $\Lambda^{(2)}_m(u)$ there,
we introduce a set ${\cal T}^{(2)}_m$ of $2\times m$ tableaux
%
$$
	\normalbaselines\m@th\offinterlineskip
	\vcenter{
   \hbox{\fsquare(0.4cm,i_1)\naga \hskip-0.4pt\fsquare(0.4cm,i_m)}
	      \vskip-0.4pt
   \hbox{\fsquare(0.4cm,j_1)\naga\hskip-0.4pt\fsquare(0.4cm,j_m)}
    }\eqno(3.25{\rm a})
$$
%
obeying the conditions
%
$$\eqalignno{
&\hbox{Every column belongs to } {\cal T}^{(2)}_1
\hbox{ (3.7) for  } C_2,&(3.25{\rm b})\cr
%
&i_1 \preceq \cdots \preceq i_m,\quad
\hbox{and }\quad j_1 \preceq \cdots \preceq j_m,
&(3.25{\rm c})\cr
%
&\hbox{The column }
	\normalbaselines\m@th\offinterlineskip
	\vcenter{
   \hbox{\Fsquare(0.5cm,1)}
	      \vskip-0.4pt
   \hbox{\Fsquare(0.5cm,\overline{1})} }
   \hbox{ is contained at most once. }&(3.25{\rm d})\cr}
$$
%
We identify each element (3.25a) in ${\cal T}^{(2)}_m$ with
the product of (3.4) with the spectral parameters
as follows.
%
$$
\prod_{k=1}^m\Fsquare(0.4cm,i_k)
\vert_{u\rightarrow u-m-{1\over 2}+2k}\,
\prod_{k=1}^m\Fsquare(0.4cm,j_k)
\vert_{u\rightarrow u-m-{3\over 2}+2k}.\eqno(3.26)
$$
%
Namely, the shifts increase by 2 from the left to the right,
decrease by 1 from the top to the bottom and their average is 0.
%
Then the result in [10] reads
%
$$
\Lambda^{(2)}_m(u) = \sum_{T \in {\cal T}^{(2)}_m} T.
\eqno(3.27)
$$
%
%
\beginsection 4. Eigenvalues for $B_r$

As in the $C_r$ case we first introduce elementary boxes
attached to the vector representation.
%
Using them as the building blocks, we will construct the DVF for
$\Lambda^{(a)}_1(u) (1 \le a \le r-1)$ and prove its pole-freeness
under the BAE.
%
We also conjecture an explicit form of
$\Lambda^{(a)}_m(u)  (1 \le a \le r-1)$
in terms of tableaux made of these boxes.
\par
%
Compared with the $C_r$ case, a distinct feature in $B_r$ (and $D_r$)
is the existence of the spin representation.
%
Any finite dimensional irreducible $B_r$-module is
generated by decomposing a tensor product of the spin representation.
%
Thus we introduce
another kind of elementary boxes
attached to the spin representation.
%
It enables a unified description of the DVFs for
all the fundamental eigenvalues
$\Lambda^{(a)}_1(u)  (1 \le a \le r)$.
%
An explicit relation will be given between the two kinds
of the elementary boxes.
%
\par\noindent
%
{\bf 4.1 Eigenvalue $\Lambda^{(1)}_1(u)$.}\par\noindent
%
Let $\epsilon_a, 1 \le a \le r$ be the orthonormal vectors
$(\epsilon_a \vert \epsilon_b) = \delta_{a b}$
realizing the root system as follows.
%
$$\eqalign{
\alpha_a &= \cases{\epsilon_a - \epsilon_{a+1} &
for $1 \le a \le r-1$\cr
\epsilon_r & for $a = r$\cr},\cr
%
\omega_a &=
\cases{
\epsilon_1 + \cdots + \epsilon_a& for $1 \le a \le r-1$\cr
%
{1 \over 2}(\epsilon_1 + \cdots + \epsilon_r)& for $a = r$\cr}.\cr}
\eqno(4.1)$$
%
The auxiliary space relevant to
$\Lambda^{(1)}_1(u)$ is
$W^{(1)}_1 \simeq V(\omega_1)$ as an $B_r$-module.
%
This is the vector representation, whose weights are
$\epsilon_a, -\epsilon_a (1 \le a \le r)$ and $0$.
%
By abbreviating them to $a, \overline{a}$
and $0$, the set of weights is given by
%
$$
J = \{1,2,\ldots,r,0,\overline{r},\ldots,\overline{1}\}.
\eqno(4.2)
$$
%
All the weights are multiplicity-free, therefore
one can determine the DVF from (2.12) and (2.14).
%
The result reads
%
$$\Lambda^{(1)}_1(u) = \sum_{a \in J}\Fsquare(0.4cm,a),
\eqno(4.3)$$
%
which is formally the same with (3.3).
%
The elementary boxes here are defined by
%
$$\eqalign{
\Fsquare(0.5cm,a)  &= \psi_{a}(u)
      {{Q_{a-1}( u+a+1  )
            Q_{a}(u+a-2 )}\over
       { Q_{a-1}(u+a-1)Q_{a}(u+a)}}
  \qquad 1\le a \le r,\cr
\Fsquare(0.5cm, 0)  &=
  \psi_0(u) {{Q_r(u+r-2) Q_{r}(u+r+1)}\over
   { Q_{r}(u+r)Q_{r}(u+r-1)}},  \cr
\Fsquare(0.5cm,\bar{a})  &=
\psi_{\bar{a}}(u)
      {{Q_{a-1}( u+2r-a-2  )
            Q_{a}(u+2r-a+1 )}\over
       { Q_{a-1}(u+2r-a)Q_{a}(u+2r-a-1)}}
  \qquad 1\le a \le r,\cr
}\eqno(4.4{\rm a})$$
%
where we have set $Q_0(u) = 1$.
%
The vacuum parts $\psi_{a}(u)$ depend on the quantum space
$\otimes_{j=1}^N W^{(p)}_1(w_j)$
and are given by
%
$$\eqalign{
\psi_a(u) =\cases{
\phi(u+p+{1\over t_p})\phi(u+2r-p-1+{1\over t_p}) \Phi^r_p(u)
&   for $1 \preceq a \preceq p$\cr
\phi(u+p-{1\over t_p})\phi(u+2r-p-1+{1\over t_p}) \Phi^r_p(u)
&   for $p+1 \preceq a \preceq \overline{p+1}$   \cr
\phi(u+p-{1\over t_p})\phi(u+2r-p-1-{1\over t_p}) \Phi^r_p(u)
&   for $\overline{p} \preceq a \preceq \overline{1}$ \cr}
}\eqno(4.4{\rm b})$$
where
$$\eqalign{
\Phi^r_p(u) &= \prod_{j=1}^{p-1}
\phi(u+p-2j-{1\over t_p})\phi(u+2r-p+2j-1+{1\over t_p})\cr
%
&= \Phi^r_p(-2r+1-u)\vert_{w_k \rightarrow -w_k}.\cr}
\eqno(4.4{\rm c})
$$
%
The common factor $\Phi^r_p(u)$ here will play a role in section 4.7,
where the boxes here are related to those in section 4.5.
%
The order $\prec$ in the set $J$ (4.2) is defined by
%
$$
1 \prec 2 \prec \cdots \prec r \prec 0 \prec
\overline{r} \prec \cdots \prec \overline{2} \prec
\overline{1}. \eqno(4.5)
$$
%
Note the top term $\Fsquare(0.4cm,1)$ (2.12),
the dress universality (2.11) and the crossing symmetry
(2.18) for the pairs
$\Fsquare(0.4cm,a) \leftrightarrow
\Fsquare(0.4cm,\overline{a})$ and
$\Fsquare(0.4cm,0) \leftrightarrow \Fsquare(0.4cm,0)$.
%
Under the BAE (2.7), (4.3) is pole-free because
the coupling rule (2.14) and (2.15) have been embodied as
%
$$\eqalignno{
&Res_{u=-b + iu^{(b)}_k}
(\Fsquare(0.4cm,b) + \Flect(0.4cm,1.0cm,b+1)) = 0\quad 1 \le b \le r-1,
&(4.6{\rm a})\cr
%
&Res_{u=-r + iu^{(r)}_k}
(\Fsquare(0.4cm,r) + \Fsquare(0.4cm,0)) = 0,
&(4.6{\rm b})\cr
%
&Res_{u=-r+1+iu^{(r)}_k}
(\Fsquare(0.4cm,0) + \Fsquare(0.4cm,\overline{r})) = 0,
&(4.6{\rm c})\cr
%
&Res_{u=-2r+b+1 + iu^{(b)}_k}
(\Flect(0.4cm,1.0cm,\overline{b+1}) + \Fsquare(0.4cm,{\bar b})) = 0
\quad 1 \le b \le r-1.
&(4.6{\rm d})\cr}
$$
%
Thus we have a diagram
%
$$
\Fsquare(0.4cm,1) {\buildrel 1 \over \longrightarrow} \Fsquare(0.4cm,2)
{\buildrel 2 \over \longrightarrow} \cdots
{\buildrel r-1 \over \longrightarrow} \Fsquare(0.4cm,r)
{\buildrel r \over \longrightarrow} \Fsquare(0.4cm,0)
{\buildrel r \over \longrightarrow} \Fsquare(0.4cm,\bar{r})
{\buildrel {r-1} \over \longrightarrow} \cdots
{\buildrel 2 \over \longrightarrow}
\Fsquare(0.4cm,\bar{2})
{\buildrel 1 \over \longrightarrow}
\Fsquare(0.4cm,\bar{1})
$$
%
This is again identical with the crystal graph [24,25].
%
For $p=1$, (4.3) has been known earlier in [21].
%
\par\noindent
{\bf 4.2 Eigenvalue $\Lambda^{(a)}_1(u)$ for $1 \le a \le r-1$.}
\hskip0.3cm
For $1 \le a \le r-1$, let ${\cal T}^{(a)}_1$ be the set of
the tableaux of the form (3.7a) with $i_k \in J$ (4.2)
obeying the condition
%
$$
i_k \prec i_{k+1} \hbox{ or } i_k = i_{k+1} = 0
\,\,\hbox{ for any } 1 \le k \le a-1.\eqno(4.7)
$$
%
Namely, the entries must increase strictly from the top to the bottom
in the sense of (4.5) except a possible segment of consecutive $0$'s.
%
We identify each element (3.7a) of ${\cal T}^{(a)}_1$ with
the product of (4.4a) with the spectral parameters
$u+a-1, u+a-3,\ldots,u-a+1$ from the top to the bottom
%
$$
\prod_{k=1}^a \Fsquare(0.4cm,i_k)
\vert_{u \rightarrow u+a+1-2k}.\eqno(4.8)
$$
%
Then the analytic Bethe ansatz yields the following DVF.
%
$$
\Lambda^{(a)}_1(u) =
{1\over F^{(p,r)}_a(u)}
\sum_{T \in {\cal T}^{(a)}_1} T\quad
\qquad 1 \le a \le r-1,\eqno(4.9{\rm a})
$$
%
where the scalar $F^{(p,r)}_a(u)$ is defined by
%
$$\eqalign{
&F^{(p,r)}_a(u) \cr
&= \prod_{j=1}^{a-1}\prod_{k=0}^{p-1}
\phi(u+p+a-1-{1\over t_p}-2j-2k)
\phi(u+2r-p-a+{1\over t_p}+2j+2k)\cr
%
&= F^{(p,r)}_a(-2r+1-u)\vert_{w_k \rightarrow -w_k}.\cr}
\eqno(4.9{\rm b})
$$
%
Notice that $F^{(p,r)}_1(u) = 1$ hence (4.9a)
reduces to (4.3) when $a=1$.
%
{}From (4.4c) and (4.27c) in section 4.5, (4.9b) can also be written as
%
$$\eqalignno{
F^{(p,r)}_a(u) &=
\prod_{j=1}^{a-1}\phi(u+p+a-1-{1\over t_p}-2j)
\phi(u+2r+p+a-2+{1\over t_p}-2j)\cr
&\times\prod_{j=1}^{a-1}\Phi^r_p(u+a-1-2j)
&(4.10{\rm a})\cr
%
&= \prod_{j=1}^{a-1}
\psi^{(p,r)}_0(u+r-a-{1\over 2}+2j)
\psi^{(p,r)}_p(u-r+a+{1\over 2}-2j).
&(4.10{\rm b})\cr}
$$
%
By using (4.10a), it can be checked that
each summand $T$ in (4.9a) contains the factor
$F^{(p,r)}_a(u)$ and $\Lambda^{(a)}_1(u)$ is homogeneous
of order $2p$ w.r.t $\phi(u+\cdots)$.
%
This will be seen more manifestly
in Theorem 4.7.1.
%
One can observe the top term and the crossing symmetry
in the DVF (4.9a) as done after (3.9).
%
The character limit (2.16) is also valid.
%
To see this, we introduce a map $\chi$ from ${\cal T}^{(a)}_1$
to the Laurent polynomials
${\bf C}[z_1,z^{-1}_1,\ldots,z_r,z^{-1}_r]$ by
%
$$
\raise 4ex \hbox{$\chi \Biggl($}
  \hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\fsquare(0.5cm,\hbox{$i_1$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\fsquare(0.5cm,\hbox{$i_a$})$}\vskip-0.4pt
        }
      }
\raise 4ex \hbox{$\Biggr) = y_{i_1} \cdots y_{i_a}$,}
\eqno(4.11{\rm a})
$$
%
where
$$
y_0 = 1,\, y_a = z_a, \, y_{\overline{a}} = z^{-1}_a, \, 1 \le a \le r.
\eqno(4.11{\rm b})
$$
%
In view of
$\lim_{u \rightarrow \infty, \vert q \vert > 1}
q^* \Fsquare(0.4cm,a) = q^{2(\omega^{(p)}_1\vert \epsilon_a)}$ for some
$*$, this corresponds to taking the limit (2.16) of
the element (3.7a).
%
Since $W^{(a)}_1 \simeq
V(\omega_a) \oplus V(\omega_{a-2}) \oplus \cdots$ from (2.4b),
%
we are to show
%
$$
\sum_{T \in {\cal T}^{(a)}_1} \chi(T) =
ch V(\omega_a) + ch V(\omega_{a-2}) + \cdots,
\eqno(4.11{\rm c})
$$
%
for $1 \le a \le r-1$.
%
Here $ch V$ denotes the classical character of the $B_r$-module
$V$ on letters $z_1, \ldots, z_r$.
%
This can be easily proved from (4.7) and the known formula
%
$$
ch V(\omega_a) =
\sum_{\scriptstyle i_1, \ldots, i_a \in J \atop
       \scriptstyle i_1 \prec \cdots \prec i_a}
       y_{i_1} \cdots y_{i_a},\eqno(4.11{\rm d})
$$
%
for $1 \le a \le r-1$.
%
Eq.(4.11d) originates in
$so(2r+1) \hookrightarrow gl(2r+1)$.
%
\par\noindent
{\bf 4.3 Pole-freeness of $\Lambda^{(a)}_1(u)$ for $1 \le a \le r-1$.}
\hskip0.3cm
%
The purpose of this section is to show
%
\proclaim Theorem 4.3.1.
$\Lambda^{(a)}_1(u) (1 \le a \le r-1)$ (4.9) is free of poles
provided that the BAE (2.7) (for $s=1$) is valid.
\par
For the proof we need
%
\proclaim Lemma 4.3.2. For $n \in {\bf Z}_{\ge 0}$, put
%
$$\eqalignno{
\prod_{j=0}^n \Fsquare(0.4cm,0)_{v-2j} &=
{Q_r(v+r+1)Q_r(v+r-2n-2)\over
Q_r(v+r)Q_r(v+r-2n-1)}X_1,&(4.12{\rm a})\cr
%
\Fsquare(0.4cm,r)_v \prod_{j=1}^n \Fsquare(0.4cm,0)_{v-2j} &=
{Q_r(v+r-1)Q_r(v+r-2n-2)\over
Q_r(v+r)Q_r(v+r-2n-1)}X_2,&(4.12{\rm b})\cr
%
\Fsquare(0.4cm,\overline{r})_{v-2n}
\prod_{j=0}^{n-1} \Fsquare(0.4cm,0)_{v-2j} &=
{Q_r(v+r+1)Q_r(v+r-2n)\over
Q_r(v+r)Q_r(v+r-2n-1)}X_3,&(4.12{\rm c})\cr
%
\Fsquare(0.4cm,r)_v \Fsquare(0.4cm,\overline{r})_{v-2n}
\prod_{j=1}^{n-1} \Fsquare(0.4cm,0)_{v-2j} &=
{Q_r(v+r-1)Q_r(v+r-2n)\over
Q_r(v+r)Q_r(v+r-2n-1)}X_4,&(4.12{\rm d})\cr}
$$
%
where the indices specify the spectral parameters attached
to the boxes (4.4).
%
Then
$$\eqalignno{
&X_i\hbox{'s do not involve } Q_r \hbox{ function},
&(4.13{\rm a})\cr
%
&{Q_r(v+r\pm 1)\over Q_r(v+r)} \hbox{ comes from the box }
\Fsquare(0.4cm,\ast)_v,
&(4.13{\rm b})\cr
%
&{Q_r(v+r-2n-1\pm 1)\over Q_r(v+r-2n-1)} \hbox{ comes from the box }
\Fsquare(0.4cm,\ast)_{v-2n},
&(4.13{\rm c})\cr}
$$
%
where $\ast = r, \overline{r}$ or $0$.
\par
%
This can be verified by a direct calculation.
%
\proclaim Lemma 4.3.3. If the BAE (2.7) ($s=1$) is valid, then
%
$$\eqalign{
&Res_{v=-r+iu^{(r)}_k}((4.12{\rm a}) + (4.12{\rm b})) =
 Res_{v=-r+iu^{(r)}_k}((4.12{\rm c}) + (4.12{\rm d})) = 0,\cr
%
&Res_{v=-r+2n+1+iu^{(r)}_k}((4.12{\rm a}) + (4.12{\rm c})) \cr
&\quad = Res_{v=-r+2n+1+iu^{(r)}_k}((4.12{\rm b}) + (4.12{\rm d})) = 0.\cr}
\eqno(4.14)
$$
%
\par
%
This follows from (4.13b,c) and (4.6b,c).
%
Now we proceed to
\par\noindent
%
{\it Proof of Theorem 4.3.1.}\hskip0.3cm
%
As remarked after (4.10), there is no pole originated from
the overall scalar $1/F^{(p,r)}_a(u)$ in (4.9a).
%
Thus one has only to show that the apparent color $b$ poles
$1/Q_b(u+\cdots)$ in
$\sum_{T \in {\cal T}^{(a)}_1} T$
are spurious for all $1\le b \le r$
under the BAE.
%
For $1\le b \le r-1$, this can be done similarly to
the proof of Theorem 3.3.1.
%
In fact the present case is much easier since (4.7) is
so compared with (3.7b,c).
%
Henceforth we focus on the $b=r$ case which needs a separate consideration.
%
{}From (4.4a), we have only to keep track of the boxes
$\Fsquare(0.4cm,r), \Fsquare(0.4cm,0)$ and
$\Fsquare(0.4cm,\overline{r})$ containing $Q_r$.
%
Let us classify the tableaux (3.7a) in ${\cal T}^{(a)}_1$ (4.7)
into the sectors labeled by
the number $n$ of $\Fsquare(0.4cm,0)$'s contained in them.
%
In each sector, we further divide the tableaux into four types
according to the entries $(u,d)$ in the boxes just above
and below the consecutive $\Fsquare(0.4cm,0)$'s.
%
$$\eqalign{
&\hbox{type } 1_n: u \neq r \hbox{ and } d \neq \overline{r},\cr
&\hbox{type } 2_n: u  =   r \hbox{ and } d \neq \overline{r},\cr
&\hbox{type } 3_n: u \neq r \hbox{ and } d  =   \overline{r},\cr
&\hbox{type } 4_n: u  =   r \hbox{ and } d  =   \overline{r}.\cr
}$$
%
Thus we have
%
$$\eqalignno{
\sum_{T \in {\cal T}^{(a)}_1} T
&= \sum_{n=0}^a \sum_{i=1}^4 S_{n,i},&(4.15{\rm a})\cr
%
S_{n,i} &= \sum_{T \in \hbox{type } i_n} T,&(4.15{\rm b})\cr
%
S_{a,2} &= S_{a,3} = S_{a,4} = S_{a-1,4} = 0.
&(4.15{\rm c})\cr}
$$
%
Consider the following quartet of the tableaux of
types $1_{n+1}, 2_n, 3_n$ and $4_{n-1}$, respectively.
%
$$
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.8cm,0.5cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\Flect(0.8cm,0.5cm,\hbox{$\eta$})$}\vskip-0.4pt

        }
      }
\quad\qquad
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.8cm,0.5cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$r$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\Flect(0.8cm,0.5cm,\hbox{$\eta$})$}\vskip-0.4pt

        }
      }
\quad\qquad
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.8cm,0.5cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$\overline{r}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.8cm,0.5cm,\hbox{$\eta$})$}\vskip-0.4pt

        }
      }
\quad\qquad
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Flect(0.8cm,0.5cm,\hbox{$\xi$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$r$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$0$})$}\vskip-0.4pt
      \hbox{$\fsquare(0.5cm,\hbox{$\overline{r}$})$}\vskip-0.4pt
      \hbox{$\Flect(0.8cm,0.5cm,\hbox{$\eta$})$}\vskip-0.4pt

        }
      }
\eqno(4.16)
$$
%
Here, $\Fsquare(0.4cm,\xi)$ and $\Fsquare(0.4cm,\eta)$
are the columns with total length $a-n-1$ and they
do not contain $\Fsquare(0.4cm,r),
\Fsquare(0.4cm,0)$ and $\Fsquare(0.4cm,\overline{r})$.
%
In view of (4.8) and (4.13a), the tableaux (4.16) are proportional
to the four terms (4.12) with some $v$ up to an overall factor
not containing $Q_r$.
%
Thus from Lemma 4.3.3, their sum is free of
color $r$ singularity.
%
This is true for any fixed
$\Fsquare(0.4cm,\xi)$ and $\Fsquare(0.4cm,\eta)$ such that
the tableaux (4.16) belong to ${\cal T}^{(a)}_1$.
%
Therefore $S_{n+1,1} + S_{n,2} + S_{n,3} + S_{n-1,4}$
is free of color $r$ singularity for each
$1 \le n \le a-1$.
%
Due to (4.15c), the remaining terms in (4.15a) are
$S_{1,1}, S_{0,1}, S_{0,2}$ and $S_{0,3}$.
%
By the definition $S_{0,1}$ is independent of $Q_r$ and
it is straightforward to check that
$S_{1,1}+S_{0,2}+S_{0,3}$ is free of color $r$ singularity
by using (4.6b,c).
%
This establishes the Theorem.
\par\noindent
{\bf 4.4 Eigenvalue $\Lambda^{(a)}_m(u)$ for $1\le a \le r-1$.}
\hskip0.3cm
Starting from (4.9) and $\Lambda^{(r)}_1(u)$
that will be described in section 4.5, we are to solve
the $T$-system (2.5a) with
%
$$\eqalign{
g^{(a)}_m(u) &= 1 \qquad \hbox{ for } 2 \le a \le r,\cr
%
g^{(1)}_m(u) &= \prod_{k=1}^m g^{(1)}_1(u+m+1-2k),\cr
%
g^{(1)}_1(u) &= F^{(p,r)}_2(u),\cr}
\eqno(4.17)
$$
where the last quantity has been given in (4.9b) and (4.10).
%
The solution will yield a DVF
for the general eigenvalue $\Lambda^{(a)}_m(u)$.
%
Here we shall present the so derived conjecture for $1 \le a \le r-1$.
%
\par
Let ${\cal T}^{(a)}_m (1 \le a \le r-1)$ be the set of the
$a \times m$ rectangular tableaux containing
$\Fsquare(0.5cm,i_{j k}), \, i_{j k} \in J$ at the
$(j,k)$ position.
%
$$
	\normalbaselines\m@th\offinterlineskip
	\vcenter{
   \hbox{$\Flect(0.7cm,0.7cm,\hbox{$i_{1 1}$})\hskip-0.4pt
         \Flect(0.7cm,1.5cm,\hbox{$\cdots$}) \hskip-0.4pt
         \Flect(0.7cm,0.7cm,\hbox{$i_{1 m}$})$}
    \vskip-0.4pt
   \hbox{$\Flect(1.2cm,0.7cm,\hbox{$\vdots$})\hskip-0.4pt
         \Flect(1.2cm,1.5cm,\hbox{$\ddots$}) \hskip-0.4pt
         \Flect(1.2cm,0.7cm,\hbox{$\vdots$})$
         }
     \vskip-0.4pt
   \hbox{$\Flect(0.7cm,0.7cm,\hbox{$i_{a 1}$})\hskip-0.4pt
         \Flect(0.7cm,1.5cm,\hbox{$\cdots$}) \hskip-0.4pt
         \Flect(0.7cm,0.7cm,\hbox{$i_{a m}$})$}
         }
$$
%
The entries are to obey the conditions
%
$$\eqalignno{
&i_{j k} \prec i_{j+1 k} \hbox{ or }
i_{j k} = i_{j+1 k} = 0 \hbox{ for any }
1 \le j \le a-1, 1 \le k \le m,
&(4.18{\rm a})\cr
%
&i_{j k} \prec i_{j k+1} \hbox{ or }
i_{j k}  = i_{j k+1} \in J \setminus \{ 0 \}\,\,
\hbox{ for any } 1 \le j \le a, 1 \le k \le m-1.
&(4.18{\rm b})\cr}
$$
%
Notice that (4.18a) is equivalent to saying that
each column belongs to ${\cal T}^{(a)}_1$
defined in (4.7).
%
We identify each element of ${\cal T}^{(a)}_m$ as above
with the following product of (4.4a):
%
$$
\prod_{j=1}^a\prod_{k=1}^m \Fsquare(0.5cm,i_{j k})
\vert_{u \rightarrow u+a-m-2j+2k}.\eqno(4.19)
$$
%
Then we conjecture that the $T$-system (2.5a) with
(4.17) and the initial condition (4.9) leads to
%
$$
\Lambda^{(a)}_m(u) =
{1\over \prod_{k=1}^m F^{(p,r)}_a(u-m-1+2k)}
\sum_{T \in {\cal T}^{(a)}_m} T\qquad
1 \le a \le r-1, m \in {\bf Z}_{\ge 1}.
\eqno(4.20)
$$
%
{}From (4.18a) and the remark after (4.10),
the rhs is homogeneous of degree $2pm$ w.r.t $\phi$.
%
The conjecture (4.20) reduces to (4.9a) when $m=1$.
%
For $B_2$, (4.20) is certainly true
because $\Lambda^{(1)}_m(u)$ of $B_2$ equals
$\Lambda^{(2)}_m(u)$ of $C_2$ given in (3.27)
under the exchange $Q_1(u) \leftrightarrow Q_2(u)$.
%
The cases $m=2, a=1,2$ have also been checked directly for
$B_3$ and $B_4$.
%
As a further support, we have verified
$\sharp {\cal T}^{(a)}_m = \hbox{dim } W^{(a)}_m$
by computer for several values of $a$ and $m$.
%
For example, both sides yields 247500 for $B_5, a=m=3$.
%
We emphasize that the set ${\cal T}^{(a)}_m$ is
specified by a remarkably simple rule (4.18).
%
It would deserve to be a proper base of the $U_q(B^{(1)}_r)$ or
$Y(B_r)$ module $W^{(a)}_m$ having the classical content (2.4b).
\par\noindent
{\bf 4.5 Eigenvalue $\Lambda^{(r)}_1(u)$.}\hskip0.3cm
%
{}From (2.4b)
the relevant auxiliary space is $W^{(r)}_1 \simeq V(\omega_r)$
as a $B_r$-module.
%
This is the spin representation, whose weights are
all multiplicity-free and given by
%
$$
{1 \over 2}(\mu_1 \epsilon_1 + \cdots + \mu_r \epsilon_r),
\qquad \mu_1, \ldots, \mu_r = \pm.\eqno(4.21)
$$
%
Thus we shall introduce another kind of
elementary boxes
$\Flect(0.4cm,2.5cm,{\mu_1,\mu_2,\ldots,\mu_r})$
by which the DVF can be written as
%
$$
\Lambda^{(r)}_1(u)=\sum_{\{\mu_j = \pm\}}
\overbrace{\Flect(0.4cm,2.5cm,{\mu_1,\mu_2, \cdots , \mu_r})}^{r}_p.
\eqno(4.22)
$$
%
We let ${\cal T}^{(r)}_1$ denote the set of
$\hbox{dim } W^{(r)}_1 = 2^r$ boxes as above.
%
The indices $r$ and $p$ here signify the rank of $B_r$ and
the quantum space
$\otimes_{j=1}^N W^{(p)}_1(w_j)$, respectively.
%
Each box is identified with a product of dress and vacuum parts
that are defined via certain
recursion relations w.r.t these indices.
%
To describe them we introduce the
operators $\tau^u_\gamma, \tau^Q$ and
$\tau^C_\gamma$ acting on the DVF (2.9b) as follows.
%
$$\eqalignno{
\tau^u_\gamma &: u \rightarrow u + \gamma,&(4.23{\rm a})\cr
%
\tau^Q &: Q_a(u) \rightarrow Q_{a+1}(u),&(4.23{\rm b})\cr
%
\tau^C_\gamma &: Q_a(u+x) \rightarrow Q_a(u+\gamma-x), \,
                 \phi(u+x) \rightarrow \phi(u+\gamma-x)
                 \hbox{ for any } x.&(4.23{\rm c})\cr}
$$
%
By the definition they obey the relations
%
$$\eqalignno{
&\tau^Q \tau^u_\gamma = \tau^u_\gamma \tau^Q,
\quad
\tau^Q \tau^C_\gamma = \tau^C_\gamma \tau^Q,&(4.24{\rm a})\cr
%
&\tau^C_\gamma \tau^C_{\gamma^\prime} =
\tau^u_{\gamma - \gamma^\prime},
\quad
\tau^u_{\gamma} \tau^u_{\gamma^\prime} =
\tau^u_{\gamma + \gamma^\prime}.&(4.24{\rm b})\cr
}$$
%
In view of (2.8), $\tau^Q$ is equivalent to
$N_a \rightarrow N_{a+1}$ and $u^{(a)}_j \rightarrow u^{(a+1)}_j$.
%
It is to be understood as replacing $Q_a(u)$ with
$1 \le a \le r-1$ for $B_{r-1}$ by $Q_{a+1}(u)$ for $B_r$.
%
The operator $\tau^C_\gamma$ will be used to describe the
transformation (2.19) concerning the crossing symmetry.
%
Now the recursion relations read,
%
$$\eqalignno{
\overbrace{\Flect(0.4cm, 1.5cm, {+,+,\xi })}^{r}_p &=
\phi(u+r+p-{3\over 2}+{1\over t_p})
\tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {+,\xi })}^{r-1}_{p-1},
&(4.25{\rm a})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {+,-,\xi })}^{r}_p &=
\phi(u+r+p-{3\over 2}+{1\over t_p})
{Q_1(u+r-{5\over 2})\over Q_1(u+r-{1\over 2})}
\tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {-,\xi })}^{r-1}_{p-1},
&(4.25{\rm b})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {-,+,\xi })}^{r}_p &=
\phi(u+r-p+{1\over 2}-{1\over t_p})
{Q_1(u+r+{3\over 2})\over Q_1(u+r-{1\over 2})}
\tau^u_2 \tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {+,\xi })}^{r-1}_{p-1},
&(4.25{\rm c})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {-,-,\xi })}^{r}_p &=
\phi(u+r-p+{1\over 2}-{1\over t_p})
\tau^u_2 \tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {-,\xi })}^{r-1}_{p-1},
&(4.25{\rm d})\cr
}$$
%
where $\xi$ denotes arbitrary sequence of $\pm$ symbols with length
$r-2$.
%
The recursions (4.25) are valid for
$1 \le p \le r$ and $r \ge 3$.
%
The initial condition is given by
%
$$\eqalign{
dr \overbrace{\Flect(0.4cm,0.8cm,{+,+})}^{2}_{p}
   &={Q_2(u-{1\over 2}) \over Q_2(u+{1\over 2})},
   \qquad\qquad  \quad\,
vac \overbrace{\Flect(0.4cm,0.8cm,{+,+})}^{2}_{p}
= \cases{\phi(u+{5\over 2}) & $p=1$\cr
         \phi(u+1)\phi(u+3) & $p=2$\cr},\cr
%
dr \overbrace{\Flect(0.4cm,0.8cm,{+,-})}^{2}_{p}
   &={Q_1(u-{1\over 2})Q_2(u+{3\over 2}) \over
      Q_1(u+{3\over 2})Q_2(u+{1\over 2})},
   \quad
vac \overbrace{\Flect(0.4cm,0.8cm,{+,-})}^{2}_{p}
= \cases{\phi(u+{5\over 2}) & $p=1$\cr
         \phi(u)\phi(u+3) & $p=2$\cr},\cr
%
dr \overbrace{\Flect(0.4cm,0.8cm,{-,+})}^{2}_{p}
   &={Q_1(u+{7\over 2})Q_2(u+{3\over 2}) \over
      Q_1(u+{3\over 2})Q_2(u+{5\over 2})},
   \quad
vac \overbrace{\Flect(0.4cm,0.8cm,{-,+})}^{2}_{p}
= \cases{\phi(u+{1\over 2}) & $p=1$\cr
         \phi(u)\phi(u+3) & $p=2$\cr},\cr
%
dr \overbrace{\Flect(0.4cm,0.8cm,{-,-})}^{2}_{p}
   &={Q_2(u+{7\over 2}) \over Q_2(u+{5\over 2})},
   \qquad\qquad  \quad\,
vac \overbrace{\Flect(0.4cm,0.8cm,{-,-})}^{2}_{p}
= \cases{\phi(u+{1\over 2}) & $p=1$\cr
         \phi(u)\phi(u+2) & $p=2$\cr}.\cr}
\eqno(4.26{\rm a})
$$
%
Note that one formally needs the dress and the vacuum parts
for $p=0$ when applying (4.25) with $p=1$.
%
As for the vacuum parts we fix this by putting
$$
vac \overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_0 = 1
\quad \hbox{ for any } r \hbox{ and }  \{ \mu_j \}.
\eqno(4.26{\rm b})
$$
%
As for the dress parts  we simply let
$dr \overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p$
be the same for any $0 \le p \le r$.
%
This is consistent with (4.26a) and the dress universality (2.11).
%
Under these setting
the recursions (4.25) and the initial condition (4.26) provide
a complete characterization of our
$\overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p$
for any $0 \le p \le r, r \ge 2$ and
$\{ \mu_j \}$.
%
Thus we have presented the DVF (4.22) for the eigenvalue
$\Lambda^{(r)}_1(u)$.
%
In the rational case ($q \rightarrow 1$)
with $p=1$, a similar recursive
description is available in [5].
\par
%
Let us observe various features of our DVF (4.22) before
proving that it is pole-free in section 4.6.
%
Firstly, it is easy to calculate the vacuum parts
explicitly.
%
$$\eqalignno{
&vac \overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p
= \psi^{(p,r)}_n(u),&(4.27{\rm a})\cr
%
&n = \sharp\{ j \mid \mu_j = -, 1 \le j \le p \},&(4.27{\rm b})\cr
%
&\psi^{(p,r)}_n(u) =
\prod_{j=0}^{n-1}\phi(u+r-p+2j+{1\over 2} - {1\over t_p})\cr
%
&\quad\qquad
\times \prod_{j=n}^{p-1}\phi(u+r-p+2j+{1\over 2} + {1\over t_p})\quad
0 \le n \le p.&(4.27{\rm c})\cr}
$$
%
This is order $p$ w.r.t $\phi$.
%
Secondly,
the top term
is given by
%
$$
\overbrace{\Flect(0.4cm,2.4cm,{+,+,\ldots, +})}^r_p =
\psi^{(p,r)}_0(u){Q_r(u-{1\over 2})\over Q_r(u+{1\over 2})}.
\eqno(4.28)
$$
%
This is consistent with (2.12) since
the above box is associated with the highest weight
$(\epsilon_1 + \cdots + \epsilon_r)/2 = \omega_r$ from (4.1) and (4.21).
%
Thirdly, the crossing symmetry
$\Lambda^{(r)}_1(u) =
(-)^{pN}\Lambda^{(r)}_1(-2r+1-u)
\vert_{w_j \rightarrow -w_j, u^{(a)}_i \rightarrow -u^{(a)}_i}$
is valid, which is precisely (2.17) with
the order $k=p$ as remarked above.
%
At the level of the boxes, this is due to
$$
\tau^C_{2r-1}
\overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p
= \overbrace{\Flect(0.4cm,2.3cm,{-\mu_1,\ldots,-\mu_r})}^r_p,
\eqno(4.29)
$$
%
where the effect of $(-)^{pN}$ has been absorbed into
$\tau^C_{2r-1}$ as explained in (2.19).
%
In the sequel, we will write such $\pm$ sequences as above
simply as $\mu$ and $\overline{\mu}$, etc.
%
As an warming-up exercise
let us show (4.29) by induction on $r$.
%
In view of $(\tau^C_\gamma)^2 = 1$, it suffices to check
the two cases $(\mu_1, \mu_2) = (+,+)$ and $(+,-)$.
%
We shall do the former case
and leave the latter to the readers.
%
Put $\mu = (+,+,\nu)$ with $\nu$ being a length $r-2$ sequence of
$\pm$.
%
Then the lhs of (4.29) becomes
%
$$\eqalign{
\tau^C_{2r-1} \overbrace{\Flect(0.4cm,2.0cm,{+,+,\nu})}^r_p
&= \tau^C_{2r-1}\Bigl(
\phi(u+r+p-{3\over 2} +{1\over t_p})\tau^Q
\overbrace{\Flect(0.4cm,1.8cm,{+,\nu})}^{r-1}_{p-1}\Bigr)\cr
%
&= \phi(u+r-p+{1\over 2}-{1\over t_p})\tau^C_{2r-1}\tau^Q
\tau^C_{2r-3}
\overbrace{\Flect(0.4cm,1.8cm,{-,\overline{\nu}})}^{r-1}_{p-1},\cr
}
$$
%
where we have used (4.25a) in the first line and
the induction assumption in the second line.
%
By means of (4.24) one may substitute
$\tau^C_{2r-1} \tau^Q \tau^C_{2r-3} = \tau^u_2 \tau^Q$
into the latter.
%
{}From (4.25d), the result is equal to
$\overbrace{\Flect(0.4cm,2.0cm,{-,-,\overline{\nu}})}^r_p$,
which is the rhs of (4.29).
%
\par\noindent
{\bf 4.6 Pole-freeness of $\Lambda^{(r)}_1(u)$.}\hskip0.3cm
%
In section 4.5, we have formally allowed $p=0$ in
the boxes that consist of the DVF (4.22).
%
Correspondingly, we find it convenient to consider
the BAE with $p=0$ as the one obtained from
(2.7) by setting its lhs always $-1$.
%
We shall quote (2.7) as $\hbox{BAE}^r_p$.
%
Our aim here is to establish
%
\proclaim Theorem 4.6.1.
For $r \ge 2$ and $0 \le p \le r$, $\Lambda^{(r)}_1(u)$ (4.22)
is free of poles provided that the $\hbox{BAE}^r_p$ (2.7)
(for $s=1$) is valid.
%
\par
%
We are to show that color $a$ poles
$1/Q_a$ are spurious for each $1 \le a \le r$.
%
The poles are located by
%
\proclaim Lemma 4.6.2.
For $1 \le a \le r-1$ the factor $1/Q_a$
is contained in the box
$\overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p$
if and only if $(\mu_a,\mu_{a+1}) = (+,-)$ or $(-,+)$.
%
Any two such boxes
$\overbrace{\Flect(0.4cm,2.0cm,{\eta,+,-,\xi})}^r_p$ and
$\overbrace{\Flect(0.4cm,2.0cm,{\eta,-,+,\xi})}^r_p$
share a common color $a$ pole $1/Q_a(u+y)$ for some $y$.
%
The factor $1/Q_r$ is contained in all the boxes.
For $\epsilon = \pm$, any two boxes
$\overbrace{\Flect(0.4cm,1.5cm,{\zeta,\epsilon,\epsilon})}^r_p$ and
$\overbrace{\Flect(0.4cm,1.5cm,{\zeta,\epsilon,-\epsilon})}^r_p$
share a common color $r$ pole $1/Q_r(u+z)$ for some $z$.
\par
%
The assertions are immediate consequences of (4.25) and (4.26).
%
If one puts
\par\noindent
$\lambda = (\eta,+,-,\xi)$,
$\mu = (\eta,-,+,\xi)$ and identifies them with the weights via (4.21),
one has $\lambda - \mu = \epsilon_a - \epsilon_{a+1} = \alpha_a$
for $1 \le a \le r-1$ by (4.21).
%
A similar relation holds for $a=r$ as well.
%
Thus the above Lemma is another example of the
coupling rule (2.14a).
%
In this view
Theorem 4.6.1 is a corollary of
%
\proclaim Theorem 4.6.3.
%
For $1 \le a \le r-1$,
let $\eta, \xi$ and $\zeta$ be any $\pm$ sequences with
lengths $a-1, r-a-1$ and $r-2$, respectively.
%
If the $\hbox{BAE}^r_p$ (2.7) (for $s=1$) is valid, then
%
$$\eqalignno{
&Res_{u=-y+iu^{(a)}_k}\Bigl(
\overbrace{\Flect(0.4cm,2.0cm,{\eta,+,-,\xi})}^r_p +
\overbrace{\Flect(0.4cm,2.0cm,{\eta,-,+,\xi})}^r_p \Bigr) = 0,
&(4.30{\rm a})\cr
%
&Res_{u=-z+iu^{(r)}_k}\Bigl(
\overbrace{\Flect(0.4cm,2.0cm,{\zeta,\pm,\pm})}^r_p +
\overbrace{\Flect(0.4cm,2.0cm,{\zeta,\pm,\mp})}^r_p \Bigr) = 0,
&(4.30{\rm b})\cr
}$$
%
where $y$ and $z$ are those in Lemma 4.6.2.
\par
%
The rest of the present subsection will be devoted to a proof
of this Theorem assuring that all the color $a$ poles are spurious.
%
In fact the proof will be done essentially by establishing
(2.14b) and (2.15).
%
It follows that the character limit (2.16) is also valid for
$\Lambda^{(r)}_1(u)$ (4.22).
%
We prepare
%
\proclaim Lemma 4.6.4.
%
Let $\xi$ be any sequence of $\pm$ with length $r-1$.
Then
%
$$\tau^C_{2r+1}
\overbrace{\Flect(0.5cm,1.3cm,{-,\overline{\xi}})}^r_p
= \Bigl({\phi(u+r+p+{1\over 2}+{1\over t_p})\over
         \phi(u+r-p+{1\over 2}-{1\over t_p})}\Bigr)^{1-\delta_{p 0}}
{Q_1(u+r-{1\over 2})\over Q_1(u+r+{3\over 2})}
\overbrace{\Flect(0.5cm,1.3cm,{-,\xi})}^r_p
\eqno(4.31)
$$
%
for $0 \le p \le r$.
%
\par\noindent
%
{\it Proof.}\hskip0.3cm
%
We show this for $\xi = (+,\nu)$.
%
The case $\xi = (-,\nu)$ is similar.
%
Suppose $p \ge 1$. Then the lhs of (4.31) is rewritten as
%
$$\eqalign{
\tau^C_{2r+1} \overbrace{\Flect(0.4cm,1.5cm,{-,-,\overline{\nu}})}^r_p
&= \tau^C_{2r+1}\tau^C_{2r-1}
\overbrace{\Flect(0.4cm,1.5cm,{+,+,\nu})}^r_p\cr
%
&= \phi(u+r+p+{1\over 2}+{1\over t_p})\tau^u_2\tau^Q
\overbrace{\Flect(0.4cm,1.5cm,{+,\nu})}^{r-1}_{p-1}\cr}
\eqno(4.32)
$$
%
by means of (4.24b), (4.25a) and (4.29).
%
In the rhs of (4.31), the box part is
$\overbrace{\Flect(0.4cm,1.5cm,{-,+,\nu})}^r_p$.
%
Replacing this by the rhs of (4.25c) with
$\xi = \nu$, one finds that the resulting
expression coincides with the last line in (4.32).
%
The case $p=0$ follows from this and (4.26b).
%
\par
Finally we give
%
\par\noindent
{\it Proof of Theorem 4.6.3.}\hskip0.3cm
%
It is straightforward to check (4.30) for $r=2$
by (4.26a).
%
We assume that Theorem is true for $B_{r-1}$
and use induction on $r$.
%
We shall verify (4.30a) only.
%
Eq. (4.30b) can be shown more easily by a similar method.
%
In the sequel the cases $a \ge 3$, $a=2$ and $a=1$
are considered separately.
%
\par
%
{\it The case $a \ge 3$.}\hskip0.3cm
%
Put $\eta = (\eta_1,\eta^\prime)$.
%
Then (4.25) transforms the sum of the two boxes
in (4.30a) into
%
$$
X_1 \tau^u_{1-\eta_1} \tau^Q\Bigl(
\overbrace{\Flect(0.4cm,2.0cm,{\eta^\prime,+,-,\xi})}^{r-1}_{p-1}
+ \overbrace{\Flect(0.4cm,2.0cm,{\eta^\prime,-,+,\xi})}^{r-1}_{p-1}
\Bigr),
$$
%
where $X_1$ involves only $\phi$ and possibly $Q_1/Q_1$.
%
Then Lemma 4.6.2 implies that
the poles $1/Q_a(u+y)$ in (4.30a) must originate in the factor
$1/Q_{a-1}(u+y-1+\eta_1)$ shared by the above two boxes.
%
Thus the lhs of (4.30a) is proportional
to $Res_{u=-y+1-\eta_1 + iu^{(a-1)}_k}$ of the sum of the above
boxes.
%
But this is a case of (4.30a) for $B_{r-1}$ hence $0$ by
the induction assumption.
\par
%
{\it The case $a = 2$.}\hskip0.3cm
%
In (4.30a) the length of $\eta$ is $1$.
%
We consider the case $\eta = +$.
%
The proof for $\eta=-$ is almost identical.
%
First we rewrite the two boxes in (4.30a) by (4.25) as follows.
%
$$
\overbrace{\Flect(0.4cm,1.8cm,{+,+,-,\xi})}^r_p = X_2
\tau^Q
\overbrace{\Flect(0.4cm,1.5cm,{+,-,\xi})}^{r-1}_{p-1},\quad
%
\overbrace{\Flect(0.4cm,1.8cm,{+,-,+,\xi})}^r_p
= X_2
{Q_1(u+r-{5\over 2})\over Q_1(u+r-{1\over 2})}
\tau^Q \overbrace{\Flect(0.4cm,1.5cm,{-,+,\xi})}^{r-1}_{p-1},
\eqno(4.33)
$$
%
where $X_2 = \phi(u+r+p-{3\over 2}+{1\over t_p})$
is independent of $Q_b$'s.
%
By further using (4.25), one finds that
%
$$
\overbrace{\Flect(0.4cm,1.5cm,{+,-,\xi})}^{r-1}_{p-1}
= X_3 {Q_1(u+r-{7\over 2})\over Q_1(u+r-{3\over 2})},\quad
%
\overbrace{\Flect(0.4cm,1.5cm,{-,+,\xi})}^{r-1}_{p-1}
= X_4 {Q_1(u+r+{1\over 2})\over Q_1(u+r-{3\over 2})},
\eqno(4.34)
$$
%
where $X_3$ and $X_4$ do not involve $Q_1$.
%
{}From the induction assumption,
the sum of these boxes must be ensured to be regular
at $u = -r+{3\over 2} +iu^{(1)}_k$
via the $\hbox{BAE}^{r-1}_{p-1}$.
%
Therefore
%
$$\eqalign{
&\overbrace{\Flect(0.4cm,1.5cm,{-,+,\xi})}^{r-1}_{p-1} /
\overbrace{\Flect(0.4cm,1.5cm,{+,-,\xi})}^{r-1}_{p-1}
\cr
%
&\quad =
{Q_1(u+r+{1\over 2})Q_2(u+r-{5\over 2})
\phi(u+r-{3\over 2} - {1\over t_p}\delta_{1 p-1}) \over
Q_1(u+r-{7\over 2})Q_2(u+r-{1\over 2})
\phi(u+r-{3\over 2} + {1\over t_p}\delta_{1 p-1})},\cr}
\eqno(4.35)
$$
should hold in order that the
lhs be evaluated as $-1$ at $u = -r + {3\over 2} + iu^{(1)}_k$
from the $\hbox{BAE}^{r-1}_{p-1}$.
%
In deriving (4.35)
we have used the fact that $t_{p-1}$ of $B_{r-1}$ is equal to
$t_p$ of $B_r$.
%
Combining (4.33) and (4.35) one deduces
%
$$\eqalign{
&\overbrace{\Flect(0.4cm,1.8cm,{+,-,+,\xi})}^{r}_{p} /
\overbrace{\Flect(0.4cm,1.8cm,{+,+,-,\xi})}^{r}_{p}
\cr
%
&\quad =
{Q_1(u+r-{5\over 2})Q_2(u+r+{1\over 2})Q_3(u+r-{5\over 2})
\phi(u+r-{3\over 2} - {1\over t_p}\delta_{2 p}) \over
Q_1(u+r-{1\over 2})Q_2(u+r-{7\over 2})Q_3(u+r-{1\over 2})
\phi(u+r-{3\over 2} + {1\over t_p}\delta_{2 p})}.\cr}
\eqno(4.36)
$$
%
Thanks to $\hbox{BAE}^r_p$ (2.7), this is indeed $-1$
at $u = -r + {3\over 2} + iu^{(2)}_k$,
proving that color $a=2$ poles are spurious.
\par
{\it The case $a=1$.}\hskip0.3cm
%
The two boxes in (4.30a) are
$\overbrace{\Flect(0.4cm,1.5cm,{+,-,\xi})}^{r}_{p}$ and
$\overbrace{\Flect(0.4cm,1.5cm,{-,+,\xi})}^{r}_{p}$.
%
{}From (4.25b,c) they share a color $a=1$ pole
at $u=-r+{1\over 2}+iu^{(1)}_k$.
%
Let us rewrite the latter as follows.
%
$$\eqalign{
\overbrace{\Flect(0.5cm,1.5cm,{-,+,\xi})}^{r}_{p}
&= \tau^C_{2r-1}
\overbrace{\Flect(0.5cm,1.5cm,{+,-,\overline{\xi}})}^{r}_{p}\cr
%
&= \phi(u+r-p+{1\over 2}-{1\over t_p})
{Q_1(u+r+{3\over 2}) \over Q_1(u+r-{1\over 2})}
\tau^Q\tau^C_{2r-1}
\overbrace{\Flect(0.5cm,1.2cm,{-,\overline{\xi}})}^{r-1}_{p-1},\cr}
$$
%
where we have used (4.29) and (4.25b).
%
In the last line,
$\tau^C_{2r-1}
\overbrace{\Flect(0.5cm,1.2cm,{-,\overline{\xi}})}^{r-1}_{p-1}$
can be further rewritten by applying Lemma 4.6.4 with
$r \rightarrow r-1, p \rightarrow p-1$.
%
Dividing the resulting expression by the rhs of
(4.25b) we obtain
%
$$\eqalign{
\overbrace{\Flect(0.4cm,1.5cm,{-,+,\xi})}^{r}_{p} /
\overbrace{\Flect(0.4cm,1.5cm,{+,-,\xi})}^{r}_{p}
=
{Q_1(u+r+{3\over 2})Q_2(u+r-{3\over 2})\over
 Q_1(u+r-{5\over 2})Q_2(u+r+{1\over2})}
\Bigl(
{\phi(u+r-p+{1\over 2}-{1\over t_p})\over
 \phi(u+r+p-{3\over 2}+{1\over t_p})}
\Bigr)^{\delta_{1 p}}.}
$$
%
At the pole location $u=-r+{1\over 2}+iu^{(1)}_k$,
this is just $-1$ owing to $\hbox{BAE}^r_p$ (2.7)
with $a=1$.
%
Therefore (4.30a) is free of color $1$ poles.
%
This completes the proof of Theorem 4.6.3 hence Theorem 4.6.1.
%
\par\noindent
{\bf 4.7 Relations between two kinds of boxes.}\hskip0.3cm
Here we clarify the relation between the two kinds
of the boxes
$\Fsquare(0.4cm,a)$ and $\Flect(0.4cm,1.8cm,{\mu_1,\ldots,\mu_r})$
introduced in section 4.1 and 4.5, respectively.
%
In terms of the relevant auxiliary spaces,
they are associated with the vector
and the spin representations.
%
To infer their relation, recall the classical tensor
product decomposition
%
$$
V(\omega_r)\otimes V(\omega_r)
= V(2\omega_r) \oplus V(\omega_{r-1}) \oplus \cdots \oplus
V(\omega_1) \oplus V(0).
\eqno(4.37)
$$
%
Correspondingly, there exists an
$U_q(B^{(1)}_r)$ quantum $R$-matrix
$R_{W^{(r)}_1, W^{(r)}_1}(u)$ [32]
acting on the $q$-analogue of the above.
%
On each component $V(\omega)$ of the rhs,
it acts as a constant $\rho_\omega(u)$ that depends on
the spectral parameter $u$.
%
A little investigation of the spectrum $\rho_\omega(u)$
in [32] tells that only
$\rho_{\omega_a}(u), \rho_{\omega_{a-2}}(u), \ldots$
are non zero
at $u=-2(r-a)+1$ for $1\le a \le r-1$.
%
{}From this and (2.4b) we see that the specialized $R$-matrix
$R_{W^{(r)}_1, W^{(r)}_1}(-2(r-a)+1)$
yields the embedding
%
$$
W^{(a)}_1(u) \hookrightarrow W^{(r)}_1(u+r-a-{1\over 2})
\otimes W^{(r)}_1(u-r+a+{1\over 2})\eqno(4.38)
$$
in the notation of [8].
%
According to the arguments there, (4.38) imposes
the following functional relation among the
transfer matrices having the relevant auxiliary spaces:
%
$$
T^{(r)}_1(u+r-a-{1\over 2})T^{(r)}_1(u-r+a+{1\over 2})
= T^{(a)}_1(u) + T^\prime(u) \quad \hbox{ for } 1 \le a \le r-1.
\eqno(4.39)
$$
%
Here $T^\prime(u)$ denotes some matrix commuting with all
$T^{(b)}_m(v)$'s.
%
When $a = r-1$, (4.39) is
just the last equation in (2.5a) with $m=0$, hence
$T^\prime(u) = T^{(r)}_2(u)$.
%
Viewed as a relation among the eigenvalues,
(4.39) implies that each term in the DVF (4.9a)
can be expressed as a product of
certain two boxes in section 4.5 with the spectral parameters
$u+r-a-{1\over 2}$ and $u-r+a+{1\over 2}$.
%
Actually we have
%
\proclaim Theorem 4.7.1.
%
For $1 \le a \le r-1, k, n, l \in {\bf Z}_{\ge 0}$ such that
$k+n+l=a$, take any integers
$1 \le i_1 < \cdots < i_k \le r$ and
$1 \le j_1 < \cdots < j_l \le r$.
%
Then the following equality holds between the elements of
${\cal T}^{(a)}_1$ and ${\cal T}^{(r)}_1$ defined in
(4.7,8) and (4.25,26), respectively.
%
$$
\raise 13ex \hbox{${1 \over F^{(p,r)}_a(u)}$}
\,
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Fsquare(0.5cm,\hbox{$i_1$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\Fsquare(0.5cm,\hbox{$i_k$})$}\vskip-0.4pt
	     \os
      \hbox{$\Fsquare(0.5cm,\hbox{$\overline{j}_{\ell}$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
      \hbox{$\Fsquare(0.5cm,\hbox{$\overline{j}_1$})$}
        }
      }
%
\,\,
%
 \raise 13ex
  \hbox{$= \bigl(\, \tau^u_{-r+a+{1\over 2}}
        \overbrace{\Flect(0.4cm,1.8cm,{\mu_1,\ldots,\mu_r})}^r_p \, \bigr)
        \bigl(\, \tau^u_{r-a-{1\over 2}}
        \overbrace{\Flect(0.4cm,1.8cm,{\nu_1,\ldots,\nu_r})}^r_p \, \bigr),
    $}\eqno(4.40{\rm a})
$$
%
where there are $n$ $\Fsquare(0.4cm,0)$'s in the lhs and
$F^{(p,r)}_a(u)$ is defined in (4.9b) and (4.10).
%
The $\pm$ sequences in the rhs are specified by
%
$$\eqalign{
\mu_b &= \cases{+ & if $b \in \{ i_1, \ldots, i_k \}$ \cr
                - & otherwise \cr},\cr
%
\nu_b &= \cases{- & if $b \in \{ j_1, \ldots, j_l \}$ \cr
                + & otherwise \cr}.\cr}\eqno(4.40{\rm b})
$$
\par
%
Note that both sides of (4.40a) are of order
$2p$ w.r.t $\phi$ and carry the same weight
$\epsilon_{i_1} + \cdots + \epsilon_{i_k}
- \epsilon_{j_1} - \cdots - \epsilon_{j_l}$.
%
The Theorem is again proved by induction on the rank $r$.
%
To do so we write the boxes (4.4a) as
$\overbrace{\Fsquare(0.4cm,a)}^r_p$ to mark the
$r, p$ dependence explicitly.
%
Then they enjoy the recursive property as follows.
%
\proclaim Lemma 4.7.2.
For $1 \le p \le r$ and $2 \le b \le r$, the boxes (4.4)
fulfill
%
$$\eqalign{
\overbrace{\Fsquare(0.4cm,b)}^r_p &= X_1
\Bigl({Q_1(u+3)\over Q_1(u+1)}\Bigr)^{\delta_{2 b}}
\tau^u_1 \tau^Q
\overbrace{\Flect(0.4cm,1.0cm,b-1)}^{r-1}_{p-1},\cr
%
X_1 &= \phi(u-p+2-{1\over t_p})\phi(u+2r+p-3+{1\over t_p}).\cr}
\eqno(4.41)
$$
%
\par
%
This can be checked directly by using the explicit form
(4.4).
%
In particular, one uses
$X_1 = \Phi^r_p(u)/\Phi^{r-1}_{p-1}(u+1)$.
%
\par\noindent
{\it Proof of Theorem 4.7.1 for $a=k=1$ and $n=l=0$.}
%
We illustrate an inductive proof w.r.t $r$ in this case.
%
General cases can be verified based on it by a similar idea
through tedious calculations.
%
Eq.(4.40) can be directly checked for $r=2$.
%
By letting $i_1 = b$ and noting that
$F^{(p,r)}_1(u) = 1$, (4.40) reads
%
$$
\overbrace{\Fsquare(0.4cm,b)}^r_p =
\bigl(\, \tau^u_{-r+{3\over 2}}
\overbrace{\Flect(0.4cm,3.7cm,{-,\cdots,-,+,-,\cdots,-})}^r_p \,\bigr)
\bigl(\, \tau^u_{r-{3\over 2}}
\overbrace{\Flect(0.4cm,1.5cm,{+,\cdots,+})}^r_p \,\bigr),
\eqno(4.42)
$$
where the $+$ symbol in the first box on the rhs is located
at $b$-th position from the left.
%
Below we focus on the case $b \ge 2$
and leave $b=1$ case to the readers.
%
Then, by applying the recursions (4.25) and
(4.24) the rhs is rewritten as
%
$$
X_1 \Bigl({Q_1(u+3) \over Q_1(u+1)} \Bigr)^{\delta_{2 b}}
\tau^u_1\tau^Q\Biggl(
\bigl(\, \tau^u_{-r+{5\over 2}}
\overbrace{
\Flect(0.4cm,3.7cm,{-,\cdots,-,+,-,\cdots,-})}^{r-1}_{p-1}\,\bigr)
\bigl(\, \tau^u_{r-{5\over 2}}
\overbrace{\Flect(0.4cm,1.5cm,{+,\cdots,+})}^{r-1}_{p-1} \,\bigr)
\Biggr),\eqno(4.43)
$$
where $X_1$ is the one in (4.41).
%
The $+$ symbol in the first box is now at the $b-1$ th position.
%
The quantity in the largest parenthesis of (4.43) is precisely
the rhs of (4.42) with $r \rightarrow r-1$ and $p \rightarrow p-1$.
%
By induction one may replace it with
$\overbrace{\Flect(0.4cm,1.0cm,b-1)}^{r-1}_{p-1}$.
%
The resulting expression is just the rhs of (4.41) hence
equal to $\overbrace{\Fsquare(0.4cm,b)}^r_p$ by Lemma 4.7.2.
%
This completes the induction step hence the proof.
%
\beginsection 5. Eigenvalues for $D_r$

Our results for $D_r = so(2r)$ are quite parallel with those
for $B_r = so(2r+1)$ in many respects.
%
In fact many formulas here
becomes those in section 4 through a formal
replacement $r \rightarrow r + {1\over 2}$.
%
Thus we shall state them without a proof,
which can be done in a similar manner to the $B_r$ case.
%
We will introduce two kinds of boxes
associated with the vector and the spin representations
and clarify their relation.
%
A distinct feature in $D_r$ is
that there are two representations of the latter kind,
$V(\omega_{r-1})$ and $V(\omega_r)$, each having
the quantum affine analogue
$W^{(r-1)}_1$ and $W^{(r)}_1$, respectively.
%
They are interchanged under the Dynkin diagram automorphism.
%
In order to respect the symmetry under it,
we modify the
quantum spaces
$\otimes_{j=1}^N W^{(p)}_1(w_j)$
for $p=r-1$ and $r$ into
%
$\otimes_{j=1}^N W^{(\pm)}_1(w_j)$ where
$W^{(\pm)}_1(w) = W^{(r)}_1(w\mp 2)\otimes W^{(r-1)}_1(w\pm 2)$.
%
Pictorially, one may view this as
arranging the vertical lines on the square lattice
endowed with the modules
$V(\omega_r), V(\omega_{r-1})$ alternately and with
the inhomogeneity as
$w_1 \mp 2, w_1 \pm 2, w_2 \mp 2, w_2 \pm 2, \ldots$.
%
This pattern has been introduced to utilize the degeneracy of
the spin-conjugate spin $R$-matrix [32]
$\hbox{Im} R_{W^{(r)}_1, W^{(r-1)}_1}(u=4) \simeq
V(\omega_r + \omega_{r-1})$,
where the image becomes manifestly symmetric under the automorphism.
%
The BAE (2.7) (with $s=1$) is thereby
unchanged as long as $p = 1,2, \ldots, r-2$.
%
Instead of $p=r-1$ and $r$, we now take
$p=\pm$, for which the BAE reads
%
$$
-{\phi^+_p(iu_k^{(a)}+\delta_{a r}) \phi^-_p(iu_k^{(a)}+\delta_{a r-1})
\over
 \phi^+_p(iu_k^{(a)}-\delta_{a r})  \phi^-_p(iu_k^{(a)}-\delta_{a r-1}) } =
\prod_{b=1}^r {{Q_b(iu_k^{(a)}+(\alpha_a|\alpha_b)) }\over
                {Q_b(iu_k^{(a)}-(\alpha_a|\alpha_b))}}.\eqno(5.1)
$$
%
Here the functions in the lhs are defined via $\phi(u)$ (1.4b) by
%
$$\phi^{\pm}_{\pm}(u) = \phi(u+2), \quad
  \phi^{\pm}_{\mp}(u) = \phi(u-2).\eqno(5.2)
$$
%
\par\noindent
{\bf 5.1 Eigenvalue $\Lambda^{(1)}_1(u)$.}\hskip0.3cm
%
Let $\epsilon_a, 1 \le a \le r$ be the orthonormal vectors
$(\epsilon_a \vert \epsilon_b) = \delta_{a b}$
realizing the root system as follows.
%
$$\eqalign{
\alpha_a &= \cases{\epsilon_a - \epsilon_{a+1} &
for $1 \le a \le r-1$\cr
\epsilon_{r-1} + \epsilon_r & for $a = r$\cr},\cr
%
\omega_a &=
\cases{
\epsilon_1 + \cdots + \epsilon_a& for $1 \le a \le r-2$\cr
%
{1 \over 2}(\epsilon_1 + \cdots + \epsilon_{r-1} - \epsilon_r)
& for $a = r-1$\cr
%
{1 \over 2}(\epsilon_1 + \cdots + \epsilon_{r-1} + \epsilon_r)
& for $a = r$\cr}.\cr}
\eqno(5.3)$$
%
The auxiliary space relevant to
$\Lambda^{(1)}_1(u)$ is
$W^{(1)}_1 \simeq V(\omega_1)$ as an $D_r$-module by (2.4b).
%
This is the vector representation, whose weights are
all multiplicity-free and given by
$\epsilon_a$ and $-\epsilon_a (1 \le a \le r)$.
%
By abbreviating them to $a$ and $\overline{a}$,
the set of weights and the DVF are given as follows.
%
$$\eqalignno{
J &= \{1,2,\ldots,r,\overline{r},\ldots,\overline{1}\},
&(5.4)\cr
\Lambda^{(1)}_1(u) &= \sum_{a \in J}\Fsquare(0.4cm,a).
&(5.5)\cr}
$$
%
This is formally the same with (3.2-3).
%
The elementary boxes are defined by
%
$$\eqalign{
\Fsquare(0.5cm,a)  &= \psi_{a}(u)
      {{Q_{a-1}( u+a+1  )
            Q_{a}(u+a-2 )}\over
       { Q_{a-1}(u+a-1)Q_{a}(u+a)}}
  \qquad 1 \le a \le r-2,\cr
%
\Flect(0.5cm,1.0cm,r-1)  &= \psi_{r-1}(u)
      {{Q_{r-2}(u+r) Q_{r-1}(u+r-3 ) Q_{r}(u+r-3 )}\over
       { Q_{r-2}(u+r-2) Q_{r-1}(u+r-1 ) Q_{r}(u+r-1 )}}, \cr
%
\Fsquare(0.5cm, r)  &=
  \psi_r(u) {{Q_{r-1}(u+r+1) Q_{r}(u+r-3)}\over
   { Q_{r-1}(u+r-1)Q_{r}(u+r-1)}},  \cr
%
\Fsquare(0.5cm,\overline{r})  &= \psi_{\overline{r}}(u)
  {{Q_{r-1}(u+r-3) Q_{r}(u+r+1)}\over
   { Q_{r-1}(u+r-1)Q_{r}(u+r-1)}},  \cr
%
\Flect(0.5cm,1.0cm,\overline{r-1})  &=
      \psi_{\overline{r-1}}(u)
      {{Q_{r-2}(u+r-2) Q_{r-1}(u+r+1 ) Q_{r}(u+r+1 )}\over
       { Q_{r-2}(u+r) Q_{r-1}(u+r-1 ) Q_{r}(u+r-1 )}}, \cr
%
\Fsquare(0.5cm,\overline{a})  &=
\psi_{\overline{a}}(u)
      {{Q_{a-1}( u+2r-a-3  )
            Q_{a}(u+2r-a)}\over
       { Q_{a-1}(u+2r-a-1)Q_{a}(u+2r-a-2)}}
  \qquad 1\le a \le r-2,\cr}\eqno(5.6{\rm a})
$$
where we have set $Q_0(u) = 1$.
%
The vacuum part $\psi_{a}(u)$ depends on the quantum space
$\otimes_{j=1}^N W^{(p)}_1(w_j)$ and is given by
%
$$\eqalign{
&\hbox{if }  1 \le p \le r-2 \cr
&\psi_a(u) =
\cases{ \phi(u+p+1) \phi(u+2r-p-1) \Phi^r_p(u)&
for $1 \preceq a \preceq p$\cr
        \phi(u+p-1) \phi(u+2r-p-1) \Phi^r_p(u)&
for $p+1 \preceq a \preceq \overline{p+1}$   \cr
        \phi(u+p-1) \phi(u+2r-p-3)\Phi^r_p(u)&
for $\overline{p} \preceq a \preceq \overline{1}$ \cr
                     }  \cr
%
&\hbox{if }  p =\pm \cr
&\psi_a(u)
=\cases{\phi^{+}_p(u+r) \phi^{-}_p(u+r) \Phi^r_{r+1}(u)&
                          for  $1 \preceq a \preceq r-1$ \cr
                   \phi^{+}_p(u+r) \phi^{-}_p(u+r-2) \Phi^r_{r+1}(u)&
                      for $a=r$ \cr
                   \phi^{-}_p(u+r) \phi^{+}_p(u+r-2) \Phi^r_{r+1}(u)&
                      for $a=\overline{r}$ \cr
                   \phi^{-}_p(u+r-2) \phi^{+}_p(u+r-2) \Phi^r_{r+1}(u)&
               for  $\overline{r-1} \preceq a \preceq \overline{1} $ \cr
           }   \cr}\eqno(5.6{\rm b})
$$
%
where
$$\eqalign{
\Phi^r_p(u) &= \prod_{j=1}^{p-1}
\phi(u+p-2j-1)\phi(u+2r-p+2j-1)\cr
%
&= \Phi^r_p(-2r+2-u)\vert_{w_k \rightarrow -w_k}.\cr}
\eqno(5.6{\rm c})
$$
%
The order $\prec$ in the set $J$ (5.4) is specified by
%
$$
1 \prec 2 \prec \cdots \prec r-1 \prec
\eqalign{&r\cr &\overline{r}\cr} \prec \overline{r-1} \prec
\cdots \overline{2} \prec \overline{1}.\eqno(5.7)
$$
We impose no order between $r$ and $\overline{r}$.
%
The DVF (5.5) possesses all the features
explained in section 2.4.
%
In particular it is pole-free under the BAE (2.7) and (5.1) thanks to
the coupling rule (2.14).
%
It can be summarized in the diagram
%
\vskip 0.5cm
\centerline{$\Fsquare(0.4cm,r)$}\par\noindent
\centerline{${\buildrel r-1 \over \nearrow} \hskip0.4cm
{\buildrel r \over \searrow}   $} \par\noindent
\centerline{$
\Fsquare(0.4cm,1) {\buildrel 1 \over \longrightarrow} \Fsquare(0.4cm,2)
{\buildrel 2 \over \longrightarrow} \cdots
{\buildrel r-2 \over \longrightarrow}
\Flect(0.4cm,1.0cm,r-1)
%
\hskip 1.2cm
\Flect(0.4cm,1.0cm,\overline{r-1})
{\buildrel r-2 \over \longrightarrow}
\cdots {\buildrel 2 \over \longrightarrow}
\Fsquare(0.4cm,\overline{2})
{\buildrel 1 \over \longrightarrow}
\Fsquare(0.4cm,\overline{1})
$}\par\noindent
\centerline{${\buildrel r \over \searrow} \hskip0.4cm
{\buildrel r-1 \over \nearrow}   $} \par\noindent
\centerline{$\Fsquare(0.4cm,\overline{r})$}\par\noindent
\vskip0.5cm\par\noindent
%
in the same sense with those in sections 3.1 and 4.1.
%
This is again identical with the crystal graph [24,25].
%
For $p=1$, the DVF (5.5-6) has been known earlier in [21].
%
\par\noindent\
{\bf 5.2 Eigenvalue $\Lambda^{(a)}_1(u)$ for $1 \le a \le r-2$.}
\hskip0.3cm
For $1 \le a \le r-2$, let ${\cal T}^{(a)}_1$ be the set of
the tableaux of the form (3.7a) with $i_k \in J$ (5.4)
obeying the condition
%
$$
i_k \prec i_{k+1} \hbox{ or }
(i_k, i_{k+1}) = (r,\overline{r}) \hbox{ or }
(i_k, i_{k+1}) = (\overline{r},r)
\,\,\hbox{ for any } 1 \le k \le a-1.\eqno(5.8)
$$
%
We identify each element (3.7a) of ${\cal T}^{(a)}_1$ with
the product of (5.6a) with the spectral parameters
$u+a-1, u+a-3,\ldots,u-a+1$ from the top to the bottom as in (4.8).
%
Then the analytic Bethe ansatz yields the following DVF.
%
$$
\Lambda^{(a)}_1(u) =
{1\over F^{(p,r)}_a(u)}
\sum_{T \in {\cal T}^{(a)}_1} T\quad
\qquad 1 \le a \le r-2.\eqno(5.9{\rm a})
$$
%
Here the function $F^{(p,r)}_a(u)$ is defined by
%
$$\eqalign{
&F^{(p,r)}_a(u)\cr
&= \cases{
\prod_{j=1}^{a-1}
\psi^{(p,r)}_0(u+r-a-1+2j)
\psi^{(p,r)}_p(u-r+a+1-2j)& for $1 \le p \le r-2$\cr
%
\prod_{j=1}^{a-1}\psi^{(p,r)}_{0,+}(u+r-a-1+2j)
\psi^{(p,r)}_{r-1,-}(u-r+a+1-2j)& for $p = \pm$\cr}\cr
%
&= F^{(p,r)}_a(-2r+2-u)\vert_{w_k \rightarrow -w_k},\cr}
\eqno(5.9{\rm b})
$$
where $\psi^{(p,r)}_n(u)$ and
$\psi^{(p,r)}_{n,\pm}(u)$ are specified in
(5.15) in section 5.4.
%
Notice that $F^{(p,r)}_1(u) = 1$ hence (5.9a)
reduces to (5.5) when $a=1$.
%
It can be shown that each summand $T$ in (5.9a)
contains the factor $F^{(p,r)}_a$.
%
This will be seen manifestly in Theorem 5.5.1.
%
The DVF (5.9) for $\Lambda^{(a)}_1(u)$ is homogeneous w.r.t $\phi$
of order $2p$ if $1\le p \le r-2$ and
order $2r+2$ if $p=\pm$.
%
\par
One can observe the top term and the crossing
symmetry in the DVF (5.9) as done after (3.9).
%
To check the character limit (2.16)
is also similar to (4.11).
%
{}From (2.4b) we must show (4.11c) again for $1 \le a \le r-2$
under the absence of $y_0$ in (4.11b).
%
But this is straightforward from (5.8) and by noting that
the character formula (4.11d) is still valid for $D_r$
if $J$ is taken as (5.4).
%
\par
%
By a similar method to Theorem 4.3.1 one can prove
%
\proclaim Theorem 5.2.1.
$\Lambda^{(a)}_1(u) (1 \le a \le r-2)$ (5.9) is free of poles
provided that the BAE (2.7) (with $s=1$)
for $1 \le p \le r-2$ and (5.1) for $p=\pm$ are valid.
\par
%
\par\noindent
{\bf 5.3 Eigenvalue $\Lambda^{(1)}_m(u)$.}
\hskip0.3cm
Starting from (5.9) and the DVFs of
$\Lambda^{(r)}_1(u), \Lambda^{(r-1)}_1(u)$
that will be given in section 5.4, we are to solve
the $T$-system (2.5c).
%
The scalar
$g^{(a)}_m(u)$ there is to be taken as (4.17)
with $F^{(p,r)}_2(u)$
determined from (5.9b).
%
The solution will yield a DVF
for the general eigenvalue $\Lambda^{(a)}_m(u)$.
%
This program is yet to be executed completely.
%
Here we shall only present a conjecture on
$\Lambda^{(1)}_m(u)$.
%
\par
For $m \in {\bf Z}_{\ge 1}$, let ${\cal T}^{(1)}_m$ denote
the set of tableaux of the form
%
$$
\Fsquare(0.4cm,i_1)\naga\Fsquare(0.4cm,i_m)
$$
with $i_k \in J$ (5.4) obeying the condition
%
$$\eqalign{
&i_k \preceq i_{k+1} \quad \hbox{ for any } 1 \le k \le m-1,\cr
%
&r \hbox{ and } \overline{r} \hbox{ do not appear simultaneously.}\cr
}\eqno(5.10)
$$
%
We identify each element of ${\cal T}^{(1)}_m$ as above with
the product of (5.6a) as follows.
%
$$
\prod_{k=1}^m \Fsquare(0.4cm,i_k)\vert_{u \rightarrow u-m-1+2k}.
$$
%
Then we have the conjecture
%
$$
\Lambda^{(1)}_m(u) = \sum_{T \in {\cal T}^{(1)}_m} T,\eqno(5.11)
$$
%
which reduces to (5.5) when $m=1$.
%
It is easy to prove
$\sharp {\cal T}^{(1)}_m = \hbox{dim } W^{(1)}_m$.
%
We have checked (5.11) up to $m=4$ for $D_4$ and
$m=3$ for $D_5$.
%
\par\noindent
{\bf 5.4 Eigenvalues $\Lambda^{(r-1)}_1(u)$ and $\Lambda^{(r)}_1(u)$.}
\hskip0.3cm
%
Now the relevant auxiliary spaces are
\par\noindent
$W^{(r-1)}_1 \simeq V(\omega_{r-1})$ and
$W^{(r)}_1 \simeq V(\omega_r)$ as $D_r$-modules.
%
They are the two spin representations, whose weights are all
multiplicity-free and given by (4.21) for
$V(\omega_{r-1})$ if $\mu_1\mu_2 \cdots \mu_r = -$ and
for
$V(\omega_r)$ if $\mu_1\mu_2 \cdots \mu_r = +$.
%
As in the $B_r$ case we shall build the boxes
$\Flect(0.4cm,2.5cm,{\mu_1,\mu_2,\ldots,\mu_r})$
by which the DVF can be written as
%
$$\eqalignno{
\Lambda^{(r-1)}_1(u) &=\sum_{\{\mu_j = \pm ; \prod_{j=1}^r \mu_j = -\}}
\overbrace{\Flect(0.4cm,2.5cm,{\mu_1,\mu_2, \cdots , \mu_r})}^{r}_p,
&(5.12{\rm a})\cr
%
\Lambda^{(r)}_1(u) &=\sum_{\{\mu_j = \pm ; \prod_{j=1}^r \mu_j = +\}}
\overbrace{\Flect(0.4cm,2.5cm,{\mu_1,\mu_2, \cdots , \mu_r})}^{r}_p.
&(5.12{\rm b})\cr}
$$
%
We let ${\cal T}^{(r-1)}_1$ and
${\cal T}^{(r)}_1$ denote the sets of
$\hbox{dim } W^{(r-1)}_1 = \hbox{dim } W^{(r)}_1 = 2^{r-1}$ boxes
in (5.12a) and (5.12b), respectively.
%
The indices $r$ and
$p \in \{1,2,\ldots,r-2,+,-\}$ signify the rank of $D_r$
and the quantum space
$\otimes_{j=1}^N W^{(p)}_1(w_j)$, respectively.
%
The boxes are again defined by the recursion relations
w.r.t these indices.
%
By using the operators (4.24), they read,
%
$$\eqalignno{
\hbox{for } 1 \le p \le r-2,&\cr
\overbrace{\Flect(0.4cm, 1.5cm, {+,+,\xi })}^{r}_p &=
\phi(u+r+p-1)
\tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {+,\xi })}^{r-1}_{p-1},
&(5.13{\rm a})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {+,-,\xi })}^{r}_p &=
\phi(u+r+p-1)
{Q_1(u+r-3)\over Q_1(u+r-1)}
\tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {-,\xi })}^{r-1}_{p-1},
&(5.13{\rm b})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {-,+,\xi })}^{r}_p &=
\phi(u+r-p-1)
{Q_1(u+r+1)\over Q_1(u+r-1)}
\tau^u_2 \tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {+,\xi })}^{r-1}_{p-1},
&(5.13{\rm c})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {-,-,\xi })}^{r}_p &=
\phi(u+r-p-1)
\tau^u_2 \tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {-,\xi })}^{r-1}_{p-1},
&(5.13{\rm d})\cr
%
\hbox{for } p = \pm,\qquad \quad&\cr
\overbrace{\Flect(0.4cm, 1.5cm, {+,+,\xi })}^{r}_p &=
\phi(u+2r)
\tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {+,\xi })}^{r-1}_{p},
&(5.13{\rm e})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {+,-,\xi })}^{r}_p &=
\phi(u+2r)
{Q_1(u+r-3)\over Q_1(u+r-1)}
\tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {-,\xi })}^{r-1}_{p},
&(5.13{\rm f})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {-,+,\xi })}^{r}_p &=
\phi(u-2)
{Q_1(u+r+1)\over Q_1(u+r-1)}
\tau^u_2 \tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {+,\xi })}^{r-1}_{p},
&(5.13{\rm g})\cr
%
\overbrace{\Flect(0.4cm, 1.5cm, {-,-,\xi })}^{r}_p &=
\phi(u-2)
\tau^u_2 \tau^Q \overbrace{\Flect(0.4cm, 1.2cm, {-,\xi })}^{r-1}_{p}.
&(5.13{\rm h})\cr
}$$
%
Here $\xi$ denotes arbitrary sequence of $\pm$ symbols with length
$r-2$.
%
The recursions (5.13) involve both boxes in
${\cal T}^{(r-1)}_1$ and ${\cal T}^{(r)}_1$ and hold for $r \ge 5$.
%
As in $B_r$ case, we formally
consider boxes with $p=0$ and fix them
by (4.26b) and the convention explained after it.
%
We are yet to specify the initial condition, i.e.,
data for $D_4$ case.
%
As for the dress parts,
they are given by
%
$$\eqalign{
dr \Flect(0.4cm, 1.6cm, {+,+,+,+}) &={{Q_4(u-1)}\over{Q_4(u+1)}},\cr
%
dr \Flect(0.4cm, 1.6cm, {+,+,-,-}) &=
  {{Q_2(u)Q_4(u+3)}\over{Q_2(u+2)Q_4(u+1)}},\cr
%
dr \Flect(0.4cm, 1.6cm, {+,-,+,-}) &=
  {{Q_1(u+1)Q_2(u+4)Q_3(u+1)}\over{Q_1(u+3)Q_2(u+2)Q_3(u+3)}},\cr
%
dr \Flect(0.4cm, 1.6cm, {+,-,-,+}) &=
  {{Q_1(u+1)Q_3(u+5)}\over{Q_1(u+3)Q_3(u+3)}},\cr
%
dr \Flect(0.4cm, 1.6cm, {-,+,+,-}) &=
  {{Q_1(u+5)Q_3(u+1)}\over{Q_1(u+3)Q_3(u+3)}},\cr
%
dr \Flect(0.4cm, 1.6cm, {-,+,-,+}) &=
  {{Q_1(u+5)Q_2(u+2)Q_3(u+5)}\over{Q_1(u+3)Q_2(u+4)Q_3(u+3)}},\cr
%
dr \Flect(0.4cm, 1.6cm, {-,-,+,+}) &=
  {{Q_2(u+6)Q_4(u+3)}\over{Q_2(u+4)Q_4(u+5)}},\cr
%
dr \Flect(0.4cm, 1.6cm, {-,-,-,-}) &={{Q_4(u+7)}\over{Q_4(u+5)}}. \cr
}\eqno(5.14{\rm a})$$
%
The other 8 are deduced from the above by
%
$$
dr \Flect(0.4cm,2.4cm,{\mu_1,\mu_2,\mu_3,\mu_4}) =
dr \Flect(0.4cm,2.4cm,{\mu_1,\mu_2,\mu_3,-\mu_4})
\vert_{Q_3(u) \leftrightarrow Q_4(u)},
\eqno(5.14{\rm b})
$$
%
which is consistent with the diagram automorphism symmetry.
%
In fact, through the recursions (5.13), the property (5.14b)
leads to
%
$$
dr \overbrace{\Flect(0.4cm,2.7cm,{\mu_1,\ldots,\mu_{r-1},\mu_r})}^r_p
=
dr \overbrace{\Flect(0.4cm,3.0cm,{\mu_1,\ldots,\mu_{r-1},-\mu_r})}^r_p
\vert_{Q_{r-1}(u) \leftrightarrow Q_r(u)}.
$$
%
As for the vacuum parts, we shall give their
general form that includes the initial condition ($r=4$)
and fulfills the recursions (5.13).
%
$$\eqalignno{
&vac \overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p
= \cases{
\psi^{(p,r)}_n(u)& for $1 \le p \le r-2$\cr
%
\psi^{(p,r)}_{n,\mu_r}(u)& for $p = \pm$\cr},
&(5.15{\rm a})\cr
%
&n = \cases{
\sharp \{ j \mid \mu_j = -, 1 \le j \le p \}
& for $1 \le p \le r-2$\cr
%
\sharp \{ j \mid \mu_j = -, 1 \le j \le r-1 \}
& for $p = \pm$\cr},&(5.15{\rm b})\cr
%
&\psi^{(p,r)}_n(u) =
\prod_{\scriptstyle j=0 \atop
       \scriptstyle j \neq n}^p \phi(u+r-p+2j-1),
&(5.15{\rm c})\cr
%
&\psi^{(+,r)}_{n,+}(u) =
\psi^{(-,r)}_{n,-}(u) =
\prod_{\scriptstyle j=0 \atop
       \scriptstyle j \neq n+1}^{r+1} \phi(u+2j-2),
&(5.15{\rm d})\cr
%
&\psi^{(+,r)}_{n,-}(u) =
\psi^{(-,r)}_{n,+}(u) = \phi(u+2n)
\prod_{\scriptstyle j=0 \atop
       \scriptstyle j \neq n, n+2}^{r+1} \phi(u+2j-2).
&(5.15{\rm e})\cr}
$$
%
By the definition, $n$ ranges over $0 \le n \le p$
in (5.15c) and $0 \le n \le r-1$ in (5.15d,e).
%
This completes the characterization of
all the $2^r$ boxes hence the DVF (5.12)
for any $r \ge 4, p \in \{0,1,\ldots, r-2, +, - \}$.
%
In the rational case ($q \rightarrow 1$) with $p=1$,
a similar recursive description is available in [5].
%
\par
%
Let us list a few features explained in section 2.4.
%
Firstly, the top term (2.12) corresponds to
%
$$\eqalign{
dr \overbrace{\Flect(0.4cm,2.4cm,{+,\ldots,+,-})}^r_p &=
{Q_{r-1}(u-1)\over Q_{r-1}(u+1)},\cr
%
dr \overbrace{\Flect(0.4cm,2.4cm,{+,\ldots,+,+})}^r_p &=
{Q_r(u-1)\over Q_r(u+1)},\cr}
\eqno(5.16)
$$
%
where the lhs' are indeed associated with the highest weights
$\omega_{r-1}$ and $\omega_r$ in view of (5.3) and (4.21).
%
Secondly, the crossing symmetry (2.18,19) holds.
%
$$\eqalign{
\tau^C_{2r-2}
\overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p
&= \overbrace{\Flect(0.4cm,2.3cm,{-\mu_1,\ldots,-\mu_r})}^r_p
\quad \hbox{ for } 1 \le p \le r-2,\cr
%
\tau^C_{2r-2}
\overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p
&= \overbrace{\Flect(0.4cm,2.3cm,{-\mu_1,\ldots,-\mu_r})}^r_{-p}
\quad \hbox{ for } p = \pm.\cr}
\eqno(5.17)
$$
%
Thirdly, the coupling rule (2.14a) is valid due to
%
\proclaim Lemma 5.4.1.
For $1 \le a \le r-1$ the factor $1/Q_a$
is contained in the box
$\overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p$
if and only if $(\mu_a,\mu_{a+1}) = (+,-)$ or $(-,+)$.
%
Any two such boxes
$\overbrace{\Flect(0.4cm,2.0cm,{\eta,+,-,\xi})}^r_p$ and
$\overbrace{\Flect(0.4cm,2.0cm,{\eta,-,+,\xi})}^r_p$
share a common color $a$ pole $1/Q_a(u+y)$ for some $y$.
%
The factor $1/Q_r$
is contained in the box
$\overbrace{\Flect(0.4cm,2.0cm,{\mu_1,\ldots,\mu_r})}^r_p$
if and only if $\mu_{r-1} = \mu_r$.
%
Any two such boxes
$\overbrace{\Flect(0.4cm,1.5cm,{\zeta,+,+})}^r_p$ and
$\overbrace{\Flect(0.4cm,1.5cm,{\zeta,-,-})}^r_p$
share a common color $r$ pole $1/Q_r(u+z)$ for some $z$.
\par
%
As introduced in the
beginning of section 4.6,
let $\hbox{BAE}^r_{p=0}$ be (2.7) with the lhs being always $-1$
%
Under the BAE,
the pair of the coupled boxes yield zero residue in total.
%
We claim this in
%
\proclaim Theorem 5.4.2.
For $1 \le a \le r-1$,
let $\eta, \xi$ and $\zeta$ be any $\pm$ sequences with
lengths $a-1, r-a-1$ and $r-2$, respectively.
%
If the $\hbox{BAE}^r_p$ (2.7) (with $s=1$)
for $0 \le p \le r-2$ and (5.1) for $p=\pm$ are valid, then
%
$$\eqalignno{
&Res_{u=-y+iu^{(a)}_k}\Bigl(
\overbrace{\Flect(0.4cm,2.0cm,{\eta,+,-,\xi})}^r_p +
\overbrace{\Flect(0.4cm,2.0cm,{\eta,-,+,\xi})}^r_p \Bigr) = 0,
&(5.18{\rm a})\cr
%
&Res_{u=-z+iu^{(r)}_k}\Bigl(
\overbrace{\Flect(0.4cm,2.0cm,{\zeta,+,+})}^r_p +
\overbrace{\Flect(0.4cm,2.0cm,{\zeta,-,-})}^r_p \Bigr) = 0,
&(5.18{\rm b})\cr
}$$
%
where $y$ and $z$ are those in Lemma 5.4.1.
\par
%
The proof is similar to that for Theorem 4.6.3.
%
In particular (2.14b) and (2.15) can be shown,
therefore the character limit (2.16) is valid for
the DVFs (5.12).
%
(When $p = \pm$, one modifies the $\omega^{(p)}_{s=1}$ in (2.16)
suitably.)
%
Notice that both of
the coupled boxes in (5.18) belong to the
same set ${\cal T}^{(r-1)}_1$ or ${\cal T}^{(r)}_1$.
%
Thus Lemma 5.4.1 and Theorem 5.4.2 lead to
%
\proclaim Theorem 5.4.3.
For $r \ge 4$ and $p \in \{0,1,\ldots, r-2,+,-\}$,
$\Lambda^{(r-1)}_1(u)$ and
$\Lambda^{(r)}_1(u)$ in (5.12)
are free of poles provided that the $\hbox{BAE}^r_p$ (2.7)
(with $s=1$) for
$0 \le p \le r-2$ and
(5.1) for $p = \pm$ are valid.
\par
%
\par\noindent
{\bf 5.5 Relations between two kinds of boxes.}\hskip0.3cm
%
The elementary boxes
$\Fsquare(0.4cm,a)$ and $\Flect(0.4cm,1.8cm,{\mu_1,\ldots,\mu_r})$
introduced in section 5.1 and 5.4 are related by
%
\proclaim Theorem 5.5.1.
%
For $1 \le a \le r-2, k, n, l \in {\bf Z}_{\ge 0}$ such that
$k+2n+l=a$, take any integers
$1 \le i_1 < \cdots < i_k \le r$ and
$1 \le j_1 < \cdots < j_l \le r$.
%
Then the following equality holds between the elements of
${\cal T}^{(a)}_1$ and
${\cal T}^{(r-1)}_1 \cup {\cal T}^{(r)}_1$ defined in
section 5.2 and (5.13-15), respectively.
%
$$
\raise 13ex \hbox{${1 \over F^{(p,r)}_a(u)}$}
\,
\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Fsquare(0.5cm,\hbox{$i_1$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\Fsquare(0.5cm,\hbox{$i_k$})$}\vskip-0.4pt
      \hbox{$\Fsquare(0.5cm,\hbox{$\overline{r}$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\Fsquare(0.5cm,\hbox{$r$})$}\vskip-0.4pt
      \hbox{$\Fsquare(0.5cm,\hbox{$\overline{j}_{\ell}$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
      \hbox{$\Fsquare(0.5cm,\hbox{$\overline{j}_1$})$}
        }
      }
%
\,\,
%
 \raise 13ex
  \hbox{$= \bigl(\, \tau^u_{-r+a+1}
        \overbrace{\Flect(0.4cm,1.8cm,{\mu_1,\ldots,\mu_r})}^r_p \, \bigr)
        \bigl(\, \tau^u_{r-a-1}
        \overbrace{\Flect(0.4cm,1.8cm,{\nu_1,\ldots,\nu_r})}^r_p \, \bigr),
    $}\eqno(5.19)
$$
%
where there are $n$
$\hbox{
   \normalbaselines\m@th\baselineskip0pt\offinterlineskip
   \vbox{
      \hbox{$\Fsquare(0.4cm,\hbox{$\overline{r}$})$}\vskip-0.4pt
	     \hbox{$\Fsquare(0.4cm,\hbox{$r$})$}\vskip-0.4pt
        }
      }
$'s in the lhs and $F^{(p,r)}_a$ is defined in (5.9b).
%
The $\pm$ sequences $\mu$ and $\nu$ in the rhs are determined
by (4.40b).
\par
%
Put
$a \equiv r + \sigma$  mod  2 where
$\sigma = 0$ or $1$.
%
Then the tableaux in the rhs of (5.19) belong to the following sets.
%
$$
\overbrace{\Flect(0.4cm,1.0cm,\mu)}^r_p \in
\cases{ {\cal T}^{(r-\sigma)}_1 &  $l$ even \cr
        {\cal T}^{(r-1+\sigma)}_1 & $l$ odd \cr},\quad
%
\overbrace{\Flect(0.4cm,1.0cm,\nu)}^r_p \in
\cases{ {\cal T}^{(r)}_1 &  $l$ even \cr
        {\cal T}^{(r-1)}_1 & $l$ odd \cr}.
\eqno(5.20)
$$
%
One can rewrite the rhs of (5.19) so as to interchange
the parity of $l$ in (5.20).
%
Given any $\pm$ sequences
$\mu = (\mu_1,\ldots,\mu_r)$ and
$\nu = (\nu_1,\ldots,\nu_r)$, we set
%
$$
\eqalignno{
e_k(\mu,\nu) &= \sharp \{ j \mid 1 \le j \le k, \mu_j = - \}
- \sharp \{j \mid 1 \le j \le k, \nu_j = - \},
&(5.21{\rm a})\cr
%
d_y(\mu,\nu) &= \hbox{min }\Bigl(
\{ \infty \} \cup \{ k \mid 1 \le k \le r-1,
e_k(\mu,\nu) = y \} \Bigr).
&(5.21{\rm b})\cr
}$$
%
Then we have
%
\proclaim Lemma 5.5.2.
For any $1 \le a \le r-2$
and any $\pm$ sequences
$\mu = (\mu_1,\ldots, \mu_r), \nu = (\nu_1,\ldots, \nu_r)$,
one has
%
$$\eqalign{
       &\bigl(\, \tau^u_{-r+a+1}
        \overbrace{\Flect(0.4cm,1.8cm,{\mu_1,\ldots,\mu_r})}^r_p \, \bigr)
        \bigl(\, \tau^u_{r-a-1}
        \overbrace{\Flect(0.4cm,1.8cm,{\nu_1,\ldots,\nu_r})}^r_p \, \bigr)
\cr
& =        \bigl(\, \tau^u_{-r+a+1}
\overbrace{\Flect(0.4cm,1.8cm,{\mu^\prime_1,\ldots,\mu^\prime_r})
}^r_p \, \bigr)
        \bigl(\, \tau^u_{r-a-1}
\overbrace{\Flect(0.4cm,1.8cm,{\nu^\prime_1,\ldots,\nu^\prime_r})
}^r_p \, \bigr),\cr}\eqno(5.22{\rm a})
$$
%
where $\mu^\prime_j$ and $\nu^\prime_j$ are determined by
%
$$
(\mu^\prime_j,\nu^\prime_j) = \cases{
(\mu_j,\nu_j) & if $1 \le j \le d_{r-a-1}(\mu,\nu)$\cr
(\nu_j,\mu_j) & otherwise \cr}.\eqno(5.22{\rm b})
$$
%
\par
%
The Lemma enables the interchange of those $\mu_j$ and $\nu_j$
with $j > d_{r-a-1}(\mu,\nu)$ in the products (5.22a).
%
In case $d_{r-a-1}(\mu,\nu) = \infty$,
the assertion is trivial.
%
One may apply Lemma 5.5.2 to rewrite the rhs of (5.19).
%
A little inspection tells that $1 \le d_{r-a-1}(\mu,\nu) \le r-1$
for any those $\mu$ and $\nu$ appearing there.
%
Moreover, for such $d = d_{r-a-1}(\mu,\nu)$ one can
evaluate the difference
%
$$
\sharp \{j \mid d < j \le r, \mu_j = - \} -
\sharp \{j \mid d < j \le r, \nu_j = - \} = 2n+1
\in 2{\bf Z} + 1,
$$
%
in terms of the $n$ in Theorem 5.5.1.
%
Thus Lemma 5.5.2 expresses the rhs of (5.19)
by the tableaux such that
%
$$
\overbrace{\Flect(0.4cm,1.0cm,\mu^\prime)}^r_p \in
\cases{ {\cal T}^{(r-1+\sigma)}_1 &  $l$ even \cr
        {\cal T}^{(r-\sigma)}_1 & $l$ odd \cr},\quad
%
\overbrace{\Flect(0.4cm,1.0cm,\nu^\prime)}^r_p \in
\cases{ {\cal T}^{(r-1)}_1 &  $l$ even \cr
        {\cal T}^{(r)}_1 & $l$ odd \cr},
\eqno(5.23)
$$
which is opposite to (5.20).
%
Based on these observations,
we can give a similar argument to section 4.7
that backgrounds Theorem 5.5.1.
%
There is a degeneracy
point $u=-2(r-a-1)$ of the $U_q(D^{(1)}_r)$ quantum $R$-matrix [32]
where it yields embedding
%
$$\eqalign{
W^{(a)}_1(u) \hookrightarrow
&W^{(r-1)}_1(u+r-a-1) \otimes W^{(r-1+\sigma)}_1(u-r+a+1),\cr
%
W^{(a)}_1(u) \hookrightarrow
&W^{(r)}_1(u+r-a-1) \otimes W^{(r-\sigma)}_1(u-r+a+1),\cr}
\eqno(5.24)
$$
%
%
According to [8], (5.24) implies the functional relations
%
$$\eqalignno{
T^{(r-1)}_1(u+r-a-1)T^{(r-1+\sigma)}_1(u-r+a+1)
&= T^{(a)}_1(u) + T^\prime(u),&(5.25{\rm a})\cr
%
T^{(r)}_1(u+r-a-1)T^{(r-\sigma)}_1(u-r+a+1)
&= T^{(a)}_1(u) + T^{\prime \prime}(u),&(5.25{\rm b})\cr}
$$
%
where $T^\prime(u)$ and $T^{\prime \prime}(u)$
are some matrices commuting with
all $T^{(b)}_m(v)$'s.
%
In particular if $a=r-2$ ($\sigma = 0$),
(5.25) is the last equation in (2.5c)
with $m=1$, hence
$T^\prime(u) = T^{(r-1)}_2(u)$ and
$T^{\prime \prime}(u) = T^{(r)}_2(u)$.
%
One may regard (5.25a,b) as equations on the eigenvalues
and substitute (5.9a) and (5.12).
%
Then Theorem 5.5.1 tells how one can pick up the
DVF for $\Lambda^{(a)}_1(u)$ from the lhs.
%
For example in (5.25a), one depicts the terms in
$\Lambda^{(a)}_1(u)$ as the lhs of (5.19).
%
Then the $l$ odd terms are indeed contained in
$\Lambda^{(r-1)}_1(u+r-a-1)\Lambda^{(r-1+\sigma)}_1(u-r+a+1)$
due to (5.19) and (5.20).
%
The $l$ even terms
can also be found by expressing the above product
in terms of the tableaux in (5.23).
%
\beginsection 6. Discussions

\noindent{\bf 6.1 Summary.}\hskip0.3cm
%
In this paper we have constructed the
dressed vacuum forms (DVFs) (2.9b) for
several eigenvalues $\Lambda^{(a)}_m(u)$
of the row-to-row transfer matrices $T^{(a)}_m(u)$ (1.2) via
the analytic Bethe ansatz.
%
Relevant vertex models are those associated
with the fusion quantum $R$-matrices
$R_{W^{(a)}_m, W^{(p)}_s}(u)$ for $Y(X_r)$ or
$U_q(X^{(1)}_r)$ with $X_r = B_r, C_r$ and $D_r$.
%
We have determined the DVFs for all the transfer matrices
$T^{(a)}_1(u)\, (1 \le a \le r)$
associated with the fundamental representations
$W^{(a)}_1$ in the sense of [28].
%
In particular, they have been proved pole-free under
the Bethe ansatz equation, a crucial
property in the analytic Bethe ansatz.
%
Once the DVFs of $\Lambda^{(a)}_1(u)$ are fixed,
those for the other eigenvalues are
uniquely determined from the $T$-system [8],
a set of functional relations
among the transfer matrices.
%
Based on this we have conjectured the DVFs for several
$\Lambda^{(a)}_m(u)$ with higher $m$.
%
These results extend earlier ones in
[5,10,11,19,20,21].
%
\par
%
The DVFs for $\Lambda^{(a)}_m(u)$
are Yang-Baxterizations of the characters
of the auxiliary spaces $W^{(a)}_m$.
%
We have found that they are described by remarkably simple rules
using analogues of the
semi-standard Young tableaux.
%
We believe that the sets of tableaux
${\cal T}^{(a)}_m$ introduced in this paper
are natural objects that label the base of
the irreducible finite dimensional modules $W^{(a)}_m$
over the Yangians or the quantum affine algebras.
%
\par\noindent
{\bf 6.2 Further extensions.}\hskip0.3cm
%
Let us indicate further applications of our approach.
%
As can be observed
through sections 2 to 5, the hypotheses called
the top term (2.12) and the coupling rule (2.14), (2.15) severely
restrict possible DVFs.
%
This is especially significant when as many
weight spaces as possible are multiplicity-free (2.13)
in the auxiliary space.
%
An interesting example of such a situation is
Yangian analogue of the adjoint representation.
%
Below we exclude the case $X_r = A_r$, where
the DVF for general eigenvalues is already available [19].
%
Then it is known [12,28] that
the Yangian $Y(X_r)$ admits the irreducible representation
$W_{adj}$ isomorphic to
$V(\theta) \oplus V(0)$ as an $X_r$-module.
%
Here $\theta$ denotes the highest root hence
$V(\theta)$ means the adjoint representation of $X_r$.
%
One can identify $W_{adj}$
in the family $\{ W^{(a)}_m \}$ by $\theta$ and
the data in appendix A of [33].
%
$$
(\theta, W_{adj}) = \cases{
(\omega_1, W^{(1)}_1) & $E_7, E_8, F_4, G_2$\cr
%
(2\omega_1, W^{(1)}_2) & $C_r$\cr
%
(\omega_2, W^{(2)}_1) & $B_r, D_r$\cr
%
(\omega_6, W^{(6)}_1) & $E_6$\cr}.
$$
%
Thus the cases $X_r = B_r, C_r$ and $D_r$ are already covered in this paper.
%
For $G_2$, the DVF of $\Lambda^{(1)}_1(u)$ has been obtained
recently [11].
%
Let us turn to the remaining cases,
$\Lambda^{(1)}_1(u)$ of $E_{7,8}, F_4$ and
$\Lambda^{(6)}_1(u)$ of $E_6$.
%
By the definition,
$\hbox{dim } W_{adj} = \hbox{dim } X_r + 1$.
%
All the weights
in $W_{adj}$ are multiplicity-free
except the null one,
$\hbox{mult}_0 W_{adj} = r + 1$.
%
Thus one may try to apply
the top term (2.12), the coupling rule (2.14,15) and
the crossing symmetry (2.18)
to possibly determine the $\hbox{dim } X_r - r$ terms
in the DVF corresponding to the root vectors.
%
We have checked that this certainly works consistently and fix
those terms uniquely.
%
Moreover, we have found that
pole-freeness under the BAE requires
exactly $r + 1$ more terms
that make the null weight contribution $(r+1)q^0$
in the character limit (2.16).
%
These features are equally valid in the trigonometric case as well.
%
Thus the resulting DVFs are candidates of
the transfer matrix eigenvalues for
the trigonometric vertex models associated with
$U_q(E_8^{(1)})$, etc.
%
The details will appear elsewhere.
%
It still remains to understand the hypotheses
(2.12), (2.14) and (2.18) intrinsically and thus to unveil the full
aspects of the analytic Bethe ansatz.
%
\beginsection Acknowledgments

The authors thank E. Date, A.N. Kirillov,
M. Kashiwara, T. Nakashima and M. Okado for discussions.
%
They also thank T. Nakashima for allowing them to use the
\TeX macros produced by K. Nakahara.
%
A part of this work is done in Department of Mathematics,
Kyushu University.
%
%
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%
\bye

