%Paper: hep-th/9506167
%From: atsuo kuniba <atsuo@hep1.c.u-tokyo.ac.jp>
%Date: Mon, 26 Jun 95 18:00:00 +0900



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%\input ksmacro.tex
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%

\vskip2.5cm
\centerline{\bf Quantum Jacobi-Trudi and Giambelli Formulae}
\centerline{\bf for $U_q(B^{(1)}_r)$ from Analytic Bethe Ansatz}
\vskip1.0cm \centerline{by}
\vskip1.0cm
\centerline{Atsuo Kuniba\footnote\dag{
E-mail: atsuo@hep1.c.u-tokyo.ac.jp}}
\centerline{Institute of Physics, University of Tokyo}
\centerline{Komaba 3-8-1, Meguro-ku, Tokyo 153 Japan}
%
\par\vskip0.3cm
\centerline{Yasuhiro Ohta\footnote\ddag{
E-mail: ohta@kurims.kyoto-u.ac.jp}}
\centerline{Faculty of Engineering, Hiroshima University}
\centerline{Higashi Hiroshima, Hiroshima 724 Japan}
%
\par\vskip0.3cm
\centerline{and}\par\vskip0.3cm
\centerline{Junji Suzuki\footnote\P{
E-mail: jsuzuki@tansei.cc.u-tokyo.ac.jp}} 
\centerline{Institute of Physics, University of Tokyo}
\centerline{Komaba 3-8-1, Meguro-ku, Tokyo 153 Japan}
\vskip5.0cm
\centerline{\bf Abstract}
\vskip0.2cm
\par
%
Analytic Bethe ansatz is executed for a wide class of
finite dimensional $U_q(B^{(1)}_r)$ modules.
%
They are labeled by skew-Young diagrams which, in general, 
contain a fragment corresponding to the spin representation.
%
For the transfer matrix spectra of the relevant vertex models,
we establish a number of formulae,
which are $U_q(B^{(1)}_r)$ analogues of the 
classical ones due to Jacobi-Trudi and Giambelli on
Schur functions.
%
They yield a full solution to the previously proposed 
functional relation ($T$-system), which is a Toda equation
on discrete space-time.


%
\vfill
\eject


\beginsection 1. Introduction

\noindent
%
In [KS1] analytic Bethe ansatz was worked out
for all the fundamental representations of the Yangians 
$Y(X_r)$ of classical types 
$X_r = B_r, C_r$ and $D_r$.
%
Namely, for any $a \in \{1,2, \ldots, r \}$,
a rational function $\Lambda^{(a)}_1(u)$ of the 
spectral parameter $u$ has been constructed, which should
describe the spectrum of the transfer matrices
of the corresponding solvable vertex models.
%
It is a Yangian analogue of the
character of the auxiliary space and satisfies a couple
of conditions required for it.
%
In particular $\Lambda^{(a)}_1(u)$ has been shown pole-free
provided that the Bethe ansatz equation (BAE) holds.
%
These results are also valid for $U_q(X^{(1)}_r)$ case
after replacing the rational functions by their natural 
$q$-analogues.
%
See [R,KS1] for general accounts on the 
analytic Bethe ansatz.
\par
%
In this paper we extend such analyses 
beyond the fundamental representations for $X_r = B_r$.
%
We introduce skew-Young diagrams $\lambda \subset \mu$ [M]
and a set of tableaux on them obeying a certain
semi-standard like conditions.
%
Then we construct the corresponding function
$T_{\lambda \subset \mu}(u)$ in terms of a sum over such 
tableaux via a certain rule.
%
The $T_{\lambda \subset \mu}(u)$ is to be regarded 
as the spectrum of the commuting
transfer matrix with
auxiliary space 
labeled by $\lambda \subset \mu$.
%
It has a dressed vacuum form (DVF) in the analytic Bethe ansatz. 
%
We shall rewrite $T_{\lambda \subset \mu}(u)$ 
in several determinantal forms,
where the matrix elements are only those $T_\mu(u)$
for the usual Young diagrams 
with shapes $\mu = (1^a), (m)$ or $(m+1,1^a)$.
%
They can be viewed as $U_q(B^{(1)}_r)$ analogues
of the classical Jacobi-Trudi and Giambelli formulae
on Schur functions [M].
%
Pole-freeness of the $T_{\lambda \subset \mu}(u)$ 
under BAE follows immediately from these formulae and our 
previous proof for the case $\mu = (1^a)$ [KS1].
%
These results correspond to
the case where the auxiliary space is even with 
respect to the tensor degree of the spin
representation.
%
We shall simply refer to such a case 
spin-even and spin-odd otherwise.
%
See the remark after (3.12) for a precise definition.
%
We will also treat the spin-odd case by 
using a modified skew-Young diagrams and 
semi-standard like conditions on them.
%
Combining these results, we obtain a full solution in terms of the DVF
to the transfer matrix functional relation ($T$-system)
proposed in [KNS].
%
This substantially achieves our program raised 
in [KS1] for $B_r$.
%
\par
A natural question here is, what is the finite dimensional 
auxiliary space labeled by those skew-Young diagrams as 
a representation space of $U_q(B^{(1)}_r)$ or $Y(B_r)$?
%
We suppose that it is an irreducible 
one in view that
all the terms in $T_{\lambda \subset \mu}(u)$ are
coupled to make the associated poles suprious 
under BAE.
%
Moreover we specify, in the Yangian 
context, the Drinfeld polynomial explicitly
based on some empirical procedure.
%
We shall also determine how the irreducible $Y(B_r)$ module
decomposes as a $B_r$ module through the embedding 
$B_r \hookrightarrow Y(B_r)$ for the spin-even case.
%
\par
The paper is organized as follows.
%
In the next section we recall the results 
in [KS1] on $U_q(B^{(1)}_r)$.
%
We then introduce the basic functions
$T^a(u)$ and $T_m(u)$ for all $a, m \in {\bf Z}_{\ge 0}$.
%
These are analogues of $a$-th anti-symmetric and 
$m$-th symmetric fusion transfer matrices (or its
eigenvalues), respectively.
%
For $1 \le a \le r-1$, we have 
$T^a(u) = \Lambda^{(a)}_1(u) = T_{(1^a)}(u)$ 
in the above.
%
The introduction of $T^a(u)$ with $a \ge r$ is a key
in this paper and we point out 
a new functional relation (2.14) among them.
%
In section 3 and 4, we treat the spin-even and odd cases,
respectively.
%
In terms of the DVFs in these sections, we give,
in section 5, a full solution
to the $T$-system [KNS] 
with an outline of the proof.
%
Until this point we will exclusively consider the situation
where the quantum space is formally trivial.
%
This means that the vacuum part in DVF is
always 1 as well as the ``left hand side'' of the BAE.
%
Section 6 includes a discussion on how to recover
the vacuum part for the non-trivial quantum spaces.
%
A prototype of them is a tensor product of 
irreducible finite dimensional modules such as (6.1).
%
The problem is essentially equivalent to 
specifying the left hand side of the BAE 
(cf section 2.4 in [KS1]) for such a general 
quantum space.
%
For the Yangian $Y(X_r)$, 
we propose quite generally for any $X_r$ that it is just given by 
a ratio of the relevant Drinfeld polynomials.\footnote\dag{
We thank E.K. Sklyanin and V.O. Tarasov for a discussion on this point.}
%
See (6.2).
%W
Then we shall briefly indicate a way to recover the vacuum parts.
%
\par
Many formulae in section 3 are formally valid also 
for $U_q(A^{(1)}_r)$ under a suitable condition.
%
In particular $\lambda = \phi$ case of (3.5) 
has appeared in [BR], for which a representation theoretical
background is available in [C].
%
\par
We hope to report similar results for $C_r$ and $D_r$ 
cases in near future.



\beginsection 2. Review of the results on fundamental representations

\noindent
%
Here we shall recall the $B_r$ case of the results in [KS1].
%
Let $\{ \alpha_1, \ldots, \alpha_r \}$
and $\{ \Lambda_1, \ldots, \Lambda_r \}$ be the set of 
the simple roots and 
fundamental weights of $B_r$ ($r \ge 2$).
%
Our normalization is 
$t_1 = \cdots = t_{r-1} = {1 \over 2} t_r = 1$ for 
$t_a = 2/(\alpha_a \vert \alpha_a)$.
%
Then $(\alpha_a \vert \alpha_b) = 
{2 \over t_a}\delta_{a, b} - \delta_{a, b-1} - \delta_{a, b+1}$
and $(\alpha_a \vert \Lambda_b) = \delta_{a b}/{t_a}$.
%
The $U_q(B^{(1)}_r)$ BAE for the trivial quantum space 
reads [RW]
%
$$\eqalignno{
-1 &= \prod_{b = 1}^r 
{Q_b(v^{(a)}_k + (\alpha_a \vert \alpha_b)) \over
 Q_b(v^{(a)}_k - (\alpha_a \vert \alpha_b)) }\quad
\hbox{ for } 1 \le a \le r, \, 1 \le k \le N_a,
&(2.1)\cr
%
Q_a(u) &= \prod_{j=1}^{N_a} [u - v^{(a)}_j], 
&(2.2)\cr}
$$
%
where 
$[u] = (q^u - q^{-u})/(q-q^{-1})$
and $N_1, \ldots, N_r$ are some positive integers.
%
Throughout the paper we assume that $q$ is generic.
%
The LHS of (2.1) is just -1 as opposed to the non-trivial quantum space
case (6.2), which will be discussed in section 6.
%
Until then we shall focus on the dress parts 
in the analytic Bethe ansatz.
%
\par
%
Following [KS1] we introduce the set $J$ and the order 
$\prec$ in it as
%
$$\eqalignno{
&J = \{1, 2, \ldots, r, 0, {\bar r}, \ldots, {\bar 1} \},
&(2.3{\rm a})\cr
%
&1 \prec 2 \prec \cdots, \prec r \prec 0 \prec 
{\bar r} \prec \cdots, \prec {\bar 1}.
&(2.3{\rm b})\cr}
$$
%
For $a \in J$, define the function
$z(a; u)$ by
%
$$\eqalign{
z(a; u)  &=
      {{Q_{a-1}( u+a+1  ) 
            Q_{a}(u+a-2 )}\over
       { Q_{a-1}(u+a-1)Q_{a}(u+a)}} 
  \qquad 1\le a \le r,\cr
z(0; u)  &= 
    {{Q_r(u+r-2) Q_{r}(u+r+1)}\over
   { Q_{r}(u+r)Q_{r}(u+r-1)}},  \cr
z({\bar a}; u)  &= 
      {{Q_{a-1}( u+2r-a-2  ) 
            Q_{a}(u+2r-a+1 )}\over
       { Q_{a-1}(u+2r-a)Q_{a}(u+2r-a-1)}} 
  \qquad 1\le a \le r,\cr
}\eqno(2.4)$$
%
where we have set $Q_0(u) = 1$.
%
$z(a, u)$ is the dress part of the box 
$\Fsquare(0.5cm,a)$ in (4.4a) of [KS1], which corresponds
to a weight in the vector representation.
%
For $(\xi_1, \ldots, \xi_r) \in \{\pm \}^r$,
define the function $sp(\xi_1, \ldots, \xi_r; u)$
by the following recursion relation with
respect to $r$ and the initial condition $r = 2$.
%
$$\eqalign{
sp(+,+,\xi_3,\ldots,\xi_r; u) &= \tau^Q
sp(+,\xi_3,\ldots,\xi_r; u),\cr
%
sp(+,-,\xi_3,\ldots,\xi_r; u) &= 
{Q_1(u+r-{5\over 2})\over Q_1(u+r-{1\over 2})}
\tau^Q sp(-,\xi_3,\ldots,\xi_r; u),\cr
%
sp(-,+,\xi_3,\ldots,\xi_r; u) &= 
{Q_1(u+r+{3\over 2})\over Q_1(u+r-{1\over 2})}
\tau^Q sp(+,\xi_3,\ldots,\xi_r; u+2),\cr
%
sp(-,-,\xi_3,\ldots,\xi_r; u) &= \tau^Q
sp(-,\xi_3,\ldots,\xi_r; u+2).\cr}
\eqno(2.5{\rm a})$$
%
%
$$\eqalign{
sp(+,+; u)
   &={Q_2(u-{1\over 2}) \over Q_2(u+{1\over 2})},\cr
%
sp(+,-; u)
   &={Q_1(u-{1\over 2})Q_2(u+{3\over 2}) \over 
      Q_1(u+{3\over 2})Q_2(u+{1\over 2})},\cr
%
sp(-,+; u)
   &={Q_1(u+{7\over 2})Q_2(u+{3\over 2}) \over 
      Q_1(u+{3\over 2})Q_2(u+{5\over 2})},\cr
%
sp(-,-; u)
   &={Q_2(u+{7\over 2}) \over Q_2(u+{5\over 2})}.\cr}
\eqno(2.5{\rm b})
$$
%
In (2.5a) $\tau^Q$ is the operation $Q_a \rightarrow Q_{a+1}$, namely,
%
$$\eqalign{
\tau^Q & F(Q_1(u+x^1_1), Q_1(u+x^1_2), \ldots,
         Q_2(u+x^2_1), Q_2(u+x^2_2), \ldots)  \cr
&=  F(Q_2(u+x^1_1), Q_2(u+x^1_2), \ldots,
         Q_3(u+x^2_1), Q_3(u+x^2_2), \ldots)}
\eqno(2.6)
$$
for any function $F$.
%
$sp(\xi_1, \ldots, \xi_r; u)$ is the dress part of the box 
$\overbrace{
\Flect(0.4cm,2.2cm,{\xi_1,\xi_2, \cdots , \xi_r})}^{r}$
in (4.25,26) of [KS1].
\par
Now we introduce the meromorphic 
functions $T^a(u)$ and $T_m(u)$ of $u$ 
for any $a, m \in {\bf Z}_{\ge 0}$
by the following ``non-commutative generating series''
%
$$\eqalignno{
&(1+z({\bar 1};u)X) \cdots (1+z({\bar r};u)X)
(1-z(0;u)X)^{-1}(1+z(r;u)X) \cdots (1+z(1;u)X)\cr
& = \sum_{a=0}^\infty T^a(u+a-1) X^a, &(2.7{\rm a})\cr
%
&(1-z(1;u)X)^{-1} \cdots (1-z(r;u)X)^{-1}
(1+z(0;u)X)
(1-z({\bar r};u)X)^{-1} \cdots (1-z({\bar 1};u)X)^{-1}\cr
& = \sum_{m=0}^\infty T_m(u+m-1) X^m, &(2.7{\rm b})\cr}
$$
%
where $X$ is a difference operator with the commutation relation
%
$$
X Q_a(u) = Q_a(u+2) X \quad \hbox{ for any } 1 \le a \le r.
\eqno(2.8)
$$
%
Thus $X z(a;u) = z(a; u+2) X$ for any $a \in J$.
%
We set $T^a(u) = T_m(u) = 0$ for $a, m < 0$.
%
An immediate consequence of the above definition is
%
$$\eqalignno{
\delta_{i j} &= \sum_{k=0}^N (-)^{i-k}T_{i-k}(u+i+k)
T^{k-j}(u+k+j) &(2.9{\rm a})\cr
%
&= \sum_{k=0}^N (-)^{i-k}T_{i-k}(u-i-k)
T^{k-j}(u-k-j) &(2.9{\rm b})\cr}
$$
%
for any $N \ge 0$ and $0 \le i, j \le N$.
%
Define $T^{(a)}_1(u)$ for $1 \le a \le r$ by
$$\eqalign{
T^{(a)}_1(u) &= T^a(u) \quad \hbox{ for } 1 \le a \le r-1,\cr
T^{(r)}_1(u) &= \sum_{\xi_1, \ldots, \xi_r = \pm }
sp(\xi_1, \ldots, \xi_r; u).\cr}
\eqno(2.10)
$$
%
Then $T^{(a)}_1(u)$ coincides with the dress part of $\Lambda^{(a)}_1(u)$
in [KS1] for all $1 \le a \le r$.
%
\proclaim Theorem 2.1. 
$T^{(r)}_1(u), T^a(u)$ and
$T_m(u) (\forall a, m \in {\bf Z})$ are pole-free
provided that the BAE (2.1) holds.
\par\noindent
%
For $T^{(r)}_1(u)$ and 
$T^a(u)$ with $a \le r-1$, this was proved in [KS1] in 
the more general setting
including the vacuum parts.
%
The other cases can be verified quite similarly.
%
$T^{(1)}_1(u)$ and $T^{(r)}_1(u)$ was considered earlier [R].
\par
The functions $z(a; u)$ and 
$sp(\xi_1, \ldots, \xi_r; u)$ are related as follows.
%
Given two sequences
$(\xi_1, \ldots, \xi_r)$ and 
$(\eta_1, \ldots, \eta_r) \in \{ \pm \}^r$, 
we define $i_1 < \cdots < i_k, I_1 < \cdots < I_{r-k}\, (0 \le k \le r)$ 
and 
$j_1 < \ldots < j_l, J_1 < \ldots < J_{r-l}\, (0 \le l \le r)$ 
by the following.
%
$$\eqalign{
&\xi_{i_1} = \cdots = \xi_{i_k} = +, \,
 \xi_{I_1} = \cdots = \xi_{I_{r-k}} = -,\cr
%
&\eta_{j_1} = \cdots = \eta_{j_l} = -, \,
 \eta_{J_1} = \cdots = \eta_{J_{r-l}} = +.\cr}
\eqno(2.11)
$$
%
Then we have
%
\proclaim Proposition 2.2.
For any $a \in {\bf Z}_{\ge 0}$, 
%
$$\eqalign{
&sp(\xi_1, \ldots, \xi_r; u-r+a+{1 \over 2})
sp(\eta_1, \ldots, \eta_r; u+r-a-{1 \over 2})\cr
&= \prod_{n=1}^a z(b_n; u+a+1-2n)\quad\quad  \hbox{ if } k + l \le a,\cr}
\eqno(2.12{\rm a})
$$
where
$$b_n = \cases{i_n & for $1 \le n \le k$\cr
               0 &   for $k < n \le a-l$\cr
               \overline{j_{a+1-n}} & for $a-l < n \le a$ \cr}.
\eqno(2.12{\rm b})
$$
%
For any $a \in {\bf Z}_{\le 2r-1}$, 
%
$$\eqalign{
&sp(\xi_1, \ldots, \xi_r; u-r+a+{1\over 2})
sp(\eta_1, \ldots, \eta_r; u+r-a-{1\over 2})\cr
&= \prod_{n=1}^{2r-1-a}
z(b^\prime_n; u+2r-a-2n)\quad \quad\hbox{ if } k + l \ge a+1,\cr}
\eqno(2.13{\rm a})
$$
where
$$b^\prime_n = \cases{J_n & for $1 \le n  \le r-l$\cr
               0 &   for $r-l < n \le r+k-1-a$\cr
               \overline{I_{2r-a-n}} & for $r+k-1-a < n \le 2r-1-a$ \cr}.
\eqno(2.13{\rm b})
$$
\par
%
This enables the evaluation of the product
$sp(\xi_1, \ldots, \xi_r; u-r+a+{1\over 2})
sp(\eta_1, \ldots, \eta_r; u+r-a-{1\over 2})$
for any $\{ \xi_i \}, \{ \eta_i \}$ and $a \in {\bf Z}$
in terms of $z$ (2.4).
%
For $1 \le a \le r-1$, (2.12) is theorem A.1 in [KS1].
%
It is straightforward to extend it to any $a \in {\bf Z}_{\ge 0}$.
%
Eq. (2.13) can be derived from (2.12) by replacing
$a$ by $2r-1-a$.
%
Note in (2.12b) that
$b_1 \prec \cdots \prec b_k \prec b_{k+1} = \cdots = 
b_{a-l} = 0 \prec b_{a-l+1} \prec \cdots \prec b_a \in J$.
%
A similar inequality holds also for $b^\prime_n$.
%
Comparing them with (2.7a) and (2.10) we get
%
\par\noindent
\proclaim Theorem 2.3.
$$
T^a(u) + T^{2r-1-a}(u) = T^{(r)}_1(u-r+a+{1 \over 2})
 T^{(r)}_1(u+r-a-{1 \over 2})\quad 
\forall a \in {\bf Z}.\eqno(2.14)
$$
%
\par
This is invariant under the exchange
$a \leftrightarrow 2r-1-a$.
%
If $a < 0$ or $a > 2r-1$, there is in fact only one term
on the LHS.
%
The new functional relation (2.14) will play an important role
in this paper.
%
It is also valid
after including the vacuum parts.
%
See section 6.



\beginsection 3. Spin-even case

\noindent
%
Let $\mu = (\mu_1, \mu_2, \ldots)$, $\mu_1 \ge \mu_2 \ge \cdots \ge 0$
be a Young diagram and 
$\mu^\prime = (\mu^\prime_1, \mu^\prime_2, \ldots)$ be its transpose.
%
We let $d_\mu$ denote the length of the main diagonal of $\mu$.
%
By a skew-Young diagram we mean a pair of 
Young diagrams $\lambda \subset \mu$.
%
It is depicted by the region corresponding to the 
subtraction $\mu -\lambda$.
%
See the Fig.3.1 for example.
%
\par
\centerline{Fig.3.1}
\par
%
For definiteness, we assume that 
$\lambda^\prime_{\mu_1} = \lambda_{\mu^\prime_1} = 0$.
%
A Young diagram $\mu$ is naturally identified with a skew-Young diagram 
$\phi \subset \mu$.
%
By an {\it admissible} tableau $b$ 
on a skew-Young diagram $\lambda \subset \mu$ we mean an assignment
of an element $b(i,j) \in J$ to 
the $(i,j)$-th box in $\lambda \subset \mu$
under the following rule:
(We locate $(1,1)$ at the top left corner of 
$\mu$, $(i+1,j)$ and 
$(i,j+1)$ to the below and the right of $(i,j)$, 
respectively.)
%
$$\eqalign{
&b(i,j) \preceq b(i,j+1), \qquad b(i,j) \prec b(i+1,j)\quad
\hbox{with the exception that}\cr
%
&b(i,j) = b(i,j+1) = 0 \, \hbox{ is forbidden}, \quad
b(i,j) = b(i+1,j) = 0 \, \hbox{ is allowed}.\cr
}\eqno(3.1)
$$
%
Without the exception this coincides with
the usual definition of the semi-standard Young tableau.
%
Denote by $Atab(\lambda \subset \mu)$ the set of 
admissible tableaux on $\lambda \subset \mu$.
\par
%
Given a skew-Young diagram $\lambda \subset \mu$,
we define the function 
$T_{\lambda \subset \mu}(u)$ as the following 
sum over the admissible tableaux.
%
$$
T_{\lambda \subset \mu}(u) = \sum_{b \in Atab(\lambda \subset \mu)}
\prod_{(i,j) \in (\lambda \subset \mu)}
z(b(i,j); u+\mu^\prime_1 - \mu_1 - 2i+2j).
\eqno(3.2)
$$
%
Comparing this with (2.7) we have
%
$$\eqalignno{
T^a(u) &= T_{(1^a)}(u): \hbox{ single column of length } a,&(3.3{\rm a})\cr
T_m(u) &= T_{(m)}(u): \hbox{ single row of length } m.&(3.3{\rm b})\cr
}$$
%
We also prepare a notation for the single hook,
%
$$T_{k,l} = T_{(l+1,1^k)}(u). \eqno(3.3{\rm c})
$$
%
Our main result in this section is
%
\par
\proclaim Theorem 3.1.
$$
T_{\lambda \subset \mu}(u) = 
det \pmatrix{
0& \cdots & 0 & R_{1 1}& \cdots & R_{1 d_\mu}\cr
\vdots & \ddots & \vdots & \vdots &   & \vdots \cr
0& \cdots & 0 & R_{d_\lambda 1}& \cdots & R_{d_\lambda d_\mu}\cr
C_{1 1} & \cdots & C_{1 d_\lambda} & 
H_{1 1} & \cdots & H_{1 d_\mu} \cr
\vdots &  & \vdots & \vdots & \ddots & \vdots \cr
C_{d_\mu 1} & \cdots & C_{d_\mu d_\lambda} & 
H_{d_\mu 1} & \cdots & H_{d_\mu d_\mu} \cr},\eqno(3.4{\rm a})
$$
%
where 
$$\eqalign{
R_{i j} &= T_{\mu_j - \lambda_i +i-j}
(u+\mu^\prime_1-\mu_1 + \mu_j + \lambda_i -i-j+1),\cr
%
C_{i j} &= - T^{\mu^\prime_i - \lambda^\prime_j -i+j}
(u+\mu^\prime_1-\mu_1 - \mu^\prime_i - \lambda^\prime_j +i+j-1),\cr
%
H_{i j} &= T_{\mu^\prime_i -i, \mu_j-j}
(u+\mu^\prime_1-\mu_1 - \mu^\prime_i +\mu_j +i-j).\cr}
\eqno(3.4{\rm b})
$$
\par
%
Two particular cases corresponding to the formal choices
$\mu_i = \lambda_i$ or 
$\mu^\prime_i = \lambda^\prime_i$
for $1 \le i \le d_\lambda = d_\mu$ yield simpler formulae.
%
In these cases, redefining $\mu_i, \mu^\prime_i, 
\lambda_i$ and $\lambda^\prime_i$ so that
$\lambda^\prime_{\mu_1} = \lambda_{\mu^\prime_1}=0$,
we have
%
$$\eqalignno{
T_{\lambda \subset \mu}(u)
&= det_{1 \le i,j \le \mu_1}
(T^{\mu^\prime_i - \lambda^\prime_j -i+j}
(u+\mu^\prime_1-\mu_1 - \mu^\prime_i - \lambda^\prime_j +i+j-1)),
&(3.5{\rm a})\cr
%
&= det_{1 \le i,j \le \mu^\prime_1}
(T_{\mu_j - \lambda_i +i-j}
(u+\mu^\prime_1-\mu_1 + \mu_j + \lambda_i -i-j+1)).
&(3.5{\rm b})\cr}
$$
%
Eq.(3.5a) can be verified, for example, by induction on $\mu_1$, i.e.,
by showing the same recursive relation for the tableau sum (3.2)
as an expansion of the determinant.
%
Then (3.5b) follows from (2.9).
%
Theorem 3.1 is proved from these results by applying 
Sylvester's theorem on determinants.
%
From (3.5a) and Theorem 2.1 one has
%
\proclaim Corollary.
$T_{\lambda \subset \mu}(u)$ is pole-free provided the BAE (2.1) holds.
\par
%
The admissibility condition (3.1) leads to the above conclusion 
although it is by no means obvious in the defining expression (3.2).
%
Despite the exception in (3.1),
our formulae (3.4) and (3.5) formally coincide with 
the classical ones due to Giambelli and Jacobi-Trudi
on Schur functions [M] if one drops
the $u$-dependence (or in the limit 
$\vert u \vert \rightarrow \infty$).
%
If $\mu^\prime_{i+1} - \lambda^\prime_i > 2r$ for some $i$,
$Atab(\lambda \subset \mu) = \phi$.
%
Correspondingly, one can show that the determinant (3.5a) is
vanishing using the fact that $T^a(u)$ factorizes
for $a \ge 2r$ due to Theorem 2.3.
%
Henceforth we assume that 
$\mu^\prime_{i+1} - \lambda^\prime_i \le 2r$
for $1 \le i \le \mu_1$.
(We set $\mu^\prime_{\mu_1 + 1} = -\infty$.)
\par
%
The $T_{\lambda \subset \mu}(u)$ (3.2)
describes the spectrum of the transfer matrix 
whose auxiliary space is labeled by the skew-Young diagram
$\lambda \subset \mu$ and $u$.
%
Denote the space by $W_{\lambda \subset \mu}(u)$.
%
We suppose it is an irreducible finite dimensional module 
over $Y(B_r)$ (or $U_q(B^{(1)}_r)$ in the trigonometric case)
in view that all the terms in (3.2) 
seem coupling to make the apparent poles suprious under BAE. 
%
Now we shall specify the Drinfeld polynomial
$P_a(\zeta)$ [D] that characterizes 
$W_{\lambda \subset \mu}(u)$ based on some 
empirical procedure.
%
Our convention slightly differs
from the original one in Theorem 2 of [D]
in such a way that
$$
1 + \sum_{k=0}^\infty d_{i k} \zeta^{-k-1} = 
{P_i(\zeta + {1 \over t_i}) \over 
P_i(\zeta - {1 \over t_i})}.
\eqno(3.6)
$$
%
For any $b \in Atab(\lambda \subset \mu)$, the 
corresponding summand (3.2) has the form
%
$$
\prod_{a=1}^r
{Q_a(u+x^a_1) \cdots Q_a(u+x^a_{i_a}) \over
Q_a(u+y^a_1) \cdots Q_a(u+y^a_{i_a})}, \eqno(3.7)
$$
%
where $x^a_j, y^a_j$ and $i_a$ are specified from $b$.
%
This summand carries the $B_r$-weight
%
$$
wt(b) = \sum_{a=1}^r \Bigl(
{t_a \over 2} \sum_{j=1}^{i_a} (y^a_j - x^a_j) \Bigr) \Lambda_a
\eqno(3.8)
$$
in the sense that
$\lim_{q^u \rightarrow \infty}
\hbox{(3.7)} = q^{-2(wt(b) \vert \sum_{a=1}^r N_a \alpha_a)}$.
%
From $Atab(\lambda \subset \mu)$, take 
such $b_0$ that $wt(b_0)$ is highest,
which corresponds to the ``top term'' in section 2.4 of [KS1].
%
In our case, such $b_0$ is unique and given as follows.
%
Fill the left most column of $\lambda \subset \mu$
from the top to the bottom by assigning the first 
$\mu^\prime_1 - \lambda^\prime_1$ letters
from the sequence
$1,2,\ldots, r,0,0,\ldots$.
%
Given the $(i-1)$-th column, the $i$-th column is built 
from the top to the bottom
by taking the first 
$\mu^\prime_i - \lambda^\prime_i$ letters
from the sequence
$1,2,\ldots,r,\overbrace{0,\ldots,0}^k,\overline{r},
\overline{r-1},\ldots,\overline{1}$, where
$k = \hbox{max}(0,\hbox{min}(
\lambda^\prime_{i-1} - \lambda^\prime_i,
\mu^\prime_i - \lambda^\prime_i - r))$.
(We set $\lambda^\prime_0 = +\infty$.)
%
See the example in Fig.3.2.
%
\par
\centerline{Fig.3.2.}
\par\noindent
%
It turns out that (3.7) for the top term $b_0$ 
can be expressed uniquely in the form
%
$$
\prod_{a=1}^r \prod_{j=1}^{M_a}
{Q_a(u+z^a_j - {1 \over t_a}) \over
Q_a(u+z^a_j + {1 \over t_a})}\eqno(3.9)
$$
for some $M_a$ and $\{ z^a_j \vert 1 \le j \le M_a \}$
up to the permutations of $z^a_j$'s for each $a$.
%
We then propose that the Drinfeld polynomial 
$P^{W_{\lambda \subset \mu}(u)}_a(\zeta)$ for 
$W_{\lambda \subset \mu}(u)$ is given by
%
$$
P^{W_{\lambda \subset \mu}(u)}_a(\zeta) = \prod_{j=1}^{M_a}
(\zeta - u - z^a_j)\quad 1 \le a \le r.
\eqno(3.10)
$$
%
In our case, it reads explicitly as follows.
%
$$\eqalign{
P^{W_{\lambda \subset \mu}(u)}_a(\zeta)
&= \prod_{ \scriptstyle 1 \le i \le \mu_1  \atop
                \scriptstyle  \mu'_i-\lambda'_i=a }
            (\zeta-u-\mu'_1 +\mu_1+1+a+2\lambda'_i-2i)\cr
& \times \prod_{ \scriptstyle 1 \le i \le \mu_1-1 \atop
                \scriptstyle  \mu^\prime_{i+1}-\lambda'_i=2r-a }
            (\zeta-u-\mu'_1 +\mu_1+2+a+2\lambda'_i-2i)\quad 1 \le a \le r-1,\cr}
\eqno(3.11{\rm a})
$$
%
%
$$\eqalign{
P^{W_{\lambda \subset \mu}(u)}_r(\zeta)
&= \prod_{ \scriptstyle 1 \le i \le \mu_1  \atop
                \scriptstyle \lambda'_i+r \le \mu'_i \le \lambda'_{i-1}+r }
            (\zeta-u-\mu'_1 +\mu_1+2\mu'_i-2i-r+{3\over 2})\cr
& \times \prod_{ \scriptstyle 1 \le i \le \mu_1 \atop
                \scriptstyle  \mu^\prime_{i+1}\le \lambda'_i+r \le \mu'_i }
            (\zeta-u-\mu'_1 +\mu_1+2\lambda'_i-2i+r+{1\over 2}),\cr}
\eqno(3.11{\rm b})
$$
where we have set 
%
$$\mu'_{\mu_1+1}=-\infty, \quad \lambda'_0=\infty.
\eqno(3.12)$$
%
We will call the irreducible finite dimensional
$Y(B_r)$ module spin-even (resp. spin-odd)
if and only if the characterizing Drinfeld
polynomial $P_r(\zeta)$ is even (resp. odd) degree.
%
The one in (3.11b) is even for any skew-Young diagram
$\lambda \subset \mu$.
%
For example, in the case of the single 
column or row (3.3), (3.11) reads
%
$$\eqalignno{
P^{W_{(1^c)}(u)}_a(\zeta) &= \cases{
(\zeta - u)^{\delta_{a c}} & $1 \le c < r$\cr
\bigl( (\zeta - u + c - r + {1 \over 2})
       (\zeta - u - c + r - {1 \over 2}) \bigr)^{\delta_{a r}}
& $c \ge r$ \cr}, &(3.13{\rm a})\cr
%
P^{W_{(m)}(u)}_a(\zeta) &= 
\bigl( (\zeta - u+m-1)(\zeta - u+m-3) \cdots (\zeta - u -m+1)
\bigr)^{\delta_{a 1}}.&(3.13{\rm b})\cr}
$$
%
\par
%
As a $B_r$ module, the $Y(B_r)$ module 
$W_{\lambda \subset \mu}(u)$ decomposes as 
%
$$
W_{\lambda \subset \mu}(u) \simeq 
\sum_{\eta} \bigl(
\sum_{\kappa, \nu} LR^\mu_{\lambda \nu}
LR^{\nu}_{(2\kappa)^\prime \eta} \bigr)
\pi_{O(2r+1)}(V_\eta),\eqno(3.14)
$$
%
which is $u$-independent.
%
Here $LR^\mu_{\lambda \nu}$ etc denote the 
Littlewood-Richardson coefficients
for the universal character ring $\Lambda$ 
of $GL$ type introduced in [KT].
%
The sums run over all the Young diagrams $\eta, \nu$ and 
$\kappa = (\kappa_1, \kappa_2, \ldots)$, 
where $(2\kappa)^\prime$
stands for the transpose of
$2\kappa = (2\kappa_1, 2\kappa_2, \ldots)$.
%
$\pi_{O(2r+1)}(V_\eta)$ is the image of the 
specialization homomorphism [KT].
%
It is equal to ($\pm 1$ or 0) ``times'' 
the irreducible $B_r$ module $V_{\eta^\ast}$
with the highest weight labeled by the Young diagram
$\eta^\ast$ with $(\eta^{\ast})^ \prime_1 \le r$.
%
They are determined according to the equality
$\pi_{O(2r+1)}(\chi(\eta)) = (\pm 1 \hbox{ or } 0)
\times \chi(\eta^\ast)$ at the character level [KT].
%


%-------------------------------------------------


\beginsection 4. Spin-odd case

\noindent
%
Consider the following subset 
$Spin \subset Atab((1^r))$.
%
$$
\hbox{
  \m@th\baselineskip0pt\offinterlineskip
   \vbox{ 
      \hbox{$\fsquare(0.5cm,\hbox{$i_1$})$}\vskip-0.4pt
      \hbox{$\vnaka$}\vskip-0.4pt
	     \hbox{$\fsquare(0.5cm,\hbox{$i_r$})$}\vskip-0.4pt
        }
      } 
\raise 4ex \hbox{$\in Spin \Leftrightarrow
\cases{
i_1 \prec \cdots \prec i_r \in J,\cr
%
0 \hbox{ is not contained},\cr
\hbox{only one of } i \hbox{ and } \overline{ i } \hbox{ is contained
for any } 1 \le i \le r.\cr}$}
\eqno(4.1)
$$
%
There is a bijection $\iota: Spin \rightarrow 
\{ (\xi_1, \ldots, \xi_r) \mid \xi_j = \pm \}$ sending
(4.1) with 
$1 \preceq i_1 \prec \cdots \prec i_k \preceq r \prec 
\overline{r} \preceq i_{k+1} \prec \cdots \prec i_r \preceq 
\overline{1}$ to
such $(\xi_1, \ldots, \xi_r)$ that
$\xi_{i_1} = \cdots = \xi_{i_k} = +,\,
\xi_{\overline{i_{k+1}}} = \cdots = \xi_{\overline{i_r}} = -$,
where we interpret $\overline{k} = i$ if $k = \overline{i}$.
%
Thus the latter of (2.10) can also be written as
$T^{(r)}_1(u) = \sum_{b \in Spin} sp(\iota(b);u)$.
%
This type of $\iota$ has also been utilized in [KN].
%
\par
%
For a skew-Young diagram $\lambda \subset \mu$ with
$\mu^\prime_1 - \lambda^\prime_1 \ge r$,
hatch the bottom $r$ boxes in the leftmost column, 
which we call an L-hatched skew-Young diagram
$\lambda \subset \mu$.
See Fig.4.1.
%
\par \centerline{Fig.4.1.}\par
\noindent
%
Consider a tableau $b$ on it, namely,
a map $b: \hbox{ L-hatched } \lambda \subset \mu 
\rightarrow J$.
%
We call a tableau $b$ on an L-hatched $\lambda \subset \mu$
{\it L-admissible} if and only if all of the following
three conditions are valid.
($n = \mu^\prime_2 - (\mu^\prime_1 - r)$ and see Fig. 4.2 
for the definitions of $i_l$ and $j_l$.)
%
$$\eqalign{
\hbox{(i)}& \hbox{ hatched part } \in Spin,
\, \hbox{ and } (3.1) \hbox{ for non-hatched part},\cr
\hbox{(ii)}&\, j_0 \prec i_1,\cr
\hbox{(iii)}&\, i_1 \preceq j_1, \ldots, i_n \preceq j_n 
\hbox{ or there exists } k \in \{1, \ldots, n \} \hbox{ such that }\cr
&\, i_1 \preceq j_1, \ldots, i_{k-1} \preceq j_{k-1} \hbox{ and }
\overline{r} \preceq j_k \prec i_k \preceq \overline{1}.\cr}
\eqno(4.2)
$$
%
Here (ii) is void when $\mu^\prime_1 - \lambda^\prime_1 = r$ and so is
(iii) for $n=0$.
%
\par \centerline{Fig.4.2.}\par
\noindent
%
Denote by $Atab_L(\lambda \subset \mu)$ the set of 
L-admissible tableaux on the L-hatched  $\lambda \subset \mu$.
%
We note that 
$Atab(\lambda \subset \mu) \not\subseteq
Atab_L(\lambda \subset \mu)$ nor
$Atab(\lambda \subset \mu) \not\supseteq
Atab_L(\lambda \subset \mu)$.
%
Given an L-hatched skew-Young diagram 
$\lambda \subset \mu$, we define the function
$S_{\lambda \subset \mu}^L(u)$ by 
%
$$\eqalign{
S_{\lambda \subset \mu}^L(u) &= 
\sum_{b \in Atab_L(\lambda \subset \mu)}
sp(\iota(\hbox{hatched part});u)\cr
%
&\times
\prod_{(i,j) \in \hbox{ non hatched part of } (\lambda \subset \mu)}
z(b(i,j); u+2\mu^\prime_1-r-2i+2j-{3 \over 2}).}\eqno(4.3)
$$
%
\par
We have an L $\leftrightarrow$ R  (left vs. right) dual of these 
definitions as follows.
%
For a skew-Young diagram $\lambda \subset \mu$ with
$\mu^\prime_{\mu_1} \ge r$ 
(remember we assumed $\lambda^\prime_{\mu_1}=0$),
hatch the top $r$ boxes in the rightmost column, 
which we call an R-hatched skew-Young diagram
$\lambda \subset \mu$.
See Fig.4.3.
%
\par \centerline{Fig.4.3.}\par
\noindent
%
Consider a tableau 
$b: \hbox{ R-hatched } \lambda \subset \mu 
\rightarrow J$.
%
We call a tableau $b$ on an R-hatched $\lambda \subset \mu$
{\it R-admissible} if and only if all of the following
three conditions are valid.
($n = r  - \lambda^\prime_{\mu_1 - 1}$ and see Fig. 4.4 
for the definitions of $i_l$ and $j_l$.)
%
$$\eqalign{
\hbox{(i)}&\, \hbox{ hatched part } \in Spin,
\, \hbox{ and } (3.1) \hbox{ for non-hatched part},\cr
\hbox{(ii)}&\, i_1 \prec j_0,\cr
\hbox{(iii)}&\, j_1 \preceq i_1, \ldots, j_n \preceq i_n 
\hbox{ or there exists } k \in \{1, \ldots, n \} \hbox{ such that }\cr
&\, j_1 \preceq i_1, \ldots, j_{k-1} \preceq i_{k-1} \hbox{ and }
1 \preceq i_k \prec j_k \preceq r,\cr}
\eqno(4.4)
$$
%
where (ii) is void when $\mu^\prime_{\mu_1} = r$ and so is
(iii) for $n=0$.
%
\par \centerline{Fig.4.4.}\par
\noindent
%
Denoting by $Atab_R(\lambda \subset \mu)$ the set of 
R-admissible tableaux on the R-hatched  $\lambda \subset \mu$,
we define 
%
$$\eqalign{
S_{\lambda \subset \mu}^R(u) &= 
\sum_{b \in Atab_R(\lambda \subset \mu)}
sp(\iota(\hbox{hatched part});u)\cr
%
&\times
\prod_{(i,j) \in \hbox{ non hatched part of } (\lambda \subset \mu)}
z(b(i,j); u-2\mu_1+r-2i+2j+{3 \over 2}).}\eqno(4.5)
$$
%
\par
Our first main results in this section is
%
\proclaim Theorem 4.1.
%
$$\eqalignno{
S_{\lambda \subset \mu}^L(u) 
&= det_{1 \le i,j \le \mu_1}({\cal S}^L_{i j}) &(4.6{\rm a})\cr
&= det_{1 \le i,j \le \mu^\prime_2}
(\overline{\cal S}^L_{i j}), &(4.6{\rm b})\cr}
$$
%
where
%
$$\eqalignno{
{\cal S}^L_{i j} &= 
  \cases{  T^{\mu^\prime_j-\lambda^\prime_i+i-j} 
(u+2\mu^\prime_1-\mu^\prime_j-\lambda^\prime_i+i+j-r-{5\over 2})
                \, \,  \, j \ge 2  \cr
           T^{(r)}_1(u+2i-2+2(\mu^\prime_1-\lambda^\prime_i-r)) 
                  \,\, \,  j=1 \cr
         },&(4.7{\rm a})\cr
%
\overline{\cal  S}^L_{i j} &= 
  \cases{  T_{\mu_i-\lambda_j-i+j} 
(u+2\mu^\prime_1+\mu_i+\lambda_j-i-j-r-{1\over2})
                   \,\,  1\le j \le \lambda'_1  \cr
           {\cal H}^L_{\mu_i+\lambda^\prime_1-i}
(u+2\mu^\prime_1-2\lambda'_1-2r) 
                   \, \,\, j=\lambda'_1+1      \cr
           T_{\mu_i-i+j-1} (u+2\mu^\prime_1+\mu_i-i-j-r+{1\over2})
                   \, \,\,  j> \lambda'_1+1 \cr
         },&(4.7{\rm b})\cr
%
{\cal H}^L_m(u) &= \sum_{l=0}^{m} (-1)^l
             T^{(r)}_1 (u+2l) T_{m-l}(u+m+r+l-{1 \over 2}).
&(4.7{\rm c})\cr}
$$
\par
%
From (4.6a), (4.7a,c) and Theorem 3.1,
${\cal H}^L_m(u)$ is equal to the L-hatched 
hook $S^L_{(m+1,1^{r-1})}(u)$.
%
\par
%
For an R-hatched diagram $\lambda \subset \mu$,
let $\xi \subset \eta$ be the sub-diagram obtained
by removing the rightmost column of $\lambda \subset \mu$.
%
See Fig. 4.5.
%
\par\centerline{ Fig. 4.5.} \par\noindent
%
Thus for example 
$\eta_i = \mu_1 -1$ for 
$1 \le i \le \mu^\prime_{\mu_1} - \lambda^\prime_{\mu_1 - 1}$.
%
Then another main result in this section is the R-hatched 
version of the previous theorem as follows.
%
\proclaim Theorem 4.2.
$$\eqalignno{
S_{\lambda \subset \mu}^R(u) 
&= det_{1 \le i,j \le \mu_1}({\cal S}^R_{i j}) &(4.8{\rm a})\cr
&= det_{1 \le i,j \le \eta^\prime_1}
(\overline{\cal S}^R_{i j}), &(4.8{\rm b})\cr}
$$
%
where
%
$$\eqalignno{
{\cal  S}^R_{i j} &= 
  \cases{T^{\mu^\prime_j-\lambda^\prime_i+i-j}
(u-2\mu_1-\mu^\prime_i-\lambda^\prime_j+i+j+r+{1\over 2})
                \,\,\,   j \le \mu_1-1  \cr
           T^{(r)}_1 (u-2\mu_1-2\mu^\prime_i+2i+2r) 
                  \,\,\,  j=\mu_1 \cr
         },&(4.9{\rm a})\cr
%
\overline{\cal S}^R_{i j} &= 
  \cases{  
T_{\eta_i-\xi_j-i+j}(u-2\lambda'_{\mu_1-1}-
                                2\eta_1+\eta_i+\xi_j-i-j+r+{1\over2})  
            \,       i\ne \eta'_1-\mu^\prime_1+\mu'_{\mu_1}    \cr
             {\cal H}^R_{\eta_i-\xi_j-i+j}(u-2\mu'_{\mu_1}+2r)    
              \,\,\,     i= \eta'_1-\mu'_1+\mu'_{\mu_1}  \cr}\cr
&  &(4.9{\rm b})\cr
%
{\cal H}^R_m(u) &= \sum_{l=0}^{m} (-1)^{l}
               T^{(r)}_1 (u-2l) T_{m-l}(u-m-r-l+{1 \over 2}).
&(4.9{\rm c})}
$$
%
\par
%
From (4.8a), (4.9a,c) and Theorem 3.1, one sees that
%
${\cal H}^R_m(u)$ is equal to the R-hatched ``dual hook''
$S^R_{(m^{r-1}) \subset ((m+1)^r)}(u)$.
%
From Theorems 2.1, 4.1 and 4.2, we have
%
\proclaim Corollary.
$S^L_{\lambda \subset \mu}(u)$ and
$S^R_{\lambda \subset \mu}(u)$ are pole-free provided that the BAE
(2.1) holds.
\par
%
Following a similar argument to the previous section, we propose the 
Drinfeld polynomials corresponding to the auxiliary
spaces $W^L_{\lambda \subset \mu}(u)$ and
$W^R_{\lambda \subset \mu}(u)$ of 
$S^L_{\lambda \subset \mu}(u)$ and 
$S^R_{\lambda \subset \mu}(u)$, respectively.
%
$$
P^{W^L_{\lambda \subset \mu}(u)}_a(\zeta)
 = P^{W_{\lambda \subset \mu}
(u+\mu^\prime_1+\mu_1-r-{3 \over 2})}_a(\zeta)
\quad 1 \le a \le r-1,\eqno(4.10{\rm a})
$$
%
$$\eqalign{
P^{W^L_{\lambda \subset \mu}(u)}_r(\zeta)
&= {1 \over \zeta - u + 1} P^{W_{\lambda \subset \mu}
(u+\mu^\prime_1+\mu_1-r-{3 \over 2})}_r(\zeta)\cr
%
&= \prod_{ \scriptstyle 2 \le i \le \mu_1  \atop
                \scriptstyle \lambda'_i+r \le \mu'_i \le \lambda'_{i-1}+r }
            (\zeta-u+3+2(\mu'_i-i-\mu^\prime_1))\cr
& \times \prod_{ \scriptstyle 1 \le i \le \mu_1 \atop
                \scriptstyle  \mu^\prime_{i+1}\le \lambda'_i+r \le \mu'_i }
           (\zeta-u+2+2(\lambda'_i-i-\mu^\prime_1+r)),\cr}
\eqno(4.10{\rm b})
$$
%
$$
P^{W^R_{\lambda \subset \mu}(u)}_a(\zeta)
 = P^{W_{\lambda \subset \mu}
(u-\mu^\prime_1-\mu_1+r+{3 \over 2})}_a(\zeta)
\quad 1 \le a \le r-1,\eqno(4.11{\rm a})
$$
%
$$\eqalign{
P^{W^R_{\lambda \subset \mu}(u)}_r(\zeta)
&= {1 \over \zeta - u - 1} P^{W_{\lambda \subset \mu}
(u-\mu^\prime_1-\mu_1+r+{3 \over 2})}_r(\zeta)\cr
%
&= \prod_{ \scriptstyle 1 \le i \le \mu_1  \atop
                \scriptstyle \lambda'_i+r \le \mu'_i \le \lambda'_{i-1}+r }
            (\zeta-u+2(\mu'_i-i+\mu_1-r))\cr
& \times \prod_{ \scriptstyle 1 \le i \le \mu_1-1 \atop
                \scriptstyle  \mu^\prime_{i+1}\le \lambda'_i+r \le \mu'_i }
           (\zeta-u-1+2(\lambda'_i-i+\mu_1)),\cr}
\eqno(4.11{\rm b})
$$
%
where we assume (3.12).
%
\par
%
In $Atab_L(\lambda \subset \mu)$
and $Atab_R(\lambda \subset \mu)$, 
we have considered the hatched part ($Spin$ (4.1))
only in the bottom left or top right position.
%
A natural question may be whether it is possible to define 
a tableau sum that becomes pole-free and 
contains $Spin$ simultaneously in various places 
in a skew-Young diagram $\lambda \subset \mu$.
%
It is indeed possible 
to include $Spin$ both at the bottom left and the top right.
%
However, we have found only few examples beyond that so far.
%

%-------------------------------------------------------------

\beginsection 5. Solution to the $T$-system

\noindent
%
The functions 
$T_{\lambda \subset \mu}(u)$ (3.2), 
$S^L_{\lambda \subset \mu}(u)$ (4.3) and 
$S^R_{\lambda \subset \mu}(u)$ (4.5) provide
the solution to the $T$-system for $B_r$, one of the 
functional relations proposed in [KNS] for any $X_r$.
(See [KS2] for the $T$-system of twisted quantum affine algebras.)
%
For $m \in {\bf Z}_{\ge 0}$, put
%
$$\eqalignno{
T^{(a)}_m(u) &= T_{(m^a)}(u) \quad 1 \le a \le r-1,
&(5.1{\rm a})\cr
%
T^{(r)}_{2m}(u) &= T_{(m^r)}(u),&(5.1{\rm b})\cr
%
T^{(r)}_{2m+1}(u) &= S^L_{((m+1)^r)}(u-m) =
S^R_{((m+1)^r)}(u+m).&(5.1{\rm c})\cr}
$$
%
The latter equality in (5.1c) can be shown easily 
by using 
(2.14), (4.7a) and (4.9a).
%
The definition (5.1) includes (2.10).
%
Moreover, from (3.11) and (4.10,11),
the Drinfeld polynomial corresponding to
$T^{(a)}_m(u)$ is given by
$P_b(\zeta)=
\bigl(\prod_{i=1}^m(\zeta-u+{m+1-2i\over t_a})\bigr)^{\delta_{b a}}$
for $1 \le b \le r$, 
in agreement with (2.3) of [KS1].
%
Thus $T^{(a)}_m(u)$ here is the DVF for the transfer matrix 
$T^{(a)}_m(u)$ considered in [KS1].
%
\proclaim Theorem 5.1.
$T^{(a)}_m(u)$ defined above satisfies the following functional relations.
%
$$\eqalign{
%
T^{(a)}_m(u-1)T^{(a)}_m(u+1) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) +
T^{(a-1)}_m(u)T^{(a+1)}_m(u)\cr
% 
&\qquad\qquad\qquad\qquad\hbox{ for }\, 1 \le a \le r-2,\cr
%
T^{(r-1)}_m(u-1)T^{(r-1)}_m(u+1) &=
T^{(r-1)}_{m+1}(u)T^{(r-1)}_{m-1}(u) + 
T^{(r-2)}_m(u)T^{(r)}_{2m}(u),\cr
%
T^{(r)}_{2m}(u-{1\over 2})T^{(r)}_{2m}(u+{1\over 2}) &=
T^{(r)}_{2m+1}(u)T^{(r)}_{2m-1}(u) \cr
%
&+ T^{(r-1)}_m(u-{1\over 2})T^{(r-1)}_m(u+{1\over 2}),\cr
%
T^{(r)}_{2m+1}(u-{1\over 2})T^{(r)}_{2m+1}(u+{1\over 2}) &=
T^{(r)}_{2m+2}(u)T^{(r)}_{2m}(u) +
T^{(r-1)}_m(u)T^{(r-1)}_{m+1}(u).\cr}
\eqno(5.2)
$$
\par 
{\it Outline of the proof.} 
We use the determinantal expressions (3.5a) and (4.6a).
%
Then the first two equations in (5.2) reduce to the Jacobi
identity.
(cf. [KNS] eqs.(2.20)-(2.22).)
%
To prove the third equation, substitute (4.6a) into
$T^{(r)}_{2m+1}(u)T^{(r)}_{2m-1}(u)$.
%
Expanding the determinants with respect to the first column,
we have
%
$$\eqalign{
&T^{(r)}_{2m+1}(u)T^{(r)}_{2m-1}(u) =
\sum_{i=0}^{m-1}\sum_{j=0}^m (-1)^{i+j}
R^{(m)}_jR^{(m-1)}_i\cr
&\times \bigl(T^{r+j-i-1}(u-m+i+j+{1\over 2})+
T^{r+i-j}(u-m+i+j+{1\over 2})\bigr).\cr}
$$
%
Here, $R^{(m)}_j$ denotes the cofactor of 
$T^{(r)}_1(u-m+2j)$ in $T^{(r)}_{2m+1}(u)$
and we have used (2.14).
%
On taking the $j$-sum, the 
$T^{r+j-i-1}(u-m+i+j+{1\over 2})$ term vanishes.
%
After taking the $i$-sum, the 
$T^{r+i-j}(u-m+i+j+{1\over 2})$ term is non-zero
only for $j=0$ or $j=m$.
%
Noting that 
$R^{(m)}_0 = T^{(r)}_{2m}(u+{1\over 2})$
and $R^{(m)}_m = T^{(r-1)}_m(u-{1\over 2})$,
one has the third equation.
%
The last equation in (5.2) can be verified quite similarly.
\par
%
The functional relation (5.2) is the unrestricted $T$-system for $B_r$, 
(3.20) in [KNS] (in a different normalization).
%
There was a factor $g^{(a)}_m(u)$ 
in each equation as
$T^{(a)}_m(u+{1 \over t_a})
T^{(a)}_m(u-{1 \over t_a}) = 
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) + g^{(a)}_m(u)(\cdots)$.
%
The $g^{(a)}_m(u)$ 
is 1 here because
we are considering the case where vacuum part = 1.
%
The choice (5.1a) has been conjectured 
in (4.20) of [KS1] including the vacuum parts.
%
The case $r = 2$ had been proved earlier [K].
%
\par
It may be interesting to regard $u$ and $m$ as discrete space-time
variables and consider (5.2) as a 
discretized Toda equation.
%
Actually, a ``continuum limit'' of (5.2) (with 
$g^{(a)}_m(u)$) under an 
appropriate rescaling of $u, m$ and $g^{(a)}_m(u)$
leads to
$$
(\partial_u^2 - \partial_m^2)\hbox{ log } \phi_a(u,m)
= \hbox{const } \prod_{b=1}^r
\phi_b(u,m)^{-A_{a b}},$$
%
where $\phi_a(u,m)$ is a scaled $T^{(a)}_m(u)$
and $A_{a b} = {2(\alpha_a \vert \alpha_b)\over
(\alpha_a \vert \alpha_a)}$ is the Cartan matrix.
%
The constant above can be made arbitrary by choosing
the $g^{(a)}_m(u)$ suitably.
%
We remark that 
the $T$-system proposed in [KNS] has this aspect
for all the classical simple Lie algebra $X_r$.


%------------------------------------------------------------

\beginsection 6. On vacuum parts and BAE in terms of Drinfeld polynomial

\noindent
%
So far we have treated the case where the quantum space is
formally trivial.
%
This corresponds to choosing the LHS of the BAE (2.1) to be just -1
and the vacuum parts in the DVFs
$T_{\lambda \subset \mu}(u)$,
$S^L_{\lambda \subset \mu}(u)$ and 
$S^R_{\lambda \subset \mu}(u)$ to be 1.
%
To recover the vacuum parts for the non-trivial quantum space
%
$$
\otimes_{i=1}^N W^{(i)},\eqno(6.1)
$$
%
one needs to know the corresponding BAE.
%
Assuming that each $W^{(i)}$ in (6.1) is a finite dimensional 
irreducible $Y(B_r)$ module characterized by the Drinfeld
polynomial $P^{(i)}_a(\zeta) \, (1 \le a \le r)$,
we conjecture the BAE:
%
$$\eqalign{
-{P_a(v^{(a)}_k+{1\over t_a}) \over
P_a(v^{(a)}_k-{1\over t_a}) } &=
 \prod_{b = 1}^r 
{Q_b(v^{(a)}_k + (\alpha_a \vert \alpha_b)) \over
 Q_b(v^{(a)}_k - (\alpha_a \vert \alpha_b)) }
\quad 1 \le a \le r, \, 1 \le k \le N_a,\cr
%
P_a(\zeta) &= \prod_{i=1}^N P^{(i)}_a(\zeta).\cr}
\eqno(6.2)
$$
%
Here we understand that $q \rightarrow 1$ in (2.2)
for $Y(B_r)$.
%
(On the other hand, for generic $q$, we suppose 
that (6.2) is the correct BAE for $U_q(B^{(1)}_r)$
if $P^{(i)}_a(\zeta)$ is replaced by a natural $q$-analogue.)
%
The equation (6.2) has been formulated purely from the 
representation theoretical data, the root system 
and the Drinfeld polynomial.
%
Thus we suppose that it is the BAE for any $Y(X_r)$
(or $U_q(X^{(1)}_r)$ in the trigonometric case).
%
This is actually true for all the known examples in 
which alternative derivations of the BAE are known such as
the algebraic Bethe ansatz.
%
It is also agreed in [ST].
%
Once (6.2) is admitted, 
the vacuum parts are determined uniquely up to an overall scalar
by requiring that the pole-freeness
is ensured by (6.2).
%
This is a straightforward task and here we shall only 
indicate the initial step concerning Theorems 2.1 and 2.3.
%
\par
Redefine $z(a;u)$ (2.4) and 
$sp(\xi_1,\ldots,\xi_r;u)$ (2.5) by multiplying 
the vacuum parts $vac (\cdots)$
(cf. (2.9a) in [KS1]):
%
$$\eqalign{
vac\, z(a;u) &= \prod_{j=1}^{a-1}P_j(u+j-1)
\prod_{j=a}^{r-1}P_j(u+j+1) 
P_r(u+r+{1\over 2})P_r(u+r-{1\over 2})\cr
&\times \prod_{j=1}^{r-1}P_j(u+2r-j) \Phi(u)\quad 1 \le a \le r,\cr
% 
vac\, z(0;u) &= \prod_{j=1}^{r-1}P_j(u+j-1)
P_r(u+r-{1\over 2})^2
\prod_{j=1}^{r-1}P_j(u+2r-j) \Phi(u),\cr
% 
vac\, z({\bar a};u) &= \prod_{j=1}^{r-1}P_j(u+j-1)
P_r(u+r-{1\over 2})P_r(u+r-{3\over 2})\cr
&\times \prod_{j=1}^{a-1}P_j(u+2r-j)
\prod_{j=a}^{r-1}P_j(u+2r-j-2) \Phi(u)\quad 1 \le a \le r,\cr
}\eqno(6.3{\rm a})$$
%
$$
\Phi(u) = \prod_{b=1}^r \prod_{j=1}^{b-1}
P_b(u+b-2j-{1\over t_b})
P_b(u+2r-b+2j-1+{1\over t_b}).\eqno(6.3{\rm b})
$$
%
$$\eqalignno{
vac\, sp(\xi_1,\ldots, \xi_r; u) &= 
\psi^{(1)}_{n_1}(u) \cdots \psi^{(r)}_{n_r}(u), &(6.4{\rm a})\cr
%
n_b &= \sharp \{ j \mid \xi_j = -, 1 \le j \le b \},
&(6.4{\rm b})\cr
%
\psi^{(b)}_n(u) &= \prod_{j=0}^{n-1}
P_b(u+r-b+2j+{1\over 2}-{1\over t_b})\cr
&\times 
\prod_{j=n}^{b-1}
P_b(u+r-b+2j+{1\over 2}+{1\over t_b}).&(6.4{\rm c})\cr}
$$
%
In terms of $z(a;u)$ involving the above vacuum parts,
redefine $T^a(u)$ by (2.7a)
assuming $X P_b(u) = P_b(u+2) X\, (1 \le b \le r)$ and modifying the 
RHS into
%
$$\eqalign{
&\sum_{a=0}^\infty
F_a(u+a-1) T^a(u+a-1) X^a,\cr
%
&F_a(u) = \prod_{b=1}^r \prod_{j=1}^{a-1}
\psi_0^{(b)}(u+r-a-{1\over 2}+2j)
\psi_b^{(b)}(u-r+a+{1\over 2}-2j).\cr}
$$
%
It is easily seen that this $T^a(u)$ is of positive order $2b$ 
with respect to the $P_b$ function (6.2).
%
One can check that Theorem 2.1 is still valid 
(for $T^{(r)}_1(u)$ and $T^a(u)$) for the BAE (6.2).
%
Relations (2.12a) and (2.13a) also hold if 
the right hand sides are divided by
$F_a(u)$ and $F_{2r-1-a}(u)$, respectively.
%
Thus Theorem 2.3 remains valid without any changes.
%
Along these lines, one can proceed further 
to include the vacuum parts for general
$T_{\lambda \subset \mu}(u)$, 
$S^L_{\lambda \subset \mu}(u)$ and 
$S^R_{\lambda \subset \mu}(u)$
so that they become pole-free under the BAE (6.2).


%
%-----------------------------------------------------------
%
\beginsection Acknowledgments

The authors thank 
E. Date, R. Hirota, J. Satsuma,
E.K. Sklyanin, V.O. Tarasov and I. Terada for discussions.
%
%
\beginsection References

\item{[BR]}{V.V. Bazhanov,V.V., N.Yu. Reshetikhin,
J.Phys.A:Math.Gen. {\bf 23} (1990) 1477}
%
\item{[C]}{I. Cherednik, in Proc. of the XVII International 
Conference on Differential Geometric Methods in Theoretical Physics,
Chester, ed. A.I. Solomon, World Scientific, Singapore, 1989}
%
\item{[D]}{V.G. Drinfel'd,
Sov.Math.Dokl.{\bf 36} (1988) 212}
%
\item{[K]}{A. Kuniba, J. Phys.A: Math.Gen.{\bf 27} (1994) L113}
%
\item{[KN]}{M. Kashiwara and T. Nakashima, ``Crystal graphs for 
representations of the}
\item{}{$q$-analogue of classical Lie algebras.''
RIMS preprint {\bf 767} (1991)}
%
\item{[KNS]}{A. Kuniba, T. Nakanishi and J. Suzuki,
Int.J.Mod.Phys. {\bf A9} (1994) 5215}
%
\item{[KS1]}{A. Kuniba and J. Suzuki, 
``Analytic Bethe ansatz for fundamental representations of 
Yangians'', hep-th.9406180, Commun. Math. Phys. in press}
%
\item{[KS2]}{A. Kuniba and J. Suzuki, J. Phys.A: Math.Gen.
{\bf 28} (1995) 711}
%
\item{[KT]}{K. Koike and I. Terada, J. Alg. {\bf 107} (1987) 466}
%
\item{[M]}{I. G. Macdonald, {\it Symmetric functions and Hall polynomials},
2nd ed., Oxford University Press, 1995}
%
\item{[R]}{N.Yu. Reshetikhin, Sov.Phys.JETP {\bf 57} (1983) 691,
Theor.Math.Phys.{\bf 63} (1985) 555, 
Lett.Math.Phys.{\bf 14} (1987) 235}
%
\item{[RW]}{N.Yu. Reshetikhin and P.B. Wiegmann,
Phys.Lett.B{\bf 189} (1987) 125}
%
\item{[ST]}{E.K. Sklyanin and V.O. Tarasov, private communication}

\vfill\eject

\beginsection Figure Captions.

%
\par\noindent 
Figure 3.1:  An example of a skew-Young diagram 
$\lambda \subset \mu$. Here $\mu=(5,4^2,1),
\lambda=(2,1), \mu'=(4,3^3,1)$ and $\lambda'=(2,1)$, respectively.
The lengths of the main diagonal are given by $d_{\mu}=3$ and 
$d_{\lambda}=1$.
%
\par\noindent
Figure 3.2:  The way to assign the letters to each box 
is explained in the text.
This is an example for $r=3, \mu'=(9,7,2), \lambda'=(3,1)$. Notice that
zeros are arranged lest they are adjacent horizontally. 
%
\par\noindent
Figure 4.1: An example of an L-hatched skew-Young diagram
$r=4, \mu=(4^3,3,2,1^2), \lambda=(3,1)$.
%
\par\noindent
Figure 4.2: The bottom left part of an
L-hatched skew-Young tableau and the assignment of 
the letters $\{i_l\}$ and $\{j_l\}$ in (4.2).
%
\par\noindent
Figure 4.3: An example of an R-hatched skew-Young diagram
$r=4, \mu=(4^5,3,1), \lambda=(3^2,2,1)$.
%
\par\noindent
Figure 4.4: The top right part of an
R-hatched skew-Young tableau and the assignment of the letters 
$\{i_l\}$ and $\{j_l\}$ in (4.4).
%
\par\noindent
Figure 4.5: An R-hatched skew-Young diagram for $r=3$ with $\mu=(5^4,3,2,1),
\lambda=(4,3,1^2)$. Broken lines are guides to eyes for defining
$\eta=(4^3,3,2,1), \xi=(3, 1^2)$.

\bye

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