%Paper: hep-th/9509039
%From: atsuo kuniba <atsuo@hep1.c.u-tokyo.ac.jp>
%Date: Fri, 8 Sep 95 18:08:28 +0900






%version 95/9/8
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\centerline{\bf Pfaffian and Determinant Solutions to}
\centerline{\bf A Discretized Toda Eqaution for $B_r, C_r$ and $D_r$}
\vskip1.0cm \centerline{by}
\vskip1.0cm
%
\par\vskip0.3cm
\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\vskip0.4cm
%
\centerline{Shuichi Nakamura%
\footnote{\ddag}{Present address: Hitachi, Ltd. Information Systems Division,
890 Kashimada, Saiwai-ku, Kawasaki, Kanagawa, Japan}}
\centerline{Department of Electronics and Communication Engineering}
\centerline{School of Science and Engineering}
\centerline{Waseda University, Tokyo 169 Japan}
\vskip0.3cm
\centerline{and}\par\vskip0.3cm
\centerline{Ryogo Hirota\footnote\P{
E-mail: roy@hirota.info.waseda.ac.jp}}
\centerline{Department of Information and Computer Science}
\centerline{Waseda University, Tokyo 169 Japan}
%
\vskip5.0cm
\centerline{\bf Abstract}
\vskip0.2cm
\par
%
We consider a class of
2 dimensional Toda equations on discrete space-time.
%
It has arisen as functional relations
in commuting family of transfer matrices in
solvable lattice models associated with any classical
simple Lie algebra $X_r$.
%
For $X_r = B_r, C_r$ and $D_r$,
we present the solution in terms of Pfaffians and determinants.
%
They may be viewed as
Yangian analogues of the classical Jacobi-Trudi formula
on Schur functions.
%
\vfill
\eject
%
\beginsection 1. Introduction

Consider the following systems of difference
equations on $T^{(a)}_m(u)$
$(m \in {\bf Z}_{\ge 0}, u \in {\bf C},
a \in \{1,2,\ldots, r \})$
%
$$\eqalignno{
B_r:(r \ge 2)\qquad\qquad\qquad\qquad &&\cr
%
T^{(a)}_m(u-1)T^{(a)}_m(u+1) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) +
T^{(a-1)}_m(u)T^{(a+1)}_m(u)&(1{\rm a})\cr
%
&\qquad\qquad\qquad\qquad\qquad 1 \le a \le r-2,&\cr
%
T^{(r-1)}_m(u-1)T^{(r-1)}_m(u+1) &=
T^{(r-1)}_{m+1}(u)T^{(r-1)}_{m-1}(u) +
T^{(r-2)}_m(u)T^{(r)}_{2m}(u),&(1{\rm b})\cr
%
T^{(r)}_{2m}(u-{1\over 2})T^{(r)}_{2m}(u+{1\over 2}) &=
T^{(r)}_{2m+1}(u)T^{(r)}_{2m-1}(u) \cr
%
&+ T^{(r-1)}_m(u-{1\over 2})
T^{(r-1)}_m(u+{1\over 2}),&(1{\rm c})\cr
%
T^{(r)}_{2m+1}(u-{1\over 2})T^{(r)}_{2m+1}(u+{1\over 2}) &=
T^{(r)}_{2m+2}(u)T^{(r)}_{2m}(u) +
T^{(r-1)}_m(u)T^{(r-1)}_{m+1}(u).&(1{\rm d})\cr}
$$
%
$$\eqalignno{
C_r:(r \ge 2)\qquad\qquad\qquad\qquad \quad&&\cr
%
T^{(a)}_m(u-{1\over 2})T^{(a)}_m(u+{1\over 2}) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) +
T^{(a-1)}_m(u)T^{(a+1)}_m(u)&(2{\rm a})\cr
%
&\qquad\qquad\qquad\qquad\qquad 1 \le a \le r-2,&\cr
%
T^{(r-1)}_{2m}(u-{1\over 2})T^{(r-1)}_{2m}(u+{1\over 2}) &=
T^{(r-1)}_{2m+1}(u)T^{(r-1)}_{2m-1}(u)& \cr
%
&+ T^{(r-2)}_{2m}(u)
T^{(r)}_m(u-{1\over 2})T^{(r)}_m(u+{1\over 2}),&(2{\rm b})\cr
%
T^{(r-1)}_{2m+1}(u-{1\over 2})T^{(r-1)}_{2m+1}(u+{1\over 2}) &=
T^{(r-1)}_{2m+2}(u)T^{(r-1)}_{2m}(u) \cr
%
&+ T^{(r-2)}_{2m+1}(u)
T^{(r)}_m(u)T^{(r)}_{m+1}(u),&(2{\rm c})\cr
%
T^{(r)}_m(u-1)T^{(r)}_m(u+1) &=
T^{(r)}_{m+1}(u)T^{(r)}_{m-1}(u) +
T^{(r-1)}_{2m}(u).&(2{\rm d})\cr}
$$
%
$$\eqalignno{
D_r: (r \ge 4)\qquad\qquad\qquad\qquad &&\cr
%
T^{(a)}_m(u-1)T^{(a)}_m(u+1) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) +
T^{(a-1)}_m(u)T^{(a+1)}_m(u)&(3{\rm a})\cr
%
&\qquad\qquad\qquad\qquad\qquad 1 \le a \le r-3,&\cr
%
T^{(r-2)}_m(u-1)T^{(r-2)}_m(u+1) &=
T^{(r-2)}_{m+1}(u)T^{(r-2)}_{m-1}(u) \cr
&+ T^{(r-3)}_m(u)T^{(r-1)}_m(u)T^{(r)}_m(u),&(3{\rm b})\cr
%
T^{(a)}_{m}(u-1)T^{(a)}_{m}(u+1) &=
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u)
+ T^{(r-2)}_m(u)\quad a = r-1, r.&(3{\rm c})\cr}
$$
%
($T^{(0)}_m(u) = 1$.)
%
We shall exclusively consider the initial condition
$T^{(a)}_0(u) = 1$ for any $1 \le a \le r$.
%
Then one can solve the systems (1), (2) and (3) iteratively
to express $T^{(a)}_m(u)$ in terms of
$T^{(1)}_1(u+ \hbox{shift}),\ldots ,T^{(r)}_1(u+ \hbox{shift})$.
%
For example,
$T^{(1)}_2(u) = T^{(1)}_1(u-1)T^{(1)}_1(u+1)-T^{(2)}_1(u)$
from (1a).
%
The purpose of this paper is to present the formulae that
express an arbitrary $T^{(a)}_m(u)$ ($m \ge 1$)
as a determinant or a Pfaffian of matrices
with elements 0 or $\pm T^{(b)}_1(u+\hbox{shift})$
($0 \le b \le r$).
%
\par
In fact, such formulae
had been partially conjectured in [KNS1],
where a set of functional relations, $T$-system,
was introduced
for the commuting family of transfer matrices
$\{ T^{(a)}_m(u) \}$ for solvable lattice models
associated with any classical simple Lie algebra
$X_r$.
%
In this context, eqs. (1), (2) and (3) correspond to
$X_r = B_r, C_r$ and $D_r$ cases of the $T$-system, respectively.
%
$T^{(a)}_m(u)$ denotes a transfer matrix (or its eigenvalue)
with spectral parameter $u$ and ``fusion type"
labeled by $a$ and $m$ [KNS1].
%
Our result here confirms all of the determinant conjectures
raised in section 5 of [KNS1].
%
Moreover it extends them
to a full solution of (1), (2) and (3), which, in general,
involves Pfaffians as well.
%
In the representation theoretical viewpoint,
this yields a Yangian analogue of the Jacobi-Trudi formula [Ma], i.e.,
a way to construct Yangian characters out of
those for the fundamental representations [CP].
%
\par
Beside the significance in the lattice model context [KNS2],
the beautiful structure in these solutions
indicate a rich content of the $T$-system
also as an example of discretized soliton equations
[AL,H1,H2,H3,H4,HTI,K,DJM,S,VF,BKP,Wa,Wi].
%
In fact, regarding $u$ and $m$
as continuous space-time coordinates,
one can take a suitable scaling limit where the $T$-system
becomes 2-dimensional Toda
(or Toda molecule) equation for $X_r$ [T,MOP,LS]:
$$
(\partial_u^2 - \partial_m^2)\hbox{ log } \phi_a(u,m)
= \hbox{const } \prod_{b=1}^r
\phi_b(u,m)^{-A_{a b}}. \eqno(4)$$
%
Here $\phi_a(u,m)$ 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.
%
In this sense, our $T$-system is a
discretization of the Toda equation
allowing determinant and Pfaffian solutions at
least for $X_r = A_r, B_r, C_r$ and $D_r$.
%
See also the remarks in section 6 concerning
the $T$-systems for twisted affine Lie algebras [KS].
\par
%
The outline of the paper is as follows.
%
In sections 2, 3 and 4, we present solutions to the
$B_r, C_r$ and $D_r$ cases, respectively.
%
Pfaffians are needed for
$T^{(r)}_m(u)$ in $C_r$ and
$T^{(r-1)}_m(u)$ and $T^{(r)}_m(u)$ in $D_r$.
%
In section 5, we illustrate a proof for the $C_r$ case.
%
The other cases can be verified quite similarly.
%
Section 6 is devoted to summary and discussion.
%
\par
Before closing the introduction, a few remarks are in order.
%
Firstly, the original $T$-system [KNS1] had a factor
$g^{(a)}_m(u)$ in front of the second term
in the rhs of
$T^{(a)}_m(u+{1 \over t_a})
T^{(a)}_m(u-{1 \over t_a}) = \cdots$.
%
Throughout this paper we shall set $g^{(a)}_m(u) = 1$.
%
To recover the dependence on $g^{(a)}_m(u)$
is quite easy as long as the relation
$g^{(a)}_m(u+{1 \over t_a})
g^{(a)}_m(u-{1 \over t_a}) =
g^{(a)}_{m+1}(u)
g^{(a)}_{m-1}(u)$ is satisfied
(cf [KNS1]).
%
Secondly,
$T$-systems (1) and (2) coincide for $r=2$
under the exchange
$T^{(1)}_m(u) \leftrightarrow T^{(2)}_m(u)$,
which reflects the Lie algebra equivalence
$B_2 \simeq C_2$.
%
In this case (9) and (12) yield two alternative
expressions for the same quantity.
%
Thirdly,
for $X_r = A_r$, the $T$-system
$T^{(a)}_m(u-1)T^{(a)}_m(u+1) =
T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u) +
T^{(a-1)}_m(u)T^{(a+1)}_m(u)$
($1 \le a \le r, T^{(0)}_m(u) = T^{(r+1)}_m(u) = 1$)
is the so-called Hirota-Miwa equation.
%
In the transfer matrix context, it
has been proved in [KNS1]
by using the determinant formula in [BR].
%
Finally,
for $X_r = B_r$, a determinant solution different from (9) has
been obtained in [KOS].
%
The relevant matrix there is not sparse as (7) and
the matrix elements are not necessarily $T^{(a)}_1(u)$
but contain some quadratic expressions of $T^{(r)}_1(u)$
in general.
%
\beginsection 2. $B_r$ Case

For any $k \in {\bf C}$, put
$$x^a_k = \cases{T^{(a)}_1(u+k) &$1 \le a \le r$,\cr
                 1              &$a=0$. \cr}\eqno(5)
$$
%
We introduce the infinite dimensional matrices
${\cal T} = ({\cal T}_{i j} ) _{i,j \in {\bf Z}}$
and
${\cal E} = ({\cal E}_{i j} ) _{i,j \in {\bf Z}}$
as follows.
%
$$\eqalign{
{\cal T}_{i j} &= \cases{
x^{{j-i \over 2}+1}_{{i+j\over 2}-1} &
if $i \in 2{\bf Z}+1$
and ${i-j \over 2} \in \{1,0,\ldots,2-r\}$,\cr
%
-x^{{i-j \over 2}+2r-2}_{{i+j \over 2}-1} &
if $i \in 2{\bf Z}+1$ and
${i-j \over 2} \in
\{1-r,-r,\ldots,2-2r\}$,\cr
%
-x^r_{r+i-{5 \over 2}} & if $i \in 2{\bf Z}$ and $j=i+2r-3$,\cr
0 &otherwise,\cr}\cr
%
{\cal E}_{i j} &= \cases{
\pm 1 & if $i = j - 1 \pm 1$ and $i \in 2{\bf Z}$,\cr
x^r_{i-1} & if $i = j - 1$ and $i \in 2{\bf Z} + 1$,\cr
0 & otherwise.\cr}\cr}\eqno(6)$$
%
For example, for $B_3$, they read
$$
({\cal T}_{i j})_{i,j \ge 1}=
\pmatrix{     x^1_0&     0&     x^2_1&      0&
             -x^2_2&     0&   -x^1_3 &      0&    -1&       \cr
%
                  0&     0&         0&      0&    -x^3_{5/2}&
                  0&     0&         0&     0&               \cr
%
                  1&     0& x^1_2&      0&         x^2_3 &  0&
               -x^2_4&   0&  -x^1_5  &\cdots                \cr
%
                   0& 0& 0& 0& 0& 0& -x^3_{9/2}& 0& 0&      \cr
%
                   0& 0& 1& 0& x^1_4& 0& x^2_5& 0& -x^2_6&  \cr
                    & &   &  &  \vdots &&&&& \ddots \cr},
\eqno(7{\rm a})
$$
$$
({\cal E}_{i j})_{i,j \ge 1}=
\pmatrix{ 0& x^3_0&  0&     0&     0&     0&     0&   \cr
          0&     1&  0&    -1&     0&     0&     0&   \cr
          0&     0&  0& x^3_2&     0&     0&     0&   \cr
          0&     0&  0&     1&     0&    -1&     0&  \cdots \cr
          0&     0&  0&     0&     0& x^3_4&     0&   \cr
          0&     0&  0&     0&     0&     1&     0&   \cr
           &      &   &    \vdots&  &      &      &\ddots \cr
         }.
\eqno(7{\rm b})
$$
%
As is evident from the above example,
for any $1 \le a \le r$ and $k$, the quantity $\pm x^a_k$
is contained in ${\cal T}\vert_{u \rightarrow u + \xi}$
once and only once as its matrix element.
%
Here $u \rightarrow u + \xi$ means the overall shift of
lower indices in accordance with (5).
%
For example, the shift $\xi = 1$ is necessary to
accomodate $x^1_1$ as the (1,1) element of
${\cal T}\vert_{u \rightarrow u + \xi}$.
%
In view of this, we shall employ the notation
${\cal T}_m(i,j,\pm x^a_k)$ to mean the
$m$ by $m$ sub-matrix of
${\cal T}\vert_{u \rightarrow u + \xi}$
whose $(i,j)$ element is
exactly $\pm x^a_k$.
%
This definition is unambiguous irrespective of various possible
choices of $\xi$.
%
For example in (7a),
%
$$\eqalign{
{\cal T}_3(1,1,x^1_0) &= \pmatrix{
x^1_0 & 0 & x^2_1 \cr
0     & 0 &   0 \cr
1     & 0 &  x^1_2 \cr},
\quad
{\cal T}_3(1,1,x^1_1) = \pmatrix{
x^1_1 & 0 & x^2_2 \cr
0     & 0 &   0 \cr
1     & 0 &  x^1_3 \cr}, \cr
%
{\cal T}_2(1,2,-x^3_{5/2}) &= \pmatrix{
0 & -x^3_{5/2} \cr
0 & x^2_3 \cr},
%
\quad
{\cal T}_2(1,2,-x^3_2) = \pmatrix{
0 & -x^3_2 \cr
0 & x^2_{5/2} \cr}.
}\eqno(8)
$$
%
We shall also use the similar notation
${\cal E}_m(i,j,\pm x^r_k)$.
%
With these notations our result in this section
is stated as
%
\proclaim Theorem 2.1.
%
For $m \in {\bf Z}_{\ge 1}$,
$$\eqalignno{
T^{(a)}_m(u) &= det\bigl(
{\cal T}_{2m-1}(1,1,x^a_{-m+1}) +
{\cal E}_{2m-1}(1,2,x^r_{-m+r-a+{1\over 2}})\bigr), \,
1 \le a < r, &(9{\rm a})\cr
%
T^{(r)}_m(u) &= (-1)^{m(m-1)/2} det\bigl(
{\cal T}_m(1,2,-x^{r-1}_{-{m\over 2}+1}) +
{\cal E}_m(1,1,x^r_{-{m\over 2}+{1\over 2}})\bigr),
&(9{\rm b})\cr}
$$
%
solves the $B_r$ $T$-system (1).
\par
Up to some conventional change, (9a) in the above
had been conjectured in
eq.(5.6) of [KNS1].
%
The formula (9b) is new.
%
\beginsection 3. $C_r$ Case

Here we introduce the inifinite dimensional
matrix ${\cal T}$ by
%
$$
{\cal T}_{i j} = \cases{
x^{j-i+1}_{{i+j\over 2}-1} & if $i - j \in \{1,0, \ldots, 1-r\}$,\cr
%
- x^{i-j+2r+1}_{{i+j\over 2}-1} & if $i - j \in
\{-1-r,-2-r, \ldots, -1-2r\}$,\cr
0 & otherwise. \cr}
\eqno(10)$$
%
For example, for $C_2$, it reads
%
$$({\cal T}_{i j})_{i,j \ge 1} = \pmatrix{
x^1_0 &  x^2_{1/2} & 0 & -x^2_{3/2} & -x^1_2 & -1 & 0 & 0 &    \cr
1 &  x^1_1  &  x^2_{3/2} & 0 & -x^2_{5/2} & -x^1_3 & -1 & 0 & \cdots \cr
0 & 1 &  x^1_2  &  x^2_{5/2} & 0 & -x^2_{7/2} & -x^1_4 & -1 &  \cr
0 & 0 & 1 & x^1_3 & x^2_{7/2} & 0 & -x^2_{9/2} & -x^1_5 & \cr
&&& & \vdots &&&& \ddots \cr}.
\eqno(11)$$
%
We keep the same notations (5) and
${\cal T}_m(i,j,\pm x^a_k)$ ($1 \le a \le r$)
as in section 2.
%
Note that ${\cal T}_m(1,2,-x^r_k)$ is an anti-symmetric matrix
for any $m$.
%
Our result in this section is
%
\proclaim Theorem 3.1.
%
For $m \in {\bf Z}_{\ge 1}$,
$$\eqalignno{
T^{(a)}_m(u) &= det\bigl(
{\cal T}_m(1,1,x^a_{-{m\over 2}+{1\over 2}}) \bigr)
\quad 1 \le a < r, &(12{\rm a})\cr
%
T^{(r)}_m(u) &= (-1)^m pf \bigl(
{\cal T}_{2m}(1,2,-x^r_{-m+1}) \bigr), &(12{\rm b})\cr}
$$
%
solves the $C_r$ $T$-system (2).
\par
%
The expression (12a) is essentially the conjecture
(5.10) in [KNS1].
%
The Pfaffian formula (12b) is new.
%
In proving the theorem in section 5, we will also
establish the relations
%
$$\eqalignno{
T^{(r)}_m(u-{1\over 2})T^{(r)}_m(u+{1\over 2}) &=
det \bigl( {\cal T}_{2m}(1,1,x^r_{-m+{1\over 2}}) \bigr),
&(13{\rm a})\cr
%
T^{(r)}_m(u) T^{(r)}_{m+1}(u) &=
det \bigl( {\cal T}_{2m+1}(1,1,x^r_{-m}) \bigr).
&(13{\rm b})\cr}$$
%
\beginsection 4. $D_r$ Case

Here we define the infinite dimensional matrices
${\cal T}$ and ${\cal E}$ by
%
$$\eqalignno{
{\cal T}_{i j} &= \cases{
x^{{j-i \over 2}+1}_{{i+j\over 2}-1} &
if $i \in 2{\bf Z}+1$ and
${i-j\over 2} \in \{1,0,\ldots,3-r\}$,\cr
%
-x^{r-1}_{i+j-1\over 2}&if $i \in 2{\bf Z}+1$ and
${i-j\over 2}={5\over 2}-r$,\cr
%
-x^{r}_{i+j-3\over 2}&if $i \in 2{\bf Z}+1$ and
${i-j\over 2}={3\over 2}-r$,\cr
%
-x^{{i-j\over 2}+2r-3}_{{i+j\over 2}-1}
&if $i \in 2{\bf Z}+1$ and
${i-j\over 2} \in \{1-r,-r,\ldots,3-2r\}$,\cr
0&otherwise,\cr}
&(14{\rm a})\cr
%
{\cal E}_{i j} &= \cases{
\pm 1 &if $i=j-2 \pm 2$ and $i \in 2{\bf Z}$,\cr
%
x^{r-1}_i&if $i=j-3$ and $i \in 2{\bf Z}$,\cr
%
x^{r}_{i-2}&if $i=j-1$ and $i \in 2{\bf Z}$,\cr
%
0&otherwise.\cr}&(14{\rm b})\cr}$$
%
For example, for $D_4$, they read
$$
({\cal T}_{i j})_{i,j \ge 1} = \pmatrix{
x^1_0&      0&    x^2_1&
   -x^{3}_2&        0&    -x^{4}_2& -x^{2}_3&
        0& -x^{1}_4&             0&    -1&
                       \cr
    0&         0&            0&
            0&        0&           0&
         0&         0&          0&             0&     0&
                          \cdots \cr
    1&      0&    x^1_2&                0&x^{2}_3&  -x^{3}_4&
        0&  -x^{4}_4& -x^{2}_5&         0&-x^1_{6}&
                     \cr
    0&      0&            0&            0&
            0&        0&           0&
      0&         0&          0&          0&
                   \cr
&      &           &              &
      &         &  \vdots          &          &
         &         &      &
                      \ddots \cr
},\eqno(15{\rm a})
$$
$$
({\cal E}_{i j})_{i,j \ge 1} =
\pmatrix{
         0&        0&        0&          0&
 0&         0&          0&
         0&        0&   &   \cr
         0&        1& x^4_{0}&         0& x^{3}_{2}&       -1&
          0&
         0&        0&   \ldots&   \cr
         0&        0&        0&          0&            0&
       0&          0&
         0&        0&   &   \cr
         0&        0&        0&          1&    x^4_{2}&
        0& x^{3}_{4}&
       -1&          0&  &  \cr
   &        &         &     &             &     \vdots
    &           &
         &    & \ddots   \cr
}. \eqno(15{\rm b})
$$
%
We keep the same notations (5),
${\cal T}_m(i,j,\pm x^a_k)$ ($1 \le a \le r-2$)
and ${\cal T}_m(i,j,-x^a_k), {\cal E}_m(i,j,x^a_k)$ ($a = r-1, r$)
as in section 2.
%
Our result in this section is
%
\proclaim Theorem 4.1.
%
For $m \in {\bf Z}_{\ge 1}$,
$$\eqalignno{
&T^{(a)}_m(u) = det\bigl(
{\cal T}_{2m-1}(1,1,x^a_{-m+1}) +
{\cal E}_{2m-1}(2,3,x^r_{-m-r+a+4}) \bigr), \,\,
1 \le a \le r-2, &(16{\rm a})\cr
%
&T^{(r-1)}_m(u) = pf \bigl(
{\cal T}_{2m}(2,1,-x^{r-1}_{-m+1})+
{\cal E}_{2m}(1,2,x^{r-1}_{-m+1}) \bigr),
&(16{\rm b})\cr
%
&T^{(r)}_m(u) = (-1)^m pf \bigl(
{\cal T}_{2m}(1,2,-x^r_{-m+1})+
{\cal E}_{2m}(2,1,x^r_{-m+1}) \bigr),
&(16{\rm c})\cr}
$$
%
solves the $D_r$ $T$-system (3).
\par
The matrices in (16b,c) are indeed anti-symmetric.
%
Eq.(16a) is essentially the conjecture (5.15) in [KNS1].
%
The Pfaffian formulae (16b,c) are new.
%
By using them one can show the relations
%
$$\eqalignno{
&T^{(r-1)}_m(u)T^{(r)}_m(u) = (-1)^m
det \bigl( {\cal T}_{2m}(1,1,-x^{r-1}_{-m+1})+
{\cal E}_{2m}(2,2,x^r_{-m+1}) \bigr),
&(17{\rm a})\cr
%
&T^{(r-1)}_m(u+1)T^{(r)}_m(u-1) = (-1)^m
det \bigl( {\cal T}_{2m}(1,1,-x^{r}_{-m})+
{\cal E}_{2m}(2,2,x^{r-1}_{-m+2}) \bigr),
&(17{\rm b})\cr
%
&T^{(r-1)}_{m+1}(u)T^{(r)}_m(u-1) = (-1)^{m+1}
det \bigl( {\cal T}_{2m+1}(1,1,-x^{r-1}_{-m})+
{\cal E}_{2m+1}(2,2,x^r_{-m}) \bigr),
&(17{\rm c})\cr
%
&T^{(r-1)}_m(u+1)T^{(r)}_{m+1}(u) = (-1)^m
det \bigl( {\cal T}_{2m+1}(2,1,x^{r-2}_{-m+1})+
{\cal E}_{2m+1}(1,1,x^r_{-m}) \bigr).
&(17{\rm d})\cr}$$
%
The proof of (17) is analogous to that of (13), which
will be explained in the next section.
%
\beginsection 5. Proof of Theorem 3.1

Here we shall outline the proof of theorem 3.1, namely
$C_r$ $T$-system (2) starting from (12).
%
As it turns out, all of the three term relations
in (2) reduce to Jacobi's identity:
%
$$
D{1 \atopwithdelims
 \lbrack \rbrack 1}
D{n \atopwithdelims
 \lbrack \rbrack n} =
D D{1,n \atopwithdelims
 \lbrack \rbrack 1,n} +
D{1 \atopwithdelims
 \lbrack \rbrack n}
D{n \atopwithdelims
 \lbrack \rbrack 1}. \eqno(18)
$$
%
Here $D$ is the determinant of any $n$ by $n$ matrix
and $D{i_i, i_2, \ldots  \atopwithdelims
\lbrack \rbrack j_1, j_2, \ldots}$
denotes its minor removing the $i_k$'s rows and
$j_k$'s columns.
%
\par
Let us prove (13a) first.
%
Taking its square and substituting (12b),
we are to show
%
$$
det\big({\cal T}_{2m}(1,2,-x^r_{-m+{1\over 2}}) \bigr)
det\big({\cal T}_{2m}(1,2,-x^r_{-m+{3\over 2}}) \bigr)
= \biggl(det\big({\cal T}_{2m}(1,1,-x^r_{-m+{1\over 2}})
\bigr)\biggr)^2.\eqno(19)
$$
%
To see this we set
%
$$D = det\big({\cal T}_{2m+1}(1,2,-x^r_{-m+{1\over 2}}) \bigr)
= det \pmatrix{
0 & -x^r_{-m+{1\over 2}} & -x^{r-1}_{-m+1} & \cr
x^r_{-m+{1\over 2}} & 0 & -x^r_{-m+{3\over 2}} & \cdots \cr
x^{r-1}_{-m+1} & x^r_{-m+{3\over 2}} & 0 & \cr
  & \vdots & & \ddots \cr}
= 0, \eqno(20)
$$
%
since this is an anti-symmetric matrix with odd size.
%
{}From (20) it is easy to see
%
$$\eqalign{
D{1 \atopwithdelims
 \lbrack \rbrack 1} &=
det\big({\cal T}_{2m}(1,2,-x^r_{-m+{3\over 2}}) \bigr),\quad
%
D{2m+1 \atopwithdelims
 \lbrack \rbrack 2m+1} =
det\big({\cal T}_{2m}(1,2,-x^r_{-m+{1\over 2}}) \bigr),\cr
%
D{1 \atopwithdelims
 \lbrack \rbrack 2m+1} &=
D{2m+1 \atopwithdelims
 \lbrack \rbrack 1} =
det\big({\cal T}_{2m}(1,1,x^r_{-m+{1\over 2}}) \bigr).\cr}
\eqno(21)
$$
%
Thus (19) follows immediately from (21) and (18).
%
In taking the square root of (19), the relative sign
can be fixed by
comparing the coefficients of
$x^r_{-m+1/2} x^r_{-m+3/2} \cdots x^r_{m-1/2}$ on both sides,
which agrees with (13a).
%
The relation (13b) can be shown similarly by setting
$D = det\big({\cal T}_{2m+2}(1,2,-x^r_{-m}) \bigr)$.
%
\par
%
Now we proceed to the proof of the $T$-system (2).
%
To show (2a), it suffices to apply (18) for
$D = det\big({\cal T}_{m+1}(1,1,x^a_{-{m \over 2}}) \bigr) =
T^{(a)}_{m+1}(u)$ and to note that
$D{1 \atopwithdelims \lbrack \rbrack 1} = T^{(a)}_m(u+{1\over 2})$,
$D{m+1 \atopwithdelims \lbrack \rbrack m+1} = T^{(a)}_m(u-{1\over 2})$,
$D{1,m+1 \atopwithdelims \lbrack \rbrack 1,m+1} = T^{(a)}_{m-1}(u)$,
$D{m+1 \atopwithdelims \lbrack \rbrack 1} = T^{(a+1)}_m(u)$ and
$D{1 \atopwithdelims \lbrack \rbrack m+1} = T^{(a-1)}_m(u)$.
%
Similarly (2b) (resp. (2c)) can be derived
by setting
$D = det\big({\cal T}_{2m+1}(1,1,x^{r-1}_{-m}) \bigr) =
T^{(r-1)}_{2m+1}(u)$
(resp. $D = det\big({\cal T}_{2m+2}(1,1,x^{r-1}_{-m-{1\over 2}}) \bigr) =
T^{(r-1)}_{2m+2}(u)$)
and using (13a) (resp. (13b)) to
identify
$D{2m+1 \atopwithdelims
 \lbrack \rbrack 1}$ with
$T^{(r)}_m(u-{1\over 2})T^{(r)}_m(u+{1\over 2})$
(resp. $T^{(r)}_m(u)T^{(r)}_{m+1}(u)$).
%
Finally to show (2d), we put
$D = det\big({\cal T}_{2m+1}(1,1,x^{r}_{-m}) \bigr)$.
Then from (12) and (13) we have
%
$$\eqalign{
D &= T^{(r)}_m(u)T^{(r)}_{m+1}(u), \quad
%
D{1,2m+1 \atopwithdelims \lbrack \rbrack 1,2m+1} =
T^{(r)}_{m-1}(u)T^{(r)}_{m}(u), \cr
%
D{1 \atopwithdelims \lbrack \rbrack 1} &=
T^{(r)}_{m}(u)T^{(r)}_{m}(u+1), \quad
%
D{2m+1 \atopwithdelims \lbrack \rbrack 2m+1} =
T^{(r)}_{m}(u-1)T^{(r)}_{m}(u), \cr
%
D{1 \atopwithdelims \lbrack \rbrack 2m+1} &=
T^{(r-1)}_{2m}(u), \quad
%
D{2m+1 \atopwithdelims \lbrack \rbrack 1} =
\bigl(T^{(r)}_{m}(u) \bigr)^2. \cr}\eqno(22)
$$
%
Substituting (22) into (18) (for $n=2m+1$)
and cancelling out the common factor
$\bigl(T^{(r)}_{m}(u) \bigr)^2$, we obtain (2d).
%
This completes the proof of theorem 3.1.
%
\beginsection 6. Summary and discussion

In this paper we have considered the
difference equations (1), (2) and (3), which
may be viewed as 2 dimensional Toda equations
on discrete space-time as argued in (4).
%
They have arisen as the
$B_r$, $C_r$ and $D_r$ cases of the $T$-system,
which are functional relations
among commuting families of transfer matrices
in the associated solvable lattice models.
%
Under the initial condition $T^{(a)}_0(u) = 1$ ($1 \le a \le r$),
we have given the solutions (9), (12) and (16) for
$T^{(a)}_m(u)$ with $m \in {\bf Z}_{\ge 1}$.
%
They are expressed
in terms of Pfaffians or
determinants of the matrices
(6), (10) and (14), which contain only
$\pm T^{(a)}_1(u+\hbox{shift})$ or $\pm 1$ as their matrix elements.
%
This confirms the earlier conjectures [KNS1]
and extends them to the full solutions.
%
\par
It will be interesting to extend a similar analysis
to the $T$-system for the exceptional
algebras $E_{6,7,8}, F_4, G_2$ [KNS1] and also
the twisted quantum affine algebras
$A^{(2)}_n, D^{(2)}_n, E^{(2)}_6$ and $D^{(3)}_4$ [KS].
%
In fact, the solutions to the
$A^{(2)}_n, D^{(2)}_n$ and $D^{(3)}_4$ cases can be
obtained just by imposing the ``modulo $\sigma$ relations"
((3.4) in [KS]) on the corresponding non-twisted cases
$A_n, D_n$ and $D_4$ treated in this paper.
%
On the other hand, to deal with the exceptional cases,
it seems necessary to introduce matrices whose
elements are some higher order
expressions in $T^{(a)}_1(u)$ 's analogous to [KOS].
\par



%
\beginsection Acknowledgements

One of the authors (A.K.) thanks
E. Date, L.D. Faddeev, K. Fujii,
Y. Ohta, J. Suzuki and P.B. Wiegmann
for helpful discussions.
%
\vfill
\eject
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\bye



