# Differential equations for the closed geometric crystal chains

Taichiro Takagi  Department of Applied Physics, National Defense Academy,
Yokosuka, Kanagawa 239-8686, Japan<EMAIL_ADDRESS>
###### Abstract

We present two types of systems of differential equations that can be derived
from a set of discrete integrable systems which we call the closed geometric
crystal chains. One is a kind of extended Lotka-Volterra systems, and the
other seems to be generally new but reduces to a previously known system in a
special case. Both equations have Lax representations associated with what are
known as the loop elementary symmetric functions, which were originally
introduced to describe products of affine type $A$ geometric crystals for
symmetric tensor representations. Examples of the derivations of the
continuous time Lax equations from a discrete time one are described in
detail, where a novel method of taking a continuum limit by assuming
asymptotic behaviors of the eigenvalues of the Lax matrix in Puiseux series
expansions is used.

##  1 Introduction

In modern mathematical physics, studies on the relations between continuous,
discrete, and ultra-discrete integrable dynamical systems have been developed
extensively [6, 3, 7, 9, 24, 27]. Motivated by many studies in this field, and
in particular by those based on the theory of crystals [13, 14] and that of
geometric crystals [1], the author and T. Yoshikawa [26] constructed a new
class of discrete integrable systems that can be viewed as a geometric lifting
of a class of integrable cellular automata known as the periodic box-ball
systems [17, 18, 19, 31, 32]. Since they are related to a realization of type
$A_{n-1}$ geometric crystals and geometric $R$-matrices by G. Frieden [4, 5],
we called the new integrable systems closed geometric crystal chains.

From a conventional viewpoint, the rank of the affine Lie algebra $A_{n-1}$ is
related to the number of solitons in the periodic box-ball system. In this
viewpoint, the geometric lifting of the system has already known and
identified with the discrete periodic Toda chain [8, 15]. On the other hand,
this rank is related to the number of degrees of freedom in each site
variables in our construction. Therefore, a closed geometric crystal chain is
not equivalent to the discrete periodic Toda chain. As a result, while the
discrete Toda chain yields the Toda equation in a continuum limit, it is
unclear what kind of differential equations will be derived from the closed
geometric crystal chains in such a limit, except for the case of $n=2$ that
was studied in §2.3.2 of reference [26].

The purpose of this paper is to give an outlook for extending this result to
the case of general $n$. We present two types of systems of differential
equations that can be derived from the chains. One is equation (1), which is a
kind of extended Lotka-Volterra systems [2, 11, 21]. The other is equation
(24) with a function $e^{(\alpha)}_{L-1}$ in (3), which seems to be generally
new but reduces to a previously known system studied in [23] in a special case
(36). An important point here is that both equations are related to Lax
equations associated with the _loop elementary symmetric functions_ , which
were originally introduced in reference [20] for a description of products of
affine type $A$ geometric crystals for symmetric tensor representations. (We
shall use its variation in reference [10]. ) Examples of the derivations of
the continuous time Lax equations from a discrete time one are described in
detail, where a novel method of taking a continuum limit by assuming
asymptotic behaviors of the eigenvalues of the Lax matrix in Puiseux series
expansions is used. In those examples, the method of imposing periodic
boundary conditions on the discrete integrable systems in [26], which uses
Perron-Frobenius theorem, turns out to be related to an unexpected way of
taking the continuum limit. We expect that it will provide a new technique for
studying the above mentioned relations between various types of integrable
dynamical systems. In this paper we restrict ourselves to the case of $n=4$
for such explicit derivations. Discussions for more general cases will be
explored in a separate publication.

The remaining part of this paper is organized as follows. In §2 we study a
system of differential equations that can be viewed as a kind of the extended
Lotka-Volterra systems. We introduce the loop elementary symmetric functions
in §2.1, and define a matrix $\mathcal{L}$ with elements given by them. We
present Theorem 1, which is the first result of this paper, showing that the
system of differential equations leads to a Lax equation satisfied by
$\mathcal{L}$ with a companion matrix $\mathcal{Y}$. A proof of this theorem
is shown in §2.2, where the Lax equation is decomposed into a set of Lax
triads [25]. In §2.3, we study several properties that can be derived from the
Lax equation. In §3 we introduce another system of differential equations
related to the loop elementary symmetric functions. The second result of this
paper, Theorem 7, is presented in §3.1 to show that the system of differential
equations leads to a Lax equation satisfied by $\mathcal{L}$ with another
companion matrix $\mathcal{Z}$. A proof of this theorem is shown in §3.2,
where the Lax equation is decomposed into another set of Lax triads. In §3.3,
we study several properties that can be derived from the Lax equation. In §4,
we present a connection of the systems of differential equations and the
closed geometric crystal chains. We give a brief review of the latter system
in §4.1 mainly for the case that can be described by $4\times 4$ matrices as
an example. In §4.2, §4.3, and §4.4, we give discussions for deriving the two
types of differential equations from the closed geometric crystal chains in
the case of $n=4$, where Theorems 13 and 14 are presented as the third and
fourth results of this paper. Several concluding remarks are given in §5 and a
few detailed calculations are shown in A and B.

##  2 Type I differential equations

###  2.1 Definitions and the first result

Let $L,n$ be a pair of coprime integers. Then there is a unique integer
$0<p<n$ such that the condition $Lp\equiv 1\,(\mod n)$ is satisfied. Let
$t\in{\mathbb{R}}$ be the time variable and $u_{i}^{(\alpha)}$ be a set of
dependent variables labeled by
$(\alpha,i)\in({\mathbb{Z}}/n{\mathbb{Z}})\times({\mathbb{Z}}/L{\mathbb{Z}})$.
Suppose that the system of differential equations

$\frac{{\rm d}u_{i}^{(\alpha)}}{{\rm
d}t}=u_{i}^{(\alpha)}\left(\sum_{j=1}^{\min(Lp-1,L(n-p))}(u_{i-j}^{(\alpha+j)}-u_{i+j}^{(\alpha-j)})\right),$
(1)

is satisfied by them.

For any
$(\alpha,i)\in({\mathbb{Z}}/n{\mathbb{Z}})\times({\mathbb{Z}}/L{\mathbb{Z}})$,
there is a unique integer $0\leq m<n$ such that the condition
$1-i-\alpha\equiv Lm\,(\mod n)$ is satisfied. Set $f(\alpha,i):=i+Lm$, which
can be viewed as an element of ${\mathbb{Z}}/(nL){\mathbb{Z}}$. By the Chinese
remainder theorem, the map sending $(\alpha,i)$ to the $f(\alpha,i)$ gives an
isomorphism of rings
$({\mathbb{Z}}/n{\mathbb{Z}})\times({\mathbb{Z}}/L{\mathbb{Z}})\simeq{\mathbb{Z}}/(nL){\mathbb{Z}}$,
for which the inverse map is given by sending $f$ to $((1-f)(\mod n),f(\mod
L))$. Thus if we write $P_{f(\alpha,i)}=u_{i}^{(\alpha)}$, then equation (1)
is written as
$\dot{P}_{f}=P_{f}\left(\sum_{g=1}^{\min(Lp-1,L(n-p))}(P_{f-g}-P_{f+g})\right)$,
which is a kind of the extended Lotka-Volterra systems.

For the set of variables $u_{i}^{(\alpha)}$ and an integer $m$, let
$e^{(\alpha)}_{m}\,(\alpha\in{\mathbb{Z}}/n{\mathbb{Z}})$ be the $m$-th loop
elementary symmetric functions defined by

$e^{(\alpha)}_{m}=\sum_{1\leq j_{1}<j_{2}<\dots<j_{m}\leq
L}u_{j_{1}}^{(\alpha+1-j_{1})}u_{j_{2}}^{(\alpha+2-j_{2})}\cdots
u_{j_{m}}^{(\alpha+m-j_{m})},$ (2)

and $e^{(\alpha)}_{0}=1$, $e^{(\alpha)}_{m}=0\,(m<0\quad\mbox{or}\quad m>L)$
[10, 20]. In particular, we have
$e^{(\alpha)}_{1}=u_{1}^{(\alpha)}+u_{2}^{(\alpha-1)}+\dots+u_{L}^{(\alpha-L+1)}$,
$e^{(\alpha)}_{L}=u_{1}^{(\alpha)}u_{2}^{(\alpha)}\dots u_{L}^{(\alpha)}$, and

$e^{(\alpha)}_{L-1}=\sum_{i=1}^{L}\left(\prod_{j=1}^{i-1}u_{j}^{(\alpha)}\prod_{k=i+1}^{L}u_{k}^{(\alpha-1)}\right).$
(3)

Let $\lambda$ be an indeterminate and $\mathcal{L}$ be the $n\times n$ matrix
defined by

$\mathcal{L}=(\mathcal{L}_{ij})_{1\leq i,j\leq
n},\qquad\mathcal{L}_{ij}=\sum_{m\geq 0}e^{(i)}_{j-i+L-mn}\lambda^{m},$ (4)

which we call a Lax matrix. We define

$y^{(\alpha)}=\sum_{j=0}^{p-1}e^{(\alpha-
jL)}_{1}-(p/n)\sum_{r=1}^{n}e^{(r)}_{1},$ (5)

and let $\mathcal{Y}$ be the $n\times n$ matrix

$\mathcal{Y}=\begin{pmatrix}y^{(1)}&&&&\lambda\\\ 1&y^{(2)}&&&\\\
&1&\ddots&&\\\ &&\ddots&\ddots&\\\ &&&1&y^{(n)}\end{pmatrix}.$ (6)

Through the variables $u_{i}^{(\alpha)}$, the elements of these matrices are
functions of the time variable $t$.

###### Theorem 1

Suppose that the variables $u_{i}^{(\alpha)}$ are satisfying the system of
differential equations (1). Then the Lax matrix satisfies the equation

$\frac{{\rm d}\mathcal{L}}{{\rm d}t}=[\mathcal{L},\mathcal{Y}].$ (7)

This result implies that conserved quantities of the dynamical system
represented by equation (1) can be given by the coefficients of the
characteristic polynomial of the Lax matrix $\mathcal{L}$, or equivalently by
$\mathrm{Tr}\,\mathcal{L}^{m}/m\,(m=1,\dots,n)$.

###  2.2 Proof of Theorem 1

We define

$\mbox{\Large{{u}}}_{i}=(u_{i}^{(1)},\dots,u_{i}^{(n)}),\quad
U=(\mbox{\Large{{u}}}_{1},\mbox{\Large{{u}}}_{2},\dots,\mbox{\Large{{u}}}_{L}),$
(8)

and $\sigma
U=(\mbox{\Large{{u}}}_{2},\dots,\mbox{\Large{{u}}}_{L},\mbox{\Large{{u}}}_{1})$,
where $\sigma$ denotes the cyclic shift to the left. For the $y^{(\alpha)}$
defined in (5), we write its dependence on the variable $U$ as
$y^{(\alpha)}(U)$, and define

$y^{(\alpha)}_{i}=y^{(\alpha)}(\sigma^{i-1}U),$ (9)

for any $i\in{\mathbb{Z}}/L{\mathbb{Z}}$. Let
$\mathcal{Y}_{i},\mathcal{M}_{i}$ denote the $n\times n$ matrices defined by

$\mathcal{Y}_{i}=\begin{pmatrix}y^{(1)}_{i}&&&&\lambda\\\ 1&y^{(2)}_{i}&&&\\\
&1&\ddots&&\\\ &&\ddots&\ddots&\\\
&&&1&y^{(n)}_{i}\end{pmatrix},\qquad\mathcal{M}_{i}=\begin{pmatrix}u^{(1)}_{i}&&&&\lambda\\\
1&u^{(2)}_{i}&&&\\\ &1&\ddots&&\\\ &&\ddots&\ddots&\\\
&&&1&u^{(n)}_{i}\end{pmatrix}.$ (10)

Using the identities $\mathcal{L}=\mathcal{M}_{1}\cdots\mathcal{M}_{L}$ (Lemma
6. 1 of [10]) and $\mathcal{Y}=\mathcal{Y}_{1}$, one sees that the assertion
of Theorem 1 follows from:

###### Proposition 2

The system of differential equations (1) is equivalent to

$\frac{{\rm d}\mathcal{M}_{i}}{{\rm
d}t}=\mathcal{M}_{i}\mathcal{Y}_{i+1}-\mathcal{Y}_{i}\mathcal{M}_{i}.$ (11)

Proof. The matrix elements of equation (11) are explicitly written as

$\displaystyle y^{(\alpha)}_{i+1}-y^{(\alpha+1)}_{i}$
$\displaystyle=u_{i}^{(\alpha)}-u_{i}^{(\alpha+1)},$ (12)
$\displaystyle\frac{{\rm d}u_{i}^{(\alpha)}}{{\rm d}t}$
$\displaystyle=u_{i}^{(\alpha)}(y^{(\alpha)}_{i+1}-y^{(\alpha)}_{i}).$ (13)

Thus the assertion of the proposition is a consequence of the following two
lemmas, which are satisfied by any set of variables $u_{i}^{(\alpha)}$ for
$(\alpha,i)\in({\mathbb{Z}}/n{\mathbb{Z}})\times({\mathbb{Z}}/L{\mathbb{Z}})$.
$\Box$

###### Lemma 3

Let $y^{(\alpha)}_{i}$ be the one defined in (9). Then the relation (12)
holds.

Proof. Using the expression
$e^{(\alpha)}_{1}=\sum_{k=0}^{L-1}u_{1+k}^{(\alpha-k)}$, we can write
$e^{(\alpha-jL)}_{1}=\sum_{k=0}^{L-1}u_{1+k}^{(\alpha-
jL-k)}=\sum_{k=0}^{L-1}u_{1+k+jL}^{(\alpha-jL-k)}$. Therefore

$y^{(\alpha)}(U)=\sum_{j=0}^{p-1}\sum_{k=0}^{L-1}u_{1+k+jL}^{(\alpha-
jL-k)}-pc_{0}=\sum_{j=0}^{Lp-1}u_{1+j}^{(\alpha-j)}-pc_{0},$ (14)

where $c_{0}:=(1/n)\sum_{r=1}^{n}e^{(r)}_{1}$. Hence we have
$y^{(\alpha)}_{i+1}-y^{(\alpha+1)}_{i}=y^{(\alpha)}(\sigma^{i}U)-y^{(\alpha+1)}(\sigma^{i-1}U)=\sum_{j=0}^{Lp-1}\left(u_{1+i+j}^{(\alpha-j)}-u_{i+j}^{(\alpha+1-j)}\right)=u_{i+Lp}^{(\alpha-
Lp+1)}-u_{i}^{(\alpha+1)}=u_{i}^{(\alpha)}-u_{i}^{(\alpha+1)}$, where the
condition $Lp\equiv 1\,(\mod n)$ is used in the last step. $\Box$

###### Lemma 4

Let $y^{(\alpha)}_{i}$ be the one defined in (9). Then the following relation
holds:

$y^{(\alpha)}_{i+1}-y^{(\alpha)}_{i}=\sum_{j=1}^{\min(Lp-1,L(n-p))}\left(u_{i-j}^{(\alpha+j)}-u_{i+j}^{(\alpha-j)}\right).$
(15)

Proof. First we suppose $p\leq\frac{n}{2}$. Then $\min(Lp-1,L(n-p))=Lp-1$.
Using (14) and the relation $u_{i+Lp}^{(\alpha-Lp+1)}=u_{i}^{(\alpha)}$, we
have

$y^{(\alpha)}_{i+1}-y^{(\alpha)}_{i}=y^{(\alpha)}(\sigma^{i}U)-y^{(\alpha)}(\sigma^{i-1}U)=\sum_{j=0}^{Lp-1}\left(u_{1+i+j}^{(\alpha-j)}-u_{i+j}^{(\alpha-j)}\right)=\sum_{j=1}^{Lp-1}\left(u_{i+j}^{(\alpha-j+1)}-u_{i+j}^{(\alpha-j)}\right).$
(16)

Replacing the index $j$ by $Lp-j$ and using the condition $Lp\equiv 1\,(\mod
n)$, we have
$\sum_{j=1}^{Lp-1}u_{i+j}^{(\alpha-j+1)}=\sum_{j=1}^{Lp-1}u_{i+Lp-j}^{(\alpha-
Lp+j+1)}=\sum_{j=1}^{Lp-1}u_{i-j}^{(\alpha+j)},$ so that (15) follows.

Second we suppose $p>\frac{n}{2}$. Then $\min(Lp-1,L(n-p))=L(n-p)$. Since $L$
and $n$ are coprime, we have
$\sum_{j\in({\mathbb{Z}}/n{\mathbb{Z}})}e^{(\alpha-
jL)}_{1}=\sum_{\alpha=1}^{n}e^{(\alpha)}_{1}=nc_{0}$. This implies that

$y^{(\alpha)}(U)=\sum_{j=0}^{p-1}e^{(\alpha-
jL)}_{1}-pc_{0}=(n-p)c_{0}-\sum_{j=1}^{n-p}e^{(\alpha+jL)}_{1}.$ (17)

Using the expression
$e^{(\alpha+L)}_{1}=\sum_{k=0}^{L-1}u_{1+k}^{(\alpha+L-k)}=\sum_{k=1}^{L}u_{1-k}^{(\alpha+k)}$,
one can rewrite the summation in the second term of the right hand side of
equation (17) as

$\sum_{j=1}^{n-p}e^{(\alpha+jL)}_{1}=\sum_{j=1}^{L(n-p)}u_{1-j}^{(\alpha+j)}=\sum_{j=1}^{L(n-p)}u_{j}^{(\alpha-j)},$

where the last expression is derived from the second one by replacing $j$ by
$L(n-p)+1-j$. Hence we have
$y^{(\alpha)}(\sigma^{i-1}U)=(n-p)c_{0}-\sum_{j=1}^{L(n-p)}u_{i-j}^{(\alpha+j)}$
and
$y^{(\alpha)}(\sigma^{i}U)=(n-p)c_{0}-\sum_{j=1}^{L(n-p)}u_{i+j}^{(\alpha-j)}$,
so that (15) follows. $\Box$

###### Remark 5

Let the loop elementary symmetric functions $e^{(\alpha)}_{i}$ be denoted by
$e^{(\alpha)}_{i}(U)$ for showing their dependence on the variable $U$.
Suppose $p=1$ or the condition $L\equiv 1\,(\mod n)$ is satisfied. Then the
set of differential equations (1) is written as

$\frac{{\rm d}u_{i}^{(\alpha)}}{{\rm
d}t}=u_{i}^{(\alpha)}\left(e^{(\alpha)}_{1}(\sigma^{i}U)-e^{(\alpha)}_{1}(\sigma^{i-1}U)\right).$
(18)

In particular, consider the case of $n=2$. By the reason that will be shown in
Remark 6, we can set $u_{i}^{(1)}u_{i}^{(2)}=1$. So if we define
$u_{i}=u_{i}^{(1)}$, then $u_{i}^{(2)}=1/u_{i}$. In this case one has
$e^{(1)}_{1}(\sigma^{i-1}U)=\sum_{j=0}^{L-1}(u_{i+j})^{(-1)^{j}}=u_{i}+\sum_{j=1}^{L-1}(1/u_{i+j})^{(-1)^{j-1}}$
and
$e^{(1)}_{1}(\sigma^{i}U)=\sum_{j=0}^{L-1}(u_{i+j+1})^{(-1)^{j}}=\sum_{j=1}^{L}(u_{i+j})^{(-1)^{j-1}}=u_{i}+\sum_{j=1}^{L-1}(u_{i+j})^{(-1)^{j-1}}$,
because $L$ is odd. Therefore equation (18) is written as

$\frac{{\rm d}u_{i}}{{\rm
d}t}=u_{i}\sum_{j=1}^{L-1}(-1)^{j-1}\left(u_{i+j}-\frac{1}{u_{i+j}}\right).$
(19)

This is the system of differential equations that we obtained in §2.3.2 of
reference [26].

###  2.3 Properties derived from the Lax equation

In this section, we study equation (7) without assuming any explicit
expression for $y^{(\alpha)}$ to find several conserved quantities. As a
byproduct, we are going to see that this equation on the Lax matrix
$\mathcal{L}$ and a matrix of the form $\mathcal{Y}$ in (6) basically
determines the expression for $y^{(\alpha)}$ uniquely.

Equation (7) with a matrix of the form $\mathcal{Y}$ in (6) is also equivalent
to the set of differential equations

$\frac{{\rm d}e_{i}^{(\alpha)}}{{\rm
d}t}=e_{i}^{(\alpha)}\left(y^{(i+\alpha-L)}-y^{(\alpha)}\right)+e_{i+1}^{(\alpha)}-e_{i+1}^{(\alpha-1)}.$
(20)

Letting $i=L$, we see that
$e_{L}^{(\alpha)}(=\prod_{i=1}^{L}u_{i}^{(\alpha)})$ is a conserved quantity,
because $e_{L+1}^{(\alpha)}=0$ for any $\alpha$. On the other hand, if we let
$i=0$ in equation (20), then

$0=\frac{{\rm d}e_{0}^{(\alpha)}}{{\rm
d}t}=\left(y^{(\alpha-L)}-y^{(\alpha)}\right)+e_{1}^{(\alpha)}-e_{1}^{(\alpha-1)},$

because $e_{0}^{(\alpha)}=1$. Since $e_{1}^{(\alpha-1)}=e_{1}^{(\alpha-Lp)}$,
this identity is equivalent to
$y^{(\alpha-L)}+e_{1}^{(\alpha)}=y^{(\alpha)}+e_{1}^{(\alpha-Lp)}$, which
determines an expression for $y^{(\alpha)}$ as

$y^{(\alpha)}=\sum_{j=0}^{p-1}e^{(\alpha-jL)}_{1}+C,$ (21)

where $C$ is a term not depending on $\alpha$. Now by letting $i=1$ in
equation (20) we have

$\frac{{\rm d}e_{1}^{(\alpha)}}{{\rm
d}t}=e_{1}^{(\alpha)}\left(y^{(1+\alpha-L)}-y^{(\alpha)}\right)+e_{2}^{(\alpha)}-e_{2}^{(\alpha-1)}.$
(22)

Using (21) and the condition $Lp\equiv 1\,(\mod n)$, one sees that

$y^{(1+\alpha-L)}=y^{(\alpha+L(p-1))}=\sum_{j=0}^{p-1}e^{(\alpha+L(p-1-j))}_{1}+C=\sum_{j=0}^{p-1}e^{(\alpha+jL)}_{1}+C.$
(23)

Then by using the identities (21), (22), (23), and
$\sum_{\alpha=1}^{n}e^{(\alpha)}_{1}e^{(\alpha+jL)}_{1}=\sum_{\alpha=1}^{n}e^{(\alpha-
jL)}_{1}e^{(\alpha)}_{1}$, we have ${\rm
d}(\sum_{\alpha=1}^{n}e^{(\alpha)}_{1})/{\rm d}t=0$, so that
$\sum_{\alpha=1}^{n}e^{(\alpha)}_{1}$ is a conserved quantity. Therefore, our
choice of $C=-(p/n)\sum_{\alpha=1}^{n}e^{(\alpha)}_{1}$ for the $y^{(\alpha)}$
above equation (6) is so chosen as to make it a conserved quantity, and to
impose the condition $\sum_{\alpha=1}^{n}y^{(\alpha)}=0$.

###### Remark 6

In the case of $p\leq\frac{n}{2}$, an expression for
$y^{(\alpha)}_{i+1}-y^{(\alpha)}_{i}$ was given by (16). A similar expression
for the case of $p>\frac{n}{2}$ can also be derived as

$y^{(\alpha)}_{i+1}-y^{(\alpha)}_{i}=\sum_{j=0}^{L(n-p)-1}\left(u_{i+j}^{(\alpha-j-1)}-u_{i+j}^{(\alpha-j)}\right)+u_{i}^{(\alpha-1)}-u_{i}^{(\alpha+1)}.$

Using these expressions, one sees that the relation
$\sum_{\alpha=1}^{n}(y^{(\alpha)}_{i+1}-y^{(\alpha)}_{i})=0$ is satisfied for
any $i$. Therefore, by using equation (13), we have ${\rm
d}(\log\prod_{\alpha=1}^{n}u^{(\alpha)}_{i})/{\rm d}t=0$, so that
$\prod_{\alpha=1}^{n}u^{(\alpha)}_{i}$ is a conserved quantity for any $i$.

##  3 Type II differential equations

###  3.1 Definitions and the second result

As in §2.1, let $t\in{\mathbb{R}}$ be the time variable and $u_{i}^{(\alpha)}$
be a set of dependent variables labeled by
$(\alpha,i)\in({\mathbb{Z}}/n{\mathbb{Z}})\times({\mathbb{Z}}/L{\mathbb{Z}})$,
but now the integers $L$ and $n$ are not necessarily coprime. Suppose that the
system of differential equations

$\frac{{\rm d}u_{i}^{(\alpha)}}{{\rm
d}t}=u_{i}^{(\alpha)}\left(\frac{1}{e^{(\alpha)}_{L-1}}\prod_{l=1}^{i-1}u_{l}^{(\alpha)}\prod_{k=i+1}^{L}u_{k}^{(\alpha-1)}-\frac{1}{e^{(\alpha+1)}_{L-1}}\prod_{l=1}^{i-1}u_{l}^{(\alpha+1)}\prod_{k=i+1}^{L}u_{k}^{(\alpha)}\right),$
(24)

is satisfied by them.

Comparing with the definition (3), one sees that part of the product in the
first term of the right hand side of equation (24) equals to the $i$-th term
of the summation in the definition of $e_{L-1}^{(\alpha)}$. Therefore, by
using this equation and the definitions of $e_{L}^{(\alpha)}$ and
$e_{L-1}^{(\alpha)}$, we have

$\frac{{\rm d}\log e_{L}^{(\alpha)}}{{\rm
d}t}=\sum_{i=1}^{L}\frac{1}{u_{i}^{(\alpha)}}\frac{{\rm
d}u_{i}^{(\alpha)}}{{\rm d}t}=1-1=0.$

Hence $e_{L}^{(\alpha)}$ is a conserved quantity for any $\alpha$. Recall that
the Lax matrix $\mathcal{L}$ was defined by equation (4), and let
$\mathcal{Z}$ be the $n\times n$ matrix

$\mathcal{Z}=\begin{pmatrix}0&z^{(2)}&&&\\\ &0&z^{(3)}&&\\\
&&\ddots&\ddots&\\\ &&&\ddots&z^{(n)}\\\ z^{(1)}/\lambda&&&&0\end{pmatrix},$
(25)

where

$z^{(\alpha)}=1/e^{(\alpha)}_{L-1}.$ (26)

###### Theorem 7

Suppose that the variables $u_{i}^{(\alpha)}$ are satisfying the system of
differential equations (24), and that the condition $e_{L}^{(\alpha)}=1$ is
satisfied for any $\alpha$. Then the Lax matrix $\mathcal{L}$ satisfies the
equation

$\frac{{\rm d}\mathcal{L}}{{\rm d}t}=[\mathcal{L},\mathcal{Z}].$ (27)

By Theorems 1 and 7, we see that the dynamical systems represented by
equations (1) and (24) share a common Lax matrix defined by (4) in some cases.
As a result, they share a common set of conserved quantities in such cases.

###  3.2 Proof of Theorem 7

For the $z^{(\alpha)}$ defined in (26), we write its dependence on the
variable $U$ in (8) as $z^{(\alpha)}(U)$, and define

$z^{(\alpha)}_{i}=z^{(\alpha)}(\sigma^{i-1}U),$ (28)

for any $i\in{\mathbb{Z}}/L{\mathbb{Z}}$. Let $\mathcal{M}_{i}$ be the matrix
in (10) and define the matrix $\mathcal{Z}_{i}$ as

$\mathcal{Z}_{i}=\begin{pmatrix}0&z^{(2)}_{i}&&&\\\ &0&z^{(3)}_{i}&&\\\
&&\ddots&\ddots&\\\ &&&\ddots&z^{(n)}_{i}\\\
z^{(1)}_{i}/\lambda&&&&0\end{pmatrix}.$ (29)

###### Proposition 8

The system of differential equations (24) with the condition
$e_{L}^{(\alpha)}=1$ is equivalent to

$\frac{{\rm d}\mathcal{M}_{i}}{{\rm
d}t}=\mathcal{M}_{i}\mathcal{Z}_{i+1}-\mathcal{Z}_{i}\mathcal{M}_{i}.$ (30)

This together with the relations
$\mathcal{L}=\mathcal{M}_{1}\cdots\mathcal{M}_{L}$ and
$\mathcal{Z}=\mathcal{Z}_{1}$ yields the assertion of Theorem 7.

Proof. The matrix elements of equation (30) are explicitly written as

$\displaystyle u_{i}^{(\alpha-1)}z^{(\alpha)}_{i+1}$
$\displaystyle=u_{i}^{(\alpha)}z^{(\alpha)}_{i},$ (31)
$\displaystyle\frac{{\rm d}u_{i}^{(\alpha)}}{{\rm d}t}$
$\displaystyle=z^{(\alpha)}_{i+1}-z^{(\alpha+1)}_{i}.$ (32)

Thus the assertion of the proposition is a consequence of the following two
lemmas, which are satisfied by any set of variables $u_{i}^{(\alpha)}$ for
$(\alpha,i)\in({\mathbb{Z}}/n{\mathbb{Z}})\times({\mathbb{Z}}/L{\mathbb{Z}})$
under the condition $e_{L}^{(\alpha)}=1$. $\Box$

###### Lemma 9

If $e_{L}^{(\alpha)}$ does not depend on $\alpha$, then the relation (31) or
equivalently the following relation holds:

$e_{L-1}^{(\alpha)}(\sigma^{i}U)\frac{u_{i}^{(\alpha)}}{u_{i}^{(\alpha-1)}}=e_{L-1}^{(\alpha)}(\sigma^{i-1}U).$
(33)

Proof. Suppose $e_{L}^{(\alpha)}=C$ for any $\alpha$. Then

$\displaystyle
e_{L-1}^{(\alpha)}(\sigma^{i}U)\frac{u_{i}^{(\alpha)}}{u_{i}^{(\alpha-1)}}$
$\displaystyle=\sum_{l=1}^{L}\left(\prod_{j=1}^{l-1}u_{j+i}^{(\alpha)}\prod_{k=l+1}^{L}u_{k+i}^{(\alpha-1)}\right)\frac{u_{i}^{(\alpha)}}{u_{i}^{(\alpha-1)}}$
$\displaystyle=\sum_{l=1}^{L-1}\left(\prod_{j=1}^{l-1}u_{j+i}^{(\alpha)}\prod_{k=l+1}^{L}u_{k+i}^{(\alpha-1)}\right)\frac{u_{i}^{(\alpha)}}{u_{i}^{(\alpha-1)}}+\frac{C}{u_{i}^{(\alpha-1)}}$
$\displaystyle=\sum_{l=1}^{L-1}\left(\prod_{j=0}^{l-1}u_{j+i}^{(\alpha)}\prod_{k=l+1}^{L-1}u_{k+i}^{(\alpha-1)}\right)+\prod_{k=1}^{L-1}u_{k+i}^{(\alpha-1)}$
$\displaystyle=\sum_{l=2}^{L}\left(\prod_{j=1}^{l-1}u_{j+i-1}^{(\alpha)}\prod_{k=l+1}^{L}u_{k+i-1}^{(\alpha-1)}\right)+\prod_{k=2}^{L}u_{k+i-1}^{(\alpha-1)}$
$\displaystyle=\sum_{l=1}^{L}\left(\prod_{j=1}^{l-1}u_{j+i-1}^{(\alpha)}\prod_{k=l+1}^{L}u_{k+i-1}^{(\alpha-1)}\right)=e_{L-1}^{(\alpha)}(\sigma^{i-1}U).$

In the second line, we wrote the term for $l=L$ separately. In the third line,
the extra factor is absorbed in the parenthesized expression, and we used
$C=e_{L}^{(\alpha-1)}$ in the separated term. In the fourth line, we replaced
$(j,k,l)$ by $(j-1,k-1,l-1)$. In the last line, we included the separated term
into the summation as the term for $l=1$. $\Box$

###### Lemma 10

Suppose $e_{L}^{(\alpha)}=1$ for any $\alpha$, and let $z^{(\alpha)}_{i}$ be
the one defined in (28). Then

$z^{(\alpha)}_{i+1}-z^{(\alpha+1)}_{i}=u_{i}^{(\alpha)}\left(\frac{1}{e^{(\alpha)}_{L-1}}\prod_{l=1}^{i-1}u_{l}^{(\alpha)}\prod_{k=i+1}^{L}u_{k}^{(\alpha-1)}-\frac{1}{e^{(\alpha+1)}_{L-1}}\prod_{l=1}^{i-1}u_{l}^{(\alpha+1)}\prod_{k=i+1}^{L}u_{k}^{(\alpha)}\right).$
(34)

Proof. By definition, we have

$\displaystyle z^{(\alpha)}_{i+1}-z^{(\alpha+1)}_{i}$
$\displaystyle=\frac{1}{e_{L-1}^{(\alpha)}(\sigma^{i}U)}-\frac{1}{e_{L-1}^{(\alpha+1)}(\sigma^{i-1}U)}$
$\displaystyle=u_{i}^{(\alpha)}\left(\frac{1}{u_{i}^{(\alpha)}e_{L-1}^{(\alpha)}(\sigma^{i}U)}-\frac{1}{u_{i}^{(\alpha+1)}e_{L-1}^{(\alpha+1)}(\sigma^{i}U)}\right),$

where we used Lemma 9 in the second line. Therefore it suffices to show that

$u_{i}^{(\alpha)}e_{L-1}^{(\alpha)}(\sigma^{i}U)=\left(\prod_{l=1}^{i-1}u_{l}^{(\alpha)}\prod_{k=i+1}^{L}u_{k}^{(\alpha-1)}\right)^{-1}e^{(\alpha)}_{L-1}(U),$

for any $\alpha\in{\mathbb{Z}}/(n{\mathbb{Z}})$ and $1\leq i\leq L$. This
relation is proved by applying Lemma 9 on the left hand side repeatedly as,

$\displaystyle u_{i}^{(\alpha)}e_{L-1}^{(\alpha)}(\sigma^{i}U)$
$\displaystyle=u_{i}^{(\alpha-1)}e_{L-1}^{(\alpha)}(\sigma^{i-1}U)$
$\displaystyle=\frac{u_{i}^{(\alpha-1)}u_{i-1}^{(\alpha-1)}\dots
u_{1}^{(\alpha-1)}}{u_{i-1}^{(\alpha)}\dots
u_{1}^{(\alpha)}}e_{L-1}^{(\alpha)}(U)$
$\displaystyle=\left(\prod_{l=1}^{i-1}u_{l}^{(\alpha)}\prod_{k=i+1}^{L}u_{k}^{(\alpha-1)}\right)^{-1}e^{(\alpha)}_{L-1}(U),$

where we used $\prod_{k=1}^{L}u_{k}^{(\alpha-1)}=e_{L}^{(\alpha-1)}=1$. The
proof is completed. $\Box$

###### Remark 11

There is a simple expression for the system of differential equations (24)
that can be compared with equation (18) in the type I case. In fact, from the
above considerations we see that equation (24) is written as

$\frac{{\rm d}u_{i}^{(\alpha)}}{{\rm
d}t}=\frac{1}{e_{L-1}^{(\alpha)}(\sigma^{i}U)}-\frac{1}{e_{L-1}^{(\alpha+1)}(\sigma^{i-1}U)}.$
(35)

In particular, consider the case of $L=2$. If we define
$u^{(\alpha)}=u_{1}^{(\alpha)}$, then we have
$u_{2}^{(\alpha)}=1/u^{(\alpha)}$ because we set $e_{L}^{(\alpha)}=1$.
Therefore one has

$\frac{{\rm d}u^{(\alpha)}}{{\rm
d}t}=\frac{u^{(\alpha)}}{u^{(\alpha)}u^{(\alpha-1)}+1}-\frac{u^{(\alpha)}}{u^{(\alpha)}u^{(\alpha+1)}+1}.$
(36)

This is basically the same differential equation that was appeared in
reference [23] as equation (13.58), where its discretization was referred to
as a lattice KdV equation.

###  3.3 Properties derived from the Lax equation

In this section, we study equation (27) without assuming any explicit
expression for $z^{(\alpha)}$ and find several conserved quantities. As a
byproduct, we are going to see that this equation on the Lax matrix
$\mathcal{L}$ and a matrix of the form $\mathcal{Z}$ in (25) basically
determines the expression for $z^{(\alpha)}$ uniquely.

In view of the definition (4), one sees that equation (27) with a matrix of
the form $\mathcal{Z}$ in (25) is equivalent to the system of differential
equations

$\frac{{\rm d}e_{i}^{(\alpha)}}{{\rm
d}t}=z^{(i+\alpha-L)}e_{i-1}^{(\alpha)}-z^{(\alpha+1)}e_{i-1}^{(\alpha+1)}.$
(37)

By letting $i=L+1$ we see that $e_{L}^{(\alpha)}=e_{L}^{(\alpha+1)}$, because
$e_{L+1}^{(\alpha)}=0$ for any $\alpha$. On the other hand, the summation
$\sum_{\alpha=1}^{n}e^{(\alpha)}_{L}$ is a conserved quantity, because the
conservation of $\mathrm{Tr}\,\mathcal{L}=\sum_{m\geq
0}\sum_{\alpha=1}^{n}e^{(\alpha)}_{L-mn}\lambda^{m}$ is derived from equation
(27). This forces each $e_{L}^{(\alpha)}$ to be a conserved quantity. Then by
letting $i=L$ in equation (37) we have

$0=\frac{{\rm d}e_{L}^{(\alpha)}}{{\rm
d}t}=z^{(\alpha)}e_{L-1}^{(\alpha)}-z^{(\alpha+1)}e_{L-1}^{(\alpha+1)}.$

This identity determines an expression for $z^{(\alpha)}$ as
$z^{(\alpha)}=C/e_{L-1}^{(\alpha)}$, where $C$ is a factor not depending on
$\alpha$

Now letting $i=L-1$ in equation (37) and dividing it by $e_{L-1}^{(\alpha)}$,
we have

$\frac{1}{e_{L-1}^{(\alpha)}}\frac{{\rm d}e_{L-1}^{(\alpha)}}{{\rm
d}t}=C\left(\frac{e_{L-2}^{(\alpha)}}{e_{L-1}^{(\alpha-1)}e_{L-1}^{(\alpha)}}-\frac{e_{L-2}^{(\alpha+1)}}{e_{L-1}^{(\alpha)}e_{L-1}^{(\alpha+1)}}\right).$

This implies that

$\frac{{\rm d}}{{\rm
d}t}\log(\prod_{\alpha=1}^{n}e_{L-1}^{(\alpha)})=\sum_{\alpha=1}^{n}\frac{1}{e_{L-1}^{(\alpha)}}\frac{{\rm
d}e_{L-1}^{(\alpha)}}{{\rm d}t}=0.$

Hence $\prod_{\alpha=1}^{n}e_{L-1}^{(\alpha)}$ is a conserved quantity.
Moreover, by using equation (33) we have
$\prod_{\alpha=1}^{n}e_{L-1}^{(\alpha)}(\sigma^{i}U)=\prod_{\alpha=1}^{n}e_{L-1}^{(\alpha)}$
for any $i$.

###### Remark 12

We have shown the conservation of
$e_{L}^{(\alpha)}=\prod_{i=1}^{L}u_{i}^{(\alpha)}$ for any $1\leq\alpha\leq n$
by using equation (24). In the same way, we can show that

$\frac{{\rm d}}{{\rm
d}t}\log(\prod_{\alpha=1}^{n}u_{i}^{(\alpha)})=\sum_{\alpha=1}^{n}\frac{1}{u_{i}^{(\alpha)}}\frac{{\rm
d}u_{i}^{(\alpha)}}{{\rm d}t}=0.$

Hence $\prod_{\alpha=1}^{n}u_{i}^{(\alpha)}$ is also a conserved quantity for
any $1\leq i\leq L$.

##  4 Connection to the closed geometric crystal chains: A case study for
$\mathbf{n=4}$

###  4.1 A review of the closed geometric crystal chains

We briefly review on the closed geometric crystal chains for the totally one-
row tableaux case ([26], §3. 1). Based on the notion of rational rectangles in
[4, 5], we introduce the set
$\mathbb{Y}_{1}=({\mathbb{R}}_{>0})^{n-1}\times{\mathbb{R}}_{>0}$. Let
$(\mathbf{x},s)$ denote an element of $\mathbb{Y}_{1}$ with
$\mathbf{x}=(x^{(1)},\dots,x^{(n-1)})$, and let $x^{(n)}:=s/(x^{(1)}\cdots
x^{(n-1)})$. Furthermore, we define $x^{(i)}$ for arbitrary $i\in{\mathbb{Z}}$
to be a variable determined from $\mathbf{x}$ by the relation
$x^{(i)}=x^{(i+n)}$. In what follows, we set $n=4$. Given
$(\mathbf{x},s)\in({\mathbb{R}}_{>0})^{3}\times{\mathbb{R}}_{>0}$, we define
the matrices $g^{*}$ and $g$ by

$\displaystyle g^{*}(\mathbf{x},s;\lambda)$
$\displaystyle=\begin{pmatrix}x^{(1)}x^{(2)}x^{(3)}&\lambda&\lambda
x^{(3)}&\lambda x^{(2)}x^{(3)}\\\
x^{(1)}x^{(2)}&x^{(4)}x^{(1)}x^{(2)}&\lambda&\lambda x^{(2)}\\\
x^{(1)}&x^{(4)}x^{(1)}&x^{(3)}x^{(4)}x^{(1)}&\lambda\\\
1&x^{(4)}&x^{(3)}x^{(4)}&x^{(2)}x^{(3)}x^{(4)}\end{pmatrix},$ (38)
$\displaystyle g(\mathbf{x},s;\lambda)$
$\displaystyle=\begin{pmatrix}x^{(1)}&0&0&\lambda\\\ 1&x^{(2)}&0&0\\\
0&1&x^{(3)}&0\\\ 0&0&1&x^{(4)}\end{pmatrix}.$ (39)

Actually, any element of the matrix $g^{*}(\mathbf{x},s;\lambda)$ is so
defined as to be an order $3$ minor of the matrix $g(\mathbf{x},s;\lambda)$.
For instance, the top-left element of the former is equal to the determinant
of the top-left $3\times 3$ submatrix of the latter.

By Theorem 16 of [26], we see that for any $s,l\in{\mathbb{R}}_{>0}$ and
$(\mathbf{b}_{1},\dots,\mathbf{b}_{L})\in({\mathbb{R}}_{>0})^{3L}$, there is a
unique positive real solution
$(\mathbf{v},\mathbf{b}^{\prime}_{1},\dots,\mathbf{b}^{\prime}_{L})\in({\mathbb{R}}_{>0})^{3(L+1)}$
to the equation

$g(\mathbf{b}_{1},s;\lambda)\cdots
g(\mathbf{b}_{L},s;\lambda)g(\mathbf{v},l;\lambda)=g(\mathbf{v},l;\lambda)g(\mathbf{b}^{\prime}_{1},s;\lambda)\cdots
g(\mathbf{b}^{\prime}_{L},s;\lambda).$ (40)

For any $|\mathbf{b}\rangle=(\mathbf{b}_{1},\dots,\mathbf{b}_{L})$, let
$\mathcal{L}(|\mathbf{b}\rangle;\lambda)$ be the matrix

$\mathcal{L}(|\mathbf{b}\rangle;\lambda)=g(\mathbf{b}_{1},s;\lambda)\cdots
g(\mathbf{b}_{L},s;\lambda),$ (41)

which is called a Lax matrix, and let
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ be the matrix

$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)=g^{*}(\mathbf{b}_{1},s;l)\cdots
g^{*}(\mathbf{b}_{L},s;l),$ (42)

which we call the monodromy matrix of the Lax matrix
$\mathcal{L}(|\mathbf{b}\rangle;\lambda)$ with $\lambda=l$. Note that every
matrix element of $\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ is an order $3$
minor of $\mathcal{L}(|\mathbf{b}\rangle;l)$. Note also that, if the
eigenvalues of $\mathcal{L}(|\mathbf{b}\rangle;l)$ are given by
$\mu_{1},\mu_{2},\mu_{3},\mu_{4}$, then the eigenvalues of
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ are given by
$\mu_{1}\mu_{2}\mu_{3},\mu_{1}\mu_{2}\mu_{4},\mu_{1}\mu_{3}\mu_{4},\mu_{2}\mu_{3}\mu_{4}$,
where the multiplicity of the eigenvalues has been taken into account
(Corollary 29 of [26]).

Let $E$ be the largest eigenvalue in absolute value of matrix
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$, and
$\vec{P}=(\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3},1)^{t}$ be an
eigenvector corresponding to $E$. By the Perron-Frobenius theorem, $E$ is real
positive, $\vec{P}$ is uniquely determined, and
$\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3}$ are all positive. Then, the
solution $\mathbf{v}\in({\mathbb{R}}_{>0})^{3}$ of equation (40) is given by
$\mathbf{v}=(\mathcal{P}_{3},\mathcal{P}_{2}/\mathcal{P}_{3},\mathcal{P}_{1}/\mathcal{P}_{2})$
(Proposition 15 of [26]). This unique solution $\mathbf{v}$ allows us to
define
$T^{(1)}_{l}:({\mathbb{R}}_{>0})^{3L}\rightarrow({\mathbb{R}}_{>0})^{3L}$ to
be the map given by

$T^{(1)}_{l}(\mathbf{b}_{1},\dots,\mathbf{b}_{L})=(\mathbf{b}^{\prime}_{1},\dots,\mathbf{b}^{\prime}_{L}),$
(43)

which is called a time evolution. Due to equation (40), this time evolution
(43) is governed by a discrete time analogue of the Lax equation

$\mathcal{L}(T^{(1)}_{l}|\mathbf{b}\rangle;\lambda)=g(\mathbf{v},l;\lambda)^{-1}\mathcal{L}(|\mathbf{b}\rangle;\lambda)g(\mathbf{v},l;\lambda).$
(44)

In what follows, we are going to show that this equation reduces to the
continuous time Lax equation (7) in the limit $l\rightarrow\infty$, or to
equation (27) in the limit $l\rightarrow 0$, by making several reasonable
assumptions.

###  4.2 Limit for the type I Lax equation

####  4.2.1 The case of p=1.

We assume $L\equiv 1(\mod 4)$, and set $L=4\kappa+1$. Recall that
$|\mathbf{b}\rangle=(\mathbf{b}_{1},\dots,\mathbf{b}_{L})$ and
$\mathbf{b}_{i}=(b_{i}^{(1)},b_{i}^{(2)},b_{i}^{(3)})$. By the reason
explained above Proposition 2, the Lax matrix (41) can be identified with the
matrix $\mathcal{L}$ defined by (4), in which the loop elementary symmetric
functions are defined by (2) but with the substitution
$u_{i}^{(\alpha)}=b_{i}^{(\alpha)}$. Using the explicit expression (4) and the
condition $L=4\kappa+1$, we can obtain the asymptotic form of the Lax matrix
$\mathcal{L}(|\mathbf{b}\rangle;l)$ under the limit $l\rightarrow\infty$ as

$\mathcal{L}(|\mathbf{b}\rangle;l)\approx\begin{pmatrix}e_{1}^{(1)}l^{\kappa}&e_{2}^{(1)}l^{\kappa}&e_{3}^{(1)}l^{\kappa}&l^{\kappa+1}\\\
l^{\kappa}&e_{1}^{(2)}l^{\kappa}&e_{2}^{(2)}l^{\kappa}&e_{3}^{(2)}l^{\kappa}\\\
e_{3}^{(3)}l^{\kappa-1}&l^{\kappa}&e_{1}^{(3)}l^{\kappa}&e_{2}^{(3)}l^{\kappa}\\\
e_{2}^{(4)}l^{\kappa-1}&e_{3}^{(4)}l^{\kappa-1}&l^{\kappa}&e_{1}^{(4)}l^{\kappa}\end{pmatrix}.$
(45)

We assume that the eigenvalues $\eta_{q}\,(q\in{\mathbb{Z}}/4{\mathbb{Z}})$ of
the Lax matrix $\mathcal{L}(|\mathbf{b}\rangle;l)$ for sufficiently large
$l$’s are given by the Puiseux series expansion

$\eta_{q}=l^{\kappa}\sum_{m=-1}^{\infty}c_{m}\exp\left(\frac{\pi\sqrt{-1}mq}{2}\right)l^{-m/4},$
(46)

where $c_{-1}=1$ and $c_{0}=(\sum_{\alpha=1}^{4}e^{(\alpha)}_{1})/4$. This
assumption is consistent with the relations

$\displaystyle(s-l)^{L}=\det\mathcal{L}(|\mathbf{b}\rangle;l)$
$\displaystyle=\prod_{q=1}^{4}\eta_{q}=-l^{4\kappa+1}+\mathcal{O}(l^{4\kappa}),$
$\displaystyle(\sum_{\alpha=1}^{4}e^{(\alpha)}_{1})l^{\kappa}+\mathcal{O}(l^{\kappa-1})=\mathrm{Tr}\,\mathcal{L}(|\mathbf{b}\rangle;l)$
$\displaystyle=\sum_{q=1}^{4}\eta_{q}=4l^{\kappa}c_{0}+\mathcal{O}(l^{\kappa-1}),$

where $\mathcal{O}$ denotes Landau’s symbol, and we used the identity $\det
g(\mathbf{b}_{i},s;l)=s-l$ for $1\leq i\leq L$ and the asymptotic form (45).
For reference, we show an explicit derivation of such series expansions for a
simpler case of $n=2$ in B. Then, the asymptotic form of the largest
eigenvalue of the monodromy matrix $\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$
is given by

$E=\eta_{1}\eta_{3}\eta_{4}=l^{3\kappa+\frac{3}{4}}+c_{0}l^{3\kappa+\frac{2}{4}}+\cdots.$
(47)

In view of (45), we see that the asymptotic form of the monodromy matrix
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ under the limit
$l\rightarrow\infty$ is given by

$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)\approx\begin{pmatrix}\mathcal{O}(l^{3\kappa})&l^{3\kappa+1}&e_{1}^{(3)}l^{3\kappa+1}&\mathcal{O}(l^{3\kappa+1})\\\
\mathcal{O}(l^{3\kappa})&\mathcal{O}(l^{3\kappa})&l^{3\kappa+1}&e_{1}^{(2)}l^{3\kappa+1}\\\
e_{1}^{(1)}l^{3\kappa}&\mathcal{O}(l^{3\kappa})&\mathcal{O}(l^{3\kappa})&l^{3\kappa+1}\\\
*&*&*&*\end{pmatrix},$ (48)

where the last row is omitted because we do not need it. Let $\mathsf{M}_{ij}$
denote the $ij$ element of $\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$. Then,
the eigenvector
$\vec{P}=(\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3},1)^{t}$
corresponding to the eigenvalue $E$ is determined by the linear equation

$\begin{pmatrix}\mathsf{M}_{11}-E&\mathsf{M}_{12}&\mathsf{M}_{13}\\\
\mathsf{M}_{21}&\mathsf{M}_{22}-E&\mathsf{M}_{23}\\\
\mathsf{M}_{31}&\mathsf{M}_{32}&\mathsf{M}_{33}-E\end{pmatrix}\begin{pmatrix}\mathcal{P}_{1}\\\
\mathcal{P}_{2}\\\
\mathcal{P}_{3}\end{pmatrix}=-\begin{pmatrix}\mathsf{M}_{14}\\\
\mathsf{M}_{24}\\\ \mathsf{M}_{34}\end{pmatrix}.$ (49)

Let $l=1/\delta^{4}$. By expressing the solution of this equation by the
Cramer formula and then substituting the asymptotic forms (47) and (48) into
it, we obtain the expressions

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=\frac{1}{\delta^{3}}+(e_{1}^{(1)}+e_{1}^{(2)}+e_{1}^{(3)}-3c_{0})\frac{1}{\delta^{2}}+\mathcal{O}\left(\frac{1}{\delta}\right),$
$\displaystyle\mathcal{P}_{2}$
$\displaystyle=\frac{1}{\delta^{2}}+(e_{1}^{(1)}+e_{1}^{(2)}-2c_{0})\frac{1}{\delta}+\mathcal{O}\left(1\right),$
$\displaystyle\mathcal{P}_{3}$
$\displaystyle=\frac{1}{\delta}+(e_{1}^{(1)}-c_{0})+\mathcal{O}\left(\delta\right).$
(50)

We show a detailed derivation of this result in A.1. Therefore, the asymptotic
form of the matrix $g(\mathbf{v},l;\lambda)$ with
$\mathbf{v}=(\mathcal{P}_{3},\mathcal{P}_{2}/\mathcal{P}_{3},\mathcal{P}_{1}/\mathcal{P}_{2})$
and $l=1/\delta^{4}$ is expressed as

$\delta\cdot
g(\mathbf{v},1/\delta^{4};\lambda)=\delta\cdot\begin{pmatrix}\mathcal{P}_{3}&&&\lambda\\\
1&\mathcal{P}_{2}/\mathcal{P}_{3}&&\\\ &1&\mathcal{P}_{1}/\mathcal{P}_{2}&\\\
&&1&1/(\delta^{4}\mathcal{P}_{1})\end{pmatrix}=\mathbb{I}_{4}+\delta\cdot\mathcal{Y}+\mathcal{O}\left(\delta^{2}\right),$
(51)

where $\mathcal{Y}$ is a matrix of the form (6) with $n=4,p=1$.

To summarize, we state the result of the above arguments in the case of
$n=4,p=1$ as:

###### Theorem 13

Let $\mathcal{L}(|\mathbf{b}\rangle;\lambda)$ and
$\mathcal{L}(T^{(1)}_{l}|\mathbf{b}\rangle;\lambda)$ be denoted by
$\mathcal{L}(t)$ and $\mathcal{L}(t+\delta)$, respectively, where
$l=1/\delta^{4}$. Assume that the eigenvalues of the matrix
$\mathcal{L}(|\mathbf{b}\rangle;l)$ for sufficiently large $l$’s are given the
expansion (46). Then, the discrete time Lax equation (44) reduces to the
continuous time Lax equation (7) in the limit $\delta\rightarrow 0$.

Proof. By using the expression (51), we obtain

$\mathcal{L}(t+\delta)=(\delta\cdot
g(\mathbf{v},1/\delta^{4};\lambda))^{-1}\mathcal{L}(t)(\delta\cdot
g(\mathbf{v},1/\delta^{4};\lambda))=\mathcal{L}(t)+\delta\cdot[\mathcal{L}(t),\mathcal{Y}]+\mathcal{O}\left(\delta^{2}\right),$

from equation (44). Hence the assertion of the theorem follows. $\Box$

####  4.2.2 The case of p=3.

We assume $L\equiv 3(\mod 4)$, and set $L=4\kappa+3$. As in the previous case,
we can obtain the asymptotic form of the Lax matrix
$\mathcal{L}(|\mathbf{b}\rangle;l)$ under the limit $l\rightarrow\infty$ as

$\mathcal{L}(|\mathbf{b}\rangle;l)\approx\begin{pmatrix}e_{3}^{(1)}l^{\kappa}&l^{\kappa+1}&e_{1}^{(1)}l^{\kappa+1}&e_{2}^{(1)}l^{\kappa+1}\\\
e_{2}^{(2)}l^{\kappa}&e_{3}^{(2)}l^{\kappa}&l^{\kappa+1}&e_{1}^{(2)}l^{\kappa+1}\\\
e_{1}^{(3)}l^{\kappa}&e_{2}^{(3)}l^{\kappa}&e_{3}^{(3)}l^{\kappa}&l^{\kappa+1}\\\
l^{\kappa}&e_{1}^{(4)}l^{\kappa}&e_{2}^{(4)}l^{\kappa}&e_{3}^{(4)}l^{\kappa}\end{pmatrix}.$
(52)

We assume that the eigenvalues $\eta_{q}\,(q\in{\mathbb{Z}}/4{\mathbb{Z}})$ of
the Lax matrix $\mathcal{L}(|\mathbf{b}\rangle;l)$ for sufficiently large
$l$’s are given by the Puiseux series expansion

$\eta_{q}=l^{\kappa+\frac{1}{2}}\sum_{m=-1}^{\infty}c_{m}\exp\left(\frac{\pi\sqrt{-1}(m-2)q}{2}\right)l^{-m/4},$
(53)

where $c_{-1}=1$ and $c_{2}=(\sum_{\alpha=1}^{4}e^{(\alpha)}_{3})/4$. This
assumption is consistent with the relations

$\displaystyle(s-l)^{L}=\det\mathcal{L}(|\mathbf{b}\rangle;l)$
$\displaystyle=\prod_{q=1}^{4}\eta_{q}=-l^{4\kappa+3}+\mathcal{O}(l^{4\kappa+2}),$
$\displaystyle(\sum_{\alpha=1}^{4}e^{(\alpha)}_{3})l^{\kappa}+\mathcal{O}(l^{\kappa-1})=\mathrm{Tr}\,\mathcal{L}(|\mathbf{b}\rangle;l)$
$\displaystyle=\sum_{q=1}^{4}\eta_{q}=4l^{\kappa}c_{2}+\mathcal{O}(l^{\kappa-1}),$

where we used the asymptotic form (52). Then, the asymptotic form of the
largest eigenvalue of the monodromy matrix
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ is given by

$E=\eta_{1}\eta_{3}\eta_{4}=l^{3\kappa+\frac{9}{4}}+c_{0}l^{3\kappa+\frac{8}{4}}+\cdots.$
(54)

In view of (52), we see that the asymptotic form of the monodromy matrix
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ under the limit
$l\rightarrow\infty$ is given by

$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)\approx\begin{pmatrix}e_{1}^{(3)}l^{3\kappa+2}&\mathcal{O}(l^{3\kappa+2})&\mathcal{O}(l^{3\kappa+2})&l^{3\kappa+3}\\\
l^{3\kappa+2}&e_{1}^{(2)}l^{3\kappa+2}&\mathcal{O}(l^{3\kappa+2})&\mathcal{O}(l^{3\kappa+2})\\\
\mathcal{O}(l^{3\kappa+1})&l^{3\kappa+2}&e_{1}^{(1)}l^{3\kappa+2}&\mathcal{O}(l^{3\kappa+2})\\\
\mathcal{O}(l^{3\kappa+1})&\mathcal{O}(l^{3\kappa+1})&l^{3\kappa+2}&e_{1}^{(4)}l^{3\kappa+2}\end{pmatrix}.$
(55)

From (54) and (55) one sees that

$0=\det(\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)-E\mathbb{I}_{4})=(4c_{o}-(e_{1}^{(1)}+e_{1}^{(2)}+e_{1}^{(3)}+e_{1}^{(4)}))l^{12\kappa+8+\frac{3}{4}}+\mathcal{O}(l^{12\kappa+8+\frac{2}{4}}),$

which forces us to set $c_{0}=(\sum_{\alpha=1}^{4}e_{1}^{(\alpha)})/4$. Let
$l=1/\delta^{4}$. By expressing the solution of equation (49) by the Cramer
formula and then substituting the asymptotic forms (54) and (55) into it, we
obtain the expressions

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=\frac{1}{\delta^{3}}+(e_{1}^{(3)}-c_{0})\frac{1}{\delta^{2}}+\mathcal{O}\left(\frac{1}{\delta}\right),$
$\displaystyle\mathcal{P}_{2}$
$\displaystyle=\frac{1}{\delta^{2}}+(e_{1}^{(2)}+e_{1}^{(3)}-2c_{0})\frac{1}{\delta}+\mathcal{O}\left(1\right),$
$\displaystyle\mathcal{P}_{3}$
$\displaystyle=\frac{1}{\delta}+(e_{1}^{(1)}+e_{1}^{(2)}+e_{1}^{(3)}-3c_{0})+\mathcal{O}\left(\delta\right).$
(56)

From this result and equation (17), one sees that the asymptotic form of the
matrix $g(\mathbf{v},l;\lambda)$ with
$\mathbf{v}=(\mathcal{P}_{3},\mathcal{P}_{2}/\mathcal{P}_{3},\mathcal{P}_{1}/\mathcal{P}_{2})$
and $l=1/\delta^{4}$ is expressed as equation (51), in which $\mathcal{Y}$ is
a matrix of the form (6) with $n=4,p=3$. Hence we obtain the continuous time
Lax equation (7) in the limit $\delta\rightarrow 0$, in the same way as for
the case of $p=1$.

###  4.3 Limit for the type II Lax equation

As in the previous subsection, we assume that $n=4$ but now let $L(\geq 2)$ be
an arbitrary integer. We define

$U_{\lambda}=\begin{pmatrix}&&&\lambda\\\ 1&&&\\\ &1&&\\\ &&1&\end{pmatrix}.$
(57)

Consider the matrix (41) with $s=1$ and rewrite equation (44) as

$\mathcal{L}(T^{(1)}_{l}|\mathbf{b}\rangle;\lambda)=(U_{\lambda}^{-1}g(\mathbf{v},l;\lambda))^{-1}\cdot(U_{\lambda}^{-1}\mathcal{L}(|\mathbf{b}\rangle;\lambda)U_{\lambda})\cdot(U_{\lambda}^{-1}g(\mathbf{v},l;\lambda)).$
(58)

Accordingly, we adopt the interpretation

$\mathcal{L}(t)=U_{\lambda}^{-1}\mathcal{L}(|\mathbf{b}\rangle;\lambda)U_{\lambda},\qquad\mathcal{L}(t+d\delta)=\mathcal{L}(T^{(1)}_{l}|\mathbf{b}\rangle;\lambda),$
(59)

where $d$ is a parameter that will be determined later. Let them to be
identified with the matrix $\mathcal{L}$ given by (4), but in which the set of
variables for the loop elementary symmetric functions (2) should be
interpreted as

$u_{i}^{(\alpha-1)}(t)=b_{i}^{(\alpha)}\quad\mbox{and}\quad
u_{i}^{(\alpha)}(t+d\delta)=(b_{i}^{\prime})^{(\alpha)},$ (60)

respectively. Here we used the notations defined by
$T^{(1)}_{l}|\mathbf{b}\rangle=(\mathbf{b}^{\prime}_{1},\dots,\mathbf{b}^{\prime}_{L})$
and
$\mathbf{b}^{\prime}_{i}=((b_{i}^{\prime})^{(1)},(b_{i}^{\prime})^{(2)},(b_{i}^{\prime})^{(3)})$.
The first equation of (60) implies that in the matrix
$\mathcal{L}(|\mathbf{b}\rangle;\lambda)$ we must replace $e^{(\alpha)}_{m}$
by $e^{(\alpha-1)}_{m}$ when we change its variables from the $b$’s to the
$u$’s. After this replacement in (4) and with the condition
$e_{L}^{(\alpha)}=1$ for all $\alpha$, the asymptotic form of the Lax matrix
$\mathcal{L}(|\mathbf{b}\rangle;l)$ under the limit $l\rightarrow 0$ is given
by

$\mathcal{L}(|\mathbf{b}\rangle;l)\approx\begin{pmatrix}1&e_{L-3}^{(4)}l&e_{L-2}^{(4)}l&e_{L-1}^{(4)}l\\\
e_{L-1}^{(1)}&1&e_{L-3}^{(1)}l&e_{L-2}^{(1)}l\\\
e_{L-2}^{(2)}&e_{L-1}^{(2)}&1&e_{L-3}^{(2)}l\\\
e_{L-3}^{(3)}&e_{L-2}^{(3)}&e_{L-1}^{(3)}&1\end{pmatrix}.$ (61)

Note that the condition $e_{L}^{(\alpha)}=1$ is preserved under the time
evolution $T^{(1)}_{l}$, which can be obtained from equation (40) with
$\lambda=0$. We assume that the eigenvalues
$\xi_{q}\,(q\in{\mathbb{Z}}/4{\mathbb{Z}})$ of the Lax matrix
$\mathcal{L}(|\mathbf{b}\rangle;l)$ under the condition $s=1$ and for
sufficiently small $l$’s are given by the Puiseux series expansion

$\xi_{q}=\sum_{m=0}^{\infty}d_{m}\exp\left(\frac{\pi\sqrt{-1}mq}{2}\right)l^{m/4},$
(62)

where $d_{0}=1$. This assumption is consistent with the relations

$\displaystyle(1-l)^{L}=\det\mathcal{L}(|\mathbf{b}\rangle;l)$
$\displaystyle=\prod_{q=1}^{4}\xi_{q}=(d_{0})^{4}+\mathcal{O}(l),$
$\displaystyle
4+\mathcal{O}(l)=\mathrm{Tr}\,\mathcal{L}(|\mathbf{b}\rangle;l)$
$\displaystyle=\sum_{q=1}^{4}\xi_{q}=4d_{0}+\mathcal{O}(l),$

where we used the identity $\det g(\mathbf{b}_{i},1;l)=1-l$ for $1\leq i\leq
L$, and the expression (61). Then, the asymptotic form of the largest
eigenvalue of the monodromy matrix $\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$
is given by

$E=\xi_{1}\xi_{3}\xi_{4}=1+d_{1}l^{\frac{1}{4}}+\cdots.$ (63)

From the expression (61), we see that the asymptotic form of the monodromy
matrix $\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ under the limit
$l\rightarrow 0$ is given by

$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)\approx\begin{pmatrix}1&\mathcal{O}(l)&\mathcal{O}(l)&e_{L-1}^{(4)}l\\\
e_{L-1}^{(3)}&1&\mathcal{O}(l)&\mathcal{O}(l)\\\
\mathcal{O}(1)&e_{L-1}^{(2)}&1&\mathcal{O}(l)\\\
\mathcal{O}(1)&\mathcal{O}(1)&e_{L-1}^{(1)}&1\end{pmatrix}.$ (64)

From (63) and (64) one sees that

$0=\det(\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)-E\mathbb{I}_{4})=((d_{1})^{4}-e_{L-1}^{(1)}e_{L-1}^{(2)}e_{L-1}^{(3)}e_{L-1}^{(4)})l+\mathcal{O}(l^{\frac{5}{4}}),$

which forces us to set $d_{1}=(\prod_{\alpha=1}^{4}e_{L-1}^{(\alpha)})^{1/4}$.
Let $l=\delta^{4}$. Again by using the solution of equation (49) expressed by
the Cramer formula and then now substituting the asymptotic forms (63) and
(64) into it, we obtain the expressions

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=\frac{1}{d_{1}}e_{L-1}^{(4)}\delta^{3}+\mathcal{O}\left(\delta^{4}\right),$
$\displaystyle\mathcal{P}_{2}$
$\displaystyle=\frac{1}{(d_{1})^{2}}e_{L-1}^{(4)}e_{L-1}^{(3)}\delta^{2}+\mathcal{O}\left(\delta^{3}\right),$
$\displaystyle\mathcal{P}_{3}$
$\displaystyle=\frac{1}{(d_{1})^{3}}e_{L-1}^{(4)}e_{L-1}^{(3)}e_{L-1}^{(2)}\delta+\mathcal{O}\left(\delta^{2}\right)=\frac{d_{1}}{e_{L-1}^{(1)}}\delta+\mathcal{O}\left(\delta^{2}\right).$
(65)

We show a detailed derivation of this result in A.2. Therefore, the asymptotic
form of the matrix $U_{\lambda}^{-1}g(\mathbf{v},l;\lambda)$ with
$\mathbf{v}=(\mathcal{P}_{3},\mathcal{P}_{2}/\mathcal{P}_{3},\mathcal{P}_{1}/\mathcal{P}_{2})$
and $l=\delta^{4}$ is expressed as

$U_{\lambda}^{-1}g(\mathbf{v},\delta^{4};\lambda)=\begin{pmatrix}1&\mathcal{P}_{2}/\mathcal{P}_{3}&&\\\
&1&\mathcal{P}_{1}/\mathcal{P}_{2}&\\\ &&1&\delta^{4}/\mathcal{P}_{1}\\\
\mathcal{P}_{3}/\lambda&&&1\end{pmatrix}=\mathbb{I}_{4}+d_{1}\delta\cdot\mathcal{Z}+\mathcal{O}\left(\delta^{2}\right),$
(66)

where $\mathcal{Z}$ is a matrix of the form (25) with $n=4$.

To summarize, we state the result of the above arguments in the case of
$n=4,s=1$ as:

###### Theorem 14

Let $U_{\lambda}^{-1}\mathcal{L}(|\mathbf{b}\rangle;\lambda)U_{\lambda}$ and
$\mathcal{L}(T^{(1)}_{l}|\mathbf{b}\rangle;\lambda)$ be denoted by
$\mathcal{L}(t)$ and $\mathcal{L}(t+d\delta)$, respectively, where
$d=d_{1}=(\prod_{\alpha=1}^{4}e_{L-1}^{(\alpha)})^{1/4}$ and $l=\delta^{4}$.
Assume that the condition $e_{L}^{(\alpha)}=1$ is satisfied for all $\alpha$,
and that the eigenvalues of the matrix $\mathcal{L}(|\mathbf{b}\rangle;l)$ for
sufficiently small $l$’s are given the expansion (62). Then, the discrete time
Lax equation (44) reduces to the continuous time Lax equation (27) in the
limit $\delta\rightarrow 0$.

Proof. By using the expression (66), we obtain

$\mathcal{L}(t+d\delta)=(U_{\lambda}^{-1}g(\mathbf{v},\delta^{4};\lambda))^{-1}\mathcal{L}(t)(U_{\lambda}^{-1}g(\mathbf{v},\delta^{4};\lambda))=\mathcal{L}(t)+d\delta\cdot[\mathcal{L}(t),\mathcal{Z}]+\mathcal{O}\left(\delta^{2}\right),$

from equation (58), which was equivalent to equation (44). Hence the assertion
of the theorem follows. $\Box$

###  4.4 Limits for the type I and II differential equations

The discussions for deriving the Lax equations in the previous two subsections
can be generalized to those for derivations of equations (1) and (24).
Consider the following matrix equation

$g(\mathbf{b},s;\lambda)g(\mathbf{a},l;\lambda)=g(\mathbf{a}^{\prime},l;\lambda)g(\mathbf{b}^{\prime},s;\lambda).$
(67)

For any $s,l\in{\mathbb{R}}_{>0}$ and
$(\mathbf{a},\mathbf{b})\in({\mathbb{R}}_{>0})^{6}$, there is a unique
solution $(\mathbf{a}^{\prime},\mathbf{b}^{\prime})\in({\mathbb{R}}_{>0})^{6}$
to this matrix equation. Let
$R^{(s,l)}:({\mathbb{R}}_{>0})^{6}\rightarrow({\mathbb{R}}_{>0})^{6}$ be a
rational map given by
$R^{(s,l)}:(\mathbf{b},\mathbf{a})\mapsto(\mathbf{a}^{\prime},\mathbf{b}^{\prime})$.
This is the geometric $R$-matrix in the present case. An explicit expression
for the rational map is written as

$\displaystyle a^{\prime(1)}$
$\displaystyle=a^{(1)}\frac{a^{(2)}a^{(3)}a^{(4)}+a^{(2)}a^{(3)}b^{(1)}+a^{(2)}b^{(4)}b^{(1)}+b^{(3)}b^{(4)}b^{(1)}}{a^{(1)}a^{(2)}a^{(3)}+a^{(1)}a^{(2)}b^{(4)}+a^{(1)}b^{(3)}b^{(4)}+b^{(2)}b^{(3)}b^{(4)}},$
$\displaystyle a^{\prime(2)}$
$\displaystyle=a^{(2)}\frac{a^{(3)}a^{(4)}a^{(1)}+a^{(3)}a^{(4)}b^{(2)}+a^{(3)}b^{(1)}b^{(2)}+b^{(4)}b^{(1)}b^{(2)}}{a^{(2)}a^{(3)}a^{(4)}+a^{(2)}a^{(3)}b^{(1)}+a^{(2)}b^{(4)}b^{(1)}+b^{(3)}b^{(4)}b^{(1)}},$
$\displaystyle a^{\prime(3)}$
$\displaystyle=a^{(3)}\frac{a^{(4)}a^{(1)}a^{(2)}+a^{(4)}a^{(1)}b^{(3)}+a^{(4)}b^{(2)}b^{(3)}+b^{(1)}b^{(2)}b^{(3)}}{a^{(3)}a^{(4)}a^{(1)}+a^{(3)}a^{(4)}b^{(2)}+a^{(3)}b^{(1)}b^{(2)}+b^{(4)}b^{(1)}b^{(2)}},$

and $b^{\prime(i)}=a^{(i)}b^{(i)}/a^{\prime(i)}$ [29]. Let the map
$R^{(s,l)}:(\mathbf{b},\mathbf{a})\mapsto(\mathbf{a}^{\prime},\mathbf{b}^{\prime})$
be depicted as

$\begin{picture}(40.0,40.0)(-20.0,-20.0)\put(-10.0,0.0){\line(1,0){20.0}}
\put(0.0,10.0){\line(0,-1){20.0}}
\put(-11.0,0.0){\makebox(0.0,0.0)[r]{$\mathbf{a}^{\prime}$}}
\put(0.0,11.0){\makebox(0.0,0.0)[b]{$\mathbf{b}$}}
\put(0.0,-11.0){\makebox(0.0,0.0)[t]{$\mathbf{b}^{\prime}$}}
\put(11.0,0.0){\makebox(0.0,0.0)[l]{$\mathbf{a}$}} \end{picture}.$

Then the solution of equation (40) satisfies the relations in the diagram

(68)

where $\mathbf{v}_{i}$’s are defined by the downward recursion relation
$R^{(s,l)}(\mathbf{b}_{i},\mathbf{v}_{i+1})=(\mathbf{v}_{i},\mathbf{b}_{i}^{\prime})$
with the initial condition $\mathbf{v}_{L+1}=\mathbf{v}$, where $\mathbf{v}$
is determined by the Perron-Frobenius eigenvector of the monodromy matrix
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ as described in §4.1. As in §2.2
let $\sigma$ denote the cyclic shift to the left, so we have
$\sigma|\mathbf{b}\rangle=(\mathbf{b}_{2},\dots,\mathbf{b}_{L},\mathbf{b}_{1})$.
It is easy to see that an obvious generalization of equation (44) is

$\mathcal{L}(T^{(1)}_{l}(\sigma^{i-1}|\mathbf{b}\rangle);\lambda)=g(\mathbf{v}_{i},l;\lambda)^{-1}\mathcal{L}(\sigma^{i-1}|\mathbf{b}\rangle;\lambda)g(\mathbf{v}_{i},l;\lambda).$
(69)

This equation implies that $\mathbf{v}_{i}$ can also be obtained in the same
way as for $\mathbf{v}$ in §4.1, by simply replacing
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ by
$\mathsf{M}_{l}^{(1)}(\sigma^{i-1}|\mathbf{b}\rangle)$. Based on this fact,
one can generalize equation (51) as

$\delta\cdot
g(\mathbf{v}_{i},1/\delta^{4};\lambda)=\mathbb{I}_{4}+\delta\cdot\mathcal{Y}_{i}+\mathcal{O}\left(\delta^{2}\right),$
(70)

where $\mathcal{Y}_{i}$ is a matrix of the form (10) with $n=4,p=1$, and
equation (66) as

$U_{\lambda}^{-1}g(\mathbf{v}_{i},\delta^{4};\lambda)=\mathbb{I}_{4}+d_{1}\delta\cdot\mathcal{Z}_{i}+\mathcal{O}\left(\delta^{2}\right),$
(71)

where $\mathcal{Z}_{i}$ is a matrix of the form (29) with $n=4$.

First we consider type I case, where $l=1/\delta^{4}$. Around each vertex in
the diagram (68) we have the relation

$g(\mathbf{b}_{i}^{\prime},s;\lambda)=(\delta\cdot
g(\mathbf{v}_{i},l;\lambda))^{-1}g(\mathbf{b}_{i},s;\lambda)(\delta\cdot
g(\mathbf{v}_{i+1},l;\lambda)).$

Then by setting

$\mathcal{M}_{i}(t+\delta)=g(\mathbf{b}_{i}^{\prime},s;\lambda),\qquad\mathcal{M}_{i}(t)=g(\mathbf{b}_{i},s;\lambda),$

and using (70), we can derive equation (11) in the limit $\delta\rightarrow
0$.

Second we consider type II case, where $l=\delta^{4}$, $s=1$, and with the
condition $e_{L}^{(\alpha)}=1$ for all $\alpha$. As in the previous case, we
have the relation

$g(\mathbf{b}_{i}^{\prime},1;\lambda)=(U_{\lambda}^{-1}g(\mathbf{v}_{i},l;\lambda))^{-1}(U_{\lambda}^{-1}g(\mathbf{b}_{i},1;\lambda)U_{\lambda})(U_{\lambda}^{-1}g(\mathbf{v}_{i+1},l;\lambda)).$

Then by setting

$\mathcal{M}_{i}(t+d_{1}\delta)=g(\mathbf{b}_{i}^{\prime},1;\lambda),\qquad\mathcal{M}_{i}(t)=U_{\lambda}^{-1}g(\mathbf{b}_{i},1;\lambda)U_{\lambda},$

and using (71), we can derive equation (30) in the limit $\delta\rightarrow
0$.

Finally, we conclude that equations (1) for $p=1$ and (24) are derived from
the closed geometric crystal chains by the above arguments and using
Propositions 2 and 8, respectively.

##  5 Concluding remarks

###  5.1 On a related discrete dynamical system

In order to avoid a potential confusion, we present a discussion to illustrate
the difference between the closed geometric crystal chain and another discrete
dynamical system in reference [12], which is also defined by using the
geometric $R$ matrix.

First we consider the closed geometric crystal chain. Let
$R^{(s,l)}:({\mathbb{R}}_{>0})^{2(n-1)}\rightarrow({\mathbb{R}}_{>0})^{2(n-1)}$
be a rational map that is essentially the same one in §4.4 but the condition
$n=4$ has been generalized to for an arbitrary positive integer $n\geq 2$. Let
$R_{i}^{(s,l)}$ be a map from
$({\mathbb{R}}_{>0})^{(L+1)(n-1)}=({\mathbb{R}}_{>0})^{n-1}\times\dots\times({\mathbb{R}}_{>0})^{n-1}$
to itself, which acts as the map $R^{(s,l)}$ on factors $i$ and $i+1$, and as
the identity on the other factors. We define
$\mathcal{R}^{(s,l)}=R_{1}^{(s,l)}\circ\cdots\circ R_{L}^{(s,l)}$, which are
maps from $({\mathbb{R}}_{>0})^{(L+1)(n-1)}$ to itself. Then the diagram (68)
for general $n$ implies that we have the relation
$\mathcal{R}^{(s,l)}(\mathbf{b}_{1},\dots,\mathbf{b}_{L},\mathbf{v})=(\mathbf{v},\mathbf{b}^{\prime}_{1},\dots,\mathbf{b}^{\prime}_{L})$,
which enables us to define the time evolution
$T^{(1)}_{l}:({\mathbb{R}}_{>0})^{(n-1)L}\rightarrow({\mathbb{R}}_{>0})^{(n-1)L}$
and to obtain its Lax representation, in the same way as in (43) and (44),
respectively. The above $\mathbf{v}$ is determined by the Perron-Frobenius
eigenvector of the monodromy matrix $\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$
as in the case of $n=4$ in §4.1. The commutativity of the time evolutions
$T^{(1)}_{l_{1}}\circ T^{(1)}_{l_{2}}=T^{(1)}_{l_{2}}\circ T^{(1)}_{l_{1}}$ is
satisfied for any $l_{1},l_{2}\in{\mathbb{R}}_{>0}$, as a consequence of the
fact that the maps $R_{i}^{(s,l)}$ are obeying the Yang-Baxter relation
$R_{i+1}^{(s,l_{2})}R_{i}^{(s,l_{1})}R_{i+1}^{(l_{2},l_{1})}=R_{i}^{(l_{2},l_{1})}R_{i+1}^{(s,l_{1})}R_{i}^{(s,l_{2})}$
and the involution $R_{L+1}^{(l_{2},l_{1})}\circ R_{L+1}^{(l_{1},l_{2})}={\rm
Id}$. The Lax equation (44) implies that the characteristic polynomial of the
matrix $\mathcal{L}(|\mathbf{b}\rangle;\lambda)$ in (41) is invariant under
the actions of the time evolutions $T^{(1)}_{l}$s for any
$l\in{\mathbb{R}}_{>0}$, hence its coefficients are the conserved quantities
of the closed geometric crystal chain.

Second we consider the discrete dynamical system in [12], but slightly
modified to be consistent with the notations used in this paper. In this case,
we choose the values of the parameter $s$ attached to each site unequally. So
let $|s\rangle=(s_{1},\dots,s_{L})$ and define
$\mathcal{L}(|\mathbf{b}\rangle,|s\rangle;\lambda)$ to be the matrix

$\mathcal{L}(|\mathbf{b}\rangle,|s\rangle;\lambda)=g(\mathbf{b}_{1},s_{1};\lambda)\cdots
g(\mathbf{b}_{L},s_{L};\lambda),$ (72)

which is called a Lax matrix. As in the previous paragraph, let
$R_{i}^{(s_{i},s_{i+1})}$ be a map from
$({\mathbb{R}}_{>0})^{(L+1)(n-1)}=({\mathbb{R}}_{>0})^{n-1}\times\dots\times({\mathbb{R}}_{>0})^{n-1}$
to itself, which acts as the map $R^{(s_{i},s_{i+1})}$ on factors $i$ and
$i+1$, and as the identity on the other factors. More precisely, the map
$R_{i}^{(s_{i},s_{i+1})}$ acts on the set $(\mathbb{Y}_{1})^{L}$. Applied to
$(|\mathbf{b}\rangle,|s\rangle)\in(\mathbb{Y}_{1})^{L}$, it sends
$|\mathbf{b}\rangle=(\mathbf{b}_{1},\dots,\mathbf{b}_{i},\mathbf{b}_{i+1},\dots,\mathbf{b}_{L})$
to
$(\mathbf{b}_{1},\dots,\mathbf{b}^{\prime}_{i+1},\mathbf{b}^{\prime}_{i},\dots,\mathbf{b}_{L})$
where
$(\mathbf{b}^{\prime}_{i+1},\mathbf{b}^{\prime}_{i})=R^{(s_{i},s_{i+1})}(\mathbf{b}_{i},\mathbf{b}_{i+1})$,
and $|s\rangle=(s_{1},\dots,s_{i},s_{i+1},\dots,s_{L})$ to
$(s_{1},\dots,s_{i+1},s_{i},\dots,s_{L})$. Let $Y_{i}\,(1\leq i\leq L)$ be the
map defined by

$Y_{i}=R_{i}^{(s_{i+1},s_{i})}\circ\cdots\circ
R_{L-1}^{(s_{L},s_{i})}\circ\sigma\circ R_{1}^{(s_{1},s_{i})}\circ\cdots\circ
R_{i-1}^{(s_{i-1},s_{i})},$

where $\sigma$ denotes the cyclic shift to the left. If viewed as a map on the
set $(\mathbb{Y}_{1})^{L}$, it does not change
$|s\rangle=(s_{1},\dots,s_{L})$. So it can be simply regarded as a map on the
set $({\mathbb{R}}_{>0})^{(L+1)(n-1)}$ and the relation
$Y_{i}|\mathbf{b}\rangle=|\mathbf{b}^{\prime}\rangle$ can be described by the
following diagram

The commutativity of the rational maps $Y_{i}\circ Y_{j}=Y_{j}\circ Y_{i}$ is
satisfied for any $1\leq i,j\leq L$. It is proved by using the property of the
maps $R_{i}^{(s,l)}$ obeying the Yang-Baxter relation, and it also reflects
the fact that the $Y_{i}$s for $1\leq i\leq L-1$ can be interpreted as
generators of the translation subgroup of the extended affine Weyl group
$\widetilde{W}(A_{L-1}^{(1)})$ [12]. The commutativity of this type of
_transfer maps_ defined as a composition of general Yang-Baxter maps was
discussed in [28], where its original idea was attributed to a result in the
classical paper [30] by C. N. Yang. Following this historical background, we
shall call the discrete dynamical system described by this set of rational
maps $Y_{i}$ a _birational Yang’s system_ , because its tropical counterpart
was referred to as a combinatorial Yang’s system [16] by the same reason.

In order to see a Lax representation of this system, we consider the map
$Y_{L}$. Let
$Y_{L}|\mathbf{b}\rangle=|\tilde{\mathbf{b}}\rangle=(\tilde{\mathbf{b}}_{1},\dots,\tilde{\mathbf{b}}_{L})$.
Then we have

$g(\mathbf{b}_{1},s_{1};\lambda)\cdots
g(\mathbf{b}_{L},s_{L};\lambda)=g(\tilde{\mathbf{b}}_{L},s_{L};\lambda)g(\tilde{\mathbf{b}}_{1},s_{1};\lambda)\cdots
g(\tilde{\mathbf{b}}_{L-1},s_{L-1};\lambda).$ (73)

Comparing this with equation (72), we obtain the relation

$\mathcal{L}(Y_{L}|\mathbf{b}\rangle,|s\rangle;\lambda)=g(\tilde{\mathbf{b}}_{L},s_{L};\lambda)^{-1}\mathcal{L}(|\mathbf{b}\rangle,|s\rangle;\lambda)g(\tilde{\mathbf{b}}_{L},s_{L};\lambda),$
(74)

which can be regarded as a Lax equation. By applying the cyclic shift operator
$\sigma$ repeatedly, one can obtain Lax representations for all the other time
evolutions $Y_{i}$. Therefore, the characteristic polynomial of the matrix
$\mathcal{L}(|\mathbf{b}\rangle,|s\rangle;\lambda)$ is invariant under the
actions of the time evolutions $Y_{i}$s for any $1\leq i\leq L$, because it is
common to that of
$\mathcal{L}(\sigma^{i}|\mathbf{b}\rangle,\sigma^{i}|s\rangle;\lambda)$ for
all $i$’s. Hence its coefficients are the conserved quantities of the
birational Yang’s system.

For the closed geometric crystal chain, the Lax matrix
$\mathcal{L}(|\mathbf{b}\rangle;\lambda)$ is independent of the parameter $l$,
which is in the companion matrix $g(\mathbf{v},l;\lambda)$ of the Lax pair in
equation (44). Therefore, without changing the Lax matrix, one can consider
the limit for taking $l\rightarrow\infty$ and that for $l\rightarrow 0$ in
equation (44) to obtain the continuous time Lax equations. As a result, we
were able to obtain integrable differential equations that share a Lax matrix
in common with a discrete dynamical system, at least for the cases treated in
§4. In contrast, for the birational Yang’s system, the Lax matrix
$\mathcal{L}(|\mathbf{b}\rangle,|s\rangle;\lambda)$ depends on the same
parameter $s_{L}$ in common with the companion matrix
$g(\tilde{\mathbf{b}}_{L},s_{L};\lambda)$ of the Lax pair in equation (74).
Therefore, one can not apply the same method to take the continuum limits.
However, it seems reasonable to expect that a generalization of the symmetric
form of the Painlevé equations $P_{\rm IV}$ and $P_{\rm V}$ in [22], which is
also regarded as a variation of extended Lotka-Volterra systems, can be
obtained from this discrete dynamical system by taking a continuum limit
through a similar method.

###  5.2 An outlook for generalizations

In §4, we restricted ourselves to the case of $n=4$ for deriving the
differential equations. It seems that the above derivations can also be
applied to the case of general $n$, but a remark on the largest eigenvalue of
the monodromy matrix may be in order. In this generalization, the matrices
corresponding to (38) and (39) are connected by the relation where elements of
the matrix $g^{*}(\mathbf{x},s;\lambda)$ are so defined as to be order $n-1$
minors of the matrix $g(\mathbf{x},s;(-1)^{n}\lambda)$ (See §3.1.1 of
reference [26]). Consider the type I case with $L=n\kappa+1$. As in equation
(46), suppose that the eigenvalues
$\eta_{q}\,(q\in{\mathbb{Z}}/n{\mathbb{Z}})$ of the Lax matrix
$\mathcal{L}(|\mathbf{b}\rangle;l)$ for sufficiently large $l$’s are given by
the Puiseux series expansion

$\eta_{q}=l^{\kappa}\sum_{m=-1}^{\infty}c_{m}\exp\left(\frac{2\pi\sqrt{-1}mq}{n}\right)l^{-m/n},$

where $c_{-1}=1$ and $c_{0}=(\sum_{\alpha=1}^{n}e^{(\alpha)}_{1})/n$. This
implies that the eigenvalues
$\eta_{q}^{\prime}\,(q\in{\mathbb{Z}}/n{\mathbb{Z}})$ of the matrix
$\mathcal{L}(|\mathbf{b}\rangle;-l)$ are given by

$\eta_{q}^{\prime}=(-l)^{\kappa}\sum_{m=-1}^{\infty}c_{m}\exp\left(\frac{\pi\sqrt{-1}m(2q-1)}{n}\right)l^{-m/n}.$

By the above relation between $g^{*}(\mathbf{x},s;\lambda)$ and
$g(\mathbf{x},s;(-1)^{n}\lambda)$, we see that the largest eigenvalue of the
monodromy matrix $\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$, which is defined
similarly by equation (42) for general $n$, is given by
$E=(\prod_{q=1}^{n}\eta_{q})/\eta_{n/2}$ for the case of even $n$, or by
$E=(\prod_{q=1}^{n}\eta_{q}^{\prime})/\eta_{(n+1)/2}^{\prime}$ for the case of
odd $n$. In both cases, the asymptotic form of $E$ is given by

$E=l^{(n-1)\kappa+\frac{n-1}{n}}+c_{0}l^{(n-1)\kappa+\frac{n-2}{n}}+\cdots.$

To summarize, depending on whether $n$ is even or odd, we have to pay an
attention to the choice of the sign of the loop parameter in the matrix
$\mathcal{L}(|\mathbf{b}\rangle;(-1)^{n}l)$ for obtaining the largest
eigenvalue of the monodromy matrix $\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$.

It also seems that the derivation is not restricted to the totally one-row
tableaux case but can be generalized to the rectangular tableaux cases in §3.2
of [26], which uses the description of the geometric R-matrices in reference
[5]. In this case, to define another time evolution $T_{l}^{(k)}$ we use the
monodromy matrix $\mathsf{M}_{l}^{(k)}(|\mathbf{b}\rangle)$ for a k-row
tableau “carrier”, and the entries of this matrix are order $n-k$ minors of
the Lax matrix $\mathcal{L}(|\mathbf{b}\rangle;(-1)^{n-k-1}l)$ with such a
prescribed sign of the loop parameter (§3.2.5 of [26]). Therefore, an
additional attention to the choice of the sign of the loop parameter must be
payed for considering its largest eigenvalue $E=E_{l}^{(k)}$. For instance, in
the case of $k=2$, the largest eigenvalue of the monodromy matrix
$\mathsf{M}_{l}^{(2)}(|\mathbf{b}\rangle)$ is given by
$E=(\prod_{q=1}^{n}\eta^{\prime}_{q})/(\eta^{\prime}_{n/2}\eta^{\prime}_{n/2+1})$
for the case of even $n$, or by
$E=(\prod_{q=1}^{n}\eta_{q})/(\eta_{(n-1)/2}\eta_{(n+1)/2})$ for the case of
odd $n$.

Anyway, we hope that we can report a result for such cases in the near future.

##  Appendix A Derivations of the asymptotic forms of the largest eigenvector
of the monodromy matrix

###  A.1 Type I case

Here we show a derivation for the case of $p=1$. (The other case for $p=3$ is
analogous.) Using the asymptotic expressions (48) for the monodromy matrix
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ and (47) for its largest eigenvalue
$E$, we obtain

$\displaystyle\det\begin{pmatrix}\mathsf{M}_{11}-E&\mathsf{M}_{12}&\mathsf{M}_{13}\\\
\mathsf{M}_{21}&\mathsf{M}_{22}-E&\mathsf{M}_{23}\\\
\mathsf{M}_{31}&\mathsf{M}_{32}&\mathsf{M}_{33}-E\end{pmatrix}$
$\displaystyle\approx(\mathsf{M}_{11}-E)(\mathsf{M}_{22}-E)(\mathsf{M}_{33}-E)+\mathsf{M}_{12}\mathsf{M}_{23}\mathsf{M}_{31}$
$\displaystyle=-l^{9\kappa+\frac{9}{4}}-(3c_{0}-e^{(1)}_{1})l^{9\kappa+\frac{8}{4}}+\dots.$

In the same way, we have

$\displaystyle\det\begin{pmatrix}\mathsf{M}_{11}-E&\mathsf{M}_{12}&-\mathsf{M}_{14}\\\
\mathsf{M}_{21}&\mathsf{M}_{22}-E&-\mathsf{M}_{24}\\\
\mathsf{M}_{31}&\mathsf{M}_{32}&-\mathsf{M}_{34}\end{pmatrix}$
$\displaystyle\approx(\mathsf{M}_{11}-E)(\mathsf{M}_{22}-E)(-\mathsf{M}_{34})$
$\displaystyle=-l^{9\kappa+\frac{10}{4}}-2c_{0}l^{9\kappa+\frac{9}{4}}+\dots,$
$\displaystyle\det\begin{pmatrix}\mathsf{M}_{11}-E&-\mathsf{M}_{14}&\mathsf{M}_{13}\\\
\mathsf{M}_{21}&-\mathsf{M}_{24}&\mathsf{M}_{23}\\\
\mathsf{M}_{31}&-\mathsf{M}_{34}&\mathsf{M}_{33}-E\end{pmatrix}$
$\displaystyle\approx(\mathsf{M}_{11}-E)(-\mathsf{M}_{24})(\mathsf{M}_{33}-E)+(\mathsf{M}_{11}-E)\mathsf{M}_{23}\mathsf{M}_{34}$
$\displaystyle=-l^{9\kappa+\frac{11}{4}}-(c_{0}+e^{(2)}_{1})l^{9\kappa+\frac{10}{4}}+\dots,$
$\displaystyle\det\begin{pmatrix}-\mathsf{M}_{14}&\mathsf{M}_{12}&\mathsf{M}_{13}\\\
-\mathsf{M}_{24}&\mathsf{M}_{22}-E&\mathsf{M}_{23}\\\
-\mathsf{M}_{34}&\mathsf{M}_{32}&\mathsf{M}_{33}-E\end{pmatrix}$
$\displaystyle\approx\mathsf{M}_{12}(\mathsf{M}_{24}(\mathsf{M}_{33}-E)-\mathsf{M}_{23}\mathsf{M}_{34})+\mathsf{M}_{13}(\mathsf{M}_{22}-E)\mathsf{M}_{34}$
$\displaystyle=-l^{9\kappa+\frac{12}{4}}-(e^{(2)}_{1}+e^{(3)}_{1})l^{9\kappa+\frac{11}{4}}+\dots.$

Therefore

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=\frac{l^{9\kappa+\frac{12}{4}}+(e^{(2)}_{1}+e^{(3)}_{1})l^{9\kappa+\frac{11}{4}}+\dots}{l^{9\kappa+\frac{9}{4}}+(3c_{0}-e^{(1)}_{1})l^{9\kappa+\frac{8}{4}}+\dots}=l^{\frac{3}{4}}\frac{1+(e^{(2)}_{1}+e^{(3)}_{1})l^{-\frac{1}{4}}+\dots}{1+(3c_{0}-e^{(1)}_{1})l^{-\frac{1}{4}}+\dots}$
$\displaystyle=l^{\frac{3}{4}}+(e_{1}^{(1)}+e_{1}^{(2)}+e_{1}^{(3)}-3c_{0})l^{\frac{2}{4}}+\dots,$
$\displaystyle\mathcal{P}_{2}$
$\displaystyle=\frac{l^{9\kappa+\frac{11}{4}}+(c_{0}+e^{(2)}_{1})l^{9\kappa+\frac{10}{4}}+\dots}{l^{9\kappa+\frac{9}{4}}+(3c_{0}-e^{(1)}_{1})l^{9\kappa+\frac{8}{4}}+\dots}=l^{\frac{2}{4}}\frac{1+(c_{0}+e^{(2)}_{1})l^{-\frac{1}{4}}+\dots}{1+(3c_{0}-e^{(1)}_{1})l^{-\frac{1}{4}}+\dots}$
$\displaystyle=l^{\frac{2}{4}}+(e_{1}^{(1)}+e_{1}^{(2)}-2c_{0})l^{\frac{1}{4}}+\dots,$
$\displaystyle\mathcal{P}_{3}$
$\displaystyle=\frac{l^{9\kappa+\frac{10}{4}}+2c_{0}l^{9\kappa+\frac{9}{4}}+\dots}{l^{9\kappa+\frac{9}{4}}+(3c_{0}-e^{(1)}_{1})l^{9\kappa+\frac{8}{4}}+\dots}=l^{\frac{1}{4}}\frac{1+2c_{0}l^{-\frac{1}{4}}+\dots}{1+(3c_{0}-e^{(1)}_{1})l^{-\frac{1}{4}}+\dots}$
$\displaystyle=l^{\frac{1}{4}}+(e_{1}^{(1)}-c_{0})+\dots.$

Hence we obtained the result (50) in §4.1.

###  A.2 Type II case

Using the asymptotic expressions (64) for the monodromy matrix
$\mathsf{M}_{l}^{(1)}(|\mathbf{b}\rangle)$ and (63) for its largest eigenvalue
$E$, we obtain

$\displaystyle\det\begin{pmatrix}\mathsf{M}_{11}-E&\mathsf{M}_{12}&\mathsf{M}_{13}\\\
\mathsf{M}_{21}&\mathsf{M}_{22}-E&\mathsf{M}_{23}\\\
\mathsf{M}_{31}&\mathsf{M}_{32}&\mathsf{M}_{33}-E\end{pmatrix}$
$\displaystyle\approx(\mathsf{M}_{11}-E)(\mathsf{M}_{22}-E)(\mathsf{M}_{33}-E)$
$\displaystyle=-d_{1}^{3}l^{\frac{3}{4}}+\dots.$

In the same way, we have

$\displaystyle\det\begin{pmatrix}\mathsf{M}_{11}-E&\mathsf{M}_{12}&-\mathsf{M}_{14}\\\
\mathsf{M}_{21}&\mathsf{M}_{22}-E&-\mathsf{M}_{24}\\\
\mathsf{M}_{31}&\mathsf{M}_{32}&-\mathsf{M}_{34}\end{pmatrix}$
$\displaystyle\approx-\mathsf{M}_{14}\mathsf{M}_{21}\mathsf{M}_{32}=-e^{(4)}_{L-1}e^{(3)}_{L-1}e^{(2)}_{L-1}l+\dots,$
$\displaystyle\det\begin{pmatrix}\mathsf{M}_{11}-E&-\mathsf{M}_{14}&\mathsf{M}_{13}\\\
\mathsf{M}_{21}&-\mathsf{M}_{24}&\mathsf{M}_{23}\\\
\mathsf{M}_{31}&-\mathsf{M}_{34}&\mathsf{M}_{33}-E\end{pmatrix}$
$\displaystyle\approx\mathsf{M}_{14}\mathsf{M}_{21}(\mathsf{M}_{33}-E)=-d_{1}e^{(4)}_{L-1}e^{(3)}_{L-1}l^{\frac{5}{4}}+\dots,$
$\displaystyle\det\begin{pmatrix}-\mathsf{M}_{14}&\mathsf{M}_{12}&\mathsf{M}_{13}\\\
-\mathsf{M}_{24}&\mathsf{M}_{22}-E&\mathsf{M}_{23}\\\
-\mathsf{M}_{34}&\mathsf{M}_{32}&\mathsf{M}_{33}-E\end{pmatrix}$
$\displaystyle\approx-\mathsf{M}_{14}(\mathsf{M}_{22}-E)(\mathsf{M}_{33}-E)=-d_{1}^{2}e^{(4)}_{L-1}l^{\frac{6}{4}}+\dots.$

Therefore

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=\frac{d_{1}^{2}e^{(4)}_{L-1}l^{\frac{6}{4}}+\dots}{d_{1}^{3}l^{\frac{3}{4}}+\dots}=\frac{1}{d_{1}}e_{L-1}^{(4)}l^{\frac{3}{4}}+\dots,$
$\displaystyle\mathcal{P}_{2}$
$\displaystyle=\frac{d_{1}e^{(4)}_{L-1}e^{(3)}_{L-1}l^{\frac{5}{4}}+\dots}{d_{1}^{3}l^{\frac{3}{4}}+\dots}=\frac{1}{(d_{1})^{2}}e_{L-1}^{(4)}e_{L-1}^{(3)}l^{\frac{2}{4}}+\dots,$
$\displaystyle\mathcal{P}_{3}$
$\displaystyle=\frac{e^{(4)}_{L-1}e^{(3)}_{L-1}e^{(2)}_{L-1}l+\dots}{d_{1}^{3}l^{\frac{3}{4}}+\dots}=\frac{1}{(d_{1})^{3}}e_{L-1}^{(4)}e_{L-1}^{(3)}e_{L-1}^{(2)}l^{\frac{1}{4}}+\dots.$

Hence we obtained the result (65) in §4.2.

##  Appendix B Explicit derivations of the Puiseux series expansions of the
eigenvalues of the Lax matrix for $\mathbf{n=2}$

First we consider type I case. In the case of $n=2$, using the explicit
expression (4) and the condition $L=2\kappa+1$, we can obtain the asymptotic
form of the Lax matrix $\mathcal{L}(|\mathbf{b}\rangle;l)$ under the limit
$l\rightarrow\infty$ as

$\mathcal{L}(|\mathbf{b}\rangle;l)=\begin{pmatrix}\mathcal{L}_{11}&\mathcal{L}_{12}\\\
\mathcal{L}_{21}&\mathcal{L}_{22}\end{pmatrix}\approx\begin{pmatrix}e_{1}^{(1)}l^{\kappa}&l^{\kappa+1}\\\
l^{\kappa}&e_{1}^{(2)}l^{\kappa}\end{pmatrix}.$

The eigenvalues of the matrix $\mathcal{L}(|\mathbf{b}\rangle;l)$ are
explicitly written as

$\eta_{\mp}=\frac{\mathcal{L}_{11}+\mathcal{L}_{22}\mp\sqrt{(\mathcal{L}_{11}-\mathcal{L}_{22})^{2}+4\mathcal{L}_{12}\mathcal{L}_{21}}}{2}.$

Since
$(\mathcal{L}_{11}-\mathcal{L}_{22})^{2}+4\mathcal{L}_{12}\mathcal{L}_{21}=4l^{2\kappa+1}+\mathcal{O}(l^{2\kappa})$,
one has the expansion of the form

$\sqrt{(\mathcal{L}_{11}-\mathcal{L}_{22})^{2}+4\mathcal{L}_{12}\mathcal{L}_{21}}=2l^{\kappa+\frac{1}{2}}\left(1+\sum_{m=1}^{\infty}c_{2m-1}l^{-m}\right).$

Also we have
$\mathcal{L}_{11}+\mathcal{L}_{22}=(e_{1}^{(1)}+e_{1}^{(2)})l^{\kappa}+\mathcal{O}(l^{\kappa-1})$,
hence the expansion of the form

$\mathcal{L}_{11}+\mathcal{L}_{22}=2l^{\kappa}\left(\sum_{m=0}^{\infty}c_{2m}l^{-m}\right),$

where $c_{2m}=0$ for sufficiently large $m$’s. Therefore, we have the Puiseux
series expansion of the eigenvalues $\eta_{1}=\eta_{-}$ and
$\eta_{2}=\eta_{+}$ for sufficiently large $l$ as

$\eta_{q}=l^{\kappa}\sum_{m=-1}^{\infty}c_{m}\exp\left(\pi\sqrt{-1}mq\right)l^{-m/2},$

where $c_{-1}=1$ and $c_{0}=(e_{1}^{(1)}+e_{1}^{(2)})/2$.

Second we consider type II case. In the case of $n=2$, using the explicit
expression (4) and with the condition $e_{L}^{(\alpha)}=1$, we can obtain the
asymptotic form of the Lax matrix $\mathcal{L}(|\mathbf{b}\rangle;l)$ under
the limit $l\rightarrow 0$ as

$\mathcal{L}(|\mathbf{b}\rangle;l)=\begin{pmatrix}\mathcal{L}_{11}&\mathcal{L}_{12}\\\
\mathcal{L}_{21}&\mathcal{L}_{22}\end{pmatrix}\approx\begin{pmatrix}1&e_{L-1}^{(2)}l\\\
e_{L-1}^{(1)}&1\end{pmatrix}.$

Since
$(\mathcal{L}_{11}-\mathcal{L}_{22})^{2}+4\mathcal{L}_{12}\mathcal{L}_{21}=4le_{L-1}^{(1)}e_{L-1}^{(2)}+\mathcal{O}(l^{2})$,
one has the expansion of the form

$\sqrt{(\mathcal{L}_{11}-\mathcal{L}_{22})^{2}+4\mathcal{L}_{12}\mathcal{L}_{21}}=2l^{\frac{1}{2}}\left(\sqrt{e_{L-1}^{(1)}e_{L-1}^{(2)}}+\sum_{m=1}^{\infty}d_{2m-1}l^{m}\right).$

Also we have $\mathcal{L}_{11}+\mathcal{L}_{22}=2+\mathcal{O}(l)$, hence the
expansion of the form

$\mathcal{L}_{11}+\mathcal{L}_{22}=2\left(\sum_{m=0}^{\infty}d_{2m}l^{m}\right),$

where $d_{2m}=0$ for sufficiently large $m$’s. Therefore, we have the Puiseux
series expansion of the eigenvalues $\xi_{1}=\eta_{-}$ and $\xi_{2}=\eta_{+}$
for sufficiently small $l$ as

$\xi_{q}=\sum_{m=0}^{\infty}d_{m}\exp\left(\pi\sqrt{-1}mq\right)l^{m/2},$

where $d_{0}=1$ and $d_{1}=\sqrt{e_{L-1}^{(1)}e_{L-1}^{(2)}}$.

## Acknowledgements

The author thanks Masatoshi Noumi and Yasuhiko Yamada for discussions and
drawing his attention to the difference between the two discrete dynamical
systems discussed in §5.1. He also thanks Kohei Motegi for the invitation of
giving a talk at the online workshop _Combinatorial Representation Theory and
Connections with Related Fields_ at RIMS, Kyoto University in November 2021,
which motivated him to initiate this work.

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