数理解析研究所講究録 1422 巻 2005 年 56-76

5\SPath description of conserved quantities ofgeneralized periodic box-ball systems
東京大学・大学院数理科学研究科 1 間田 潤 (Jun Mada)
根大学・教育学部 数学教室 2 泉 誠 (Makoto Idzumi)
東京大学・大学院数理科学研究科 1 時弘 哲治 (Tetsuji Tokihiro)
1 Graduate School of Mathematical Sciences, University of Tokyo

2 Department of Mathematics, Faculty of Education, Shimane University

Abstract
The present article is a review on the conserved quantities of periodic box-ball systems (PBBS) with arbitrary kinds of balls and box
capacity greater than or equal to one. By introducing the notion ofnonintersecting paths on the two dimensional array of boxes, we givea combinatorial formula for the conserved quantities of the generalized
PBBS using these paths.

1

Introduction

The box-ball system (BBS) is a reinterpretation of a soliton cellular automaton proposed by Takahashi-Satsuma [1] as a dynamical system of balls in a
one dimensional array of boxes [2], Hence, the BBS shows both a feature ofcellular automata (CAs) and that of solitons.
The periodic box-ball system (PBBS) is the BBS in which the updating
rule is extended to be compatible with a periodic boundary condition [8].
Let us consider a one-dimensional array of $N$ boxes. A periodic boundary
condition is imposed by assuming that thebox is adjacent to the firstone. (We may imagine that the boxes are arranged in a circle.) In thegeneralized PBBS (GPBBS), the capacity of the nth $(1 \leq n\leq N)$ box isdenoted by a positive integer , and we suppose that there are $M$ kinds ofballs distinguished by an integer index $j(1\leq j\leq M)$ . Whenand
$M=1$ , the GPBBS coincides with the PBBS. Then, the rule for the time
evolution of the GPBBS from time step to $t+1$ is given as follows:$N\mathrm{t}\mathrm{h}$$\theta_{n}$$\forall n\theta_{n}=1$$t$

1. At each box, create the same number of copies of the balls with index
1.
2. Choose one of the copies arbitrarily and move it to the nearest box
with an available space to the right of it.
3. Choose one of the remaining copies and move it to the nearest available
box on the right of it.

57
4. Repeat the above procedure until all the copies have been moved.
5. Delete all the original balls with index 1.

6. Perform the same procedure for the balls with index 2.
7. Repeat this procedure successively until all of the balls are moved.
An example of the time evolution of the GPBBS according to this rule isshown in Fig. 1.

$t=\theta$$\#$

Copy :$[egg1]$$\{$

Copy: 02

$\{$

$t=l$

$N=6$ , $M=2$ , $\theta_{1}=\theta_{2}=\theta_{4}=3$ , $\theta_{3}=2$ , $\theta_{5}=\theta_{6}=1$ .

Figure 1: Time evolution rule for PBBS.
In this article, we investigate the conserved quantities of the GPBBS for
arbitrary $M$ . In section 2, we derive the path description of characteristic
polynomial of particular matrices. In section 3, we briefly summarize theresults of Ref. [9], which we will use in the subsequent sections. In section 4,
we treat the ndKP equation which corresponds to the GPBBS. We shallobtain an explicit expression for the conserved quantities of the ndKP equation. Using the results in section 4, we construct the conserved quantitiesof the GPBBS in section 5. In section 6, we discuss algebraic aspects of the
GPBBS with respect to the Affine Weyl group and the crystals of quantum
affine algebra. Section 7 is devoted to the concluding remarks.

58
2Path description of a particular determinant

For a particular matrix which contains a parameter in upper half elements, we give a combinatorial description for coefficients of the characteristic polynomial $\det(\lambda I-B)$ in A and in terms of nonintersecting paths
$B$

$\mu$

$\mu$

(theorem 2.1).

Let$B:=(D_{0}-1)(D_{1}-\mathrm{Y})\cdots$ $(D_{M}-’\Gamma)$

where$D_{i}$$(\mathrm{i}=0,1, \ldots, M)$

(2.1)

,

are diagonal matrices,.

$x_{1,i}$$x_{2,i}$$x_{3,i}$$D_{i}:=\ovalbox{\tt\small REJECT}$$x_{N,i}\ovalbox{\tt\small REJECT}$

,

and
$1$

$\prime \mathrm{r}$$:=\ovalbox{\tt\small REJECT}$

1$\mu\ovalbox{\tt\small REJECT}$

1
If we put

$D_{i}^{(0)}$$:=D_{i}$and$x_{N-r+1,i}$$x_{N-r+2,i}$..
$D_{i}^{(r)}:=\{$$x_{N-r,i}\ovalbox{\tt\small REJECT}$$x_{N,i}$$x_{1,i}$..

for $0<r<N$ , we have$D_{i}^{(r+1)}’ T$

Hereafter, for$\mathrm{i}=0$$=\mathrm{f}\prime D_{i}^{(r)}$.

, we define$\sum_{c_{1}<c_{2}<\cdot<\mathrm{c}_{i}}..\cdots$$:=1$ .

.

59
Then, using the notation$D_{i}^{(r)}$, we obtain$B= \sum_{\ell=0}^{M+1}(-1)^{p}(\ldots 0\leq h_{1}<\sum_{<h_{M-\ell+1}\leq M}D_{h_{1}}^{(h_{1})}D_{h_{2}}^{(h_{2}-1)}\cdots D_{h_{NI-\mathit{1}+1}}^{(h_{M-t+1}-M+\ell))}h_{2}<\cdot\cdot 1^{l}(2.2)$

and$\det(\lambda I-B)=\Sigma(-1)^{N-k}\lambda^{k}\Sigma\mu^{j}NNk$

$k=0$

I

$\Sigma$$j=0X\subset\{1,2,..N\}J\subset X\# X=N-\acute{k}\# J=j$$\sum_{\sigma\in S_{X}^{J}}\mathrm{s}\mathrm{g}\mathrm{n}(\sigma)(\prod_{n\in J}(...<h_{M-N-n+\sigma(n)+1}0\leq h_{1}<h_{2}<\sum_{\leq M}\ldots\prod_{i=1}^{M-N-n+\sigma(n)+1}x_{n-h_{i}+i-1,h_{i))}}$$\mathrm{x}$$( \prod_{n\in X-J}(\cdots<h_{M-n+\sigma(n)+1}\sum_{\leq M}\prod_{i0\leq h_{1}<h_{2}<\cdots=1}^{M-n+\sigma(n)+1}x_{n-h_{i}+i-1,h_{i))}}$(2.3)

A combinatorial description of the coefficients is possible.
$(M+1)$ boxes in Fig. 2 and by $(n, m)$ -box
By $C_{N,M+1}$ we denote therow. We assume that thethe box at the nth column in thecolumn is adjacent to the first one.
$N\cross$$N\mathrm{t}\mathrm{h}$$(m+1)\mathrm{t}\mathrm{h}$$M$.

$\cdot$$\Phi$.

$\cdot$1

0
1

Figure 2:2...

$N\mathrm{x}$

$B$

...

$N$

$(M+1)$ boxes.

Let and be column indices $(a, b=1,2, \ldots, N)$ . A path connecting
the initial point and the end point is (continuous) polygonal line ffom
the initial point to the end point which consists of (i), (ii) or (iii) in Fig. 3
$a$

$b$

$b$

$a$

$a$

$b$

$\mathrm{a}$eolocally; here by the initial point we mean the middle point of the southedge of-box and by the end point the middle point of the north edgeof. For example, the left part in Fig. 4 shows a path connecting
the initial point 1 and the end point 1, and the right part shows a path from
5 to 2.
$a$$b$$(a, \mathrm{O})$$(b, M)- \mathrm{b}\mathrm{o}\mathrm{x}$.

.

$\omega.$$\{\mathrm{i}\}$$(..\omega$

Figure 3: A line can pass through a box in three possible ways.

There is a natural correspondence between$\prod_{i=1}^{M-\ell+1}x_{n-h_{i}+i-1,h_{i}}(0\leq h_{1}<$and a path on $C_{N,M+1}$ . To put it concretely, wedraw the line (i) on $(n-h_{i}+\mathrm{i}-1, h_{i})- \mathrm{b}\mathrm{o}\mathrm{x}(i=1,2, \ldots, M-f +1)$ . For
each , $h_{i}<r<h_{i+1}$ , we draw the line (ii) on-box and the line(iii) on $(n+\mathrm{i}-r-1, r)$ box where $h_{0}=-1$ and $h_{M-l+2}=M+1$ ; then we
obtain the path. For example, forand $M=5$ ,andcorrespond to the paths in Fig. 4 respectively.
$h_{2}<\cdots<h_{M-\ell+1}\leq M)$$(n+\mathrm{i}-r_{7} r)$$r$$N=\mathrm{S}$$x_{1,0}x_{1,1}x_{1,2}x_{1,3}x_{1,4}x_{1,5}$$x_{5,0}x_{4,2}x_{2,5}$

1

2

5

5

4

4

3

3

2

2

1

1

0

0
123

5

4

$\epsilon$

7

8

$1$

$\mathrm{x}_{1.0}$$\mathrm{x}_{1.1}$

X1.2

1

2

3

4

5

6

7

8

$1$

$\mathrm{x}1.3$

X1.4 Xl.b

Figure 4: Paths corresponding to$\mathrm{X}\epsilon,0\mathrm{X}4,2\mathrm{X}\mathrm{z}.\epsilon$$\mathrm{X}1.\mathrm{O}x_{1,1}x_{1,2}x_{1,3}x_{1,4}x_{1,5}$

and$x_{5,0}x_{4,2}x_{2,5}$.

Let $X=\{d_{1}, d_{2}, \ldots, d_{N-k}\}(1\leq d_{1}<d_{2}<\cdots<d_{N-k}\leq N)$ , and wedenote by
the set of all paths which connect the initial point andthe end point( $d\in X$ , $\sigma\in Sx$ ; cf. Fig. 5). Define:$\mathcal{P}(d;\sigma)$$\sigma(d)$$d$$\xi_{n,m}$$\mathcal{P}(d;\sigma)arrow$Gl$\sigma(\mathrm{d}\}$$M$$\mathrm{n}$10$\mathrm{d}$Figure 5: A path$\{x_{n,m}, 1\}$

$P\in \mathcal{P}(d;\sigma)$.

as$\xi_{n,m}(P):=\{$$x_{n,m}$$(\begin{array}{lll}P \mathrm{h}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}11\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}(n,m)- \mathrm{b}\mathrm{o}\mathrm{x} \mathrm{o}\mathrm{f} C_{N,M+1}\end{array})$

,

(2.4)

1(otherwise).
where
We introduce$P\in \mathcal{P}(d;\sigma)$$\mathcal{P}^{(j)}(d_{1}, \ldots, d_{N-k})$.
$:=\{$$(P_{1}, \ldots\} P_{N-k})|\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}\iota’ \mathrm{a}.\mathrm{n}\mathrm{d}j,\mathrm{i}\neq jP_{i}\mathrm{a}\mathrm{n}\mathrm{d}Pj\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}.\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}P_{i}\in \mathcal{P}(d_{i},\sigma)(\mathrm{i}=\mathrm{l},2,\ldots,N-k)$

The upper bound of the summation over

$j$

is

$\ovalbox{\tt\small REJECT}\frac{N-k}{N}(M+1)\ovalbox{\tt\small REJECT}$

where

denotes the largest integer which does not exceed .
Then, we obtain the following theorem.
$p$

Theorem 2.1
For defined in (2.1), it holds that$B$$\det(\lambda I-B)$$[ \frac{(N-k)(M+1)}{N}]$$(-1)^{j(N-k-1)}\mu^{j}$$= \sum_{k=0}^{N}(-1)^{N-k}\lambda^{k}$$\ovalbox{\tt\small REJECT}\sim 5$$\sum_{j=0}$$1. \leq..d_{1}<d_{2}<\sum_{<d_{N-k}\leq N}\cdots(F_{1,..\prime}P_{N-k})\sum_{\in P^{(j)_{(d_{1\prime}..,d_{N-k})}}}\prod_{i=1}^{N-k}\prod_{n=1}^{N}\prod_{m=0}^{M}\xi_{n,m}(P_{i})$)$[\ell]$82
wheretively.

and$\xi_{n,m}$$\mathcal{P}^{(j)}(d_{1}$, . . . , $d_{N-k})$ are defined in (2.4) and (2.5) respec$\square$

GPBBS and ndKP equation

3

In order to describe the dynamics of the GPBBS, we introduce a new independent variable. As any integer can be uniquely expressed
as $s=Mt+j(t\in \mathbb{Z}, 1\leq j\leq M)$ , we denote by
the number of balls$s(s\in \mathbb{Z})$$s$$u_{n}^{s}$with index$j\equiv s$

mod $M$ in the nth box at time step$t \equiv\ovalbox{\tt\small REJECT}\frac{s-1}{M}\ovalbox{\tt\small REJECT}$, wheredenotes the largest integer which does not exceed . In other words, the
new time variable is a refinement of the original time, indicating explicitly
when balls with index move.
We assume thatsatisfy the relation
and
$[x]$

$x$

$s$

$j$

$\theta_{n}$$u_{n}^{s}$$\sum_{n=1}^{N}\theta_{n}-\sum_{j=1n}^{M}\sum_{=1}^{N}u_{n}^{j}\geq\sum_{n=1}^{N}u_{n}^{k}(k=1,2,\ldots,M)$

.

(3.1)

The first and second terms of the left-hand side of (3.1) represent the numberof spaces and the number of balls in the GPBBS respectively, hence the lefthand side is the total number of free spaces of the GPBBS. The right-handside of (3.1) is the number of balls with index . Thus (3.1) requires thetotal number of free spaces of the GPBBS to be larger than the number ofcopies of any tyPe of ball in the time evolution process.
Let us consider the process at time ,, the movement of the ballswith index at time step where $s=Mt+j_{1}$. we often use instead of,. we treat the indices modulo $M$ . If we define , which denotes thenumber of spaces of the nth box at , by
$k$

$s$

$j$

$j$

$\mathrm{i}.e.$

$t$

$s$

$\mathrm{i}.e$$\kappa_{n}^{s}$

$s$

is:

$:=\theta_{n}-(u_{n}^{s}+u_{n}^{s-1}+\cdots+u_{n}^{s-M+1})$,

condition (3.1) is rewritten as$\sum_{n=1}^{N}\kappa_{n}^{s}\geq\sum_{n=1}^{N}u_{n}^{s-M+k}$

Then we have the following theorem$(k=1,2, -\cdot.

)$ $M)$ .

63
Theorem 3.1 ([9])
The time evolution of the GPBBS is described by a ultradiscrete equation:$u_{n}^{s}- \kappa_{n}^{s-1}=\max_{=k1,\ldots,N}\ovalbox{\tt\small REJECT}_{j=1}\sum^{k}u_{n-j}^{s-M}-\kappa_{n-j+1}^{s-1}\ovalbox{\tt\small REJECT}$$- \max[0,\mathrm{m}\mathrm{a}\mathrm{a}\mathrm{c}k=\mathrm{l},\ldots,N-1[\sum_{j=1}^{k}u_{n-j}^{s-M}-\kappa_{n-j+1}^{s-1}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$(3.2)$\square$

The ndKP equation is obtained bom the generating formula of the KP
hierarchy $[12, 13]$ . The ndKP equation is given as$(b(m)-c(n))\tau(l+1, m, n)\tau(l, m+1, n+1)$$+$$(c(n) – a(l))\tau(l, m+1, n)\tau(l+1, m, n+1)$$+(a(l)-b(m))\tau(l, m, n+1)\tau(l+1, m+1, n)=0$ ,(3.3)

function :are independent variables, thewhere 1, $m$ ,(or C) is dependent variable and the coefficients $a(l)$ , $b(m)$ , $c(n)$,are arbitrary functions which depend on the independent variables
respectively.
In order to relate the ndKP equation to the GPBBS, we take $a(l)=$, $b(m)=1$ , $c(n)=1[perp]\delta_{n}$ and impose the following constraint on $\tau(l, m, n)$ :$n\in \mathbb{Z}$$\mathrm{t}\mathrm{a}\mathrm{u}$$\tau$$\mathbb{Z}\mathrm{x}$$\mathbb{Z}\mathrm{x}$$\mathbb{Z}arrow \mathbb{R}$$l_{\gamma}m$

$n$

$0$

$\tau(l, m, n)=\tau(l-M, m-1, n)$ .

If we define $\sigma_{n}^{s}:=\tau(s-1,0, n)$ , (3.3) turns into.

$\frac{\sigma_{n+1}^{s+M-1}\sigma_{n+1}^{s}}{\sigma_{n+1}^{s+M}\sigma_{n+1}^{s-1}}-(1+\delta_{n+1})\frac{\sigma_{n}^{s-1}\sigma_{n+1}^{s}}{\sigma_{n+1}^{s-1}\sigma_{n}^{s}}=-\delta_{n+1}\frac{\sigma_{n}^{s+M}\sigma_{n+1}^{s}}{\sigma_{n}^{s}\sigma_{n+1}^{s+M}}$

Furthermore, we define$U_{n}^{s}$and$K_{n}^{s}$as$U_{n}^{s}.= \frac{\sigma_{n+1}^{s}\sigma_{n}^{s+1}}{(1+\delta_{n+1})\sigma_{n}^{s}\sigma_{n+1}^{s+1}}$,$\frac{1}{K_{n}^{s}}:=\delta_{n+1}\cdot\prod_{j=1}^{M}U_{n}^{s-j+1}$

and impose the following periodic condition on$U_{n}^{s}=U_{n+N}^{s}$.

$U_{n}^{s}$

:

(3.4)

84
Then, ffom (3.4), we have$\frac{U_{n}^{s}}{K_{n}^{s-1}}=\frac{\sum_{k=1j}^{N}\prod_{=1}^{k}\frac{U_{n-j}^{s-M}}{K_{n-j+1}^{s-1}}}{1+\sum_{k=1}^{N-1}\prod_{j=1}^{k}\frac{U_{n-j}^{s-M}}{K_{n-j+1}^{s-1}}}$

To take the ultradiscrete limit, we put. Then we found

(3.5)

.

$U_{n}^{s}=e^{u_{n}^{s}/\epsilon}$,$K_{n}^{s}=e^{\kappa_{n}^{s}/\epsilon}$,$1/\tilde{\delta}_{n+1}=$$e^{\theta_{n}/\epsilon}$

Theorem 3.2 ([9])
The ultradiscrete limit of the constrained ndKP equation
with the periodic boundary condition coincides with the time evolution equation of the
GPBBS (3.2),$(3\mathrm{A})$$\square$4

Conserved quantities of ndKP equation

In Ref. [9] we derived the Lax representation for the ndKP equation when
it has period $N$ in the spatial variable . In short, the equation (3.5) isequivalent to the matrix equation$n$$\overline{M}(s)L(M;s)=L(M;s-1)\overline{M}(s)$where$\overline{M}(s)=G_{U_{j}s}-\overline{\mathrm{T}}$,$L(M;s)=(-G_{K;s}+’\overline{\Gamma})(G_{U_{j}s-M+1}-\overline{\prime\Gamma})(G_{U_{j}s-M+2}-’\overline{Y})\cdots(G_{U;s}-\overline{\prime\Gamma})$

,
(4.1)

65
$\frac{1}{K_{1}^{s}}$$\frac{1}{K_{2}^{s}}$$G_{K;s}$$\frac{1}{K_{3}^{s}}$$:=$$\ovalbox{\tt\small REJECT}$$\frac{1}{U_{1}^{\epsilon}}$$\frac{1}{U_{2}^{s}}$$G_{U;s}$$\frac{1}{U_{3}^{s}}$$:=$$\ovalbox{\tt\small REJECT}$

11$\delta_{1}$$1+\mathit{5}_{2}$$’\overline{\mathrm{r}}$

$:=$

$\{$

.
.
..

,

$\frac{1}{K_{N}^{s}}\ovalbox{\tt\small REJECT}$$\frac{1}{U_{N}^{s}}\ovalbox{\tt\small REJECT}$$7$$(1+\delta_{N})\cdot\eta\ovalbox{\tt\small REJECT}$;$1+\delta_{N-1}$

here, is an arbitrary parameter.
This means$\eta$$\det(\lambda I+L(M;s))=\det(\lambda I+L(M;s-1))$ ;therefore, the coefficients $ek$ of the characteristic polynomial$\det(\lambda I+L(M;s))$$=\lambda^{N}+e_{N-1}\lambda^{N-1}+e_{N}$$-2\lambda^{N-2}+\cdots+$ $e_{1}\lambda+e_{0}$are conserved in time . Furthermore, since is arbitrary and$s$

if we define$e_{k}^{[j]}$$\eta$$[j]$$e_{k}$

Let

contain ,$\eta$by$e_{k}= \sum_{j}e_{k}^{[j]}\eta^{j}$

then

$ek$

(4.2)

,

are also conserved.

$\Delta:=\prod_{i=1}^{N}(1+\delta_{i})$,1$\prime \mathrm{r}$$:=\{$

1.

$\eta\Delta\ovalbox{\tt\small REJECT}$

1

,

68
and

.

$1$

$1+\delta_{1}$$(1+\delta_{1})(1+\delta_{2})$$D_{\delta}:=\ovalbox{\tt\small REJECT}$$\prod_{i=1}^{N-1}(1+\delta_{i})\ovalbox{\tt\small REJECT}$

It follows that$\prime \mathrm{r}$$=(D_{\delta})^{-1}\overline{\mathit{1}}D_{\delta}$.

Hence, we have the following Lemma.

Lemma 4.1 ([9])
Let$L_{0}(M;s):=(-GK;s+\mathrm{Y})$ $(Gu;s-M+1-\mathrm{T})$ $(GU_{j}s-M+2-\mathrm{T})$-.

$(Gu_{js}-\mathrm{T})$.

Then, it holds that$\det(\lambda I+L(M;s))=\det(\lambda I+L_{0}(M; s))$

(4.3)

.

sr
From theorem 2.1, we obtain a combinatorial formula for$e_{k}^{[j]}$immedi-ately.

Theorem 4. 1
Put$x_{n,0}= \frac{1}{K_{n}^{s}}7$in (2.4) and put
it holds that$\mu=\eta\Delta$$x_{n_{\}}m}= \frac{1}{U_{n}^{s-M+m}}$. Then, for $k=0$ , 1,$e_{k}^{\lceil j]}=(-1)^{l(k,j)}\Delta^{j}$$\ldots$$(m\neq 0)$, $N$ and $j=0,1$ ,$\ldots$, $[ \frac{(N-k)(M+1)}{N}]$ ,,$1. \leq..d_{1}<d_{2}<\sum_{<d_{N-k}\leq N}\cdots(P_{1,.\prime}P_{N-k})\sum_{\in \mathcal{P}^{(j)_{(d_{1},.,d_{N-h})}}}\prod_{i=1}^{N-k}\prod_{n=1}^{N}\prod_{m=0}^{M}\xi_{n_{2}m}(P_{i})$wizere $\ell(k,j):=(j+1)N-(k +i+kj)$ .

$\square$87
Conserved quantities of GPBBS

5

Using the results in section 4, we construct the conserved quantities of the

GPBBS
For $k=0$ , 1, . . . , $N$ and $j=0$ , 1,

limit of$ue_{k}^{[j]}$Since$e_{k}^{[j]}$-$\lim_{\epsilonarrow+0}\epsilon\log((-1)^{\ell(k,j)}e_{k}^{[j]})$$=$-$\lim_{\epsilonarrow+0}\epsilon\log(\Delta^{j}$$\lim_{\epsilonarrow+0}$, $[ \frac{(N-k)(M+1)}{N}]$ , the ultradiscrete

is

$:=$

$\theta_{n}$$\ldots$$1. \leq..d_{1}<d_{2}<\sum_{<d_{N-k}\leq N}\cdots\in p^{(\mathrm{j})_{(d_{1},\ldots d_{N-k})}}\sum_{(P_{1\prime}..,P_{N-k})},\prod_{i=1}^{N-k}\prod_{n=1}^{N}\prod_{m=0}^{M}\xi_{n,m}(P_{\dot{\mathrm{z}}}))$

is the capacity of the nth box,
elog$\Delta^{j}$$=$$j \cdot\lim_{\epsilonarrow+0}\epsilon\log\prod_{j=1}^{N}(1+e^{-\theta_{j}/\epsilon})=j\cdot\sum_{j=1}^{N}\max[0, -\theta_{j}]$

$=$

0.

Therefore, from theorem 4.1,

$ue_{k}^{[j]}$izss given by

Theorem 5.1
Put$x_{n,0}=\kappa_{n}^{s}$,in (2.4). Then, for $k=0,1$ ,that$ue_{k}^{[j]}=$$1. \leq..d_{1}<d_{2}<\min_{<d_{N-k}\leq N}\ldots\{$$x_{n,m}=u_{n}^{s-M+m}(m\neq 0)$$\ldots$, $N$ and $j=0,1$ ,$\ldots$, $[ \frac{(N-k)(M+1)}{N}]$ , it holds$\epsilon r^{(j)_{(d_{1\prime}..d_{N-k})}}\min_{(P_{1},\ldots,P_{N-k})},\ovalbox{\tt\small REJECT}\sum_{i=1}^{N-k}\sum_{n=1}^{N}\sum_{m=0}^{M}\xi_{n,m}(P_{i})]\ovalbox{\tt\small REJECT}$Cl

Remark 5. 1
The conserved quantity $ue_{k}^{[0]}(0\leq k\leq N)$ is trivial Since $j=0$ , all paths
are vertical lines. Hence we have$ue_{k}^{[0]}$$=$.

$.N \ovalbox{\tt\small REJECT}\sum_{i=1}^{N-k}(\kappa_{d_{i}}^{s}+\sum_{m=1}^{M}u_{d_{i}}^{s-M+m})]$$1. \leq..d_{1}<d_{2}<\min_{<d_{N-k}\leq}$$=$$1. \leq..d_{1}<d_{2}<\min_{<d_{N-k}\leq N}\ldots\ovalbox{\tt\small REJECT}\sum_{i=1}^{N-k}\theta_{d_{i}}\ovalbox{\tt\small REJECT}$.

gg
As is the capacity of the $nth$ box,
So we are not interested in them.
$\theta_{n}$Remark 5.2
If a quantity

dose not depend on the time steps.

$ue_{k}^{[0]}$is conserved in term of the original time step , that$=A_{1}(0)$ , we can always associate it with the quantities $B_{k}(s)$is, ,$(k=1,2, \ldots, M)$ which axe conserved in , that is,, $B_{k}(s)=Bk(0)(k=$1, 2, . . . , $M$ ). The reason is as follow$ws$ . First we note that $A_{1}(t)=F(\{u_{n}^{Mt+s}\}_{s=1}^{M})=$$:=F(\{u_{n}^{Mt+s}\}_{s=j}^{j+M})$
since the time evolution
If we putrule for the GPBBS at time step is just the same as that at $s+1$ , $Aj(t)$is also conserved in terms of the original time step . Hence if we denote by$B_{k}(s)$ the $kth$ elementary symmetric polynomial of, $B_{k}(s)$ isconserved quantity in terms of . We should also note that the number of.
is equal to that ofindependent conserved quantities$\forall t$$A_{1}(t)$

$t$

$s$

$A_{1}(t)$$\forall_{S}$$s$$F(\{u_{n}^{s}\}_{s=1}^{M})$$A_{j}(t)$

$s$

$t$

$\{Aj(t)\}_{j=1}^{M}$

$a$

$s$

$\{Bk\}_{k=1}^{M}$$\mathrm{i}_{I1}\{Aj\}_{j=1}^{M}$

An easy way to read off

is as follows: Associate$\sum_{i=1}^{N-k}\sum_{n=1}^{N}\sum_{m=0}^{M}\xi_{n,m}(P_{i})$with boxes of, $M+1$ as shown in Fig. 6. For$(P_{1}, \ldots, P_{N-k})\in \mathcal{P}^{(j)}(d_{1}, \ldots, d_{N-k})$ , summing up
the values corresponding

values$\kappa_{n}^{s}$,$u_{n}^{s}$,... ,$u_{n}^{s-M+1}$$C_{N}$

to the vertical lines of the paths, we get the value$\mathrm{u}\mathrm{l}\mathrm{s}$...

$\mathrm{U}2\mathrm{s}$$\mathrm{u}^{\mathrm{s}}\mathrm{s}$$\mathrm{u}^{5}1^{1}$$\mathrm{u}^{\mathrm{b}}2^{1}$$\mathrm{u}^{\backslash }.-3^{1}$$\ldots$

.

.

.

.

$.\cdot$$.\cdot$$.\cdot$$.\cdot$$\sum_{i=1}^{N-k}\sum_{n=1}^{N}\sum_{m=0}^{M}\xi_{n,m}(P_{i})$$\mathrm{U}\mathrm{N}\cdot 1\epsilon$.

$\mathrm{U}\mathrm{N}\mathrm{s}$.
$\mathrm{u}^{\mathrm{s}\mathit{1}}\mathrm{N}\cdot 1$.

$\mathrm{u}^{\mathrm{s}_{\mathrm{N}}1}$.

.
$\cdot$$\mathrm{h}\int_{2}+2$$\mathrm{u}^{\mathrm{s}^{\sim}\mathrm{b}\mathrm{I}+2}.1$$\mathrm{u}^{\iota^{-}\mathrm{M}+2}3$$\mathrm{u}^{\mathrm{b}}$$\mathrm{u}^{\mathrm{s}\mathrm{b}}.1^{\mathit{1}+1}$$\mathrm{u}..2\mathrm{h}i+\mathrm{I}$$2\epsilon$$\mathrm{s}1$$\kappa$$\mathrm{u}^{\epsilon\cdot \mathrm{M}+1}\mathrm{a}$$\kappa$$3\mathrm{s}$$\kappa$

123

...
...
...
...

$.\cdot$$\mathrm{u}^{\epsilon\cdot \mathrm{M}+2}\mathrm{N}\cdot 1$$\mathrm{u}^{\epsilon}\mathrm{M}+2\mathrm{N}$$\mathrm{u}^{\epsilon \mathrm{M}+1}\mathrm{N}\cdot 1$$\mathrm{u}^{\epsilon\cdot \mathrm{M}+1}\mathrm{N}$$\mathrm{N}^{8}.1$$\kappa$$N-\mathit{1}$$\mathrm{N}\epsilon$$\kappa$$N$

Figure 6:

Example 5.1
For a state in Fig. 7 (N$=10,$M $=5)$ , we obtain a table in Fig. 8.

For

69

2

1

3

5

4

7

6

8

9

10

Figure 7: A state of the GPBBS.

5
4
3
2
1

0
1

2

3

4

5

$\epsilon$

Figure 8:

7

8

9

10

70
paths shown in Fig. 9,$\sum_{i=1}^{N-k}\sum_{n=1}^{N}\sum_{m=0}^{M}\xi_{n,m}(P_{i})$$=(0+0+1)+(0+1+0+0)+(0+0+0+0)$$+(0+1+0)+(0+0+1)+(0+1+0)+(1+0)$$=6$.

Occasionally these paths minimize$\sum_{i=1}^{N-k}\sum_{n=1}^{N}\sum_{m=0}^{M}\xi_{n,m}(P_{i})$; thus,$ue_{3}^{[2]}$

is 6.

5
4
3
2
1
0

1

2

3

4

5

6

7

9

8

10

Figure 9:6

AfSne Wyle group, integrable lattice models and
GPBBS

Let be the set of
and $sl$$\mathcal{T}$$N\mathrm{x}$$(\ell\in \mathbb{Z},/(M+1)\mathbb{Z})$.

$(M+1)$ rectangular tableaux with integer entries,

and

$\pi$

be mappings:$y1_{2}M$$y_{2,M}$.

.

$y_{1,1}$$y_{2,1}$$y_{1,0}$$y_{2,\circ}$.

$\cdot$.

$\cdot$$\ldots$...
$\ldots$...

$\mathcal{T}$$arrow \mathcal{T}$$\prime yN,M$$y_{N,1}$$\prime y_{N,0}$. For a tableau

$Y$

,

71
these mappings are given as...

$s_{l}(y_{2,M})$$s_{\ell}(y_{1},M)$

.

.

.

$s_{\ell}(y_{1,1})$$s_{\ell}(y_{2,1})$$s_{\ell}(y_{1,0})$$s_{l}(y_{2,0})$

.

..

.

$\pi(y_{1,1})$$\pi(y_{2,1})$$\pi(y_{1},0)\backslash$$\pi(y_{2},0)$$s_{\ell}(y_{N,1})$$s_{l}(y_{N,0})$$\pi(y_{N,M})$

.
.

...
...
...

$\cdot$$\cdot$$\cdot$

...
...
...

$\pi(y_{2,M})$$\pi(y_{1,M})$

.
.

...

.

$\cdot$$\cdot$$s_{\ell}(y_{N,M})$$\cdot$$\pi(y_{N,1})$$\pi(y_{N},0)$

where
mod $M+1$ )$s_{l}(y_{n,m})=y_{n,m+1}+Q_{n,m}-Q_{n-1,m}$($s_{\ell}(y_{n,m+1})=y_{n,m}+Q_{n-1,m}-Q_{n,m}$$(m\equiv\ell \mathrm{m}\mathrm{o}\mathrm{d} M+1)$$s\ell(y_{n,m})=y_{n,m}$$m\equiv\ell$$(m\not\equiv l, P+1 \mathrm{m}\mathrm{o}\mathrm{d} M+1)$$\pi(y_{n,m})=y_{n,m+1}$

$2$

,

,

,

and
$Qn$ , $m$$=$$y_{n}+k$ $)m$ $+$ $1$ $+$$[ \sum_{k=1}^{h-1}$$\leq h\leq N\mathrm{m}\mathrm{a}\mathrm{x}$$k \sum_{=h+}^{N}1$$y_{n}+k)m$

$]$

.

$1$

Here we extend the indices$y_{n+N,m}=y_{n,m+M+1}=y_{n,m}$.

$n$, $m$ offor

$y_{n,m}$$n$,$m\in \mathbb{Z}$by the condition

For example,$y_{1,M}$$y2,M$

. .

.

...

.
.

.

$y_{1,2}$$y2_{1}2$$\cdot$$\cdot$$y_{1,0}$$y_{1}$, $1+Q_{1_{\}0}$ $-Q_{N}$ ,$y1,M-1$.

$y2,M-1$..

.

$\cdot$$\cdot$$\pi(Y)=$$y_{2},0+Q_{1,0}-Q_{2,0}$$+Q_{N,0}$ $-Q_{1,0}$$y_{1},0$$y_{2,0}$$y1,M$$y2,M$$y2,1$ $+Q_{20}\{$ $-Q_{1,0}$

$0$

...

...
...
...

$yN,M$$\alpha$

...
...
...

.
.

$\cdot$$yN,2$$y_{N,0}+Q_{N-1,0}-Q_{N,0}$$y_{N}$, $1+Q_{N,0}$ $-Q_{N-1}$ ,$yN,M-1$.
.

.

$\cdot$$\cdot$$y_{N,0}$$yN,M$

The following theorem is proved by direct calculations.

Theorem 6.1 ([14, 15])
The mappingsof the affine Weyl group$s_{\mathit{1}}$$(l \in \mathbb{Z}/(M+1)\mathbb{Z})$$W-(A_{M}^{(1)})$

.

and

$\pi$

defined as above give a realization$\square$$0$

72
Remark 6,1
The affine Weyl groupsimple reflections $s_{0,1)}s"$ . . 7
fundamental relations$\overline{W}(A_{n-1}^{(1)})$$s_{i}^{2}=1$$s_{n-1}$

is defined as the group generated by theand diagram rotation subject to the$\pi$,$(j\not\equiv \mathrm{i}, \mathrm{i}\pm 1 \mathrm{m}\mathrm{o}\mathrm{d} n)$$s_{i}s_{j}=s_{j}s_{i}$$s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j}$$\pi s_{i}=s_{i+1}\pi$,$(j\equiv \mathrm{i}\pm 1 \mathrm{m}\mathrm{o}\mathrm{d} n)$,,wizere we understand the indices for$s_{i}$

as elements of$\mathbb{Z}/(M+1)\mathbb{Z}$.

When we put$y_{n,0}=\kappa_{n}^{s}$,$y_{n,m}=u_{n}^{s-M+m}$$(m\neq 0)$

(6.1)

,

we get the following theorem which gives a relation between the GPBBS
and the affine Weyl group.
Theorem 6.2

7TSM-ISM-2$\ldots$$s_{0}$

gives the time evolution which concerns with time ,$t$$u_{2}^{s}$$u_{1}^{s}$$\pi s_{M-1^{S}M-2}\cdot$ . .

.

.
.

.
$\cdot$$s_{0}$$\cdot$

...
...
...
...

$\mathrm{i}.e$.

$u_{N}^{s}$.

.

$\cdot$$(\begin{array}{llllll}u_{1}^{s} u_{2}^{s} \cdots u_{N}^{s} \vdots \vdots \cdots \vdots u_{1}^{s-}\kappa_{1}^{s} + u_{2}^{s-}\kappa_{2}^{s} +\mathrm{l}\cdots\cdots u_{N}^{s-}\kappa_{N}^{s} +1\end{array})$

$+1$

$+$
$u_{1}^{s-}$$\kappa_{1}^{s}$$u_{2}^{s-}$$\kappa_{2}^{s}$$+1$$u_{N}^{s-}$$\kappa_{N}^{s}$.

$=.\cdot.\cdot\ovalbox{\tt\small REJECT}_{u_{2}^{s+}.\cdot u_{N}^{s}}^{s+}u_{1}^{s+}\kappa_{1}^{s+}\kappa_{2}^{s+}.\cdot.\cdot.\cdot\kappa_{N}^{s+}u_{2}.\cdot..u_{N}^{s+}u_{1}^{s+}\cdot\cdot.\cdot$.

$\square$

The BBSs can be reformulated as integrable lattice models at temperature zero from the crystal theory and the combinatorial $R$ matrix $[17, 18]$ .
The PBBS with one kind of ball and box capacity one has also been reformulated into two types of lattice models, a periodic
crystal lattice andcrystal chain, where $N$ denotes the number of the boxesa twiisstted
in the system [8]. It is straightforward to extend this result to the case of
the GPBBS. Here, we will briefly show how the GPBBS is reinterpreted assome integrable lattice systems. Since the proofs for the statements below
are almost the same as those in Ref. [8], we will omit them here$A_{1}^{(1)}$$A_{N-1}^{(1)}$73
be the classical crystal ofcorresponding to the fc-foldLetsymmetric tensor representation of $U_{q}(A_{M})$ . As a set it consists of the single
row semistandard tableaux of length on letters $\{1, 2, \ldots, M+1\}$ :$U_{q}’(A_{M}^{(1)})$$B_{k}$$k$$B_{k}:=\{$

An element$|1\leq \mathrm{i}_{1}\leq \mathrm{i}_{2}\leq$. . . $\leq i_{k}\leq M+1\}$ .

$b$

$b=$

$\in B_{k}$$(x^{(M+1)}, x^{(M)}, \ldots, x^{(2)}, x^{(1)})$ ,is also denoted as a series of $M+1$ integersof the GPBBS is
whereis the number of letters in . A state
naturally identified with$b\equiv$$x^{(j)}$$j$$|\psi\rangle_{t}$$b$$|\psi\rangle$

$t$

..

$\cong$$b_{1}^{t}\otimes b_{2}^{t}\otimes\cdot$$\in$$B_{\theta_{1}}\otimes B_{\theta_{2}}\otimes$$\otimes b_{N}^{t}$$\cdots\otimes B_{\theta_{N}}$

,

where
$b_{n}^{t}=$$(n=1, 2, \ldots, N)$ .

$(\kappa_{n}^{s}, u_{n}^{s-M+1}, u_{n}^{s-M+2}, \ldots, u_{n}^{s})$For the BBS without the periodic boundary condition, time evolution is
given by the isomorphism induced by the combinatorial matrices:$R$$\mathcal{T}$$\mathcal{T}$

:
:

where
the

$B_{\infty}\otimes$$(B_{\theta_{1}}\otimes B_{\theta_{2}}\otimes \cdots\otimes B_{\theta_{N}})arrow(B_{\theta_{1}}\otimes B_{\theta_{2}}\otimes\cdots \otimes B_{\theta_{N}})\otimes B_{\infty}$,$|\{0\}\rangle\otimes|\psi\rangle_{t}arrow|\psi\rangle_{t+1}\otimes|\{0\}\rangle$

is the highest weight vector of,of the auxiliary state in$|\{0\}\rangle$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$B_{\varpi}$$B_{\infty}$. For the GPBBS, by taking, we have the time

$T:=\mathrm{b}_{B_{\infty}}\mathcal{T}$evolut ion
$T$

$T$

:
:

$B_{\theta_{1}}\otimes B_{\theta_{2}}\otimes\cdots\otimes$ $B_{\theta_{N}}arrow B_{\theta_{1}}\otimes B_{\theta_{2}}\otimes\cdots\otimes B_{\theta_{N)}}$$|\psi\rangle_{t}arrow|\psi\rangle_{t+1}$.

to the unique tensor productcrystal, the operator maps
As thecrystal that exactly corresponds to the state of the GPBBS at $t+1$ .
of
The GPBBS is also reformulated as a twisted lattice of $M$ vertical axes
is identified
crystals. In this case, a state
in terms of$T$$A_{1}^{(1)}$$|\psi\rangle$$t$$A_{M}^{(1)}$$U_{q}’(A_{N-1}^{(1)})$$|\psi\rangle_{t}$$|\psi\rangle_{t}$$\cong$6$b_{\kappa}^{t}$&$(b_{u_{1}}^{t}\otimes b_{u_{2}}^{t}\otimes\ldots b_{u_{M}}^{t})$$B_{k}\otimes(B_{n_{1}}\otimes B_{n_{2}}\otimes \ldots B_{n_{M}})$where$b_{\kappa}^{t}$$b_{u_{j}}^{t}$,$=$$(\kappa_{N}^{s}, \kappa_{N-1}^{s}, \ldots, \kappa_{1}^{s})$$=$$(u_{N}^{s-M+j}, u_{N-1}^{s-M+j}, \ldots, u_{1}^{s-M+j})$$(j=1,2, \ldots, M)$ ,74
and $n_{j}:= \sum_{n=1}^{N}u_{n}^{s-M+j}$ . The time evolution is determined by
crystal:the isomorphism induced by the combinatorial -matrix for$k:= \sum_{n=1}^{N}\kappa_{n}^{s}$$R$$b_{\kappa}^{t}\otimes$$A_{N-1}^{(1)}$.

$(b_{u_{1}}^{t}\otimes b_{u_{2}}^{t}\otimes\ldots b_{u_{M}}^{t})\cong(b_{u_{1}}^{t+1}\otimes b_{u_{2}}^{t+1}\otimes\ldots b_{u_{M}}^{t+1})\otimes b_{\kappa}^{t+1}$

In Fig. 10, we schematically show the twisted crystal lattice associated with
the GPBBS.

ウ

Figure 10: The twisted crystal lattice associated with the GPBBS.

7

Concluding remarks

In this article, using a path description of characteristic polynomial of particular matrices and an algorithm to construct the conserved quantities using
the Lax representation of the ndKP equation, we showed explicit form of
the conserved quantities of the GPBBS. Relations to the affine Weyl group
action and the crystal theory were also clarified. An advantage to reformulate the PBBS as crystal lattices is that we can extend it to the crystals
associated with other root systems.
Since the GPBBS is composed of a finite number of boxes and balls, it
can only take on a finite number of patterns. Hence its trajectory is always
periodic and a fundamental cycle,. the shortest period of the periodic
motion, exists for any given initial state. In the case where the box capacity
is one everywhere and only one kind of ball exists, the formula used to
calculate the fundamental cycle is explicitly obtained using the conserved
quantities and some rescaling properties of the states [16]. Hence, using theresults in this article, we may get the formula to calculate the fundamental
cycle for the GPBBS, which is a problem we wish to address in the future$\mathrm{i}.e$75References[1] D. Takahashi and J. Satsuma, ”A soliton cellular automaton,” J. Phys.
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