$\bullet$ $\wedge$ $\Lambda$ Combinatorial aspects of box-ball systems - (Kaori Fukuda) Graduate School of Science and Technology

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1 Title 箱玉系の組合せ論的側面 ( 組合せ論的表現論とその周辺 ) Author(s) 福田香保理 Citation 数理解析研究所講究録 (2003) 1310: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

16-28 Issue Date 2003-04 URL http://hdlhandlenet/2433/42896

2 $\bullet$ $\wedge$ $\Lambda$ Combinatorial aspects of box-ball systems - (Kaori Fukuda) Graduate School of Science and Technology Kobe University ( ) [1] (RSK ) $(P Q)$ 2 (i) $P$ (ii) $Q$ $P$ $Q$ 1 $[3][5][6][7][12]$ $n$ $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{l}\geq 0)$ $k$ $\lambda_{k}$ ( ) 1 1 $n$ 1 1 $\lambda=(431)$ ( ) $\leq\}$ $<\}$ 1

$$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{l}\geq 0)$ $k$ $\lambda_{k}$ ( ) 1 1$

3 17 $T$ $I\prime V^{\tau}(T)$ $i$ $i$ $i$ $i$ $j$ $i$ $j$ ( ) $(ux v)xarrow x (uxv)$ $(u\leq x<x \leq v)$ $\text{ }$ $yzxarrow yxz$ $(x<y\leq z)$ $xzyarrow zxy$ $(x\leq y<z)$ ( ) : $i$ $(\begin{array}{l}ij\end{array})$ $j$ 2 $w^{*}$ $w$ $(P Q)$ Robinson-Schensted-Knuth (RSK ) $P$ $i$ $(\begin{array}{l}ij\end{array})$ $j$ 1 $Q$ $(QP)$ 3

$) : 5152431245 5415213245 2 $i$ $(\begin{array}{l}ij\end{array})$ $j$ 2 $w^{*}$ $w$ $(P Q)$$

4 $\bullet$ $\emptyset$ $arrow $j_{k^{\circ}}i_{k}$ $\emptyset$ 18 $w=\{$ $i_{1}$ $i_{2}$ $j_{n}i_{n})$ $j_{1}$ $j_{2}$ $\{\begin{array}{l}i_{1}\leq i_{2}\backslash _{\backslash }\leq\cdots\leq i_{n}j_{k}\leq j_{k+1}\mathrm{i}\mathrm{f}i_{k}=i_{k+1}\end{array}$ $(k=1 \ldots n-1)$ $w=(\begin{array}{l} \end{array})$ $w^{*}=(\begin{array}{llllll} \end{array})$ $rightarrow 1\cdot 1$ RSK { } $\{(P Q)\}$ $w=(\begin{array}{llllll} \end{array})$ $arrow$ $(P Q)$ $P_{0}=$ 3$ $Q_{0}=$ $P_{1}=\fbox 3$ $arrow 1$ $Q_{1}=$ $P_{2}=\mathrm{H}_{3}^{1}$ $arrow 5$ $Q_{2}=\mathrm{H}_{2}^{1}$ $P_{3}=\overline{\ovalbox{\tt\small REJECT}^{15}}$ $arrow 2$ $Q_{3}=\overline{\mathrm{E}^{1\underline{2}}}$ $P_{4}=\ovalbox{\tt\small REJECT}$ $arrow 2$ $Q_{4}=\overline{\ovalbox{\tt\small REJECT}_{4}^{12}}$ $P_{5}=$ $arrow 1$ $Q_{5}=$ $P=P_{6}=$ $Q=Q_{6}=$ 2 [1] 1 $n$

$rightarrow 1\cdot 1$ RSK { } $\{(P Q)\}$ $w=(\begin{array}{llllll}1 2 2 4 5 73 1 5 2 2 1\end{array})$ $arrow$ $(P Q)$ $P_{0}=$ 3$ $Q_{0}=$ $P_{1}=\fbox 3$ $arrow 1$ $Q_{1}=$

5 $\Downarrow$ 19 $n$ $\ldots n$ $e(=n+1)$ ( ) (1) $t\mathrm{b}^{\mathrm{a}}$ $t+1$ (2) 1 (3) $23$ $\ldots$ n 4 $n=5$ (1 ) 5 $n=9$ 2 -- $ -^{15789_{-----}236_{---}4_{ }}$ $ ^{15789_{---}236_{-}4_{ }}$ $ ^{15789_{-}2634_{ }}$ $ -^{ _{ }}$ $ -^{7_{-}568_{-}12349_{ }}$ $ ^{7}$ $ ^{7_{-----}568_{-----}12349_{ }}$ $i$ $a_{i}$ $\cdot$ $a_{i}=e$ 1 $n$ $e=n+1$ $a_{-1}a_{0}a_{1}\cdots$ $i_{1}$ $i_{2}$ $\ldots$ $i_{n}$ $a_{i_{1}}$ $a_{i_{2}}$ $\ldots$ $a_{i_{n}}$ $w$ $i_{k}$ $(\begin{array}{l}i_{k}a_{i_{k}}\end{array})$ a 2

$^{15789_{-}2634_{------------------------------}}$ $ -^{157682349_{--------------------------}}$ $ -^{7_{-}568_{-}12349_{---------------------}}$ $ ^{7}$ 568 - $$

6 $n$ $n$ $b=(b_{1} b_{2} \ldots b_{n})$ $i_{1}$ $i_{2}$ $\ldots$ $i_{n}$ ( ) $b_{1}$ $\ldots$ b $\langle$ RSK $(P Q)$ $a_{-1}a_{0}a_{1}\cdots$ $(P Q)$ 21 $(P Q)$ (i) $P$ (ii) $Q$ $P$ 6 4 $w=(\begin{array}{lllll} \end{array})$ $\Rightarrow$ $w =(\begin{array}{lllll} \end{array})$ $Q$ $\Rightarrow$ $P $ $Q $ $P=P $ $P$ $Q$ $Q $ $Q$ $w=(w_{1} w_{2} \ldots w_{n})$ $w =(w_{1} w_{2} \ldots w_{n} )$ $C=(c_{1} \ldots c_{m})$ $w$ $w_{k}$ $C_{k-1}=(c_{1}^{(k-1)} c_{2}^{(k-1)} \ldots c_{m}^{(k-1)})$ $(c_{1}^{(k-1)}\leq c_{2}^{(k-1)}\leq\cdots\leq c_{m}^{(k-1)})$ $w_{k}$ $w_{k}$ $C_{k-1}$ $w_{k}$ $C_{k-1}$ $w_{k}$ $C_{k-1}$ $w_{k}$

$\Rightarrow$ $P $ $Q $ $P=P $ $P$ $Q$ $Q $ $Q$ $w=(w_{1} w_{2} \ldots w_{n})$ $w =(w_{1} w_{2} \ldots w_{n} )$ $C=(c_{1} \ldots c_{m})$ $w$ $w_{k}$

7 $\{$ $\ddagger^{k}\tilde{\mathrm{b}}$ $\overline{7}^{\grave{\mathrm{a}}^{\backslash }]\triangleright}\cdot$ $\cdot$ $arrow w_{n} w_{n}\cdot\cdot$ 21 $w_{1}$ $w_{2}$ $w_{3}$ $\cdot$ $\cdot$ $C=C_{0}$ $arrow$ $C_{1}$ $arrow$ $C_{2}$ $arrow$ $C_{3}$ $\cdot$ $\cdot$ $\cdot$ $C_{n-1}$ $C_{n}=C \}$ $w_{1} $ $w_{2} $ $w_{3} $ $\min\{c_{i}^{(k-1)}\in C_{k-1} c_{i}^{(k-1)}>w_{k}\}$ $w_{k} $ $=$ $\{$ $\min\{c_{i}^{(k-1)}\in C_{k-1}\}$ if $\{c_{i}^{(k-1)}\in C_{k-1} c_{i}^{(k-1)}>w_{k}\}\neq\emptyset$ otherwise $C_{k}$ $=$ the sequence of numbers obtained from $C_{k-1}$ by replacing a $w_{k}$ by $w_{k}$ 2 $C=(c_{1} c_{2} \ldots c_{m})(c_{1}\leq\cdots\leq c_{m})$ $w=$ ( ) $w_{1}$ $w_{2}$ $\ldots$ $w_{n}$ $C_{0}=C$ $C =C_{n}$ $w $ $(C w)arrow(c w )$ $Cw$ $=$ $C_{0}w_{1}w_{2}w_{3}\cdots w_{n}$ $w_{1} C_{1}w_{2}w_{3}\cdots w_{n}$ $w_{1} w_{2} C_{2}w_{3}\cdots w_{n}$ $\cdot$ $w_{1} w_{2} w_{3} \cdots w_{n} C_{n}$ $=$ $w C $ 2 $\overline{f}\mathrm{i}\mathrm{j}\ldots$ $a_{-1}$ $a_{0}$ $a_{1}$ $\ldots$ $[p q]$ $p= \min\{i\in \mathrm{z} a_{i}\neq e\}$ $\int p$ $q]$ : $\{$ $q= \max\{i\in \mathrm{z} a_{i}\neq e\}+n$ 7 $\ldots$ 0 $\ldots$ $e$ $e$ $p$ $q$ $\ldots$ $1$ $e$ 1 5 $e$ $e$ $e$ $e$ $e$ $e$ $\ldots$ 2 3 $e$ $e$ $e$ $e$ $e$ $e$ $e$

$$\min\{c_{i}^{(k-1)}\in C_{k-1}\}$ if $\{c_{i}^{(k-1)}\in C_{k-1} c_{i}^{(k-1)}>w_{k}\}\neq\emptyset$ otherwise $C_{k}$ $=$ the sequence of numbers obtained from $C_{k-1}$ by replacing a $w_{k}$ by$

8 22 $e$ 31 $p$ $q$ $A=(a_{p} a_{p+1} \ldots a_{q-1} a_{q})$ $t$ # $t+1$ $Aarrow A $ n] $e$ $C=(e e \ldots e)$ 8 $\ldots$ $e$ $e$ 1 5 $e$ $e$ $e$ $e$ $e$ $e$ $\cdots$ $e$ $e$ $e$ $e$ 2 3 $e$ $e$ $e$ $e$ $\cdots$ $e$ $e$ $e$ $e$ $e+_{e}^{2}2$ $e$ $e$ $e$ $e+_{c}^{3}23$ $e$ $e$ $e+_{e}^{4}234$ $e$ $+_{2}^{e}34$ $e$ $e$ $e+_{3}^{1}14$ $e$ $e$ $e+_{c}^{5}arrow$ $eeeeee$ $arrow 145$ $e$ $+45$ $e$ $e$ $e+5$ $e$ $e$ $e$ $e+e$ $e$ $e$ $e$ $e+e$ $e$ $e$ $e$ $e+e$ $e$ $e$ $e$ $e$ $e$ $(\begin{array}{lllll}\mathrm{l} \end{array})$ $arrow$ $(\begin{array}{lllll} \end{array})$ $(\begin{array}{ll} \end{array})arrow(\begin{array}{l} \end{array})$ $b=(51236)arrow b =(74589)$ $t\mathrm{b}^{\mathrm{a}}$ 32 $t+1$ $barrow b $ $C=(l_{1} l_{2} \ldots l_{m})$ $C$ $[p q]$ 10 $b=(51236)arrow b =(74589)$ $ _{7} _{4} _{5} _{8} _{9} $ $T^{*}$ $\cdot$ $b =T^{*}(b)$

$$+45$ $e$ $e$ $e+5$ $e$ $e$ $e$ $e+e$ $e$ $e$ $e$ $e+e$ $e$ $e$ $e$ $e+e$ $e$ $e$ $e$ $e$ $e$ 14 5 9 6 $(\begin{array}{lllll}\mathrm{l} 2 3 5 62 3 4 1 5\end{array})$ $arrow$ $(\begin{array}{lllll}4 5$

9 4 ( ) $a_{i}$ 1 $a$( $a_{i}$ 1 $e$ $n=5$ $a_{7}$ $e$ 2 $a_{4}$ $e$ 3 1 ( 5) $a_{5}$ ( ) 31 3 $a_{1}=2$ 2 $a_{2}=3$ 3 $a_{3}=4$ 4 $a_{4}=e$ $3C=\{e e e e e\}$

$5 23 31 ( 2 3 4 1 5 ) 31 3 $a_{1}=2$ 2 $a_{2}=3$ 3$

10 24 $e$ 2 $a_{5}=1$ $\{3 4 e e \ldots\}$ $a_{i}$ $i$ ( ) $a_{5}=1$ $a_{5}=1$ a2 $=3$ $\ldots$ $\{ \ldots\}$ $C=\{34 e e e\}$ 5 [1]

$10 $a_{5}=1$ 7 7 5 a2 $=3$ 5 8 9 10 5 8 9 10$

11 25 21 [1] 21 (i) [3] 1 $w$ $w $ $N$ $w_{0}$ $w_{0} $ $w_{0}$ $w_{0} $ 31 $CA$ $=$ $C_{p}(a_{p} a_{p+1} \ldots a_{q-1} a_{q})$ $\cdot$ $a_{p} C_{p+1}(a_{p+1} a_{p+2} \ldots a_{q-1} a_{q})$ $(a_{p} a_{p+1} \ldots a_{q-1} a_{q} )C $ $=$ $A C$ $e$ 1 2 $n$ $CA$ $A C$ $e$ ffl A $A_{e} $ 1 (ie $A_{e}\approx A_{e} $ ) $A_{e} $ A $P$ $P$ A (ii) 2 [1] 2 $a$ $b$ $T^{*}(a)$ $T^{*}(b)$ $b$ 3 $T^{*}(b)$ 2 21 $P_{1}$ $T^{*}(Q_{1})$

$\ldots a_{q-1} a_{q} )C $ $=$ $A C$ $e$ 1 2 $n$ $CA$ $A C$ $e$ ffl A $A_{e} $ 1 (ie $A_{e}\approx A_{e} $$

12 $\bullet$ $b\approx $\bullet$ $\bullet$ $W(Q_{1})$ 26 W$ (Ql)\rightarrow T*(b)\approx T*( (Ql))( $\cdot\cdot$ 2) $T^{*}(b)=b $ and $b \approx W(Q_{2})arrow W(Q_{2})\approx T^{*}(W(Q_{1}))$ $arrow T^{*}(W(Q_{1}))$ $W(Q_{2})=T^{*}(W(Q_{1}))$ 3) ( $\cdot\cdot$ $\mathrm{i}\mathrm{j}$ $b$ $b $ ( ) $w_{1}^{*}$ $w_{2}^{*}$ $b\approx W(Q_{1})$ $b$ $Q_{1}$ 2 $(T^{*})$ $T^{*}(b)\approx T^{*}(W(Q_{1}))$ $T^{*}(b)=b $ $b$ $Q_{2}$ $T^{*}(b)\approx W(Q_{2})$ $T^{*}(b)\approx T^{*}(W(Q_{1}))$ $W(Q_{2})\approx T^{*}(W(Q_{1}))$ $W(Q_{1})$ 3 $T^{*}(W(Q_{1}))$ 2 $W(Q_{2})=T^{*}(W(Q_{1}))$ $Q_{2}$ $Q_{1}$ 21 $Q$ $w$ Tab(w) $T^{*}(Q)=\mathrm{T}\mathrm{a}\mathrm{b}(T^{*}(W(Q)))$ $[p q]\subset \mathrm{z}$ 41 $Q$ $[p q]$ $C=(l_{1} \ldots l_{m})$ $Q$ $(P Q)$ $Q$ $(Q P)$ $\llcorner$ 6 ( ) $Q$ 41 6 $P$

$T^{*}(W(Q_{1}))$ $T^{*}(b)=b $ $b$ $Q_{2}$ $T^{*}(b)\approx W(Q_{2})$ $T^{*}(b)\approx T^{*}(W(Q_{1}))$ $W(Q_{2})\approx T^{*}(W(Q_{1}))$ $W(Q_{1})$ 3 $T^{*}(W(Q_{1}))$ 2$

13 equation 5 [1] $m$ [1] K Fukuda Box-ball systems and Robinson-Schensted-Knuth correspondence (math $\mathrm{c}\mathrm{o}/ \mathrm{v}3$ ) [2] K Fukuda M Okado and Y Yamada Energy functions in box ball systems Internat J Modern Phys A15 (2000) no [3] W Fulton Young Tableaux London Mathematical Society Student Texts 35 Cambridge University Press (1997) [4] G Hatayama A Kuniba and T Takagi Soliton cellular automata associated with finite crystals Nulclear Phys $\mathrm{b}577(2000)$ [5] $\mathrm{d}\mathrm{e}$ Knuth The art of computer programming Volume 3 Sorting and searching Addison-Wesley Series in Computer Science and Information Processing Addison- Wesley Publishing Co Reading Mass-London-Don Mills Ont 1973 [6] IG Macdonald Symmetric functions and Hall $polynomials\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}$ University Press 1995 Edition Oxford [7] $\mathrm{b}\mathrm{e}$ Sagan The symmetric group Representations combinatorial algorithms and symmetric functions Second edition Graduate Texts in Mathematics 203 Springer- Verlag New York 2001 [8] D Takahashi On some soliton systems defined by using boxes and balls Proceedings of the International Symposium on Nonlinear Theory and Its Applications (NOLTA 93) 1993 pp [9] D Takahashi and J Matsukidaira Box and ball system with acarrier and ultradiscrete $\mathrm{a}^{\cdot}30$ J Phys (1997) $\mathrm{l}733-\mathrm{l}739$ modified $\mathrm{k}\mathrm{d}\mathrm{v}$ $\mathrm{j}\mathrm{p}\mathrm{n}$ [10] D Takahashi and J Satsuma Asoliton cellular automaton J Phys Soc (1990) $59$ 7 [1]

$Hatayama A Kuniba and T Takagi Soliton cellular automata associated with finite crystals Nulclear Phys $\mathrm{b}577(2000)$ 619-645 [5] $\mathrm{d}\mathrm{e}$ Knuth The art of computer programming$

14 28 [11] T Tokihiro A Nagai and J Satsuma Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization Inverse Problems 15 (1999) no [12] - $-$ sugaku books [13] NO 435 SEPTEMBER 1999 [14] pp32-41

ultra-discretization Inverse Problems 15 (1999) no 6 1639-1662

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~ ご再 ~ Title 經濟法令 Author(s) Citation 經濟論叢 (1925), 20(5): 925-942 Issue Date 1925-05-01 URL http://dx.doi.org/10.14989/128271 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University ~ ご再

$\bullet$ $\wedge$ $\Lambda$ Combinatorial aspects of box-ball systems - (Kaori Fukuda) Graduate School of Science and Technology

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