Box-Ball Systems and Robinson-Schensted-Knuth Correspondence

We study a box-ball system from the viewpoint of combinatorics of words and tableaux. Each state of the box-ball system can be transformed into a pair of tableaux (P, Q) by the Robinson-Schensted-Knuth correspondence. In the language of tableaux, the P-symbol gives rise to a conserved quantity of the box-ball system, and the Q-symbol evolves independently of the P-symbol. The time evolution of the Q-symbol is described explicitly in terms of the box-labels.