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Factorization of Combinatorial R matrices and Associated Cellular Automata

Goro Hatayama·Atsuo Kuniba·Taichiro Takagi
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Abstract

Solvable vertex models in statistical mechanics give rise to soliton cellular automata at q=0 in a ferromagnetic regime. By means of the crystal base theory we study a class of such automata associated with non-exceptional quantum affine algebras U'_q(\hat{\geh}_n). Let B_l be the crystal of the U'_q(\hat{\geh}_n)-module corresponding to the l-fold symmetric fusion of the vector representation. For any crystal of the form B = B_{l_1} \otimes ... \otimes B_{l_N}, we prove that the combinatorial R matrix B_M \otimes B \xrightarrow{\sim} B \otimes B_M is factorized into a product of Weyl group operators in a certain domain if M is sufficiently large. It implies the factorization of certain transfer matrix at q=0, hence the time evolution in the associated cellular automata. The result generalizes the ball-moving algorithm in the box-ball systems.

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Goro Hatayama, Atsuo Kuniba, Taichiro Takagi. 2000-08-16. Factorization of Combinatorial R matrices and Associated Cellular Automata. https://doi.org/10.1023/a%3A1004803003717

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