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′\(_q\)(\(\widehat {\mathfrak{g}}\) \(_n\)). Let B\(_l\) be the crystal of the U′\(_q\)(\(\widehat {\mathfrak{g}}\) \(_n\))-module corresponding to the l-fold symmetric fusion of the vector representation. For any crystal of the form B = \(B_{l_1 }\) ⊗ ...⊗\(B_{l_N }\), we prove that the combinatorial R matrix B\(_M\)⊗B\(\widetilde \to\)B⊗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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Hatayama, G., Kuniba, A. & Takagi, T. Factorization of Combinatorial R Matrices and Associated Cellular Automata. Journal of Statistical Physics 102, 843–863 (2001). https://doi.org/10.1023/A:1004803003717
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DOI: https://doi.org/10.1023/A:1004803003717