Factorization of Combinatorial R matrices and Associated Cellular Automata

Details Factorization of Combinatorial R matrices and Associated Cellular Automata Factorization of Combinatorial R matrices and Associated Cellular Automata

2000-03-25

arxiv.org/abs/math/0003161v3

J. Stat. Phys. 102 (2001) 843-863

Mathematics

Quantum Algebra

19pages, LaTeX2e, no figure. For proceedings of The Baxter revolution in mathematical physics. (revised version)

Scientific paper

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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