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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 (ĝn). Let B l be the crystal of the U ′ q (ĝn)-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 BM ⊗ B ∼ − → B ⊗ BM 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.

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for An(1)

“…n . The origin of such a factorization is the factorization of the combinatorial R itself in a certain asymptotic domain, which has been proved uniformly for all non-exceptional types in [8]. ∞ .…”

“…In the infinite (non-periodic) lattice case, it was first invented by Takahashi [37] for type A (1) n . The origin of such a factorization is the factorization of the combinatorial R itself in a certain asymptotic domain, which has been proved uniformly for all non-exceptional types in [8].…”

“…. , u n+1 ) ∈ (Z ≥0 ) n+1 | u 1 +· · ·+u n+1 = l}, where u i denotes the number of letter i contained in a length l row shape semistandard tableau u [31,8]. The cyclic automorphism σ acts as σ((u 1 , .…”

“…Given the level set P(µ) with µ satisfying the condition (2.14), we are going to introduce the bijection Φ to the set of angle variables J (µ). Recall that Σ(p) denotes the connected component of P(µ) that involves the path p. This implies that any path p can be expressed in the form p = t • p + by some highest path p + ∈ P + (µ) and time evolution t = (T (r) 8 Although such an expression is not unique in general, one can go from P(µ) to J (µ) along the following scheme:…”