12 research outputs found
Simple Algorithm for Factorized Dynamics of g_n-Automaton
We present an elementary algorithm for the dynamics of recently introduced
soliton cellular automata associated with quantum affine algebra U_q(g_n) at
q=0. For g_n = A^{(1)}_n, the rule reproduces the ball-moving algorithm in
Takahashi-Satsuma's box-ball system. For non-exceptional g_n other than
A^{(1)}_n, it is described as a motion of particles and anti-particles which
undergo pair-annihilation and creation through a neutral bound state. The
algorithm is formulated without using representation theory nor crystal basis
theory.Comment: LaTex2e 9 pages, no figure. For proceedings of SIDE IV conferenc
Soliton Cellular Automata Associated With Crystal Bases
We introduce a class of cellular automata associated with crystals of
irreducible finite dimensional representations of quantum affine algebras
U'_q(\hat{\geh}_n). They have solitons labeled by crystals of the smaller
algebra U'_q(\hat{\geh}_{n-1}). We prove stable propagation of one soliton for
\hat{\geh}_n = A^{(2)}_{2n-1}, A^{(2)}_{2n}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n
and D^{(2)}_{n+1}. For \gh_n = C^{(1)}_n, we also prove that the scattering
matrices of two solitons coincide with the combinatorial R matrices of
U'_q(C^{(1)}_{n-1})-crystals.Comment: 29 pages, 1 figure, LaTeX2
Combinatorial R matrices for a family of crystals : B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} cases
For coherent families of crystals of affine Lie algebras of type B^{(1)}_n,
D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} we describe the combinatorial R
matrix using column insertion algorithms for B,C,D Young tableaux.Comment: 39 pages, LaTeX. This is a continuation of the authors' work appeared
in "Physical Combinatorics", ed. M.Kashiwara and T.Miwa, Birkha"user, Boston,
200
Character Formulae of -Modules and Inhomogeneous Paths
Let B_{(l)} be the perfect crystal for the l-symmetric tensor representation
of the quantum affine algebra U'_q(\hat{sl(n)}). For a partition mu =
(mu_1,...,mu_m), elements of the tensor product B_{(mu_1)} \otimes ... \otimes
B_{(mu_m)} can be regarded as inhomogeneous paths. We establish a bijection
between a certain large mu limit of this crystal and the crystal of an
(generally reducible) integrable U_q(\hat{sl(n)})-module, which forms a large
family depending on the inhomogeneity of mu kept in the limit. For the
associated one dimensional sums, relations with the Kostka-Foulkes polynomials
are clarified, and new fermionic formulae are presented. By combining their
limits with the bijection, we prove or conjecture several formulae for the
string functions, branching functions, coset branching functions and spinon
character formula of both vertex and RSOS types.Comment: 42 pages, LaTeX2.0