Affine Lie algebra characters and Bethe Ansatz

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In this research, we have investigated affine Lie algebra characters using a method in solvable lattice models, Bethe Ansatz. We also obtained important results on cellular automata which had not been predicted at the beginning of the project.1. Fermionic formula. Fermionic formula is a polynomial with positive integer coefficients arising from combinatorics of Bethe Ansatz. We conjectured that this polynomial gives the branching function of an integrable representation of an affine Lie algebra, and considered its evidence using the crystal theory in quantum group. We also proved this conjecture in several cases.2. Combinatorics of Bethe Ansatz. Besides fermionic formulae, there is an important sysmtem of algebraic equations, called Q-system, in combinatorial studies of Bethe Ansatz. Kuniba, with Nakanishi et al., obtained a solution of this Q-system from Bethe equations at q = 0.3. Soliton cellular automaton. Although this research was not in our mind at the beginning, there was a new progress by us in the studies of soliton sellular automata. A cellular automaton is defined from the crystal of a finite dimensional representation of a quantum affine algebra. We showed that the motion of solitons in this cellular automaton factorizes into the product of 2-body ones and their scattering rule is explicitly given using the combinatorial R of finite crystals.4. Discrete integrable systems. The above mentioned soliton cellular automaton in the case of affine Lie algebra An^<(1)> has been known to be obtained from the ultra discrete limit of the nonautonomous discrete KP equation. Nagai et al. proved the solitonical nature and constructed conserved quantities from this approach.