Algebraic Analysis of Quantum Integrable Systems

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Solvable Lattice Models We constructed a free field realization of the ABF models in regime II and the higher spin face models, and a a new type of free field realization for the eight-vertex model at a special value of the parameters.Theory of crystals and applications We established an isomorphism between crystals constructed from inhomogeneous paths and crystals for tensor products of integrable highest weight modules of quantum affine algebras, thereby obtaining fermionic character formulas in many cases. We studied the Bethe equation in the limit q = 0 and derived a new representation for weight multiplicities for tensor products of Kirillov-Reshetikhin modules. As another application, we constructed soliton cellular automata from affine crystals, and described time evlolutions and scattering rules in terms of combinatorial R. We further derived a piecewise-linear formula for the latter and showed that its de-ultra-discretizatiohn affords the non-autonomous KP equations.Conformal field theory We considered a vertex operator algebra associated with the Virasoro minimal series M(3, p), derived a fermionic formula for the character of the subspace generated by an abelian current, and obtained a monomial basis thereof. We introduced a differential ideal of symmetric polynomials spanned by Jack polynomials with a negative rational value of the parameter. In a special case it coincides with the set of all correlation functions for the current mentioned above. We also obtained a bosonic character formula for a similar subspace of integrable representations of s1.