Combinatorial Studies of Demazure Modules

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The affine Lie algebra is of great importance in mathematical physics. We investigate representation theory of affine Lie algebras (or quantum affine algebras) from the combinatorial viewpoint.1. Demazure modules and pathsFor a quantum affine algebra there exist perfect crystals, and the crystal base of an integrable repre-sentation is described by the semi-infinite tensor product of perfect crystals (path). On the other hand, there also exists a Demazure module, which can be thought of as a finite truncation of the integrable representation. We presented a criterion for the Demazure module to have the tensor product structure in general setting. We also checked that this criterion is satisfied for almost all known perfect crystals.2. Fermionic formulaAn expression without minus sign of the affine Lie algebra character is sometimes called fermionic formula. We proved such fermionic formulae for the string and branching function when the affine Lie algebra is of type A and the highest weight of the representation is lALAMBDA_0. We also conjectured fermionic formulae of one dimensional configuration sums of classically restricted paths for an arbitrary untwisted affine Lie algebra. It is an important open problem to prove them.3. Solvable lattice modelsKuniba et al. considered a solvable vertex model of type A and performed the spectral decomposition of its corner transfer matrix. They also derived an integral equation for the transverse, longitudinal correlation length for the XXZ model at q a root of unity via quantum transfer matrix method.4. Nearly-integrable systemsOgawa considered an equation with perturbation effect from the KdV equation. To understand theoretically the selectivity of wave numbers, he used the eigenfunctions of periodic traveling wave solutions for the KdV equation, and determined the spectra of the wave numbers by perturbative calculation.