Crystal basis in quantum groups and its applications

这两年主要有两大成就。第一个是关于费米子公式和相关的话题。我们将费米子公式推广到所有扭曲量子仿射代数，并对路径实现中的张量积定理、自旋子特征公式、二加和规则、Kirillov-Reshetikhin猜想、Bethe ansatz在q = 1和q = 0处的完备性等方面给出了统一的描述。所有非例外代数的组合R也用显式插入算法得到。第二部分是关于由晶体基构造的孤子元胞自动机。我们证明了小秩代数的解的散射规则与组合R是相同的。我们证明了在无限载波情况下，时间演化被分解为Weyl群算子的乘积。这极大地简化了动力学描述为粒子和反粒子的运动，它们经历了对的产生和湮灭。对于An^<(1)>的情况，我们通过探索与离散KP方程的联系构造了n孤子解。

There are two main achievements in the two years. The first one is about the fermionic formula and related topics. We generalized the fermionic formula to all the twisted quantum affine algebras and gave a unified description on various aspects including the tensor product theorem in path realization, spinon character formulae, dilogarith sum rules, the Kirillov-Reshetikhin conjecture, completeness of Bethe ansatz both at q = 1 and q = 0 and so forth. Combinatorial R for all non exceptional algebras is also obtained in terms of an explicit insertion algorithm.The second one is about the soliton cellular automata constructed from the crystal basis. We proved that the scattering rule of solitions is identical with the combinatoral R for smaller rank algebra. We proved in the infinite carrier case that the time evolution is factorized into a product of Weyl group operators. This greatly simplified the description of the dynamics into the motion of particles and antiparticles that undergo pair creation and annihilation. For An^<(1)> case, we constructed N-soliton solution by exploring the connection to the discrete KP equation.