Combinatorial Representation Theory of Affine Lie Alqebras and Symmetric Groups

我的第一个尝试是描述几个典型的仿射李代数的基本表示的重量基础。特别地，对于最简单的仿射李代数A^<（1）>_1，我考虑了基本表示的两种实现，并发现Schur函数和Schur Q-函数的模形式分别作为权基出现。通过分析这两种实现，发现了对称群的自旋分解矩阵的初等因子的一个有趣现象。即素数为2的自旋分解矩阵的元素因子都是2的幂。虽然这一事实实际上可以用模表示的一般理论来证明，但我可以利用仿射李代数A^<（1）>_1的表示来给出一个简单的证明。在研究Jack多项式的特殊化的带状多项式时，我发现了对称群的特征标表中一个有趣的事实。后来我认识到，这个事实已经发现了50多年前由利特尔伍德，其证明是有点复杂。我和研究生水川浩（Hiroshi Mizukawa）一起给出了这个事实的一个简单证明及其旋转版本。主要工具的证明再次舒尔职能和舒尔的Q-functions.In联合工作与池田武我可以获得所有的齐次多项式解决方案的非线性薛定谔层次。在这一理论中，由矩形Young图表示的Schur函数起着至关重要的作用。

My first attempt was to describe the weight basis of the basic representations of several typical affine Lie algebras. In particular, for the simplest affine Lie algebra A^<(1)>_1, I considered two realizations of the basic representation and found that the modular version of the Schur functions and Schur's Q-functions occur as weight basis, respectively. Analysing these two realizations, I found an interesting phenomenon for the elementary divisors of the spin decomposition matrices for the symmetric group. Namely the elemntary divisors of the spin decomposition matrices for prime 2 are all powers of 2. Though this fact actually can be proved by a general theory of modular representations, I could give a simple proof of this by utilizing representations of the affine Lie algebra A^<(1)>_1.Studying the zonal polynomials, which are a specialization of the Jack polynomials, I found an interesting fact in the character tables of the symmetric group. Later I recognizes that this fact had been found more than 50 years ago by Littlewood, whose proof is a bit complicated. I gave a simple proof of this fact as well as its spin version with Hiroshi Mizukawa, a graduate student. The main tools for the proof are again Schur functions and Schur's Q-functions.In the joint work with Takeshi Ikeda I could obtain all the homogeneous polynomial solutions for the nonlinear Schrodinger hierarchy. The schur functions indexed by the rectangular Young diagrams play an essential role in this theory.