Combinatorics and Representation Theory of Nonlinear Differential Equations

我专注于对称群的表示论在某些非线性微分方程组中的应用。更确切地说，我调查了嘉当矩阵的对称群发挥了重要作用，在模块化表示理论。已知2-约化Schur函数展开式中出现的Q-函数的系数为非负整数。这些被称为Stembridge系数。我注意到Stembridge系数的矩阵与对称群的2-模表示的分解矩阵“相似”。我证明了它们可以通过简单的列运算相互转换，并且证明了嘉当矩阵的基本因子和所谓的“嘉当矩阵”的基本因子是一致的。接下来，我介绍了对称函数空间的“复合基”和根据我们的新基扩展（非约化）的舒尔函数。我发现出现的系数都是整数。这种复合基很自然地产生了，至少对我来说，是从某些仿射李代数的表示论中产生的，我已经研究了很多年。目前，我们的基仅针对特征2的情况获得，但对于任何特征p都存在这一点是合理的。自然会出现一个问题：两个基之间的转移矩阵是什么，即，Schur函数基与我们的复合基？在与水川和Aokage的联合工作中，证明了这个转移矩阵的行列式是2的幂。这是一个重要的事实。

I focused on the applications of the representation theory of the symmetric groups to certain nonlinear systems of differential equations. More precisely I investigated the Cartan matrices of the symmetric groups which play an important role in modular representation theory. It has been known that the coefficients of Q-functions appearing in the expansion of 2-reduced Schur functions are non-negative integers. These are called the Stembridge coefficients. I noticed that the matrices of Stembridge coefficients are "similar" to the decomposition matrices for the 2-modular representations of the symmetric groups. I proved that they are transformed to each other by simple column operations, and that the elementary divisors of the Cartan matrices and those of the so-called "Gartan matrices" coincide. Next I introduced the "compound basis" for the space of the symmetric functions and expanded (non-reduced) Schur functions in terms of our new basis. I found that the appearing coefficients are all integers. This compound basis arose naturally, at least for me, from representation theory of certain affine Lie algebras, which I have been studying for many years. At the present moment our basis is obtained only for the case of characteristic 2, but it is plausible that this exists for any characteristic p. A natural problem occurs: What is the transition matrix between the two bases, i.e., Schur function basis and our compound basis ? In a joint work with Mizukawa and Aokage, it is proved that the determinant of this transition matrix is a power of 2. This is a non-trivial fact.