Combinatorics and Representation Theory of Nonlinear Differential Equations

我的重点是对称群的表示理论在某些非线性微分方程组中的应用。更确切地说，我研究了在模表示理论中起重要作用的对称群的Cartan矩阵。已知在2-约化Schur函数的展开式中出现的Q-函数的系数是非负整数。这些系数被称为斯坦布里奇系数。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Cartan矩阵的初等因子和所谓的“Gartan矩阵”的初等因子重合。接下来，我引入了对称函数空间的“复合基”，并根据我们的新基对舒尔函数进行了扩展(非简化)。我发现出现的系数都是整数。至少对我来说，这种复合基是从某些仿射李代数的表示理论中自然产生的，这是我研究了多年的。目前，我们的基只针对特征2的情况而得到，但对于任何特征p都可能存在。自然出现了一个问题：这两个基即Schur函数基和我们的复合基之间的转移矩阵是什么？在与Mizukawa和Aokage的合作中，证明了这个转移矩阵的行列式是2的幂，这是一个不平凡的事实。

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.