Mathematical Sciences: Matrices and Their Invariants

本文研究域K上一组一般nxn矩阵的不变量环，以及由不变量环与一般矩阵生成的非交换环——迹环。最根本的问题是找到这些代数的表示，这将解决各种显式组合问题。这些组合问题有自己的权利，目前似乎比寻找表示的一般问题更有可能得到解决。其他要考虑的问题是C（n,r）的商域是否纯超越于K，以及群G在迹环中的各种表示的结构是什么。研究支持矩阵理论、环理论和不变量的计算。特别是，这涉及到一个老问题，即确定在各种变换下保留哪些数学函数。这在许多科学领域和数学领域都是令人感兴趣的。

This project concerns the ring of invariants of a set of generic nxn matrices over a field K, and the trace ring, which is the noncommutative ring generated by the ring of invariants and the generic matrices. The fundamental problem is to find presentations for these algebras, which would solve various explicit combinatorial problems. These combinatorial problems have interest in their own right, and currently seem more likely to be solved than the general problem of finding presentations. Other questions to be considered are whether the quotient field of C(n,r) is purely transcendental over K and what is the structure of the variety of representations of a group G in the trace ring. The research supported concerns matrix theory, ring theory and the calculation of invariants. In particular, this involves an old problem of determining what mathematical functions are preserved under various transformations. This is of interest in many areas of science as well as in mathematics.