Mathematical Sciences: Matrices and Their Invariants

这个项目涉及一组不变量的环， 域K上的一般nxn矩阵，以及迹环， 由不变量环生成的非交换环和 通用矩阵 最根本的问题是找到 对于这些代数，这将解决各种显式 组合问题 这些组合问题有兴趣 而且目前看来更有可能得到解决 than the general一般problem问题of finding寻找presentations介绍. 其他问题 要考虑的是C（n，r）的商域是否纯 K上的超越，以及 在迹环中的群G的表示。 研究支持的问题涉及矩阵理论、环理论和 不变量的计算 特别是，这涉及到一个古老的 确定哪些数学函数被保留的问题 在各种变换下。 这在许多领域都很有趣 在科学和数学上都是一样的。

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.