Finite Reflection and General Linear Groups

这项研究集中在有限反射群-这些对称性，例如，使五个柏拉图固体已知的希腊人（立方体，四面体，八面体，十二面体，二十面体）如此吸引人和迷人。他们的神秘和美丽的一部分是有点令人沮丧的;我们想了解他们的共同特点，而不仅仅是五个不相关的对象的列表。人们早就知道，有足够的反射对称性是这个故事的一部分。 这个项目是关于理解反射组，通过它们产生的反射和控制反射如何组成的代数，在更广泛的背景下。工作的结果预计有应用随机排序网络和更好地理解的表示理论的有限一般线性群。经典的反射群不仅是统一的，因为它们是由反射产生的-它们还与有限域上的一般线性群共享共同的特征。特别令人欣慰的是，当把有限的一般线性群看作反射群时，人们会对它的表示论和不变理论提出问题，这些问题（在理论上）有着令人惊讶的简单和优雅的答案。在这个项目中，有限一般线性群的普通表示理论将使用Okounkov和Vershik开创的研究对称群的方法进行探索。此外，有限的一般线性群的特征p不变理论将使用加泰罗尼亚组合学和真实的反射群的特征零不变理论之间的联系所产生的思想进行研究。

This research centers on finite reflection groups -- these are the symmetries that, for example, make the five Platonic solids known to the Greeks (the cube, tetrahedron, octahedron, dodecahedron, and icosahedron) so irresistibly attractive and fascinating. Part of their mystery and beauty is a bit frustrating; we want to understand their common features, not just as a list of five unrelated objects. It has long been known that having enough reflection symmetries is part of this story. This project is about understanding reflection groups, via their generating reflections and the algebra that governs how the reflections compose, in a broader context. Results of the work are expected to have application to random sorting networks and to better understanding of the representation theory of the finite general linear groups. Classical reflection groups are not only unified in that they are generated by reflections -- they also share common features with general linear groups over a finite field. Particularly gratifying is that, when viewing the finite general linear group as a reflection group, one is led to questions about its representation theory and its invariant theory that (conjecturally) have surprisingly simple and elegant answers. In this project, the ordinary representation theory of finite general linear groups will be explored using the method pioneered by Okounkov and Vershik for studying symmetric groups. Also, the characteristic-p invariant theory of finite general linear groups will be studied using ideas arising from the connection between Catalan combinatorics and the characteristic zero invariant theory of real reflection groups.