Finite Reflection and General Linear Groups

这项研究以有限反射群为中心——这些对称性，例如，使希腊人所知的五个柏拉图体（立方体、四面体、八面体、十二面体和二十面体）如此具有不可抗拒的吸引力和迷人。它们的神秘和美丽部分让人有点沮丧；我们想要了解它们的共同特征，而不仅仅是五个互不相关的物体的列表。人们早就知道，有足够的反射对称性是这个故事的一部分。这个项目是关于在更广泛的背景下，通过它们产生的反射和控制反射组成的代数来理解反射组。研究结果有望应用于随机排序网络，并有助于更好地理解有限一般线性群的表示理论。经典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具有共同的特征。特别令人欣慰的是，当把有限一般线性群看作一个反射群时，人们会被引导出关于它的表示理论和不变理论的问题，这些问题（从推测上）有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为研究对称群所首创的方法进行探索。同时，利用Catalan组合学与实反射群的特征零不变量理论之间的联系所产生的思想，研究有限一般线性群的特征零不变量理论。

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.