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.
