Combinatorics and Geometry in Representation Theory

该奖项支持组合学和几何学在表示理论中的应用研究。该项目研究反射群的几何和不变理论。PI探索复多面体、仿柯赛特复合体、hecke代数和协不变代数之间的关系。技术可能有助于发展复杂反射群的aKazhdan-Lusztig细胞理论。反射群以对称群的形式出现在自然界中。例如，立方体或十二面体（或任何柏拉图立体）的对称性形成反射群。这些组是由镜像反射（关于超平面）产生的，出现在物理、生物、化学和计算机科学中。反射群在连接数学的许多不同领域方面也有着丰富的历史：离散几何、拓扑、奇点理论、超平面的排列、李氏论、组合学和不变理论。

This award supports research in combinatorics and geometry withapplications to representation theory. The project investigates thegeometry and invariant theory of reflection groups. The PI exploresrelations between complex polytopes, Coxeter-like complexes, Heckealgebras, and the coinvariant algebra. Techniques may help develop aKazhdan-Lusztig theory of cells for complex reflection groups.Reflection groups arise in nature as symmetry groups. For example, thesymmetries of the cube or dodecahedron (or any Platonic solid) forma reflection group. These groups are generated by mirror reflections(about hyperplanes) and appear in physics, biology, chemistry, andcomputer science. Reflection groups also have a rich history ofconnecting many different areas of mathematics: discrete geometry,topology, singularity theory, arrangements of hyperplanes, Lietheory, combinatorics, and invariant theory.