Combinatorics and Geometry in Representation Theory

University Of North Texas, Denton TX

Investigators

Abstract

This award supports research in combinatorics and geometry with applications to representation theory. The project investigates the geometry and invariant theory of reflection groups. The PI explores relations between complex polytopes, Coxeter-like complexes, Hecke algebras, and the coinvariant algebra. Techniques may help develop a Kazhdan-Lusztig theory of cells for complex reflection groups. Reflection groups arise in nature as symmetry groups. For example, the symmetries of the cube or dodecahedron (or any Platonic solid) form a reflection group. These groups are generated by mirror reflections (about hyperplanes) and appear in physics, biology, chemistry, and computer science. Reflection groups also have a rich history of connecting many different areas of mathematics: discrete geometry, topology, singularity theory, arrangements of hyperplanes, Lie theory, combinatorics, and invariant theory.