Combinatorics and Geometry in Representation Theory

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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.