Reflection Group Combinatorics

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This proposal is to investigate various question in the combinatorics of finite reflection groups, at different levels of generality. A significant fraction of effort in the subject of algebraic combinatorics is devoted toward finding the right level of generality to define various combinatorial objects. Reflection groups are often the key to finding the "correct" generality. Some of the groups appearing within the projects are -- the finite general linear groups, -- the groups of symmetry of regular polytopes (both the classical real ones, as well as the complex regular polytopes considered by G.C. Shephard and by H.S.M. Coxeter), and -- the Weyl groups, arising in the theory of algebraic groups and Lie algebras.One reason to study these symmetry group comes from their transparent beauty, known in part already to the ancient Greeks, as the symmetries of the five regular solids: the cube, tetrahedron, octahedron, dodecahedron, and icosahedron. At the same time, this beauty frustrates us. One wants to comprehend these objects not as unrelated items on a list of five things: one wants to understand the features they have in common, and how they can be understand all at once, in terms of unified principles. This is one of the main goals of this project.