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.
" is a project that funded work done by V. Reiner and his colleagues and students, connecting the algebra behind highly symmetric objects, and their beautiful counting formulas. For example, the first image shows the 9 reflection symmetry planes of a cube. These reflections, when one composes them in various orders, give _all_ the cube's symmetries. This is a special property shared by all 5 of the regular polyhedra (tetrahedron, cube, octahedron, dodecahedron, icosahedron) known to the ancient Greeks, and shown in the second image. It gives particular beauty, for example, to the formulas counting their faces of various dimensions. Many of these formulas are closely related to the way in which the symmetries act on the (x,y,z)-coordinates of 3-dimensional space, and particularly how they transform polynomials in those coordinates. In fact, knowing which polynomials are _invariant_, that is, fixed by the all symmetries, tells one a surprising amount about these counting formulas. This proposal made great progress in understanding why this happens, including understanding the analogous properties for regular polyhedra in higher dimensions than 3, and also for polyhedra not in _real_ space but in _complex_ space! One can classify these objects and then observe that their counting formula obey certain mystical-seeming patterns of numerology, simply by checking it for every object in the classification. This is unsatisfying and unenlightening-- the goal of this proposal was to uncover the fundamental reasons behind these patterns. One of the major outcomes was the realization that similar patterns exist also when looking not only at real spaces and complex spaces, but also spaces over what are called _finite fields_, that is, where there are only finitely many numbers in the system. This involves a topic called the _finite general linear groups_ and their _modular invariant theory_. It also has connections and application to other areas of mathematics, including representation theory, geometry, and topology.
