Canonical bases of representations were originally introduced by Lusztig. They were later studied from a different perspective by Kashiwara, and this discovery was quite influential. Mirkovic and Vilonen constructed canonical bases of algebraic representations as certain irreducible subvarieties of a loop Grassmannian attached to the duel group. This leads to nice invariants of elements of canonical bases. The first such invariant are the moment map images of varieties, we will call them the Anderson Polytopes. The work will involve the study of features of Anderson polytopes that suggest new organizational features for canonical bases. The first invariant to be studied we will call the shape of the Anderson polygon. It encodes the directions directions that the polytope spreads in from the verticies. It can be viewed algebraically as a partition of the Weyl group or in a duel geometric picture as a partition of a vector space into cones. One would like to understand what are the possible shapes of Anserson polyhedra and how they transform when one moves in the crystal graph.
The problem is mathematically interesting and quite accessible (Anderson polytopes can be introduced directly without using all the background material). The work involves different levels: planar and spatial geometry, crystallographic groups, classical groups, algebraic varieties. It should also be approachable by computer visualizations and other forms of computation.
By Ross Murray rossm (at) student (dot) umass (dot) edu