Brief Research Description – Cathy Kriloff

I use primarily algebraic and combinatorial methods to study the representation theory of graded (also called degenerate affine) Hecke algebras that are built from the groups of symmetries of

• regular polygons in the plane,
• the regular icosahedron (or its dual the dodecahedron) in 3-space, and
• the two related dual regular polyhedra in 4-dimensional space with five-fold symmetry (the 120-cell and 600-cell).

The algebras are infinite-dimensional but their representation theory (the ways they can act as linear transformations on finite-dimensional vector spaces) is tightly controlled by combinatorial properties of the finite symmetry group from which they are built.  These non-crystallographic cases are less commonly considered than the crystallographic cases that arise within Lie theory, but the differences and similarities in their representation theory provide intriguing hints at possible new underlying objects and directions for unifying currently separate theories.  Recent generalizations of graded Hecke algebras include the specific examples listed above, providing new avenues of exploration and further motivation to study these cases.  Various appearances of non-crystallographic objects in the mathematics and physics literature (e.g., in the study of representation theory, in connection with moment graphs in geometry, in several combinatorial contexts, and in quasicrystals, amorphous solids, and wavefronts) also indicate tantalizing connections and directions for further study.