RUI-Combinatorial Representation Theory

Macalester College, Saint Paul MN

Investigators

Abstract

The investigator will continue his collaborations with undergraduate researchers on combinatorial aspects of the representation theory of a collection of algebraic structures: Chevalley groups, Iwahori-Hecke algebras, affine Hecke algebras, centralizer algebras, and quasi hereditary algebras. This research is unified by its methodology: using discrete, combinatorial structures and algorithms to give explicit answers to algebraic questions. Combinatorial analyses will be performed on topics such as irreducible representations and characters, unipotent representations of Chevalley groups, flag varieties, the Chevalley-Hecke correspondence, Springer representations, and Langlands parameters. The goal of this project is to use discrete structures to model and analyze abstract properties of algebraic systems that are used in particle physics, data analysis, data encryption, engineering, statistical mechanics, and geometry. These discrete structures are concrete objects that can be enumerated and investigated on a computer. Using computer explorations, investigators have made startling new theoretical discoveries, and computation has become a significant method of mathematical investigation. The work in this project furthers the understanding of these algebraic systems and makes them more accessible to computational studies. Furthermore, these methods allow the investigator to fully involve undergraduate students in research. Young researchers can construct, manipulate, and analyze these objects without needing a huge amount of advanced training. At the same time these manipulations reveal deep truths about fundamental systems from algebra, geometry, and topology.