Grant: Discovery Grants Program - Individual

Many of the modern applications of algebra in areas including physics, coding theory, and cryptology depend on a detailed understanding of algebraic structures we call groups, rings, and algebras. In many of these applications, the algebraic structures were understood well before their potential for applications was realized. "Schur Indices and Representation Theory" is a curiousity-driven research program that seeks to increase our understanding of how groups and algebras can be represented as matrices with entries in non-algebraically closed fields such as the field of rational numbers. The Schur index is the invariant which holds the key to a detailed understanding of these representations. One of the goals of the program is the development of a computer software package that would make the difficult process of Schur index calculation for representations of finite groups accessible to the mathematical research community. The theoretical research goals for this program include the study of a new theoretical tool in the area of Schur indices known as "equivalence of G-algebras", and investigations into a conjecture concerning the possible Schur indices for representations of Coxeter groups that are important in representation theory and discrete geometry.