VIGRE seminar, summer 2000: Computational Algebra and Representation Theory

Instructors

Ed Letzter, Jon McCammond

Students enrolled

Chase Franks, Larry Holifield (undergraduate mathematics students); Scott Johnson, David Miller (undergraduate physics students); Robert Main, Alicia Russell (graduate mathematics students)

Description

This seminar also had the participation of the students in our REU Program (Research Experiences for Undergraduates). In the title, "representation theory" refers broadly to the mathematics involved in finding matrix solutions to systems of noncommutative polynomial equations. Such systems typically define more abstract structures, often expressing some type of physical symmetry. This area of investigation began with Dirac's approach to quantum mechanics, and matrix representations of algebraic systems have since played a key role in mathematical physics. "Computational Algebra" refers generally to the algorithmic approach to algebraic structures, which began with Dehn's solution to the word problem for surface groups. These methods have since played a key role in the study of hyperbolic geometry. An example of the type of equation studied is provided by the "quantum Dirac-Heisenberg-Weyl commutation equation": XY - qYX = I, where X and Y are matrices, I denotes an identity matrix and q is a scalar.
Students were exposed to decidability and undecidability results for algebraic systems and a quick tutorial was provided in some of the key algorithms currently in use. These algorithms include the Buchberger algorithm for computing in Gröbner bases, and the Knuth-Bendix procedure for creating complete rewriting systems. The students were introduced to some of the implementations of these algorithms, for possible use with their projects. Projects included: Irreducible representations of algebras, semi-invariants of adjoint representations, growth and combinatorics of algebraic structures, generic matrices, and algorithmic methods.