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# 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.