This course is about finite groups of matrices. Examples include symmetries of regular polygons and the Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron), as well as permutation matrices and other more combinatorial objects. Groups useful in the sciences often naturally arise as symmetries in space, which can be written down using matrices.
We will cover the basic machinery: irreducible representations; decompositions of representations; character theory; orthogonality relations; applications to the structure of groups; group algebras; induced representations; and representations over various fields. We will also study special classes of groups, like rotation groups, linear groups over finite fields (e.g. SL2(Fq)), and especially symmetric groups.
The main reference is J.P. Serre, Linear Representations of Finite Groups, which has been ordered from the bookstore. Other relevant texts are J. Harris and W. Fulton, Representation Theory and B. Sagan, The Symmetric Group.
Prerequisites: A firm grasp of linear algebra is essential. Some experience with abstract algebra will be useful.
Assessment: Weekly problem sets (40%): Due each Tuesday starting September 1. They should be turned in during class. Homework assignments are not pledged. You are strongly encouraged to work together, though each student should write up her/his own submission.
Midterm exam (20%): A closed-book 80-minute take-home test, to be taken sometime between Thursday, October 1 and Tuesday, October 6.
Final exam (40%): A closed-book three-hour take-home exam.
ADA statement: If you have a documented disability that will impact your work in this class, please contact me to discuss necessary accommodations. Additionally, you will need to register with the Disability Support Services Office in the Ley Student Center.
Office: Herman Brown 402
Phone: (713) 348-5261