Undergraduate Representation Theory: Professor Karen E. Smith
Jyvaskylan Yliopisto MaD, T 2-4 and W 12-2.
Plus an exercise session run by Lauri Kahanpaa
Course assistant: Lauri Kahanpaa
Intended audience: advanced undergraduate math and physics majors.
Course Objective: To give students a concrete introduction to groups theory through their representations. The course has two distinct parts: the first focuses mostly on finite groups on finite dimensional vector spaces over the complex numbers, and the second on Lie Groups. Representation theory is a standard subject all mathematicians (and physicists) should know. I hope to remedy what I see as a serious gap in the standard math curricula, which never even defines a group representation at the advanced undergraduate and even beginning graduate level!
Prerequisites: For the first half, a good linear algebra course (eg Michigan's 217), a basic algebra course (Michigan's 412 is sufficient, or some exposure to elementary number theory or some other abstract course), prior exposue to proofs. The second half is more demanding, incorporating more (basic point-set) topology (Michigan 's Math 590) and advanced calculus (essentially elementary differential geometry in R^n). *NOTE: The first half of the course is a stand-alone mini-course on the representations of finite groups.
Text: None. I hope to produce my own notes. One good text which will have much of the background material is Dummit and Foote's Algebra book. Students will be assigned some reading from that book. When we are ready, the early chapters of Fulton and Harris's Representation theory book will be our guide, but because that book is quite a bit more advanced than this course, I will of course fill in many details.
The developing course notes for the first half of the course. PLEASE TELL ME YOUR COMMENTS! Note: due to the arrival of my PhD students, the notes are now being typed by Lauri Kahanpaa (in Finnish). See his webpage.
Some topics I hope to cover:
First Half: Lots of examples of groups (beginning with dihedral, then symmetric group, other symmetry groups), lots of actions of groups on sets (proving Cayley's theorem that every group is isomorphic to some transformation group for instance), orbits, class formula, subgroups and quotient groups, Cayley graphs, lots of review of linear algebra (including tensor and exterior products), and some groups of linear transformations (SL, GL, O, etc). Linear representations of groups, the regular representation, irreducible representations, complete reduciblity, Schur's lemma, characters. Classification of representations of finite groups over C.
Basics of Lie groups (including definintion of manifolds in R^N: charts and parametrizations, smooth functions, the tangent space to a point on a manifold,
the lie algebra as tangent space at identity with many explicit computations of tangent space to identity matrix for small Lie groups (GL2, SL2, SO2, S^1. etc). Abstract Lie algebras, the adjoint representation of a Lie group and lie algebra, the classical and modern points of view on the exponential map.
The lie algebra as invariant vector fields and derivations.
The fundamental theorem that representations of a simply connected and connected group is equivalent to representations of its lie algebra.
Explicit classification of the representations of SL_2.
PLANNED LECTURE TOPICS
SECOND PART OF COURSE: LIE GROUPS