Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Description

This is a new course, whose goal is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces. It is a beautiful subject by itself and has many applications in other areas, ranging from number theory and combinatorics to geometry, quantum mechanics and quantum field theory.

Topics

1. Basic objects and notions of representation theory: Associative algebras. Algebras defined by generators and relations. Group algebras. Quivers and path algebras. Lie algebras and enveloping algebras. Representations. Irreducible and indecomposable representations. Schur's lemma. Representations of sl(2).
2. Basic general results of representation theory. The density theorem. Representations of finite dimensional algebras. Semisimple algebras. Characters of representations. Jordan-Holder and Krull-Schmidt theorems. Extensions of representations.
3. Representations of finite groups, basic results. Maschke's theorem. Sum of squares formula. Duals and tensor products of representations. Orthogonality of characters. Orthogonality of matrix elements. Character tables, examples. Unitary representations. Computation of tensor product and restriction multiplicities from character tables. Applications of representation theory of finite groups.
4. Representations of finite groups, further results: Frobenius-Schur indicator. Frobenius determinant. Algebraic integers and Frobenius divisibility theorem. Applications to the theory of finite groups: Burnside's theorem. Induced representations and their characters (Mackey formula). Frobenius reciprocity. Representations of GL(2; Fq). Representations of the symmetric group and the general linear group. Schur-Weyl duality. The fundamental theorem of invariant theory.
5. Representations of quivers. Indecomposable representations of quivers of type A1,A2,A3,D4. The triple of subspaces problem. Gabriel's theorem. Proof of Gabriel's theorem: Simply laced root systems, reflection functors.

Prerequisites

The prerequisites for the course are the standard algebra sequences Algebra I and II (18.701, 18.702) or Linear Algebra and Modern Algebra (18.700, 18.703). This means that to understand this course, it is necessary and sufficient to have a strong background in linear algebra and a decent understanding of basic algebraic structures, such as groups, rings, and fields. We will prove some general results, but a lot of the attention will be paid to examples, and there will be many hands-on exercises illustrating the course.

Textbooks

Besides the lecture notes, we will also use the beginning part of the books:

Fulton, William, and Joe Harris. Representation Theory: A First Course. Graduate texts in mathematics. Vol. 129. New York, NY: Springer, 2008. ISBN: 9780387974958.

Serre, Jean Pierre. Linear Representations of Finite Groups. Graduate texts in mathematics. Vol. 42. New York, NY: Springer-Verlag, 1996. ISBN: 9780387901909.

Grading

To pass the course, it will be required to solve homework assignments which will be assigned every Thursday and due the following Thursday. The homeworks are 75% of the grade. It is ok to collaborate on homework if you creatively participate in solving it and understand what you write. Also there will be a take-home final assignment at the end of the term, which will weigh 25% of the grade.