Synopsis for B2a: Introduction to Representation Theory

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2A41)

Recommended Prerequisites:

All second year algebra.

Overview

This course gives an introduction to the representation theory of finite groups and finite dimensional algebras. Representation theory is a fundamental tool for studying symmetry by means of linear algebra: it is studied in a way in which a given group or algebra may act on vector spaces, giving rise to the notion of a representation.

We start in a more general setting, studying modules over rings, in particular over euclidean domains, and their applications. We eventually restrict ourselves to modules over algebras (rings that carry a vector space structure). A large part of the course will deal with the structure theory of semisimple algebras and their modules (representations). We will prove the Jordan-Hölder Theorem for modules. Moreover, we will prove that any finite-dimensional semisimple algebra is isomorphic to a product of matrix rings (Wedderburn's Theorem over ).

In the later part of the course we apply the developed material to group algebras, and classify when group algebras are semisimple (Maschke's Theorem).

Learning Outcomes

Students will have a sound knowledge of the theory of non-commutative rings, ideals, associative algebras, modules over euclidean domains and applications. They will know in particular simple modules and semisimple algebras and they will be familiar with examples. They will appreciate important results in the course such as the Jordan-Hölder Theorem, Schur's Lemma, and the Wedderburn Theorem. They will be familiar with the classification of semisimple algebras over and be able to apply this.

Synopsis

Noncommutative rings, one- and two-sided ideals. Associative algebras (over fields). Main examples: matrix algebras, polynomial rings and quotients of polynomial rings. Group algebras, representations of groups.

Modules over euclidean domains and applications such as finitely generated abelian groups, rational canonical forms. Modules and their relationship with representations. Simple and semisimple modules, composition series of a module, Jordan-Hölder Theorem. Semisimple algebras. Schur's Lemma, the Wedderburn Theorem, Maschke's Theorem.

1. K. Erdmann, B2 Algebras, Mathematical Institute Notes (2007).
2. G. D. James and M. Liebeck, Representations and Characters of Finite Groups (2nd edition, Cambridge University Press, 2001).

Further Reading

1. J. L. Alperin and R. B. Bell, Groups and Representations, Graduate Texts in Mathematics 162 (Springer-Verlag, 1995).
2. P. M. Cohn, Classic Algebra (Wiley & Sons, 2000). (Several books by this author available.)
3. C. W. Curtis, and I. Reiner, Representation Theory of Finite Groups and Associative Algebras (Wiley & Sons, 1962).
4. L. Dornhoff, Group Representation Theory (Marcel Dekker Inc., New York, 1972).
5. I. M. Isaacs, Character Theory of Finite Groups (AMS Chelsea Publishing, American Mathematical Society, Providence, Rhode Island, 2006).
6. J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42 (Springer-Verlag, 1977).