This is a second course on ordinary representations of finite groups, which only assumes the basics covered in Groups and Representations.Representation Theory studies ways in which an abstract group can act on vector spaces by linear transformations. This has important applications in algebra, in number theory, in geometry, in topology, in physics, and in many other areas of pure and applied mathematics.In this course we will begin by reviewing the basics of representation and character theory, covered in MA3E1. Then, we will begin introducing new powerful representation theoretic techniques, including:
* Symmetric and alternating powers, Frobenius-Schur indicators, and definability over R. For example we will be able to study the following questions:
- Given an element g in a finite group G, count the number of elements x in G whose square is g.
- Given a complex representation of G, is there a change of basis after which all matrices are defined over the reals?
* More general rationality questions, central simple algebras, Schur indices.
* Induction theorems, Brauer induction and Artin induction.
* If time permits: representation theory of particular families of groups, e.g. of GL_2(F_p), of symmetric groups, and some related families of groups.
To introduce some techniques in the the theory of ordinary representations of finite groups that go beyond the basics and that are important in other areas of mathematics.
By the end of the module the student should be able to:
- quickly compute the full character table of some important groups
- investigate fields of definition of representations
- reduce many questions about a group to those about its proper subgroups
- apply their knowledge of group representations in several contexts outside of representation theory, e.g. in number theory
Isaacs, Character Theory of Finite Groups
Curtis and Reiner, Methods of Representation Theory, with Applications to Finite Groups and Orders, Vols. 1 and 2