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 a 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. We will begin by reviewing the basics of representation and character theory, covered in MA3E1. Then, we will introduce 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?
* Representations of the symmetric groups following Vershik-Okounkov approach.
* Schur-Weyl duality and representations of the general linear groups.
* If time permits: induction theorems, Brauer induction and Artin induction.
To introduce some techniques in 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 real, complex and quaternionic fields representations
- understand characters of symmetric and general linear groups
Isaacs, Character Theory of Finite Groups
Curtis and Reiner, Methods of Representation Theory, with Applications to Finite Groups and Orders, Vols. 1 and 2
Fulton, Harris, Representation Theory: a first course
Ceccherini-Silberstein, Scaraborti, Tolli, Representation Theory of the Symmetric Groups: the Okounkov–Vershik Approach, Character Formulas, and Partition Algebras