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MA3E1 Groups & Representations

Lecturer: Miles Reid

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 one-hour lectures

Assessment: Assigned work/tests 15%. Three-hour written exam 85%

Prerequisites: The Group theory and linear algebra taught in core modules

Leads To:

Content: The concept of a group is defined abstractly (as set with an associative binary operation, a neutral element, and a unary operation of inversion) but is better understood through concrete examples, for instance

  • permutation groups
  • matrix groups
  • groups defined by generators and relations. All these concrete forms can be investigated with computers. In this module we will study groups by
  • finding matrix groups to represent them
  • using matrix arithmetic to uncover new properties. In particular, we will study the irreducible characters of a group and the square table of complex numbers they define. Character tables have a tightly-constrained structure and contain a great deal of information about a group in condensed form. The emphasis of this module will be on the interplay of theory with calculation and examples.

Aims: To introduce representation theory of finite groups in a hands-on fashion.

Objectives: To enable students to:

  • understand matrix and linear representations of groups and their associated modules,
  • compute representations and character tables of groups, and
  • know the statements and understand the proofs of theorems about groups and representations covered in this module.

Books:

We will work through printed notes written by the lecturer.
A nice book that we shall not use is:
G James & M Liebeck, Representations and Characters of Groups, Cambridge University Press, 1993. Second edition, 2001. (IBSN: 052100392X).

Additional Resources

yr1.jpg
Year 1 regs and modules
G100 G103 GL11 G1NC

yr2.jpg
Year 2 regs and modules
G100 G103 GL11 G1NC

yr3.jpg
Year 3 regs and modules
G100 G103

yr4.jpg
Year 4 regs and modules
G103

Archived Material
Past Exams
Core module averages