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Unit of study_

MATH4314: Representation Theory

Representation theory is the abstract study of the possible types of symmetry in all dimensions. It is a fundamental area of algebra with applications throughout mathematics and physics: the methods of representation theory lead to conceptual and practical simplification of any problem in linear algebra where symmetry is present. This unit will introduce you to the basic notions of modules over associative algebras and representations of groups, and the ways in which these objects can be classified. You will learn the special properties that distinguish the representation theory of finite groups over the complex numbers, and also the unifying principles which are common to the representation theory of a wider range of algebraic structures. By learning the key concepts of representation theory you will also start to appreciate the power of category-theoretic approaches to mathematics. The mental framework you will acquire from this unit of study will enable you both to solve computational problems in linear algebra and to create new mathematical theory.

Code MATH4314
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Assumed knowledge:
Familiarity with abstract algebra, specifically vector space theory and basic group theory, e.g., MATH2922 or MATH2961 or equivalent

At the completion of this unit, you should be able to:

  • LO1. Demonstrate a coherent and advanced understanding of the key concepts of representations of associative algebras, groups and other algebraic structures, and how they provide a unified approach to the study of symmetry.
  • LO2. Apply the fundamental principles and results of representation theory to solve given problems.
  • LO3. Distinguish and compare the properties of different types of representations, analysing them into constituent parts.
  • LO4. Rephrase algebraic problems in representation-theoretic terms and determine the appropriate framework to solve them.
  • LO5. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.
  • LO6. Devise computational solutions to complex problems in representation theory.
  • LO7. Compose correct proofs of unfamiliar general results in representation theory.