MAGIC075: Representation Theory of Groups

Course details

A core MAGIC course

Semester

Spring 2021
Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
80

Timetable

Mondays
11:05 - 11:55
Fridays
10:05 - 10:55

Course forum

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Description

Please see the syllabus.

Prerequisites

A course on group theory and a good background in linear algebra.

Syllabus

Outline of course

This course is an introduction to the representation theory of finite groups.

We will develop the theory of finite dimensional representations of finite groups over the field of complex numbers.

The main results are that such representations are completely reducible, and completely determined by their characters. 

We will also see how characters are used to effectively calculate such decompositions.

We will finish the course by studying the representation theory of the symmetric group in more detail. 

 Course Plan

  • Review of linear algebra (complex inner product spaces, spectral theorem) 
  • General linear group 
  • (Complex) representations of groups 
  • Maschke's Theorem 
  • Morphisms of representation, Schur's Lemma 
  • Orthogonality relations 
  • Class functions, characters, and character tables 
  • Regular representation 
  • Representations of abelian groups 
  • Representations of the symmetric group 
 Reference Texts:

  1.  W. Fulton, J. Harris; Representation Theory - A First Course 
  2.  P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina; Introduction to representation theory 
  3.  J.-P. Serre; Linear representations of finite groups 
  4.  C. Curtis, I. Reiner; Methods of Representation Theory, John Wiley & Sons, New York 1981. 
  5.  I. M. Isaacs; Character Theory of Finite Groups, Academic Press, New York 1976. 
  6.  G. James, M. Liebeck; Representations and Characters of Groups, 2nd Edition, Cambridge University Press, 2001. 

Lecturer

  • MB

    Dr Martina Balagovic

    University
    University of Newcastle

Bibliography

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  • Representations and Characters of Groups (G. James, M. Liebeck, book)
  • Character Theory of Finite Groups (Isaacs, book)
  • Methods of Representation Theory ( C. Curtis, I. Reiner, book)
  • Introduction to representation theory (Etingof, )
  • Linear representations of finite groups (Serre, )

Assessment

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Recorded Lectures

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