MAGIC075: Representation Theory of Groups

Course details

A core MAGIC course

Semester

Spring 2022
Monday, January 31st to Friday, March 25th; Monday, April 25th to Friday, May 6th

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

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, that there are finitely many irreducible representations, and that such representations are completely determined by their characters. We will also see how characters are used to effectively calculate such decompositions. We will finish the course by looking at induction and restriction functors which let us move between the categories of representations of different groups, looking briefly at what happens over other fields, and considering representations of the symmetric group. 

Prerequisites

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

Syllabus


  • Introduction, definitions and main questions
  • Review of linear algebra (complex inner product spaces, spectral theorem) 
  • Maschke's Theorem 
  • Schur's Lemma 
  • Representations of abelian groups
  • Examples (symmetric, dihedral groups)
  • The group algebra and the regular representation
  • Characters
  • Induction, coinduction, restriction
  • Other fields
  • 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

No bibliography has been specified for this course.

Assessment

Attention needed

Assessment information will be available nearer the time.

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Lectures

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