MAGIC075: Representation Theory of Groups

Course details

A core MAGIC course

Semester

Spring 2025
Monday, January 27th to Friday, April 4th

Hours

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

Timetable

Mondays
11:05 - 11:55 (UK)
Fridays
11:05 - 11:55 (UK)

Course forum

Visit the https://maths-magic.ac.uk/forums/magic075-representation-theory-of-groups

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

  • Dr Martina Balagovic

    Dr Martina Balagovic

    University
    University of Newcastle

Bibliography

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Assessment

The assessment for this course will be released on Tuesday 22nd April 2025 at 00:00 and is due in before Friday 2nd May 2025 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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