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

Visit the MAGIC075 forum

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

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Assessment

Description

The assessment for this course will be via a single take-home paper in the Spring. You will need to answer 4 questions (5 or 6 might be offered), each carrying 25 marks, and you will need 50 marks to pass. 

Files

Files marked Lecture are intended to be displayed on the main screen during lectures.

File Associated lecture(s)
RepnsLec1.pdf Lecture
  • 25/01/2021 11:00
RepnsLec2.pdf Lecture
  • 29/01/2021 10:00
RepnsLec3.pdf Lecture
  • 01/02/2021 11:00
RepnsHW1.pdf
RepnsLec4.pdf Lecture
  • 05/02/2021 10:00
RepnsHW2.pdf
RepnsLec7.pdf Lecture
  • 15/02/2021 11:00
RepnsLec8.pdf Lecture
  • 19/02/2021 10:00
RepnsLec9.pdf Lecture
  • 22/02/2021 11:00
RepnsHW3.pdf
RepnsLec10.pdf Lecture
  • 26/02/2021 10:00
RepnsLec11.pdf Lecture
  • 01/03/2021 11:00
RepnsLec12.pdf Lecture
  • 05/03/2021 10:00
RepnsHW4.pdf
RepnsLec13.pdf Lecture
  • 12/03/2021 10:00
RepnsLec14.pdf Lecture
  • 15/03/2021 11:00
RepnsHW5.pdf
RepnsLec15.pdf Lecture
  • 19/03/2021 10:00
RepnsLec16.pdf Lecture
  • 26/04/2021 11:00
MAGIC075-notes.pdf

Lectures

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