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 (UK)
- Fridays
- 10:05 - 10:55 (UK)
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:
- W. Fulton, J. Harris; Representation Theory - A First Course
- P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina; Introduction to representation theory
- J.-P. Serre; Linear representations of finite groups
- C. Curtis, I. Reiner; Methods of Representation Theory, John Wiley & Sons, New York 1981.
- I. M. Isaacs; Character Theory of Finite Groups, Academic Press, New York 1976.
- G. James, M. Liebeck; Representations and Characters of Groups, 2nd Edition, Cambridge University Press, 2001.
Lecturer
-
Dr Martina Balagovic
- University
- University of Newcastle
Bibliography
Follow the link for a book to take you to the relevant Google Book Search page
You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.
- Introduction to representation theory (Etingof, web)
- A New Approach to the Representation Thoery of the Symmetric Groups. 2 (A. M. Vershik and A. Yu. Okounkov, arxiv)
- Character Theory of Finite Groups (I. Isaacs, book)
- Linear Representations of Finite Groups (Jean Pierre Serre, book)
- Methods of Representation Theory (C. Curtis, I. Reiner, book)
- Representation Theory (William Fulton and Joe Harris, book)
- Representations and Characters of Groups (Gordon James and Martin Liebeck, book)
Assessment
The assessment for this course will be released on Monday 10th May 2021 at 00:00 and is due in before Monday 24th May 2021 at 11:00.
The assessment for this course will be via a single take-home paper in the Spring. You will need to answer 4 questions (5 will be offered), each carrying 25 marks, and you will need 50 marks to pass.
Please note that you are not registered for assessment on this course.
Files
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Lectures
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