• Term taught
•             Hilary term 2016-2017
• Lecturer
•             Dr. Victoria Lebed
E-mail: lebed (at) maths.tcd.ie
Office: 1.5, School of Mathematics
Any questions or remarks or calls for help are welcome!
• Schedule
•             Mon 2pm Sal1
Wed 4pm Sal1
Thu 11am Syn2
• Tutorials and homework
•           Tutorials take place on Mondays and Tuesdays at 11am, in the Old Seminar room. You are supposed to have a look at the problem sheets before the tutorial.
Tutor: Christian Marboe
E-mail: marboec (at) maths.tcd.ie

Homework 3 is already marked. You are welcome to collect your scripts from my office!

Problem sheets: T1, T2, T3, T4.
Problem sheets with partial solutions: T1A, T2A, T3A. Corrections, suggestions and questions are welcome!
Past homework and quizzes, with solutions: HW1, HW1A, Q1, Q1A, HW2, HW2A, Q2, Q2A, HW3, HW3A.
A remark on HW3: if you want to understand where the 3-dimensional representations of A5 come from, have a look here and here.

• Continuous assessment
•            3 home assignments + 2 quizzes
= 20% of the final mark

• Main references:
• J.-P. Serre, "Linear representations of finite groups"
Clear, concise and efficient. Several copies are provided by the School of Mathematics. You can borrow them from me when necessary.
• W. Fulton, J. Harris, "Representation theory: a first course"
An example-driven approach. More detailed, and might be easier to read.

• P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina, "Introduction to representation theory" [PDF]
• C. Curtis, I. Reiner, "Representation Theory of Finite Groups and Associative Algebras"

• Previous versions
•           Here you will find all the documents for the 2014-2015 version of this course, taught by Vladimir Dotsenko.

• Plan (a draft version)
1. The philosophy of representation theory. First examples. Notes
2. Representations of S3 seen geometrically and algebraically. Notes
3. Building blocks: irreducible representations. Notes
4. Characters as a tool for tracing representations: introduction. Notes
5. Characters as a tool for tracing representations: basic results. Notes
6. Character tables: an example and an algorithm. Notes
7. Representations of abelian and "close to abelian" groups. Examples: cyclic groups. Notes
8. Relations between irreducible representations. Schur's lemma.
9. Irreducible characters form a basis of the space of class functions: proof.
10. Schur's orthogonality relations. Isotypic components. Notes for lectures 8-10.
11. Tensor products of vector spaces.
12. Tensor products of representations. Notes for lectures 11-12 and 15.
13. Representations of symmetric groups and Young diagrams: introduction.
14. Representations of symmetric groups and Young diagrams: more techniques and examples.
15. Representations of S5. Notes for lectures 13-15.
16. Representations of symmetric groups and Young diagrams: some proofs.
17. An algorithm for constructing the character table of Sn for general n. Notes for lectures 16-17.
18. More on characters of Sn: Frobenius formula. Notes
19. Relating representations of different groups: induction and restriction.
20. Induction and restriction for subgroups of index 2.
21. Representations of the alternating groups An. Notes for lectures 19-21.
22. Braid groups.
23. Burau representations for braid groups.
24. How much do we know about the representation theory of braid groups?
25. Character varieties. Applications in topology: invariants of braids and knots. Notes for lectures 22-25.
26. Representations of the general linear groups GLn.
27. Schur-Weyl duality. Notes for lectures 26-27.