• 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
  • Suggested reading

  • Main references:
    Clear, concise and efficient. Several copies are provided by the School of Mathematics. You can borrow them from me when necessary.
    An example-driven approach. More detailed, and might be easier to read.

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

  • Representations in other sciencesPlan (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.