Contact | Description | Topics | Texts | Notes | Lectures | Assessment and Assignments |

mulogo

MAST90017
(620-619)
Representation Theory

Semester II 2011

Lecturer: Arun Ram, 174 Richard Berry, phone: 8344 6953, email: aram@unimelb.edu.au

Time and Location:
       Lecture Tuesday 9:00 - 10:00 Richard Berry 215
       Lecture Friday 9:00-10:00 Richard Berry 215
       Lecture/Practical Monday 12:00-1:00 Richard Berry 215

Announcements

  • No books, notes, calculators, ipods, ipads, phones, etc at the exam.
  • Tips to avoid freaking out:
    • The assignments are designed to take "an average of 7 hours per week. This is an average.
  • Tips for time management:
    • It is much easier (and safer) to run 45 min per day to attain 12 hours in 4 weeks, than to run for 12 hours solid every fourth week on Sunday.
    • To actually run 45 min, it takes me at least 15 min to psyche myself up and convince myself that it is actually not raining and so therefore I should go running, and after a 45 min run I always walk for 5 min and I always go home and have a glass of milk and tell my wife (at length) how cool I am for running 45 min per day. All in all, I waste a good 40 min when I go running for 45 min. If I were more efficient (and every so often, but rarely, I am) then it would only takes me 50 min.
    • Measurement of time is a tricky thing and requires real discipline. Teaching and research faculty at University of Melbourne recently had to complete a survey on distribution of their time on the various activities of the job: Do I count the 6 times I had to go check my email and the weather and my iPhone in the time that I spend preparing my Representation Theory lecture?
  • Tips for exam preparation:
    • The time that a 100m olympic runner (who wins a medal) is actually competing at the olympics is say (5 heats, 7sec each) 40 seconds. Successful performance during these 40 sec is impossible without adequate preparation.
    • The time that a Representation Theory student spends on the final exam is 3 hours. Successful performance during these 3hours is .....
  • Consultation hours are by appointment. Talk to me before or after class to make an appointment.
  • Prof. Ram reads email but generally does not respond.
  • The start of semester pack includes: Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file), SSLC (pdf file), SSLC responsibilities (pdf file), SSLC timelines (pdf file), Beyond third year (pdf file), Vacation scholarships (pdf file), Local third ann (pdf file).
  • It is University Policy that:           

    “a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results.  Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).

    Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).
  • Printing arrangements from computer Lab G70: Students must use their Unicards to print documents. Locations for Unicard uploaders can be found at: http://www.studentit.unimelb.edu.au/images/facilities/autoloaders.gif

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2011/MAST90017. The subject overview that one finds there:

Symmetries arise in mathematics as groups and Representation Theory is the study of groups via their actions on vector spaces. It has important applications in many fields: physics, chemistry, economics, biology and others. This subject will provide the basic tools for studying actions on vector spaces. The course will focus on teaching the basics of representation theory via favourite examples: symmetric groups, diagram algebras, matrix groups, reflection groups. In each case the irreducible characters and irreducible modules for the group (or algebra) will be analysed, developing more and more powerful tools as the course proceeds. Examples that will form the core of the material for the course include SL2, cyclic and dihedral groups, diagram algebras: Temperley-Lieb, symmetric group and Hecke algebras, Brauer and BMW algebras, compact Lie groups. Among the tools and motivation that will play a role in the study are characters and character formulas, induction, restriction and tensor products, and connections to statistical mechanics, mathematical physics and geometry.

If time permits, there may be some treatment of loop groups, affine Lie algebras and Dynkin diagrams.

Main Topics

  • (1) Categories
  • (2) Modules
  • (3) Algebras
  • (4) Lie groups and Lie algebras
  • (5) Enveloping algebras and quantum groups
  • (6) Roots and weight and Weyl's theorem
  • (7) Flag varieties and the Borel-Weil theorem
  • (8) Affine Hecke algebras
  • (9) Quiver Hecke algebras
  • (10) Cohomology of flag varieties
  • (11) Representations of symmetric groups
  • (12) Loop groups and affine Grassmannians

Assessment

Assessment will be based on three written assignments due at regular intervals during semester amounting to a total of up to 50 pages (50%), and a 3-hour written examination in the examination period (50%).

The plagiarism declaration is available here. The homework assignments are as follows:

  • Assignment 1 -- Due 5 September 2011: Write a thorough, complete and readable exposition of one of the following theorems:
    1. The Bruhat decomposition
    2. Presentation of GL_n -- David Kirszenblat and Michael Kipp
    3. Presentation of the symmetric group -- Kayed Al Qasemi
    4. Classification of the conjugacy classes in GL_n
    5. The Cartan decomposition -- Seb Strugnell
    6. The equivalence of categories between complex reductive algebraic groups and compact Lie groups
    7. The presentation and the representations of the Temperley-Lieb algebra -- Daniel Schepisi
    8. The Weyl character formula
    9. The Borel-Weil-Bott theorem -- Oliver Leigh
    10. The classification of the representations of sl_2 -- Adam Ong
    11. The classification of vector bundles on an elliptic curve
    12. The Littlewood-Richardson rule -- Sam Chow
    13. The Kirillov classification of representations of nilpotent groups
    14. The Peter-Weyl theorem -- Jackson Clarke
    15. Tannaka-Krein duality
    16. The existence and uniqueness of Drinfeld-Jimbo quantum groups -- Richard Hughes
    17. The seminormal representations of symmetric groups
    18. The configuration space for the braid arrangement is K(pi,1) -- Tom Lanigan
    19. The Artin-Wedderburn theorem -- Stephen Lonsdale
    20. The Harish-Chandra isomorphism
    21. The BGG resolution
    22. The Chevalley-Shephard-Todd theorem -- Ahmad Issa

    You must inform me, before 5 August, which theorem you are going to write about. I am open to substitutions, but you must inform me, before 5 August, which theorem you are going to write about.

  • Assignment 2: Due 31 October: Write a "referee report" on the 5 September paper of another student, and write an improved exposition.
  • The final exam will be 3 hours: To write a thorough, complete and readable exposition of the Jordan normal form theorem.

Resources part I: recommended Texts

  • see http://researchers.ms.unimelb.edu.au/~aram@unimelb/Teaching/RepThy2010/RepThy2010.html
  • see http://researchers.ms.unimelb.edu.au/~aram@unimelb/notes.html
  • Wikipedia

Resources part II: Lectures and lecture notes

  • Lecture 1, 26 July 2011: Categories and algebras -- handwritten lecture notes (pdf file)
    Math Grammar: Definitions, Theorems and How to do Proofs (pdf file)
    Examples of proofs written in proof machine (pdf file)

  • Lecture 2, 29 July 2011: Free objects, presentations, adjoint functors, Restriction and Induction - handwritten lecture notes (pdf file)
  • Lecture 3, 1 August 2011:Lecture 1. The Weyl Character formula
    Presenting the affine Hecke algebra: Iwahori and Bernstein presentations and the path model - handwritten lecture notes (pdf file)

  • Lecture 4, 2 August 2011: Restriction and Induction, vector spaces as a category (pdf file)
  • Lecture 5, 5 August 2011: Lecture 2. The Weyl Character formula
    Schur functions and generalizations: The Weyl character formula from the affine Hecke algebra point of view - hand written lecture notes (pdf file)
  • Lecture 6, 8 August 2011: The category of C[t]-modules (pdf file)

  • Lecture 7, 9 August 2011: Lecture 3. Representations of symmetric groups
    Quiver Hecke algebras: The Brundan-Kleshchev presentation of the affine Hecke algebra (pdf file)
  • Lecture 8, 12 August 2011: Lecture 4. Representations of symmetric groups
    Representations of Quiver Hecke algebras: Bead diagrams as generalizations of standard Young tableaux (pdf file)
  • Lecture 9, 15 August 2011: Sarnak's Monash lecture

  • Lecture 10, 16 August 2010: Lecture given by Craig Westerland: Lecture 5. Quantum groups and representations
    Shuffle algebras and characters of quiver Hecke algebra representations: PBW bases and canonical bases
    (pdf file)
  • Lecture 11, 19 August 2011: Cyclic, simple, and indecomposable modules and composition serie.
  • Lecture 12, 22 August 2011: Finitely generated modules over PIDs

  • Lecture 13, 23 August 2011: Smith Normal form
  • Lecture 14, 26 August 2011: Generators and relations for GL_n (pdf file)
  • Lecture 15, 29 August 2011: Jordan Normal form

  • Lecture 16, 30 August 2011:
  • Lecture 17, 2 September 2011: Norman Do: Jones polynomials from representation theory
  • Lecture 18, 5 September 2011: What do we need to learn in grad school? (pdf file)

    FIRST ASSIGNMENT DUE

  • Lecture 19, 6 September 2011: Branching rules (pdf file)
  • Lecture 20, 9 September 2011:Norman Do: The cohomology of G/B (pdf file)
  • Lecture 21, 12 September 2011: Stephen Lonsdale -- The Artin-Wedderburn theorem (pdf file)

  • Lecture 22, 13 September 2011: Daniel Schepisi -- The presentation and the representations of the Temperley-Lieb algebra (pdf file)
  • Lecture 23, 16 September 2011: Kayed Al Qasemi -- Presentation of the symmetric group (pdf file)
  • Lecture 24, 3 October 2011: Homework 1 feedback and Homework 2 organisation
    Lecture 25, 4 October 2011: Adam Ong -- The classification of the representations of sl_2
  • Lecture 26, 7 October 2011: Jackson Clarke -- The Peter-Weyl theorem

  • Lecture 27, 10 October 2011: Sam Chow -- The Littlewood-Richardson rule
  • Lecture 28, 11 October 2011: Seb Strugnell -- The Cartan decomposition
  • Lecture 29, 14 October 2011: Ahmad Issa -- The Chevalley-Shephard-Todd theorem
  • Lecture 30, 17 October 2011: Oliver Leigh -- The Borel-Weil-Bott Theorem
  • Lecture 31, 18 October 2011: Tom Lanigan -- The configuration space for the braid arrangement is K(π,1)
  • Lecture 32, 21 October 2011: Richard Hughes -- The existence and uniqueness of Drinfeld-Jimbo quantum groups
  • Lecture 33, 24 October 2011: Lecture 8. Cohomology of flag varieties
    Moment graphs for cohomology computations: i.e. the GKM technique for Schubert calculus
  • Lecture 34, 25 October 2010: Lecture 9. Loop groups and affine flag varieties
    Definition of MV cycles
  • Lecture 35, 28 October 2010: Lecture 10. Loop groups and affine flag varieties
    Crystals and Schur functions from MV-cycles: i.e. MV polytopes and their crystal structure