Arun Ram: Representation Theory 2011

MAST90017
Representation Theory

Semester II 2015

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

Time and Location:
       Lecture Tuesday 11:00 - 12:00 Physics Podium 211
       Lecture Thursday 11:00-12:00 Physics Podium 211
       Lecture/Practical Friday 12:00-13:00 Richard Berry 213 and then from 28 August Physics Podium 211

Consultation hours will be Wednesdays 14:00-17:00 in Room 174 Richard Berry.

Announcements

Prof. Ram reads email but generally does not respond.
The lectures will not be recorded.
No books, notes, calculators, ipods, ipads, phones, etc at the exam.
The start of semester pack includes: Housekeeping (pdf file), Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file), SSLC responsibilities (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).

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2015/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 submission of one problem per week (rather than three written assignments due at regular intervals during semester) (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 6 August 2015: One problem from Assignment 1 (pdf file)
Assignment 2 -- Due 13 August 2015: One problem from RepThy2015Ass2.pdf: (pdf file)
Assignment 3 -- Due 20 August 2015: One problem from RepThy2015Ass3.pdf: (pdf file)
Assignment 4 -- Due 27 August 2015: One problem from RepThy2015Ass4.pdf: (pdf file)
Assignment 5 -- Due 3 September 2015: One problem from RepThy2015Ass5.pdf:
Assignment 6 -- Due 10 September 2015: One problem from RepThy2015Ass6.pdf:
Assignment 7 -- Due 17 September 2015: One problem from RepThy2015Ass7.pdf:
Assignment 8 -- Due 24 September 2015: One problem from RepThy2015Ass8.pdf:
Assignment 9 -- Due 8 October 2015: One problem from RepThy2015Ass9.pdf:
Assignment 10 -- Due 15 October 2015: One problem from RepThy2015Ass10.pdf:
The final exam will be 3 hours. The problems on the exam will be taken from the list RepThyExamproblemlist.pdf.

Resources part I: recommended Texts

see Notes from 2010
see A Notes page

Wikipedia

Resources part II: Lectures and lecture notes

Additional Notes 2015 pdf file

Week 1: What is representation theory? (Tutorial Sheet for Lecture 1)

Lecture 1, 28 July 2015: Proof machine and Representation theory -- 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, 30 July 2015: Irreducible and indecomposable representations (pdf file)
Lecture 3, 31 July 2015: Linear algebra theorem 3 (Jordan normal form) (pdf file)

Week 2: Presentations

Lecture 4, 4 August 2015: $ℂ [t]$ -modules, sl₂ and U_qsl₂ (pdf file)
Lecture 5, 6 August 2015: Presenting the Temperley-Lieb algebra -- handwritten lecture notes (pdf file)
Lecture 6, 7 August 2011: Linear algebra theorem 1 (presentation of GL_n and Chevalley groups) (pdf file)

Week 3: Crystals

Lecture 7, 11 August 2015: Paths and crystals (pdf file)
Lecture 8, 13 August 2015: Tensor products and highest paths (pdf file)
Lecture 9, 14 August 2011: Characters and categorifications (pdf file)

Week 4: Crystals for GL_n

Lecture 10, 18 August 2015: Initial data, the tensor power crystal, and column strict tableaux (pdf file)
Lecture 11, 20 August 2015: Schur functions and the Weyl character formula (pdf file)
Lecture 12, 21 August 2011: S_k-crystals and the RSK bijection (pdf file)

Week 5: S_k-crystals and equivalences of categories

Lecture 13, 25 August 2015: S_k-crystals and the Robinson-Schensted-Knuth correspondence (pdf file)
Lecture 14, 27 August 2015: Five equivalences of categories (pdf file)
Lecture 15, 28 August 2011: Tangent vectors, one-parameter subgroups and enveloping algebras (pdf file)

Week 6: Equivalences of categories

Lecture 16, 1 September 2015: Maximal compact subgroups and Tannaka-Krein duality
Lecture 17, 3 September 2015: ??????
Lecture 18, 4 September 2015: Weights and the adjoint representation

Week 7: Conjugacy theorems--Borel subgroups, maximal compact subgroups, maximal tori and Sylow subgroups

Lecture 19, 8 September 2015: Root data for GL_n (pdf file)
Lecture 20, 10 September 2015: Conjugacy theorems--Borel subgroups, maximal compact subgroups, tori, Sylow subgroups (pdf file)
Lecture 21, 11 September 2015: Conjugacy theorems continued and the order of GL_n(F_q)

Week 8: Flag varieties and positive roots

Lecture 22, 15 September 2015: Symmetric spaces and Flag varieties
Lecture 23, 16 September 2015: Flag varieties, partial flag varieties and projective space (pdf file)
Lecture 24, 17 September 2015: Roots, positive roots and simple roots (pdf file)

Week 9: Verma modules and modules for sl₂

Lecture 25, 22 September 2015: Roots and simple roots for GL_n (pdf file)
Lecture 26, 24 September 2015: The Poincare-Birkhoff-Witt theorem, Induction and Verma modules for sl₂ (pdf file)
Lecture 27, 25 September 2015: Irreducible highest weight modules for sl₂ (pdf file)

Week 10: Representations of sl₂

Lecture 28, 6 October 2015: Verma modules for sl₂ (pdf file)
Lecture 29, 8 October 2015: Decomposition of S⁽²¹⁾⊗S⁽²¹⁾ as an S₃-module (pdf file)
Lecture 30, 9 October 2015: Weyl's character formula

Week 11: Chevalley groups and Hecke algebras

Lecture 31, 13 October 2015: The BGG-resolution
Lecture 32, 15 October 2015: Chevalley group relations and G/B for GL₂ (pdf file)
Lecture 33, 16 October 2015: Hecke algebras (pdf file)

OLD Week 4: Equivalences of categories

Lecture 10, 18 August 2015: complex reductive algebraic groups, compact Lie groups, reductive Lie algebras, Chevalley's theorem
Lecture 11, 20 August 2015: enveloping algebras, maximal compact subgroups, p-compact groups
Lecture 12, 21 August 2011: Linear algebra theorem 2 (Bruhat decomposition and flag varieties)

OLD Week 5: Conjugacy theorems and homogeneous spaces

Lecture 13, 25 August 2015: Sylow subgroups, maximal compact subgroups, maximal tori, Borel subgroups
Lecture 14, 27 August 2015: Projective spaces, Grassmannians, Flag varieties, Schubert varieties
Lecture 15, 28 August 2011: Hecke algebras

OLD Week 6: Roots, weights and Dynkin diagrams

Lecture 16, 1 September 2015: Weights, roots, Weyl groups (algebraic groups,Lie algebras) and Chevalley's theorem
Lecture 17, 3 September 2015: Weyl's theorem, Weyl's character formula
Lecture 18, 4 September 2015: Demazure character formula and Borel-Weil-Bott

OLD Week 7: Representations of symmetric groups

Lecture 19, 8 September 2015: partitions, induction and restriction and Jucys-Murphy elements
Lecture 20, 10 September 2015: The affine Hecke algebra of type GL_n
Lecture 21, 11 September 2015: Quiver Hecke algebras and their representations

OLD Week 8: The orbit method

Lecture 22, 15 September 2015: Heisenberg groups, quantum mechanics and the Stone-von Neumann theorem
Lecture 23, 16 September 2015: Nilpotent Lie algebras and Kirillov's classification by coadjoint orbits
Lecture 24, 17 September 2015: Monomial groups

OLD Week 9: The category O

Lecture 25, 22 September 2015: Verma modules and Shapovalov forms
Lecture 26, 24 September 2015: Kazhdan-Lusztig polynomials and decomposition numbers
Lecture 27, 25 September 2015: Soergel bimodules

OLD Week 10: Reflection groups

Lecture 28, 6 October 2015: Coxeter's presentations and the Chevalley-Shephard-Todd-Serre theorem
Lecture 29, 8 October 2015: Cohomology of projective spaces, Grassmannians and flag varieties
Lecture 30, 9 October 2015: Configuration spaces, Artin groups and K(π,1)

OLD Week 11: Tantalizer algebras

Lecture 31, 13 October 2015: Schur-Weyl duality, Brauer, BMW and Temperley-Lieb algebras
Lecture 32, 15 October 2015: Translation functors
Lecture 33, 16 October 2015: Conformal field theory

OLD Week 12: Preparation for the exam

Lecture 34, 20 October 2015: Vocabulary
Lecture 35, 22 October 2015: Theorems
Lecture 36, 23 October 2015: Computations and examples

Some advice and thoughts

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 .....
The start of semester pack includes: Housekeeping (pdf file - outdated link removed), Plagiarism (pdf file - outdated link removed), Plagiarism declaration (pdf file - outdated link removed), Academic Misconduct (pdf file - outdated link removed), SSLC responsibilities (pdf file - outdated link removed).
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).