
620619
Representation Theory

Lecturer: Arun Ram, 174 Richard Berry, phone: 8344 6953, email: aram@unimelb.edu.au
Time and Location:
620619 L01/01 Wednesday 11:00am1:00pm Richard Berry215 2 hours (L=Lecture)
620619 P01/01 Thursday 9:00pm  10:00pm Richard Berry215 1 hour (P=practical)
Subject Outline
Representation Theory is the art of studying complex structures and symmetries by "representing" them as matrices. Viewing the matrices as linear transformations makes the subject into the study of modules. A general representation/module is a large and complex structure like a molecule and is composed of smaller "irreducible components'' which are analogous to atoms. Often, even the "atomic" representations/modules are intricate and the goal of Combinatorial Representation Theory (and this course) is to explore elementary models which allow us to (more) easily determine properties of these representations/modules: size, number of components, splitting and combination rules, and character. The type of models that are the most useful have the flavor of games for children, like legos or erector sets, and yet these models enable one to obtain very explicit information about the fine structure of the corresponding representations, which may have complexities on the order of the microscopic structure of living cells.
Main Topics
 Modules
 "Small" examples: SL2, cyclic and dihedral groups,
 diagram algebras: TemperleyLieb, the symmetric group, Hecke algebras, and Brauer algebras,
 compact Lie groups, Dynkin diagrams
 characters and character formulas,
 Induction, restriction, and tensor products: branching rules
 loop groups, affine Lie algebras
 connections to statistical mechanics, mathematical physics and geometry.
References
Week by week topics
 Week 1, 15 March 2010: Algebras and presentations: Problem sheet 1 is here (pdf file).
 Week 2, 812 March 2010: Modules and examples
 Week 3, 1519 March 2010: Schur's lemma and ArtinWedderburn
 Week 4, 2226 March 2010: Crystals
 Week 5, 29 March  2 April: Symmetric functions and Weyl's character formula
 Week 6, 1216 April: Lie groups and Lie algebras
 Week 7, 1923 April: Symmetric groups and Hecke algebras
 Week 8, 2630 April: BorelWeilBott
 Week 9, 48 May: Fourier analysis
 Week 10, 1115 May: Double coset decompositions
 Week 11, 1822 May: Examples
 Week 12, 2529 May: Examples
Lectures
Lectures week 1: Problem sheet 1 is here (pdf file)
The first homework is found here (pdf version)  due 19 April.
Problem sheet 1 is here (pdf file)
The second homework is found here (pdf version)  due 31 May.
Lectures week 2
Lectures week 3
 Handwritten lecture notes from 2008.
 Typed lecture notes from Emily Peters.
 A. Ram, Dissertation Chapter 1, Unpublished chapter of Ph.D. Thesis
 F. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter graphs and Towers of Algebras, MSRI publications 14, SpringerVerlag New York, 1989. ISBN 0387969799.
Lectures week 4
Lectures week 5
Lectures week 6
Lectures week 7
Lectures week 8
Lectures week 9
 Handwritten lecture notes from 2008.
 Typed lecture notes from Tharatorn Supasiti.
 Lectures on Lie Groups and Lie Algebras, Carter, Segal, and MacDonald, London Math Society Student Texts 32 University of Cambridge Press, 1995.
Lectures week 10
Lectures week 11
Lectures week 12
 Handwritten lecture notes.
 Class notes taken by Tharatorn Supasiti.
 Lectures on Lie Groups and Lie Algebras, Carter, Segal, and MacDonald, London Math Society Student Texts 32 University of Cambridge Press, 1995.
 R. Slansky article, Group Theory for unified model building, PHYSICS REPORTS(Review Section of Physics Letters) 79, No. 1 (1981) 1—128. NorthHolland Publishing Company.
Assessment
Assessment will be based on two assignments to be handed in during semester (worth 50%) and a final 3hour exam at the end of semester (worth 50%). Problem sheet 1 is here (pdf file).
The plaigiarism declaration is available
here.
Problem sheet 1 is
here (
pdf file).The first homework is found
here (
pdf version)  due 19 April. The second homework is found
here (
pdf version)  due 31 May.
Problem sheet 1 is
here (
pdf file)
.