Title: Symmetric groups, Iwahori-Hecke algebras and knot invariants

Lecturer: Christian Kassel

Date: April 7 - 27, 2011

Time: Every Thursday, 9:00 am to 12:00 am

Place: Room 1328 at BICMR, Resource Plaza, Peking University

Abstract: We will underline two aspects of quantum groups, namely a representation-theoretic aspect and a topological aspect, by concentrating on Iwahori-Hecke algebras, which form an important class of finite-dimensional associative algebras. We start the lectures by giving a simple combinatorial construction of representations of the symmetric groups using partitions, Young diagrams and standard tableaux. We next show that these representations can be quantized into representation of Iwahori-Hecke algebras: this involves a parameter q. We finally give a short introduction to knot theory and show how to construct topological invariants of knots with the help of Iwahori-Hecke algebras.

Plan:
1. The combinatorics of partitions and Young diagrams

• - partitions, diagrams, a Heisenberg relation
• - linear algebra on standard tableaux and construction of representations of the symmetric groups
• - (if time permits) proof that these representations are irreducible and we obtain all of them

2. Quantization

• - one-parameter deformation of the previous representations
• - Iwahori-Hecke algebras: definition

3. Introduction to knot theory

• - knots, links and the classification problem
• - braid groups and relation to links and to Iwahori-Hecke algebras
• - a method to construct topological invariants from representation theory

4. Construction of the Jones polynomial, a famous knot invariant

• - basis for Iwahori-Hecke algebras
• - construction of Markov traces
• - the Jones polynomial