Combinatorics of Coxeter groups

콕세터 군의 조합론

May 15, May 16, May 22, May 23, 2015

10:30AM-12:30PM, 2:30PM-4:30PM

Lecture Hall: 2412, Bldg #E6-1.

Department of Mathematical Sciences, KAIST

Lecturer: Philippe Nadeau, CNRS & Université Claude Bernard Lyon 1, France

Abstract: Coxeter groups are fundamental structures, given by generators and relations, which are closely related to transformation groups of quadratic spaces. In fact finite Coxeter groups are precisely the finite groups of isometries of an euclidean space which are generated by reflections. More generally, the repeated occurrence of Coxeter groups in various domains of algebra, geometry or combinatorics motivates their study.

In these lectures, we will focus on the underlying combinatorial and enumerative questions raised by these groups. We will start with the study of finite reflection groups, and give the classification result based on their Coxeter presentation. Then we will give the main properties of general Coxeter groups, based mainly from the point of view of words. In the last lectures, time permitting, we willl study the specific combinatorics of affine Coxeter groups, and give an introduction to the active domain of noncrossing partitions associated to finite Coxeter groups.

Lecture 1. Motivating example: the symmetric group

Lecture 2. Finite reflection groups and their classification

Lecture 3. Coxeter groups and their geometric representation

Lecture 4. Basic properties of Coxeter groups

Lecture 5. Weak order and reduced decompositions

Lecture 6. Affine Coxeter groups

Lecture 7. Generalized noncrossing partitions

Lecture 8. Exam