A:C&T

Time and place: Wednesdays, 13-17 , U Fribourg Pérolles, PER08 (Physics) 2.52. First meeting: February 24.

Description: Since Coxeter’s seminal work in the Thirties, the class of abstract groups which bears his name has become a fundamental topic at the crossroads of algebra, geometry and topology. In fact, at the roots of the importance of Coxeter groups in each of these fields (and beyond) lies their rich combinatorial structure. This lecture will offer an introduction to some of the combinatorial objects associated to Coxeter groups. We will talk about polytopes (permutohedra and associahedra); posets associated to normal forms (Bruhat order, Weak order, etc.); arrangements of hyperplanes; non-crossing partitions; Garside structures of Artin groups. If time permits, we will also address Kazhdan-Lusztig polynomials and Deligne’s theorem on classifying spaces for Artin groups of finite type. In general, the syllabus can be adapted to the interests and the background of the participants. The only prerequisite is some basic algebra.

Main literature:

• [BB] A. Björner, F. Brenti; Combinatorics of Coxeter groups; Springer GTM.
• [H] J.E. Humphreys; Reflection groups and Coxeter groups; Cambridge UP.
• [B] K. Brown; Buildings; Springer Monographs.

Format: Lecture with some exercises. For those who need credit points, an oral examination. The exact format will be discussed during the first meeting.

Contact: SNSF-Prof. Emanuele Delucchi, emanuele.delucchi "at" unifr.ch

Lecture 1, February 24. - Introduction

Coxeter Systems, representations, roots, Coxeter complexes
References: [BB] Section 1.1, 1.2; [H] Sections 5.1, 5.3, 5.13; [B] Section 1.5H
Lecture notes: [download pdf]

Lecture 2, March 2. - Combinatorics of words

A permutation representation, reduced words, characterizations through exchange and deletion properties
References: [BB] Section 1.3 and 1.4
Lecture notes: [download pdf]

Lecture 3, March 9. The weak order

Definition, lattice property, connection to posets of regions of arrangements.
References: See lecture notes.
Lecture notes: [download pdf]

Lecture 4, March 16. Bruhat order I

Definition, properties, parabolics

Lecture 5. Review

Lecture 6. Topological properties of posets

Primer in simpicial complexes and poset topology

Lecture 7. Brieskorn's conjecture for finite type Artin groups - I

CW complexes, Fundamental group(oid)s, the Salvetti complex.

Lecture 8. Brieskorn's conjecture for finite type Artin groups - II

Quotient topology, Artin groups, the arrangement graph

Lecture 9. Brieskorn's conjecture for finite type Artin groups - III

Universal cover of the Salvetti complex, positive paths in central arrangements

Lecture 10. Brieskorn's conjecture for finite type Artin groups - IV

Property ``D'' for simplicial arrangements

Lecture 11. Brieskorn's conjecture for finite type Artin groups - V

Deligne's proof concluded - Contractibility of the universal cover: simpicial arrangements are K(π,1).

Lecture 12. TBA

Lecture 12. Preparation for the exams