# Topics in Combinatorics IV

## Epiphany 2024

 Time and place: Lectures: Mon, Th 16:00 CG218 Problems classes: Fri 16:00 TLC123, Weeks 12,14,16,18
Instructor: Pavel Tumarkin
e-mail: pavel dot tumarkin at durham dot ac dot uk
Office: MCS3009; Phone: 334-3085
Office hours: Mon 14:15-15:00 and by appointment

 Textbooks:

The content of the second term is (mostly) covered by
All sources above contain much more material than will be covered.

Preliminary course content (subject to change): Root systems, Dynkin diagrams, the Cartan-Killing classification; generalized Cartan matrices, affine Dynkin diagrams and Weyl groups; reflection groups and Coxeter groups, word problem in Coxeter groups, the Bruhat order; Coxeter elements, polynomial invariants, exponents; Coxeter-Catalan combinatorics, the root poset, the Catalan arrangement, generalised associahedra.

Schedule:

• Week 11: Linear reflections, finite reflection groups
• Week 12: Classification of finite reflection groups. Problems class
• Week 13: Classification of finite reflection groups (cont.); Coxeter groups
• Week 14: Exchange and Deleting Conditions; word problem in Coxeter groups. Problems class
• Week 15: Word problem in Coxeter groups (cont.); Coxeter groups and reflection groups
• Week 16: Root systems: classification, positive and negative roots, root poset, highest root. Problems class
• Week 17: Explicit construction of root systems
• Week 18: Non-crystallographic finite Coxeter groups, affine Dynkin diagrams. Problems class
• Week 19: Coxeter element, exponents, Coxeter plane
• Week 20: Statistics on Weyl groups; non-crossing partitions, associahedra

Handouts: (lecture notes, intended to be more or less up to date)

Problems classes:

How an exam could look like: A compilation of homework problems which could be considered as an example of an exam paper. Solutions can be found in the HW solutions sheets.

Homeworks: There will be weekly homework assignments. Selected exercises are to be handed in on weeks 13, 15, 17, and 19