Topics in Combinatorics IV

Epiphany 2026

Time and place:   Lectures: Tue 9:00, Fri 17:00 E101
Problems classes:   Wed 11:00 CG218, 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: Fri 14:30-15:30 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.)
  • Week 14: Coxeter groups; Exchange and Deleting Conditions. Problems class
  • Week 15: Word problem in Coxeter groups; 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; non-crystallographic finite Coxeter groups
  • Week 18: Affine Dynkin diagrams; Coxeter element, exponents. Problems class
  • Week 19: Exponents (cont.), Coxeter plane
  • Week 20: Statistics on Weyl groups; non-crossing partitions, associahedra

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