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: tba

Textbooks:

The content of the second term is (mostly) covered by

James E. Humphreys, Reflection groups and Coxeter groups (the access is free from DU network)
Anders Björner, Francesco Brenti, Combinatorics of Coxeter groups (available online through DU library)
Nicolas Bourbaki, Elements of mathematics: Lie groups and Lie algebras, Chapters IV--VI
Sergey Fomin, Nathan Reading, Root systems and generalized assosiahedra

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)

Problems classes:

Homeworks: There will be weekly homework assignments. Selected exercises are to be handed in on weeks 14, 16, 18, and 20