Topics in Combinatorics IV

Epiphany 2025

Time and place:   Lectures: Mon 9:00, Tue 11:00 TLC116
Problems classes:   Fri 16:00 TLC116, 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: Tue 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)

    Problems classes:

    How an exam could look like: A compilation of homework problems which could be considered as an example of an exam paper (many of these are actually exam questions from previous years). 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 14, 16, 18, and 20