18.217 Fall 2025: Coxeter Combinatorics
Bert Kostant’s game, regular and excited sponsor games.
Chip firing game, recurrent configurations and steady states, start of the proof of confluence
Proof of Roman Lemma (used to show confluence), weak Bruhat order and Young’s lattice, labeled chip firing game.
Classification of finite type graphs for Kostants game. Pattern avoidance, forbidden patterns, and allowed finite type graphs (types A_n, B_n, E_6, E_7, E_8)
Standard definition of root systems.
Review of root systems. Coxeter arrangement, Weyl chambers, permutohedron, related Newton polytopes
More on type A root sysems and the standard permutohedron. Defined △n,k .The correspondence between Weyl chambers and the Weyle group: how everything is expressed in terms of simple reflections. Properties of the Cartan Matrix.
Proof about expression of Weyl chambers/roots/reflections in terms of simple roots. Proof of equivalence between classical definition of root systems and Kostant’s game definition. Examples of Weyl chambers being simplicial cone.
Some review. More on why Weyl chambers are simplicial cones. The root poset.
Finite type, affine type, and wild type/indefinite graphs or matrices. Chip firing rules for what type a graph is, and Vinbergs theorem for what type a matrix is.
18.217 Fall 2024: Combinatorics and Geometry
Notes will be added as lectures occur. If you catch typos or errors let me know and I can add comments underneath with corrections.
Lecture 20. Wed 10/22: Problem set presentations. No notes, sorry
Lectures 31&32 guest lectures by Colin Defant. Given from powerpoint, so no written notes.
Previous semesters of 18.217 notes
One day I’ll get around to uploading these too. The main priority to start is making sure current lectures are uploaded as they occur so people taking the class now can reference them, but eventually you’ll be able to find old lectures from Fall 2022 and 2023 here as well