Teaching

Syllabus

Instructor: Christian Gaetz (Office: Evans 1083; Contact: email) 

Meetings: Tu/Th 2:10-3:30pm in Evans 736; office hours by appointment

Description: Coxeter groups appear in many areas of mathematics including representation theory, algebraic geometry, geometric group theory, convex geometry, invariant theory, and more. This class will cover aspects of Coxeter groups which have a combinatorial flavor.  I plan to cover Bruhat order, weak order, Matsumoto's theorem, root systems, finite type classification, regular polytopes, the word problem, Kazhdan-Lusztig polynomials/representations, and some enumerative results. I'll talk about connections to invariant theory, representation theory, geometry, etc. as they come up (but background in these areas won't be required).

References (available electronically from library): 

• Combinatorics of Coxeter groups by Bjorner and Brenti

• Reflection groups and Coxeter groups by Humphreys

Grading: I will post four problem sets. To get an "A" you should either turn in at least three predominantly correct problem sets, or turn in two predominantly correct problem sets and give a 40-minute lecture on an additional topic related to the course. Problem sets must be written using LaTeX and turned in via email.

Problem Sets:

1. [Due 9/17] Problem Set 1 

2. [Due 10/24] Problem Set 2

• [By 11/7] Email me if you'd like to give a lecture in lieu of a problem set

3. [Due 11/14] Problem Set 3 

4. [Due 12/5] Problem Set 4 

Extra Reading:

• Brion's lectures on the geometry of flag varieties (especially Sections 1.1-1.3)

• Stanley's paper and survey on combinatorial applications of the hard Lefschetz theorem

• Barkley & Speyer's lattice structure on affine extended weak order

• Rhoades' cyclic sieving paper 

Weekly Schedule:

More detail and references will be added as we progress through the material. 

• 8/29: Overview; sources of Coxeter groups in mathematics

• 9/3 & 9/5: Reduced words: exchange property, deletion property

• 9/10 & 9/12: Bruhat order: subword property, chain property, lifting property, automorphisms

• 9/17 & 9/19: Parabolic subgroups and quotients, tableau criterion, Stanley's proof of Erdos-Moser conjecture

• 9/24 & 9/26: Weak order: lattice property, Tits' lemma / Matsumoto's theorem, finite type classification

• 10/1 & 10/3: The geometric representation, root systems

• 10/8: Biclosed sets, extended weak order, the root poset

• 10/15: Poset topology of Bruhat intervals [Guest lecture by Michelle Wachs] 

• 10/17: Structure of Bruhat intervals, connection to BGG resolution

• 10/22 & 10/24: Hecke algebras and Kazhdan-Lusztig polynomials

• 10/29 & 10/31: Kazhdan-Lusztig basis, Deodhar decompositions, reflection orderings

• 11/5 & 11/7: Kazhdan-Lusztig cells and representations 

• 11/12 & 11/14: Robinson-Schensted correspondence, cells for the symmetric group

• 11/19: Promotion, evacuation, and the KL basis; cyclic sieving and Rhoades' theorem; fake degrees

• 11/21: Student presentations: Yu Huang (Coxeter matroids) & Xiangru Zeng (Kazhdan-Lusztig-Stanley polynomials)

• 11/26: Student presentations: John Lentfer (the Coxeter complex)

• 12/3: Student presentations: Cameron Chang (Chevalley-Shephard-Todd) & Peisheng Yu (Chevally-Solomon-Bott)

• 12/5: Student presentations: Gabriel Raposo (Jucys-Murphy elements) & Joe Hlavinka