Fall 2022 Math UN3951 Undergraduate Seminars: The Combinatorics of Coxeter Groups

Instructor: Cailan Li
E-mail: ccl2166@columbia.edu
Classes:Tuesdays 6:10 PM - 8:10 PM in Math 507
Office hours:Please email me to make an appointment.
I will usually be in my office on Tuesdays 5-6 PM if you have any last minute questions/concerns.

Description

Coxeter groups appear at the intersection between combinatorics, geometry, representation theory and topology. Examples of Coxeter groups include Weyl groups which are the combinatorial backbone of semisimple lie algebras and their representation theory. A slight generalization of Weyl groups are reflection groups (which are also Coxeter groups), which arise as the symmetries of regular polytopes (loosely speaking, these are polytopes with very high degrees of symmetry). In three dimensions these are exactly the Platonic solids. Below is an image of the hyperdodecahedron, or “120-cell”, a regular polytope in 4 dimensions.

The hyperdodecahedron has Coxeter group H₄ with Coxeter graph

Now, although we will spend a lecture or two on explaining what the previous sentence means, the primary focus of the seminar is on the algebraic and combinatorial aspects of Coxeter groups. In particular a large portion of the seminar focuses on the Bruhat order, one of the most important posets in combinatorics and representation theory. An (almost) complete outline of the semester with references, can be found in the Seminar Outline. Please skim the relevant sections of the references to see if the topic is something you want to present.

Prerequisites

The only formal requirement will be MATH UN2000: Intro to Higher Mathematics. MATH UN2010: Linear Algebra might be useful for one or two lectures. Similarly, MATH GU4041: Intro Modern Algebra I, might be useful for material related to groups, but we will only need the definition of a group for the seminar, and I will review/explain this in the introductory/background talk(s).

Grading

Grading will be based on participation, attendance, and effort. Specifically,

At the end of each class, on a piece of paper please submit to me (with your name and date) two "things" that you got from the talk. For more information on what a "thing" is, please see Ravi Vakil's description: Three Things. This will also serve as your attendance record for the lecture. (Note, you can write your two things down during the lecture.)
If you are speaking, then please submit to me your notes for the talk via email after the talk. If you have good handwriting, then you can handwrite and submit them. Otherwise I expect a pdf (you can write your notes on a tablet and convert to pdf for instance) or word document.
Each unexcused absence will result in the deduction of a half a letter grade (e.g. A to A-). If you have a valid reason for missing class then please let me know via email before class and attach some sort of evidence.
If you have unexcused absences dragging your grade down, you can make up for it by asking enough questions during lecture.
Any student that puts in a nontrivial amount of effort into learning the material in the seminar will get an A+, regardless of whether they understand the material well.

Expectations

Audience members are expected to actively engaged during the talk. In particular I want to emphasize that if you are confused or need an additional explanation at any point during a talk, PLEASE ASK A QUESTION . There will be no dumb questions in this seminar, so please feel free to ask anything you want (pertaining to the lecture).

Speakers are expected to put effort into making their lecture as clear, cohesive and engaging as possible. In particular, please put in more effort than just copying what is written in the references on the blackboard. There should be a logical flow to your talk and you should explain how the different things you are writing down relate to each other. Here are some guidelines/suggestions for your talk

You should present the information in the order of logical flow (if Theorem B is needed in the proof of Theorem A, then present Theorem A first) which does not always coincide with the order that the information is presented in the references.
You should present information that helped you understand the material. For example, you came up with example C that helped you understand concept D, so present example C.
Some things you should just say instead of writing the entire thing on the board such as a block of text. In this case you should write 1-3 key words relating to the block of text and say the rest.
REMINDER that instead of presenting the proof of a theorem, you can always do examples demonstrating the theorem.
Print out the notes you use for the lecture in a large font/enlarge them. Otherwise it will be hard for you to read when presenting.

We will need to meet once before your talk (typically on Sunday on Zoom), and also once again after your talk (directly after, in my office) where I will give you feedback on the talk.

Seminar Schedule

Dates	Speaker	Topic	Supplementary Images
9/6		Organizational Meeting
9/13	Cailan Li	Crash Course on Groups and Linear Algebra: Notes
9/20	Elizabeth Marks	Definitions and Examples: Notes	Figure 1.1, Figure 1.2, Figure 1.3
9/20	Tuan Dolmen	Finite Coxeter Groups and Root Systems: Notes	E_8 projection
9/27	Nirvaan Iyer	Lengths, Descents and Exchange Conditions: Notes
9/27	Hengzhi Zhang	The Longest element and Matsumoto’s theorem: Notes	Tetrahedron Axis of Symmetries
10/4	Olivia Solanot	Principle of Inclusion Exclusion: Notes
10/4	Talia Fine	Posets and Lattices: Notes
10/11	Savik Kinger	The Bruhat Order (of Sn): Notes
10/18	Ahkeel Timothy	Properties of the Bruhat Order: Notes
10/18	Cailan Li	Review
10/25	CJ Magleby	Parabolic Subgroups and the Tableau Criterion: Notes
10/25	Dawson Franz	Generating Functions: Notes
11/1	Gabi D'Agostino	q−analogs: Notes
11/1	Charlotte Coats	The Mobius Function: Notes
11/15	Sarah Kuriyama	(Semi)Standard Young Tableaux: Notes
11/15	Eli Baucom-Hays	Catalan Numbers and Fully Commutative Elements: Notes
11/22	Param Gujral	Young’s Lattice and Differential Posets: Notes
11/22	Jacob Daum	Poincare Series: Notes
11/29	Polina Zakharov	Eulerian Polynomials: Notes
11/29	Berkley Fang	Log Concavity, Unimodality and Matroids: Notes
12/6	Zara Hall	The Work of June Huh: Notes
12/6	Cailan Li	Closing Remarks and Q&A