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.

120-cell
The hyperdodecahedron has Coxeter group H4 with Coxeter graph
h4
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,

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

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