General Information

Title: Reflection Groups and Coxeter Groups
Course Number: MAT 690
Semester: Spring 2015
Credits: 3
Section: 1
Time: TTh at 9:35-10:50AM
Location: AMB 207

Instructor Information

Instructor: Dr. Dana C. Ernst
Office: AMB 176
Office Phone: 928.523.6852
Office Hours: 10:00-11:15AM on Mondays, Wednesdays, Fridays (or by appointment)

Course Information and Policies

Course Description

This course will develop the theory of Coxeter groups. The beauty of Coxeter groups lies in the rich interplay between geometry, combinatorics, and lattice theory. Most proofs about Coxeter groups exploit a combination of methods. The methods that play a role include the combinatorics of words, the geometry of arrangements of reflecting hyperplanes (or the geometry of root systems), order/lattice theory, and linear algebra. We will begin our study with geometry, closely mixed with order/lattice theory, and progress towards an understanding of the combinatorics of finite Coxeter groups. With the geometric intuition in mind, we will proceed to study the more combinatorial aspects of (not necessarily finite) Coxeter groups.

Learning Management System

We will make limited use of BbLearn this semester, which is Northern Arizona University’s default learning management system (LMS). Most course content (e.g., syllabus, course notes, homework, etc.) will be housed here on our course webpage that lives outside of BbLearn. I suggest you bookmark this page. In addition, we will utilize a Google Group to facilitate out of class discussion. I will send the class an invite to our Google Group and briefly discuss its use. The only thing I will use BbLearn for is to communicate grades.


Our textbook will be Reflection Groups and Coxeter Groups (Cambridge Studies in Advanced Mathematics) by James E. Humphreys. You can purchase the textbook through NAU’s bookstore or via any other source you wish. You can find the errata for the most recent edition of the book here.

I expect you to be reading the textbook. The only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the material in the textbook and notes whenever necessary by asking questions in class or posting questions to the course Google Group. Here’s what Paul Halmos has to say about reading mathematics.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof
use the hypothesis?


There will be 4 midterm exams and a cumulative final exam. Exam 1 and Exam 3 will be written exams and may include a take-home portion. These exams are tentatively scheduled for Thursday, February 19 and Thursday, April 2 Tuesday, April 7, respectively, and each is worth 17.5% of your overall grade. Exam 2 and Exam 4 will be oral exams taken individually in my office. The questions for the oral exam will predominately come from homework problems. Exam 2 will last 20-25 minutes and will be worth 5% of your overall grade. Students will schedule Exam 2 during the week of March 2-6. Exam 4 will last 30-40 minutes and will be worth 10% of your overall grade. Students will schedule Exam 4 during the week of April 20-24 or April 27-May 1. The final exam will be on Tuesday, May 5 at 7:30-9:30AM and is worth 20% of your overall grade. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.


You are allowed and encouraged to work together on homework. Yet, each student is expected to turn in his or her own work. In general, late homework will not be accepted. However, you are allowed to turn in up to 5 homework assignments late with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. Your overall homework grade will be worth 20% of your final grade.

Generally, homework will fall into 2 categories.

Daily Homework: Homework will be assigned each class meeting, and you are expected to complete each assignment before walking into the next class period. All assignments should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation and spelling. You will almost always write a draft of a given solution before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment.

Daily Homework will be graded on a $\checkmark$-system. You are allowed (in fact, encouraged!) to modify your written solution in light of presentations made in class; however, you are required to use the colored marker pens provided in class. I will provide more guidance with respect to this during the first couple weeks of the semester.

Weekly Homework: In addition to the Daily Homework, you will also be required to submit 2-3 formally written solutions/proofs each week. Many of the problems chosen for the Weekly Homework will be a designated subset of the Daily Homework, but there may also be a few new problems thrown into the mix. You are required to type your submission using $\LaTeX$. I will walk you through how to do this.

The Weekly Homework assignments described above are subject to the following rubric:

Grade Criteria
4 This is correct and well-written mathematics!
3 This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.
2 There is some good intuition here, but there is at least one serious flaw.
1 I don’t understand this, but I see that you have worked on it; come see me!
0 I believe that you have not worked on this problem enough or you didn’t submit any work.

Please understand that the purpose of the written assignments is to teach you to solve problems. It is not expected that you started the class with this skill; hence, some low grades are to be expected. However, I expect that everyone will improve dramatically. Improvement over the course of the semester will be taken into consideration when assigning grades.

Class Presentations

A portion of each class meeting will be devoted to students presenting their proposed solutions to a subset of the Daily Homework. Presenters will be chosen at random, but students are allowed/encouraged to volunteer to present problems. (The following is adopted from Chapter Zero Instructor Resource Manual) Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken very seriously since they spearhead the work of the class. Here are some of my expectations:

  • In order to make the presentations go smoothly, the presenter needs to have written out the solution in detail and gone over the major ideas and transitions, so that he or she can make clear the path of the proof to others.
  • The purpose of class presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the solution clear to the other students.
  • Presenters should explain their reasoning as they go along, not simply write everything down and then turn to explain.
  • Fellow students are allowed to ask questions at any point and it is the responsibility of the person making the presentation to answer those questions to the best of his or her ability.
  • Since the presentation is directed at the students, the presenter should frequently make eye contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.

Presentations will be graded using the rubric below.

Grade Criteria
4 Completely correct and clear proof or solution. Yay!
3 Proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct.
2 A partial explanation or proof is provided but a significant gap still exists to reach a full solution or proof.
1 Minimal progress has been made that includes relevant information & could lead to a proof or solution.
0 You were completely unprepared.

Your overall performance during presentations, as well as your level of interaction/participation during class, will be worth 10% of your overall grade.

Basis for Evaluation

Your final grade will be determined by your scores in the following categories.

  • Homework: 20%
  • Presentations/Participation: 10%
  • Exam 1 (written): 17.5%
  • Exam 2 (oral): 5%
  • Exam 3 (written): 17.5%
  • Exam 4 (oral): 10%
  • Final Exam (written): 20%

Determination of Course Grade

In general, you should expect the grades to adhere to the standard letter-grade cutoffs: A 100-90%, B 80-89%, C 70-79%, D 60-69%, F 0-59%.

Rules of the Game

You should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course. On the other hand, you may use each other, the course notes, your own intuition, and me.


Regular attendance is expected and is vital to success in this course, but you will not explicitly be graded on attendance. Yet, repeated absences may impact your participation grade (see above). Students can find more information about NAU’s attendance policy on the Academic Policies page. Of course, institutional excuses will be honored.

Class Etiquette

You are expected to treat each other with respect. You are also expected to promote a healthy learning environment, as well as minimize distracting behaviors. In particular, you should be supportive of other students while they are making presentations. Moreover, every attempt should be made to arrive to class on time. If you must arrive late or leave early, please do not disrupt class. Please turn off the ringer on your cell phone. I do not have a strict policy on the use of laptops, tablets, and cell phones. You are expected to be paying attention and engaging in class discussions. If your cell phone, etc. is interfering with your ability (or that of another student) to do this, then put it away, or I will ask you to put it away.

Department and University Policies

You are responsible for knowing and following the Department of Mathematics and Statistics Policies and other University policies listed here. More policies can be found in other university documents, especially the NAU Student Handbook (see appendices) and the website of the Office of Student Life.

Important Dates

Here are some important dates:

  • Monday, January 19: Martin Luther King Day (no classes)
  • Thursday, January 22: Last day to drop/add (no W appears on transcript)
  • Thursday, February 19: Exam 1
  • March 2-6 or March 9-13: Exam 2
  • Friday, March 13: Last day to withdraw from a course (W appears on transcript)
  • Monday, March 16-Friday, March 20: Spring Break!
  • *Thursday, April 2 Tuesday, April 7: Exam 3
  • April 20-24 or April 27-May 1: Exam 4
  • Tuesday, May 5: Final Exam