Assignments MA796, Combinatorics of Coxeter groups, Nathan Reading

MA796, Combinatorics of Coxeter groups, Homework Assignments

These are subject to change.

General comment about assignments: You are allowed to collaborate on solving the assigned problems as long as everyone in the group gains a thorough understanding of the solution. Furthermore, each student must write up the solution in their own words based on their own understanding.

Reading assignments and exercises from the Bjorner and Brenti book are marked "BB". Reading assignments and exercises from my book chapters are marked "R".

Assignment 1, due January 28

Long before the due date, carefully read BB sections 1.1 and 1.2 and R section 10.1.

Do BB Chapter 1, exercises 1, 2, and 3.

For Problem 1: It will be useful to look back at what I called "braid relations" in class.

For Problem 2: You should prove the isomorphism, not just quote it as a known fact.

For problem 3: The group H₃ is a Coxeter group whose description you can find in Appendix A1. The alternating subgroup of a Coxeter system (W,S) is the subgroup of W consisting of elements that can be written as words in an even number of generators. Thus the alternating subgroup of S₅ is the "alternating group" on 5 symbols, which you should be familiar with. You are free to use the information about H₃ in Appendix A1. As a hint for the last assertion: Consider how the homomorphism acts on the element s₁s₂s₁s₂s₁s₃s₂s₁s₂s₁s₃s₂s₁s₂s₃, where the s_i are indexed left-to-right in the diagram shown in Appendix A1. (Why magically choose this element? It's what we will later identify as the "longest" element x₀ of H₃, so it was a good element to consider. For now, knowing why we considered it is not essential.)

Additional Problem: Suppose that R₁ and R₂ are orthogonal reflections (in the sense of the usual "dot product") in Rⁿ. Suppose the associated reflecting hyperplanes meet at an angle θ. Prove that the composition of R₁ and R₂ is a rotation through an angle 2θ. (Much of Rⁿ is fixed by these transformations, so if you set things up right, you can reduce this to a two-dimensional problem where it is just a simple computation.)

Assignment 2, due February 25

Long before the due date, carefully read BB sections 1.3-1.5.

Do BB Chapter 1, exercises 6ab, 8, 13.

Assignment 3, due March 25

Long before the due date, carefully read BB sections 2.1-2.3.

Do BB Chapter 2, exercises 1, 3b, 10, and 14.

In Problem 3, note that we did part a in class, except that we had k = 2, but then general k is an easy induction. For part b, you will want to think about what the specific isomorphism is in part a, and you can feel free to restrict your attention to the k = 2 case. If necessary, remind me, and I'll give the definition of a direct product of posets in office hours or by email. It seems to have been left out of Appendix A2.

For Problem 1, you can't use the isomorphism w mapsto w^-1, which is proved using the subword property.

Assignment 4, due April 8

Long before the due date, carefully read BB sections 2.4-2.6.

Do BB Chapter 2, exercises 5, 11, and 12.

Assignment 5, due April 22

Do BB Chapter 3, exercises 1, 5.
Helpful tools for one possible approach to Problem 1: a photocopier and 3 colored pencils or the digital equivalents.

Also do the following problems:

Problem A
Find and prove necessary and sufficient conditions on a Coxeter group W to ensure that the right weak order on W coincides with the Bruhat order on W.

Problem B
Use the Word Property to show (without appealing to the classification of finite Coxeter groups) that the Coxeter graph of a finite irreducible Coxeter group has the following properties:
(0): It is a tree. A tree means a connected graph with no cycles. Since the Coxeter group is irreducible, the graph is connected. We showed in class that the graph has no cycles. Summary: You don't need to do Part 0, just understand how it works in preparation for the other parts.
(1): It has no vertex of degree >3. The degree of a vertex is the number of edges incident to that vertex.
(2): It has at most one vertex of degree 3.
(3): It has at most one edge with a label on it. (i.e. at most one m(s,t) is >3).
(4): It does not have both a vertex of degree 3 and an edge label.

Problem C (This is a continuation of Problem B)
Now look at the classification of finite irreducible Coxeter groups (Appendix A in the text). Half of the classification is showing that no connected graph not on the list can be the Coxeter graph of a finite Coxeter group. (The other half, which we won't consider in this problem, is to verify that each graph on the list encodes a finite Coxeter group.) Problem B goes a long way towards the first half of the classification. Find a finite list of graphs that you would have to rule out to complete the first half of the classification. (To make your list finite, you will have to put a label "m" on some of your graphs and say something like "We have to rule out this graph for m>9.")

Comment on Problem C
Note that Problem C is only asking you to identify the list of graphs that must be ruled out. Probably these remaining graphs can be ruled out by the method we used for Problem B. But for a few of the graphs, that might be complicated. If you get thinking about this, let me know. I'm curious about how it could be done.