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.)