Homework for Week 2

Part 1

    The permutations of different Lie types are often called Weyl groups (of those types).
  1. What is the order of the Weyl group of each type (An, Bn, Cn, and Dn)?
  2. Prove that the Weyl group of type Bn is isomorphic to the Weyl group of type Cn.
  3. Describe generators and relations for the Weyl group of type B2, C2, and D4. Can you describe generators and relations for the Weyl group of type Bn, Cn, and Dn?

Part 2

  1. Describe the permutation matrices that satisfy the equation MTJM=J where T denotes transpose and J is the square N x N matrix with:
    • (type B) J has ones along the cross-diagonal (i,N+1-i) and zeroes elsewhere (and J has an odd number of rows/columns);
    • (type D) J has ones along the cross-diagonal (i,N+1-i) and zeroes elsewhere (and J has an even number of rows/columns);
    • (type C) J has ones along the cross-diagonal in rows n+1, n+2, ..., 2n, negative ones along the cross-diagonal in rows 1, 2, ..., n, and zeroes elsewhere (and J has an even number of rows/columns).
    Start by considering these matrices when N=5 (type B), N=4 (type C), N=6 (type D). What can you say about general N?
  2. Consider any (or all!) of the Weyl groups. Can you describe a group action on the numbers {1,2,...,N}? (What should N be for each Weyl group?) How many permutations fix the number 1? Can you list the permutations that fix the number 1?
  3. Consider any (or all!) of the Weyl groups. Can you describe a group action on the binary strings of length N with exactly 1 one? (What should N be for each Weyl group?) How many permutations fix the string 10000...? Can you list the permutations that fix the string 10000...? How does this question change if you have binary strings with exactly 2 ones? Or 3 ones? (Any other thoughts?)