Math 820 - Reflection Groups: Syllabus and Homework

Syllabus

Textbook for assignments: J. Humphreys. Reflection Groups and Coxeter Groups.

All homework problems are worth 10 points unless otherwise specified.

• Due on 2/2: Derive the Coxeter presentation of a dihedral group. Hint: Consider the map from the free group, inducing a surjective map from the corresponding quotient; then do a counting argument to show that the latter is a bijection.
• Due on 2/9:
  • p.5/ex. 2 (Hint: take arbitrary w written as product of disjoint cycles; what does w^2=1 tell us?), p.11/ex. 3 (Hint: review the proof of the fact that any root can be sent to a simple one via simple reflections).
  • complete the proof on p.9, using the cases indicated there.
• Due on 2/16: p.11/ex. 2 (see the hint in the book).
• Due on 2/23: p.15/ex. 1; p. 16/ex. 2; p. 119/exercise (involving v^{-1} and w^{-1}) and example 3 (involving w_0v and w_0w).
• Due on 3/2: formula for l(wt_{ij}), w a permutation in S_n; p.122/ex. 1.
• Due on 3/9:
  • p.24/ex. 4;
  • Show that the highest root lies in the fundamental domain D.
• Due on 3/23:
  • p.24/exercise in Section 1.13;
  • Show that wC_I and w'C_I are disjoint unless w,w' are in the same coset W/W_I, in which case they coincide; also show that if I and J are different, then wC_I and w'C_J are disjoint for any w,w'.
• Due on 4/6: p.30-31/ex., p.44/ex.
• Due on 4/13: Show in detail the construction of the alcoves, particularly the fundamental one, in types B_2 and G_2.
• Due on 4/27: p.95/ex.
• Due on 5/4: p.150, section 7.4/ex. Also, compute R_{x,w} for (x,w) in S_3 equal to (1,s_1), (1,s_1s_2), (1,s_1s_2s_1), (s_1,s_1s_2), (s_1,s_2s_1), (s_1s_2,s_1s_2s_1) via the algorithm in class, and compare with Deodhar's formula (15 points).
• Due on 5/12 in my mailbox: p.156/ex.