# Math 820 - Reflection Groups and Coxeter Groups: Syllabus and Homework

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

All homework problems are worth 10 points unless otherwise specified.

• Due on 1/30: 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 2/5:
• p.5/ex. 2
• complete the proof on p.9, using the cases indicated there.
• Due on 2/12: p.11/ex. 2,3
• Due on 2/19: p.15/ex. 1; p. 16/ex. 2
• Due on 2/26:
• p.24/ex. 4
• Show that the highest root lies in D.
• Due on 3/12: p.30-31/ex., p.44/ex.
• Due on 3/24: Hong-Kang, ex. 2.1.
• Due on 4/9: Hong-Kang, ex. 3.6.
• Due by the last day of classes: Hong-Kang, ex. 3.2 (a), (b); ex. 3.3; derive relations (4.7) from the general ones, and verify that (4.8) define a representation; ex. 4.2.; the three things to check in the proof of the tensor product rule for V(a) tensor V(1).

Cristian Lenart, Department of Mathematics, ES 116A, SUNY at Albany