Summary: Seminar in Algebra and Number Theory Reflection Groups and Hecke Algebras
Fall 2005 P. Achar
Problem Set 1b
Due: September 6, 2005
1. (Humphreys, Exercise 1.5.2) Let W be a reflection group with simple system . Prove that no proper
subset of the set of simple reflections can generate W.
2. We have seen that an essential reflection group on an n-dimensional real vector space can be generated
by just n reflections (namely, the simple reflections with respect to some simple system). This is not
always true over other fields.
Let V = C2
, and consider the matrices
, t =
, u =