 
Summary: Seminar in Algebra and Number Theory Reflection Groups and Hecke Algebras
Fall 2005 P. Achar
Problem Set 2a
Due: September 27, 2005
1. Recall that a parabolic subgroup of a Coxeter group is a subgroup generated by a subset of the Coxeter
generators. Similarly, a subgroup of a reflection group is said to be parabolic if there exists a choice of
simple system with respect to which it is generated by a subset of the simple reflections.
Let W be a reflection group acting on the real vector space V . Prove that a subgroup W W is
parabolic if and only if there is a subspace V V such that W is the pointwise stabilizer of V in W.
(That is, W = {w W  wv = v for all v V }.)
2. A (true) root system (as opposed to what I have called a "pseudoroot system") is called crystallo
graphic if 2 , / , Z for all roots , . A reflection group is called crystallographic if it
has a crystallographic root system.
(a) Find two inequivalent (i.e., not scalar multiples of one another) crystallographic root systems for
the group Bn. This proves not only that Bn is a crystallographic reflection group, but also An and
Dn, since those are subgroups of Bn. (Historical note: Traditionally, the name "Bn" is attached
to just one of these root systems (namely, the one in which the roots for transpositions are longer
than the roots for sign changes). The other root system is called Cn. Indeed, the names "An" and
"Dn" should also properly be thought of as names of root systems rather than reflection groups,
which leads in to the next question . . . )
