Seminar in Algebra and Number Theory Reflection Groups and Hecke Algebras Fall 2005 P. Achar 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 "pseudo-root 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 . . . ) Collections: Mathematics