 
Summary: Seminar in Algebra and Number Theory Reflection Groups and Hecke Algebras
Fall 2005 P. Achar
Problem Set 5
Due: November 22, 2005
Let W be the cyclic group of order n, acting as a complex reflection group on C. Let s be a generator of W,
acting on C by a primitive nth root of unity .
Let q1, . . . , qn be a set of indeterminates, and let A = Z[][q±1
1 , . . . , q±1
n ]. Finally, let
H = A[Ts]/((Ts  q1) · · · (Ts  qn)
1. Consider the case n = 2, where W is actually a finite Coxeter group. Then, the definition above doesn't
quite match up with the Coxetergroup definition of the Hecke algebra. What is the relationship
between the two? (Are they isomorphic? If not, can they be made isomorphic by some sort of change
of scalars?)
2. If : A C is the map defined by (qi) = i
, show that H C C[W].
Note: For Hecke algebras of Coxeter groups, we decided at some point to assume that we were working with
algebras that possessed a specialization in which as 0 and bs 1. We then remarked that after a suitable
extension of scalars and change of variables, we could assume that as = bs  1; we also then changed the
name of bs to qs. The analogue of this for the Hecke algebra of the cyclic group is to specialize qi i
