Summary: Seminar in Algebra and Number Theory Reflection Groups and Hecke Algebras
Fall 2005 P. Achar
Problem Set 4
Due: November 3, 2005
1. (Geck-Pfeiffer, Exercise 7.1) Let k be a field, and let Mn(k) be the algebra of n × n matrices over k.
Show that the only symmetrizing traces on Mn(k) are scalar multiples of the ordinary trace function.
2. (Geck-Pfeiffer, Exercise 7.6) Suppose H is a symmetric algebra over k, and assume the symmetrizing
trace is a linear combination of characters of simple modules for H. Show that H is semisimple.
3. In class, I mentioned the following algebra and claimed that it was not semisimple:
H = C[x]/(x2
), : H C defined by (1) = (x) = 1.
Find a simple module for this algebra whose Schur element is 0.
4. Let H be the generic Hecke algebra for a finite Coxeter group W, and let V be a simple module for H.
The generic degree of V , denoted dV , is defined to be the following quotient of Schur elements:
dV = cind/cV .
Let 1 : A C be the usual specialization sending all indeterminates to 1, so that H1 C[W]. What
can you say about 1(dV )? (Hint: It may be helpful to know the following fact about characters of
finite groups: if V is an irreducible representation of the finite group G, then
V (g)V (g-1