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 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 gG V (g)V (g-1 Collections: Mathematics