189-571B: Higher Algebra II

Assignment 5

To be presented: Wednesday, February 14.

The problem marked with a (*) is for everyone to do and to hand in on the day of the problem session. I will return this problem to you on Friday of the same week, so late problems will not be accepted!

If you are stuck on your problem, come see me during my Monday office hours (or make an appointment) to get a hint...

1. (*) Let G be a finite group and let
r:G --> GL_n(C)
be an irreducible representation of G over C. Extend r by C-linearity to a C-algebra homomorphism
r:C[G] --> M_n(C).

Let A be the image of the group ring R[G] under r. The goal of this exercise is to determine the structure of A (as an R-algebra.)

a) The algebra C[G] is equipped with a natural R-linear involution c (which we will call complex conjugation) acting trivially on G and extending the usual complex conjugation on C. A complex subspace of C[G] is said to be defined over R if it is globally preserved by c. Show that the kernel of r is defined over R if and only if the character attached to r is real valued.

b) If the character attached to r is real-valued, show that A is isomorphic either to M_n(R) or to M_n/2(H). Give examples to show that both possibilities can occur. (Hint: think of a previous problem session.)

c) If the character attached to r is not real-valued, show that A is isomorphic to M_n(C).

2. (Clotilde Paris de la Bollardiere) Let V be a finite-dimensional vector space over a field F. Define a function t from End_F(V) x End_F(V) to F by the rule
t(a,b) = trace(ab).
(Note that this is well defined since the trace of an endomorphism is well defined, independently of a choice of basis for V over F.) Show that t(a,b) is a non-degenerate symmetric bilinear form on the F-vector space End_F(V). (Recall that "non-degenerate" means that the orthogonal complement of the entire space with respect to the bilinear form is the 0 vector space.)

3. ( Mahta Kashravi) Let A be a division algebra with a countable base over an uncountable algebraically closed field F. Show that A=F. (Hint: one way to approach this problem is to use an idea similar to our proof of the the Hilbert Nullstellensatz of last semester.)