How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change?
(@Marek: @rschwieb: Yes well, given one represenation (with det=1), I should be able to find others. And if I take the abstract definition of the representation being a group homomorphism, then I don't immediatenly see why the value of det stays equal.)
Is the trace of the representation of a generator maybe automatically zero too?
(I added this suggestion because I thought maybe for the linear trace-operation, it's easier to push the value forward. And if the new generators have trace=0 too, then the representation of the group would have det=1 too.)