The Weyl Character Formula is a result in the theory of compact Lie groups that expresses the character of a continuous, irreducible unitary representation in terms of a "highest weight". A proof will be provided that illustrates this remarkable result for the special case of U(n), where the characters and "highest weights" turn out to be familiar combinatorial objects.

I also intend to discuss how the Cayley transform,
z -> (1+z)/(1-z)
which conformally maps the imaginary axis onto the
unit circle can be generalised to map the space of 2 X 2 complex, trace zero, skew hermitian matrices onto SU(2). The pull-back of the characters of SU(2) will then be examined.