Abstract: I will review the classification of representations of the symmetric and unitary groups, and how they are related to each other. In particular, I will describe the Young projection operators whose images give the irreducible representations. Then I will give new formulas which use the Young projection operators to construct a family of orthogonal projections which are convenient for computations. Finally, I will describe how computations of traces of maps of symmetric group representations can be used to compute traces of maps of $U(n)$ representations for all $n$ at once. If I have time, I hope to package this up in the language of traced monoidal categories.