In 2015, Marcus, Spielman, and Srivastava realized that the expected characteristic polynomials of sums and products of randomly rotated matrices, like $A + U B U^*$ and $A U B U^*$ where $U$ is a random unitary matrix, behave in some ways like "finite" versions of the additive and multiplicative convolution operations in free probability. In this talk, I will show how techniques from combinatorial representation theory can help to understand these so-called finite free convolutions, focusing on the problem (which I solved in arXiv:2209.00523) of describing the commutator of randomly rotated matrices in this context.