In free probability, a fundamental result of Voiculescu is that random unitary matrices are asymptotically free. A representative special case is the fact that sums $A + U B U^*$ and products $A U B U^*$ of large randomly rotated matrices approximate free additive and multiplicative convolutions. In 2015, Marcus, Spielman, and Srivastava realized that in the non-asymptotic setting, one can recover "finite" analogues of these free convolutions by looking at the expected characteristic polynomials of $A + U B U^*$ or $A U B U^*$. After reviewing these finite free convolutions, I will show how techniques from combinatorial representation theory can help to understand finite free convolutions, focusing on the problem (which I recently solved in arXiv:2209.00523) of describing the commutator of randomly rotated matrices in this context. Time permitting, I will mention some potential lines of development in finite free probability (finite versions of free cumulants, $R$-diagonality, etc.) which are inspired by comparison of this result with the situation in free probability.