Finite free probability is a field concerned with the eigenvalues of random matrices, especially under the operation of matrix addition. The matrix A+UBU* is of particular importance, where U is a Haar-distributed random unitary matrix, and A and B are deterministic matrices. The "first-order" behaviour of this random matrix's characteristic polynomial, i.e., its expected value, is well understood. I am working to develop the theory of its second-order behaviour, i.e., its covariance matrix. Narrowing in on the variance of the determinant of A+UBU*, I will describe how Weingarten functions connect Haar unitary matrices with the representation theory of the symmetric group.