Averages of traces and ratios of characteristic polynomials of random unitary matrices

In this work we explore the extent to which the Ratios Theorem for averages over $U(N)$, with respect to Haar measure, of ratios of characteristic polynomials can be used to evaluate averages of traces of unitary matrices. We derive concrete expressions for all averages involving up to 4 traces, valid for any matrix size, $N$. We also calculate the so-called zero-swap and one-swap contributions to an average with any number of traces, which amounts to determining general trace moments in a range double the size of the renowned result of Diaconis and Shahshahani. The expression is tractable enough to use in applications, for example to determine the parameter range where certain trace averages assume negative values.