Fluctuation moments induced by conjugation with asymptotically liberating random matrix ensembles

G. Anderson and B. Farrel showed that conjugation of constant matrices by asymptotically liberating random unitary matrices give rise to asymptotic free independence. Independent Haar-unitary random matrices and independent Haar-orthogonal random matrices are examples of asymptotically liberating ensembles. In this paper, we investigate the fluctuation moments, and higher order moments, induced on constant matrices by conjugation with asymptotically liberating ensembles. In particular, we determine fluctuation moments induced by an ensembles related to the Discrete Fourier Transform matrix.