  Symmetries of the auto-cumulant function (the generalization of the
auto-covariance function) of a kth-order stationary time series are derived
through a connection with the symmetric group of degree k. Using theory of
group representations, symmetries of the auto-cumulant function are demystified
and lag-window functions are symmetrized to satisfy these symmetries. A
generalized Gabr-Rao optimal kernel, used to estimate general kth-order
spectra, is also derived through the developed theory.
