$Z$ polarization at an $e^+e^-$ collider and properties of decay-lepton angular asymmetries

$Z$ production at an $e^+e^-$ collider, associated with production of other particles, can be an accurate source of information of details of electroweak interactions, including possible interactions beyond the standard model. We discuss from a general physical point of view the properties of the density matrix as well as lepton angular asymmetries. While most considerations will be applicable to processes of $Z$ production associated with any other particle or particles, for some discussions, we specialize to a $HZ$ final state. While many of the results can be found in earlier literature, especially for the process $e^+e^- \to HZ$, we give details of the reasoning, which are not always found. We discuss the properties of the spin density matrix under C, P and T transformations, and combinations thereof, as also the predictions of these for the corresponding leptonic asymmetries. The specific transformations P, CP, T and CPT are of special importance and we discuss the consequences of these symmetries or their absence for the leptonic asymmetries. A specific issue which has been given attention to is the role of beam polarization, and how one can infer on general grounds which asymmetries get enhanced by the use of beam polarization. Similarly, we also discuss which asymmetries would be sensitive to the measurement of tau polarization in $Z$ decay in to $\tau^+\tau^-$.