Identifying Unconventional E$_{\bf 6}$ Models at $e^+ e^-$ Colliders

Recently it was shown that, in the framework of superstring inspired $\E$ models, the presence of generation dependent discrete symmetries allows us to construct a phenomenologically viable class of models in which the three generations of fermions do not have the same embedding within the fundamental {\bf 27} dimensional representation of E$_6$. In this scenario, these different embeddings of the conventional fermions imply that the left-handed charged leptons and the right-handed $d$-type quarks are coupled in a non--universal way to the new neutral gauge bosons $(Z_\theta)$ present in these models. It was also shown that a unique signature for this scenario, would be a deviation from unity for the ratio of cross sections for the production of two different lepton species in $e^+e^-$ annihilation. However, several different scenarios are possible, depending on the particular assignment chosen for $e_L$, $\mu_L$ and $\tau_L$ and for the right-handed $d$-type quarks, as well as on the type of $Z_\theta$ boson. Such scenarios can not be disentangled from one another by means of cross section measurements alone. In this paper we examine the possibility of identifying the pattern of embeddings through measurements of polarized and unpolarized asymmetries for fermion pair-production at the 500 GeV $e^+e^-$ Next Linear Collider (NLC). We show that it will be possible to identify the different patterns of unconventional assignments for the left-handed leptons and for the $b_R$ quark, for $Z_\theta$ masses as large as $\sim 1.5$ TeV.