)}. 
\label{u1charges}$$

In the light–cone gauge, the free fermionic heterotic–string models in four dimensions require $20$ and $44$, left–moving and right–moving real world–sheet fermions respectively, to cancel the conformal anomaly. In the usual notation these are denoted as: $\psi^\mu, \chi^{1,\dots,6},y^{1,\dots,6}, \omega^{1,\dots,6}$ and $\overline{y}^{1,\dots,6},\overline{\omega}^{1,\dots,6}$, $\overline{\psi}^{1,\dots,5}$, $\overline{\eta}^{1,2,3}$, $\overline{\phi}^{1,\dots,8}$.

The $SU(4) \times SU(2) \times U(1)$ Gauge Group
------------------------------------------------

In the following we set up the necessary ingredients for the classification of the SU421 free fermionic heterotic–string models. The analysis is along similar lines to the one performed in the classification of the $SO(10)$ [@fknr]; heterotic–string Pati–Salam models [@acfkr]; and flipped $SU(5)$ models [@frs]. The novelty compared to these cases is that the SU421 models employ two basis vectors that break the $SO(10)$ symmetry, whereas the HSPSM and FSU5 models use only one. However, we argue below that this class of heterotic–string vacua cannot in fact produce phenomenologically viable models. The basis vectors that generate our $SU(4) \times SU(2) \times U(1)$ heterotic–string models are given by the following 14 basis vectors $$\begin{aligned}
 \label{421}
v_1={\bf1}&=&\{\psi^\mu,\
\chi^{1,\dots,6},y^{1,\dots,6}, \omega^{1,\dots,6}|\overline{y}^{1,\dots,6},
\overline{\omega}^{1,\dots,6},
\overline{\eta}^{1,2,3},
\overline{\psi}^{1,\dots,5},\overline{\phi}^{1,\dots,8}\},\nonumber\\
v_2=S&=&\{{\psi^\mu},\chi^{1,\dots,6}\},\nonumber\\
v_{2+i}={e_i}&=&\{y^{i},\omega^{i}|\overline{y}^i,\overline{\omega}^i\}, \
i=1,\dots,6,\nonumber\\
v_{9}={b_1}&=&\{\chi^{34},\chi^{56},y^{34},y^{56}|\overline{y}^{34},
\overline{y}^{56},\overline{\eta}^1,\overline{\psi}^{1,\dots,5}\},\label{basis}\\
v_{10}={b_2}&=&\{\chi^{12},\chi^{56},y^{12},y^{56}|\overline{y}^{12},
\overline{y}^{56},\overline{\eta}^2,\overline{\psi}^{1,\dot