Tetrahedral Symmetry A$_{bf 4}$ in Anti-SU(5) GUT

We construct a flavor model in an anti-SU(5) GUT with a tetrahedral symmetry $A_4$. We choose a basis where $Q_{text{em}}=-frac13$ quarks and charged leptons are already mass eigenstates. This choice is possible from the $A_4$ symmetry. Then, matter representation $overline{10}_{-1}^{rm, matter}$ contains both a quark doublet and a heavy neutrino $N$, which enables us to use the $A_4$ symmetry to both $Q_{text{em}}=+frac23$ quark masses and neutrino masses (through the see-saw via $N$). This is made possible because the anti-SU(5) breaking is achieved by the Higgs fields transforming as anti-symmetric representations of SU(5), $overline{10}_{-1}^Hoplus 10_{+1}^H$, reducing the rank-5 anti-SU(5) group down to the rank-4 standard model group smg. For possible mass matrices, the $A_4$ symmetry predictions on mass matrices at field theory level are derived. Finally, an illustration from string compactification is presented.