Flavor Symmetry on Non-Commutative Compact Space and $SU(6) \times SU(2)_R$ Model

In the four-dimensional effective theory from string compactification discrete flavor symmetries arise from symmetric structure of the compactified space and generally contain both the $R$ symmetry and non-$R$ symmetry. We point out that a new type of non-Abelian flavor symmetry can also appear if the compact space is non- commutative. Introducing the dihedral group $D_4$ as such a new type of flavor symmetry together with the $R$ symmetry and non-$R$ symmetry in $SU(6) \times SU(2)_R$ model, we explain not only fermion mass hierarchies but also hierarchical energy scales including the breaking scale of the GUT-type gauge symmetry, intermediate Majorana masses of R-handed neutrinos and the scale of $\mu$-term.