  Using the density-functional-theory based electronic structure calculations,
we study the electronic state of recently discovered mixed-valent manganese
oxides $A$Mg$_4$Mn$_6$O$_{15}$ ($A=$ K, Rb, Cs), which are fully spin-polarized
ferromagnetic insulators with a cubic crystal structure. We show that the
system may be described as a three-dimensional arrangement of the
one-dimensional chains of a $2p$ orbital of O and a $3d$ orbital of Mn running
along the three axes of the cubic lattice. We thereby argue that in the ground
state the chains are fully spin polarized due to the double-exchange mechanism
and are distorted by the Peierls mechanism to make the system insulating.
