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

Ferromagnetic Peierls insulator state in $\mathit{A}$Mg$_4$Mn$_6$O$_{15}$ ($\mathit{A}$ = K, Rb, Cs)