Weyl-Heisenberg group and magnetic translations in finite phase space

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Abstract
The symmetry of an electron in a magnetic field is given by the Weyl-Heisenberg group. When a periodic potential is added, the symmetry is lowered and is given by the magnetic-translation group. Irreducible representations and the group-subgroup relationship of these groups are investigated in finite phase space. Explicit formulas are derived for the splitting of magnetic states under periodic perturbations. A discussion is presented for extended and localized magnetic states. In particular, it is shown that despite the fact that the translation group and the magnetic translation group have the same number of elements in finite phase space, they nevertheless induce, in general, different numbers of localized orbitals (the latter fewer than the former).

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