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The exact symmetry of hexagonal quantum dots (QDs) made of materials with the wurtzite structure such as GaN/AlN QDs for example, is described by the C 3v point group and does not depend on the existence of a wetting layer. We have determined the possible exact symmetries of electron states and vibration modes in the dots and derived the optical selection rules. The vibration modes involved in the Frölich interaction are totally symmetric with respect to the C 3v group and can induce transitions only between states with the same symmetry. The not totally symmetric modes provide other channels for lowering the energy of excited carriers and excitons by connecting states with symmetries different one from another. The rapid decay of created polarons, due to the short lifetime of vibration modes, releases the carriers and excitons into ground levels. In the envelope function approximation (EFA), the symmetry of the dots is represented by the C 6v point group. Interband transitions are allowed only between states whose envelope functions have the same symmetry. EFA artificially increases the number of dark exciton symmetries.

PACS 73.21.-b, 75.90.+ wWe present a group-theory analysis of the electron states in bulk and low-dimensional semiconductor structures under a uniform magnetic field. The analysis takes into account the gauge transformations under the symmetry operations. It is shown that the symmetry operations commute in dots, rods and tubes whatever is the orientation of the magnetic field whereas they commute in layers only when the field is in the plane. The commutations properties allow using conventional symmetry groups. In bulk materials, the confinement of electrons within rods whose axes are parallel to the field makes it possible to derive approximate wavefunction symmetries from the rod group involving the geometrical symmetry operations common to the field and the crystal but excluding the translations that are not parallel to the field. An approximation of the same kind can also be made for low-dimensional structures. Applications to bulk materials with the wurtzite or zinc blende structure as well as related nanostructures are presented.