Two dimensional interacting electrons on a torus in a strong magnetic field: Symmetry properties and the effect of disorder

We discuss the symmetry properties of two dimensional interacting electrons on a torus in a strong magnetic field. Invariance under magnetic translations of the center of mass leads to a decomposition of the Hilbert space into a direct sum of subspaces. The degenerate ground states are out of one of these subspaces. This is used to calculate the energy splitting of the ground state energy due to a disordered background potential. The energy splitting is small and vanishes in thermodynamic limit if the denominator of the filling factor is odd and small. Numerical calculations confirm our results.