Product of projective representations in description of multi-electron states in an external magnetic field

In this paper all inequivalent irreducible projective representations of th e two-dimensional translation group for a given factor system are determine d. A normalized, i.e. corresponding to the Landau gauge, factor system is c onsidered. Obtained representations directly lead to concept of magnetic ce lls and to periodicity with respect to the charge of a moving particle. It is also shown that the quantization condition is imposed on the product qH of the charge q and the magnitude of magnetic field H. The Kronecker produc t of such representations is considered and it is proven that the multiplic ation of representations corresponds to the addition of charges of particle s moving in it given external magnetic field. In general, coupling of d rep resentations corresponds to d-particle. states. Presented results can be ap plied in any problem related to two-dimensional electron gas in a magnetic field, for example in the fractional quantum Hall effect or high temperatur e superconductivity.