  A low energy effective Hamiltonian for the fractional quantum Hall effect is
obtained by using irreducible representations of the symmetry group. It is
found that the model described by the effective Hamiltonian is similar to the
Heisenberg spin chain. Symmetries of the effective Hamiltonian are studied in
order to decompose the relevant Hilbert space. This decomposition will be
useful for further numerical analysis on the gap of the quantum Hall system.
