  An eigenfunction method is applied to reduce the regular projective
representations (Reps) of finite groups to obtain their irreducible projective
Reps. Anti-unitary groups are treated specially, where the decoupled factor
systems and modified Schur's lemma are introduced. We discuss the applications
of irreducible Reps in many-body physics. It is shown that in symmetry
protected topological phases, geometric defects or symmetry defects may carry
projective Rep of the symmetry group; while in symmetry enriched topological
phases, intrinsic excitations (such as spinons or visons) may carry projective
Rep of the symmetry group. We also discuss the applications of projective Reps
in problems related to spectrum degeneracy, such as in search of models without
sign problem in quantum Monte Carlo Simulations.
