  As is known, the irreducible projective representations (Reps) of
anti-unitary groups contain three different situations, namely, the real, the
complex and quaternion types with torsion number 1,2,4 respectively. This
subtlety increases the complexity in obtaining irreducible projective Reps of
anti-unitary groups. In the present work, a physical approach is introduced to
derive the condition of irreducibility for projective Reps of anti-unitary
groups. Then a practical procedure is provided to reduce an arbitrary
projective Rep into direct sum of irreducible ones. The central idea is to
construct a hermitian Hamiltonian matrix which commutes with the representation
of every group element $g\in G$, such that each of its eigenspaces forms an
irreducible representation space of the group $G$. Thus the Rep is completely
reduced in the eigenspaces of the Hamiltonian. This approach is applied in the
$k\cdot p$ effective theory at the high symmetry points (HSPs) of the Brillouin
zone for quasi-particle excitations in magnetic materials. After giving the
criterion to judge the power of single-particle dispersion around a HSP, we
then provide a systematic procedure to construct the $k\cdot p$ effective
model.
