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As is known, the irreducible projective representations (Reps) ofanti-unitary groups contain three different situations, namely, the real, thecomplex and quaternion types with torsion number 1,2,4 respectively. Thissubtlety increases the complexity in obtaining irreducible projective Reps ofanti-unitary groups. In the present work, a physical approach is introduced toderive the condition of irreducibility for projective Reps of anti-unitarygroups. Then a practical procedure is provided to reduce an arbitraryprojective Rep into direct sum of irreducible ones. The central idea is toconstruct a hermitian Hamiltonian matrix which commutes with the representationof every group element $g\in G$, such that each of its eigenspaces forms anirreducible representation space of the group $G$. Thus the Rep is completelyreduced 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 Brillouinzone for quasi-particle excitations in magnetic materials. After giving thecriterion to judge the power of single-particle dispersion around a HSP, wethen provide a systematic procedure to construct the $k\cdot p$ effectivemodel.

A Hamiltonian Approach for Obtaining Irreducible Projective Representations and the $k\cdot p$ Perturbation for Anti-unitary Symmetry Groups