As is known, the irreducible projective representations (Reps) of anti-unitary groups contain three different situations, namely, the real, the complex and quaternionic 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 ∈ 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 ⋅ 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 an HSP, we then provide a systematic procedure to construct the k ⋅ p effective model.

众所周知，反酉群的不可约射影表示（Reps）包含三种不同的情况，即分别具有扭数为1、2、4的实数、复数和四元数类型。这种微妙之处增加了获得反单一群体的不可约射影代表的复杂性。在目前的工作中，引入了一种物理方法来推导出反酉群的射影代表的不可约条件。然后提供了一个实用的程序来将任意投影 Rep 减少到不可约的直接和。中心思想是构造一个 Hermitian Hamiltonian 矩阵，该矩阵与每个群元素g ∈ G的表示交换，使得它的每个特征空间形成群G 的一个不可约表示空间。因此，Rep 在哈密顿量的特征空间中完全减少。这种方法应用于布里渊区高对称点 (HSP)的k ⋅ p有效理论，用于磁性材料的准粒子激发。在给出判断 HSP 周围单粒子分散能力的标准后，我们​​提供了一个系统的程序来构建k ⋅ p有效模型。