In the first part of the talk, the eigenfunction method will be applied to
obtain all of the irreducible projective representations (Reps) of finite groups, especially of anti-unitary groups. To this end, we first solve the cocycle equations to obtain the factor system of each class of projective Reps, from which we construct the regular projective Rep and then reduce it into irreducible ones. The second part is focus on the applications of irreducible Reps in condensed matter physics. It will be shown that the spectrum degeneracy induced by irreducible projective Reps may give rise to Dirac cones in Magnetic materials whose symmetry groups are magnetic space groups. We provide the condition for the emergence of Dirac cones on the high symmetrypoints in the Brillouin zone, and enumerate all the type-III and type-IV the magnetic space groups that may host Dirac semimetals.
