  To realize novel topological phases and to pursue potential applications in
low-energy consumption spintronics, the study of magnetic topological materials
is of great interest. Starting from the theory of nonmagnetic topological
quantum chemistry [Bradlyn et al., Nature 547, 298 (2017)], we have obtained
irreducible (co)representations and compatibility relations (CRs) in momentum
space, and we constructed a complete list of magnetic band (co)representations
(MBRs) in real space for other MSGs with anti-unitary symmetries (i.e. type-III
and type-IV MSGs). The results are consistent with the magnetic topological
quantum chemistry [Elcoro et al., Nat. Comm. 12, 5965 (2021)]. Using the CRs
and MBRs, we reproduce the symmetry-based classifications for MSGs, and we
obtain a set of Fu-Kane-like formulas of symmetry indicators (SIs) in both
spinless (bosonic) and spinful (fermionic) systems, which are implemented in an
automatic code - TopMat - to diagnose topological magnetic materials. The
magnetic topological materials, whose occupied states can not be decomposed
into a sum of MBRs, are consistent with nonzero SIs. Lastly, using our online
code, we have performed spin-polarized calculations for magnetic compounds in
the materials database and find many magnetic topological candidates.
