Unified understanding of symmetry indicators for all internal symmetry classes

The interplay between symmetry and topology in electronic band structures has been one of the central subjects in condensed-matter physics. Recently, it has been getting clear that a wide variety of useful information about the band topology can be extracted by focusing the symmetry representations of valence bands without computing Wilson loops. In this paper, we extend the previous studies on this subject to all ten Altland-Zirnbauer symmetry classes in each of 230 space groups. We derive various general statements that should be useful in the search for topological superconductors and topological semimetals.