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The electronic properties in a solid depend on the specific form of thewave-functions that represent the electronic states in the Brillouin zone.Since the discovery of topological insulators, much attention has been paid tothe restrictions that the symmetry imposes on the electronic band structures.In this work we apply two different approaches to characterize all types ofbands in a solid by the analysis of the symmetry eigenvalues: the inductionprocedure and the Smith Decomposition method. The symmetry eigenvalues orirreps of any electronic band in a given space group can be expressed as thesuperposition of the eigenvalues of a relatively small number of building units(the \emph{basic} bands). These basic bands in all the space groups areobtained following a group-subgroup chain starting from P1. Once the whole setof basic bands are known in a space group, all other types of bands (trivial,strong topological or fragile topological) can be easily derived. Inparticular, we confirm previous calculations of the fragile root bands in allthe space groups. Furthermore, we define an automorphism group of equivalencesof the electronic bands which allows to define minimum subsets of, forinstance, independent basic or fragile root bands.

physical review. b./physical review. b

Application of the induction procedure and the Smith Decomposition in  the calculation and topological classification of electronic band structures  in the 230 space groups