The crossing points in band structures play an important role in several physical phenomena. In this paper, we focus on the group information of these crossing points, rather than on the specific $k·p$ models. Specifically, each crossing point can be described by a matrix group and the corresponding set of basis functions. We identify the minimal abstract group that captures all the essential information regarding the crossing point, and then decompose the matrix representation into irreducible representations (irreps) using orthogonality relations. The abstract group and irrep coefficients can be regarded as crossing point indicators. Such indicators are independent of the specific representation matrices and lattice coordinates. As an example, we applied our method to charge-1 Weyl points in 1651 magnetic space groups. We found that although as many as 9988 distinct Hamiltonians may exist before classification, they can ultimately be reduced to only nine abstract groups and 18 distinct symmetry indicators. This significantly reduces the complexity of identifying and characterizing crossing points in band structures. Our findings provide an algebraic classification of crossing points in band structures, which significantly simplifies the analysis and prediction of topological semimetals.

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• Band gap
• Symmetry protected topological states
• Group theory