Research on topological phases of matter is a core field in modern condensed matter physics. Free fermion systems, such as topological insulators and superconductors, have been studied using the “tenfold way” and K-theory. Building on Kitaev's idea of $\mathrm{\Omega }$-spectrum and classifying space, as well as Freed–Moore's K-theory, this work demonstrates that free fermionic systems form a genuine $G\text{−}\mathrm{\Omega }$-spectrum and clarifies its connection to several distinct classification schemes appearing in the physical literature. By introducing the ${\mathbb{Z}}_2$-graded algebra $A_{\mathrm{sym}}^V$, the classification problem for systems with general symmetries, including antilinear symmetries, antisymmetries, projective representations, and point group symmetries, is turned into an extension problem in representation theory. To solve this, a computational method for the ${\mathbb{Z}}_2$-graded Wedderburn–Artin decomposition of $A_{\mathrm{sym}}^V$ is developed. This decomposition not only yields a classification but also enables the explicit construction of the corresponding Dirac Hamiltonian. Furthermore, a GAP programming package has been developed to automate these calculations.

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• Dirac fermions
• Fermions
• Topological phases of matter