nLab fermionic group

Idea

A fermionic group in the sense of Benson 1988 is a (topological) group $G$ equipped with

1. a central element $-1 \in Z(G)$ squaring to $\mathrm{e}$,

2. a group homomorphism $\theta \colon G \to \mathbb{Z}_{/2}$ such that $\theta(-1) = [0]$.

This is related to the notion of quantum symmetries and as such appears in the K-theory classification of topological phases of matter and the classification of invertible field theories.

References

The notion goes back to:

• Dave Benson: Spin modules for symmetric groups, J. London Math. Soc. 2-38 2 (1988) 250–262 [doi:10.1112/jlms/s2-38.2.250]

Discussion in the context of topological phases of matter (10-fold way) and invertible field theories:

• Arun Debray, Sanath K. Devalapurkar, Cameron Krulewski, Yu Leon Liu, Natalia Pacheco-Tallaj, Ryan Thorngren: The Smith Fiber Sequence of Invertible Field Theories, Comm. Math. Phys. 407 2 (2026) 25 [doi:10.1007/s00220-025-05505-0, arXiv:2405.04649]

• Luuk Stehouwer: Free phases of Majorana fermions: Tenfold ways compared [arXiv:2507.08694]