Nontrivial cocycle extensions in discrete gauging

Investigate discrete H-gauging for nontrivial group-extension cocycles, including extensions such as the quaternion-group Q8 extension of G=Z4 by H=Z2, rather than restricting to the semidirect-product case G⋊H determined by a trivial cocycle.

Background

The framework developed in the paper assumes that the transformations induced by H satisfy a trivial cocycle condition, so that the combined symmetry is the semidirect product G⋊H. The authors explain that more general extensions can arise when the composition of H transformations differs from the transformation associated with the product in H by an element of G.

For G=Z4 and H=Z2, the two possible lifts of the nontrivial automorphism produce the dihedral group D4 when the square of the lift is trivial and the quaternion group Q8 when it equals the order-two element of Z4. Extending the projected-character and non-invertible-selection-rule formalism to such nontrivial cocycles is explicitly left unresolved.

References

We will study the H-gauging of G whose transformation is consistent with the semidirect product of G \rtimes H in the following subsection, leaving the case of nontrivial cocycles, such as the Q_8 extension above, for future work.

— Non-invertible Selection Rules from Generalized Discrete Gauging of Finite Non-Abelian Symmetries  (2609.11895 - Ohki et al., 10 Sep 2026) in Section 2.1, subsection “Generalized Field Transformations in Discrete Gauging”