Dimensions of minimal symmetric forbidden subspaces for general symmetry groups

Characterize which dimensions occur for minimal symmetric forbidden subspaces when the symmetry group has richer second cohomology than \(SO(3)\).

Background

For an $SO(3)$-symmetric monitored system, the record's projective symmetry class determines the multiplet structure of the averaged filtered state. The smallest symmetric forbidden subspaces are single irreducible multiplets, whose dimensions are even in the Haldane class and odd in the trivial class.

The paper explicitly states that groups with richer second cohomology than $SO(3)$ may produce a more complicated invariant than parity, but it does not determine the possible dimensions of their minimal symmetric forbidden subspaces.