Discrete gauging and Hasse diagrams

We analyse the Higgs branch of 4d $mathcal{N}=2$ SQCD gauge theories with non-connected gauge groups $widetilde{mathrm{SU}}(N) = mathrm{SU}(N) rtimes_{I,II} mathbb{Z}_2$ whose study was initiated in arXiv:1804.01108. We derive the Hasse diagrams corresponding to the Higgs mechanism using adapted characters for representations of non-connected groups. We propose 3d $mathcal{N}=4$ magnetic quivers for the Higgs branches in the type $I$ discrete gauging case, in the form of recently introduced wreathed quivers, and provide extensive checks by means of Coulomb branch Hilbert series computations.