Orthosymplectic quivers: indices, Hilbert series, and generalised symmetries

We investigate generalised global symmetries in 3d N=4 orthosymplectic quiver gauge theories. Using the superconformal index, we identify a D8 categorical symmetry web in a class of theories featuring so2N×usp2N gauge algebra (at zero Chern-Simons levels) and n bifundamental half-hypermultiplets, analogous to ABJ-type models. As a distinct contribution, we improve the prescription, previously studied in the literature, for computing Coulomb branch Hilbert series of SO(N) gauge theories with Nf vector hypermultiplets. Our improved prescription extends these methods by incorporating fugacities for discrete zero-form symmetries — specifically charge conjugation and magnetic symmetries — and properly treating background magnetic fluxes for the flavour symmetry. This refinement enables calculations for various global forms (O(N)±, Spin(N), Pin(N)) and ensures consistency with the Coulomb branch limit of the superconformal index and known dualities. The proper treatment of fluxes is particularly essential for analysing orthosymplectic quivers where such a flavour symmetry is gauged. We verify our methods through several examples, including an analysis of the mapping of discrete symmetries under mirror symmetry for T[SO(N)] and T[USp(2N)] theories. The analysis also readily generalises to the Tρ[SO(N)] and Tρ[USp(2N)] theories associated with partition ρ.