Coulomb branch Hilbert series and Hall-Littlewood polynomials

There has been a recent progress in understanding the chiral ring of 3d $\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques. We find a simple formula of the Hilbert series with background magnetic charges for $T_\rho(G)$ theories in terms of Hall-Littlewood polynomials. Here $G$ is a classical group and $\rho$ is a certain partition related to the dual group of $G$. The Hilbert series for vanishing background magnetic charges show that Coulomb branches of $T_\rho(G)$ theories are complete intersections. We also demonstrate that mirror symmetry maps background magnetic charges to baryonic charges.