T {/ρ σ }( G) theories and their Hilbert series

We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3 d superconformal gauge theories T {_/ρ ^σ }( G) corresponding to a set of D3 branes ending on NS5 and D5-branes, with or without O3 planes. Here G is a classical group, σ is a partition of G and ρ a partition of the dual group G ^∨. In deriving such a formula we make use of the recently discovered formula for the Hilbert series of the quantum Coulomb branch of superconformal theories. The result can be expressed in terms of a generalization of a class of symmetric functions, the Hall-Littlewood polynomials, and can be interpreted in mathematical language in terms of localization. We mainly consider the case G = SU( N) but some interesting results are also given for orthogonal and symplectic groups.