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Coulomb branch Hilbert series and Three Dimensional Sicilian Theories

We evaluate the Coulomb branch Hilbert series of mirrors of three dimensional Sicilian theories, which arise from compactifying the $6d$ $(2,0)$ theory with symmetry $G$ on a circle times a Riemann surface with punctures. We obtain our result by gluing together the Hilbert series for building blocks $T_{\mathbfρ}(G)$, where $\mathbfρ$ is a certain partition related to the dual group of $G$, which we evaluated in a previous paper. The result is expressed in terms of a class of symmetric functions, the Hall-Littlewood polynomials. As expected from mirror symmetry, our results agree at genus zero with the superconformal index prediction for the Higgs branch Hilbert series of the Sicilian theories and extend it to higher genus. In the $A_1$ case at genus zero, we also evaluate the Coulomb branch Hilbert series of the Sicilian theory itself, showing that it only depends on the number of external legs.