Discrete quotients of 3-dimensional N = 4 Coulomb branches via the cycle index

The study of Coulomb branches of 3-dimensional N=4 gauge theories via the associated Hilbert series, the so-called monopole formula, has been proven useful not only for 3-dimensional theories, but also for Higgs branches of 5 and 6-dimensional gauge theories with 8 supercharges. Recently, a conjecture connected different phases of 6-dimensional Higgs branches via gauging of a discrete global $S_n$ symmetry. On the corresponding 3-dimensional Coulomb branch, this amounts to a geometric $S_n$-quotient. In this note, we prove the conjecture on Coulomb branches with unitary nodes and, moreover, extend it to Coulomb branches with other classical groups. The results promote discrete $S_n$-quotients to a versatile tool in the study of Coulomb branches.