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The strongly interacting 4d N=2 SCFT's of type $(A_n,A_m)$ are the simplestexamples of models in the $(G,G^\prime)$ class introduced by Cecotti, Neitzke,and Vafa in arXiv:1006.3435. These systems have a known 3d N=4 mirror only if$h(A_n)$ divides $h(A_m)$, where $h$ is the Coxeter number. By 4d/2dcorrespondence, we show that in this case these systems have a nontrivialglobal flavor symmetry group and, therefore, a non-trivial Higgs branch. As anapplication of the methods of arXiv:1309.2657, we then compute the refinedHilbert series of the Coulomb branch of the 3d mirror for the simplest modelsin the series. This equals the refined Hilbert series of the Higgs branch ofthe $(A_n,A_m)$ SCFT, providing interesting information about the Higgs branchof these non-lagrangian theories.

Complete Graphs, Hilbert Series, and the Higgs branch of the 4d N=2 $(A_n,A_m)$ SCFT's