Incidence geometry in a Weyl chamber I: GL

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of gl n restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs branches of a five-dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a U ( n ) gauge group as described by the Intriligator-Morrison-Seiberg superpotential.