Incidence Geometry in a Weyl Chamber I: $GL_n$

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of $\mathfrak{gl}_n$ 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.

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