Incidence Geometry in a Weyl Chamber I: $GL_n$

EL Abstract. We study the central hyperplane arrangement whose hyperplanes are the van- ishing loci of the weights of the rst and the second fundamental representations of gln restricted to the dual fundamental Weyl chamber. We obtain generating functions that count ats 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 ve 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.