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.

中文翻译：

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Weyl 室 I 中的入射几何：GL

我们研究了中心超平面排列，其超平面是 $\mathfrak{gl}_n$ 的第一和第二基本表示的权重的消失轨迹，仅限于双基本外尔室。我们获得了计算给定维度的平面和面的生成函数。这种计数在物理学中被解释为在 U(n) 规范群的基本和反对称表示中存在超多重变换的情况下，对五维规范理论的库仑和混合库仑-希格斯分支的相位的枚举，具有 8 个超电荷由 Intriligator-Morrison-Seiberg 超电势描述。