We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of ${\mathfrak{sl}}_n$ inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the flats and the faces of a fixed dimension. In addition, we describe the extreme rays of the incidence geometry and classify simplicial faces.

From the perspective of supersymmetric gauge theories with 8 supercharges in five-dimensional spacetime, the poset of flats is isomorphic to the network of mixed Coulomb-Higgs branches. On the other hand, the poset of faces is conjectured to be isomorphic to the network of crepant partial resolutions of an elliptic fibration with gauge algebra ${\mathfrak{sl}}_n$ and “matter representation” given by the sum of the first two fundamental representations.

我们研究了正交于第一和第二基本表示的权重的超平面的多面体几何 ${\mathfrak{sl}}_{ñ}$在双重基本Weyl室内。我们获得了生成函数，这些函数枚举了固定尺寸的平面和面。此外，我们描述了入射几何的极端光线，并对简单面进行了分类。

从五维时空中具有8个增压的超对称轨距理论的角度来看，平面的波塞特与混合库仑-希格斯混合支路的网络同构。另一方面，脸部的姿态被推测为与规范代数的椭圆形纤维的新的部分分辨率网络同构${\mathfrak{sl}}_{ñ}$ 和由前两个基本表示之和给出的“事项表示”。