Incidence Geometry in a Weyl Chamber II: $SL_n$

Understand

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of $sl_n$ 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.