Incidence Geometry in a Weyl Chamber II: $SL_n$
  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. 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 $sl_n$ and "matter representation" given by the sum of the first two fundamental representations. 
Mboyo Esole, Steven Glenn Jackson, Ravi Jagadeesan, and Alfred G.   No\"el
math.RT hep-th math.CO