M Theory Lesson 76

Recall that Postnikov computed the volumes and number of lattice points for general permutohedra in a variety of ways. In particular, volumes for standard permutohedra can be computed using Weyl group lattices. For $S_{3}$ in dimension 2 one compares the area of 6 (the order of the group) triangles, such as the blue triangle shown, to the sum of the areas of the three hexagons (permutohedra). The hexagons are drawn via a specific algorithm. First pick a point in the blue triangle, the central vertex. This point has an image in the other triangles. The convex hulls of these points give the collection of hexagons. Note that these hexagons look like the central piece of a honeycomb for 3x3 matrices. Sorry about the wonky picture.