SL_3 MV-polytopes

Kamnitzer mentions that a polytope is an MV-polytope if and only if all its 2-faces are MV-polytopes. This distinguishes an important class of examples, namely the 2 dimensional polytopes corresponding to , , and . The last class are merely rectangles, the middle one is a bit more complicated, but the first class I will describe here.

For the relevant Plucker relation states that the distance between the “middle” sides of the hexagon should be equal to the maximum of the two distances between the other opposing sides. By “middle” I mean with respect to the highest and lowest weight end of the polytope.

Polytopes satisfying this property split into two symmetric cases depending on which of the two non-middle pairs is further apart. So Without loss of generality I will draw diagrams only for one case. Next I divide them into cases depending on what large distance is. For example in the picture above the black lines form a rhombus corresponding to a distance 3.

The vertical lines represent possibilites for the remaining pair of, but the distance between these faces must be no greater than 3, and one edge must be to the left of the red line, one to the right. The non degenerate (I mean, fully hexagonal) cases are pictured below.

We can take both blue edges, or both green edges, or the two interior edges, to get hexagons as pictured. We also get the triangles on either side of the red line, trapezoids with the red line as their base, and the red line itself. (The red line is included in all diagrams for reference, some explanation below.)

Below are the non-degenerate cases when the distance is 4.

That last hexagon shows what it looks like when the other non-middle pair achieves the maximum distance.  Kamnitzer observed that all MV-2-polytopes will have some diagonals (connecting vertices) parallel to the Weyl-faces.  This is the red line I’ve drawn in all my polytopes.  For there are two kinds of red lines depending on the one statement in the one pertinent relation.  For polyopes there are two red lines per polytope. These lines aid in the combinatorial breakdown of the different types of polytopes.