Tensoring Polytopes, Minkowski Sum Method (pictures)

Here’s a case in we should all be familiar with.

If we tensor the standard representation with the standard dual we get a nine-dimensional representation:

Say the red polytopes come from Standard and the blue come from Standard dual (black is the overlap).  In the Minkowski sum method the MV-polytopes are “added” head-to-tail style.

In this method the action on a basis vector is given by so the adjoint representation inside (recall ) looks like this:

And the trivial representation inside looks like:

The signs will be explained in the next post when I go into what I call “Anderson method”.  For now, notice that this makes and orthogonal sum with respect to the tensor basis.