Special case reflecting onto a vertical edge

Here is another example of a polytope construction, using the same colour scheme as last time. (There should be an arrow on the bottom row, sorry that’s missing.)

But this time we see something funny happen on the last move (into the purple box).  If you simply follow steps described in the previous post then the right side of that figure should have a red dot, a dark blue dot (reflected from the lower left corner) and two light blue dots (from the previous figure).  What happens instead is that the light blue dot from the previous diagram slides along the vertical edge.

Why does this happen?  Well returning to universal enveloping algebras, unless .  So really we shouldn’t strictly mark any points on the vertical edges but instead label those edges with partions. Then points that reflect onto a side merely specify something about the number of pieces in the partition.

I’m still unclear precisely how this works.  I’m looking into what happens further in the crystal, but calculations there take longer to verify.