A little more on Lsl_3

Red points are the root lattice. Alpha’s are the rows of the cartan matrix. The Double bold outline around the indicates that it is displaced in one extraplanar dimension. The dashed border around the points label Lambda indicates displacement in the other extraplanar dimension. The lambda’s are the fundamental weights.

The reflection associated to each α will be reflection, except for which will be a shear-reflection.

Chamber weights, that is the image of the Lambda’s under the reflections and shear reflection will have have dashed displacement of one and double-bolded displacement forming a paraboloid shape.

Recall that in the case of we also add an imaginary fundamental weight. Pseudo-Weyl polytopes are given by associating a number to each chamber weight or imaginary weight. The number associated to the imaginary weight defines the level of the polytope, the other numbers tell how far hyperplanes are displaced to cut faces in the three dimensional polytope:

These will be unboinded, paraboloid like shapes.

We will want to cap these, as I did for , with parabolas going the opposite direction.