More on D_\gamma

Today and tomorrow I’ll post some pictures illustrating more about the functions on my “towers”. If I remember correctly and have the same Lie algebra, at least the same chamber weights (those that are indexed over) so I think my diagrams below are good.

I was worried at first that the towers in the previous post are not “balanced” and so don’t represent elements of . If we think of them as elements of , are only defined modulo ?

To preview my analysis: for of level 1 describe a “pure” tower that covers the given one; for of level 2 describe when subtracted from the valuation of the determinant (which is zero for )describes a “pure” tower that lives inside.

UPDATE:

In this picture is a subspace of containing and . Its the image of under the matrix .

The red outlines those standard coordinates contained in the subspace — this is the “inner” tower and is calculated from the values. The blue outline is the “outer” tower and is calculated from the values.

In this picture we have the space corresponding to the matrix .

The green lines indicate the generating vector . The black line represents which is also in the space. Again the red outlines the “inner” tower and the blue the “outer” tower.

Sp_4 weight diagrams

Woot! I understand the weight system. My filler material this week will be to make tensor calculations like the ones I did for If ever run dry of ideas, or get tired of other things, I’ll do a few of them.

I’ll make pictures too. They don’t merit their own page so I’ll just tack them onto this post.

Polytope-to-Cycle conversion

As for examples in

1. I’ll consider Weyl polytopes. This should be review for me, so its a good place to start.
2. Then I’ll do a trivial pseudo-Weyl (but non-Weyl) polytope. The pseudo-Weyl polytopes, I don’t know if I’ve worked with their counterparts, so I just say I’ll try a trivial one (not a full hexagon, for example a single point or a line segment or maybe a triangle).
3. Then I don’t see much point in converting any other pseudo-Weyl polytope unless its an MV-polytope — especially since I have them all listed anyway.

SL_3 MV-polytopes

Kamnitzer’s (BZ’s?) Tropical Plücker relations for only imply one thing: that the distance between the two “middle” sides of the polytope is the maximum of the distances between the other two pairs of opposing sides. Such a simple relation! I should be able to quickly jot down a nice large list of them.

This may not teach me anything new, but doing this will stoke my interest!

Polytope-to-Cycle conversion

Kamnitzer specifies a formula for converting a polytope, given either by weights or by collections of integers , into subsets of the grassmanian:

Even though the formula makes sense to me I don’t have much of a sense what this means. I’m going to try to do a few examples. Even if I don’t get any answers at least I’ll have some questions which is more than I have now.