The purple box is one polytope colored two different ways. In general a polytope is colored according to the following scheme:

• Light blue dots are from the previous polytope.
• The red dot is the “moving” dot.
• Dark Blue dots come from a reflection (connected by a blue arc to its preimage).
• Green dots are added so the shape will be “Pseudo-Weyl”.

Where “Pseudo-Weyl”, in this case, means that non-vertical edges have integer slope and any point in the closed polytope minus its vertical lines must be included.

The process for constructing a new polytope goes in that order:

1. Include all previous points
2. Move the “moving point” in the desired direction
3. Reflect any corners on the appropriate side of the reflection line and include those, and
4. Include any additional points necessary to make it Pseudo-Weyl

Where the reflection line is placed so that the first corner before the red dot (in the direction it moved) would be reflected onto the red dot. Only corners on the side opposite the red dot get reflected.

Note: I use “reflect” loosley. In this case all reflection lines will be vertical, but when moving diagonally the reflection will be shear-reflection.

The following picture has the first few levels of a crystal, like those discussed in the 2008 Kamnitzer paper, that constructs a class of MV Plytopes.

The two examples at the top indicate how this method differentiates the elements and . In a sense its as though the weight in in the middle of that line is sometimes included in the polytope and sometimes not. (Though, how this plays out for more complicated partitions I don’t yet know.)

Let me describe the inductive process. For a polytope with highest weight we define new polytopes with highest weight (where is a fundamental coroot.) is characterized by the fact that it is the smallest PW-Polytope containing all the weights of as well as .

For example the purple box in the picture above outlines the polytopes with highest weight . If you test, you will see that only two of them will fit into the parabola for the basic representation.

Reading a more recent paper by Kamnitzer (arXiv:math.QA/0505398), he mentions a conjecture by Anderson-Miković which would inductively construct MV-polytopes without reference to the tropical Plücker relations. The conjecture is not true in general, but is true for for example. It may be something to look into for since none of the relations listed in Kamnitzer’s first paper apply and I don’t yet understand the mechanism by which they arise.

Here are some conjectures which should not be difficult to prove or disprove about the 1-skeleton of these polytopes.

• Its an infinite tree (i.e. acyclic) allowing that some edges (I’ll call them “leaves”)will go to infinity and therefore have only one vertex.
• For generic polytopes every vertex has order three.
• Each edge divides the tree into finite and infinite parts, thus giving a natural orientation for each edge pointing toward the infinite part.
• With the edges so oriented every vertex will have two incoming and on outgoing edge, and there will be a bijection between cells and vertices given as: the edges of a particular cell all flow toward one of its vertices and for that vertex its two incoming edges both border on that cell.
• Starting from any edge traversing around the finite trees finite side back to that edge will take you through consecutively numbered cells (numbering the cells, as in the previous post, by the portion of the root they are perpendicular to)

In Kamnitzer we consider the cell .

For , permutes the diagonal entries of .  When applied to L this will favor one column over another and in the limit will transform L’s tower into a the non-leaning tower with sillouette .

For , but what is ?

Part 1: About this post

This post to be updated throughout the day today, and finished by this evening. UPDATE: Finished with pictures by this weekend.

Based on a conversation I had a few weeks ago, I thought it worthwhile to give an outline the inductive method for calculating of intersection homology I was using last year.

Briefly, we allow closed chains living in the smooth part of the stratified space, and need only conisder whether they should be allowed to “cap off” to the lower strata, which is determined inductively and based on dimension: an already allowable chain, living in the cone over a lower strata is allowed to cap down to the strata if it is the product of an allowable lower strata chain, and the cone of a link of dimension better than half the dimension of the link.

More elaboration on what that means, and some examples later today.

Part 2: Stratified spaces

We consider a topological space such that

1. each is closed,
2. is a manifold of dimension k, and
3. latex X_{n-2}$.

We also may write the space in terms of open pieces where .

We also require that each strata is covered by open sets in such that each open set is of the form where L (called the “link”) is a stratified space depending only on the strata (or possibly on the component of the strata) and indicates the open cone .

Part 3: Admissible (co)chains

First, any closed chain that lives entirely in the “smooth” part of our stratified space, is called admissible. A chain, , that intersects will be called admissible if it can be written as the product where is an admissible chain (defined inductively) for the space , and is a chain in L with sufficiently large dimension (small co-dimension). Sufficiently large dimension isn’t mysterious; for most cases (the standard case I think) we require it to have half the dimension of the link.

Part 4: Eg. Banana Space

The banana space is the torus with one of its belts pinched to a point. So called because one way of drawing it looks like a banana bending around so its tips meet. Also you may call it a circle with two of its antipodes identified.

It has two stratum. One the singular point, and the other of dimension 2 (everything else).

The link, L, over the singular point consists of two circles (one on each side of the banana). No 1-chains can hit the singularity. Only two chains can meet the singularity.

Part 5: Eg. Three Complex 2-Planes

Next we consider the case of three complex hyperplanes complex 3-space. Or rather the one-point compactification (for technical reasons we like working on compact spaces only).

Here the “smooth part” consists of three copies of , the next strata consists of three copies of at their intersections, and the final strata consists of two points, the origin and point of compactification.

Over any point in the the link is two circles, one for each hyperplane. Once again, 1-chains cannot cross the singular stratum. Also no 2-chain can touch unless it wraps around the singular part.

[I need to dig up in my notes, I don't remember what happens near the origin.]

Part 6: Eg. Suspended 3-Torus

This example is the simplest where the link is more than one dimensional. In this case the smooth part is just a thickened three torus. The singular stratum consists of two points, one at each end of the suspension. The link around either of these points is simply a 3-torus.

We allow ourselves to cap off a chain in the 3-torus only if its dimension is 2 or 3. In other words a 1-chain (which is the cone of a 0-chain) is not allowed, only certain 3-chains (the cones of 2-chains) and 4-chains are allowed.

When we take the (co)homology of the resulting intersection (co)chain complex we will get:

0
1
2			0
3
4

Let me recap a bit. Let where w is an element of the Weyl group and is a fundamental weight. Before calculating we’ll need to choose weight vectors such that where indicates the lift of w.

For W is basically the set of permutation matrices only I feel like there is a trouble with signs. Ignoring signs for now think of it as generated by

and

.

In that case we take

.

(Note/check that when .)

This information is more completely presented in a set of diagram I have in my notes — The fastest way to get it up will probably be to scan it on Monday.

I’ve made a similary diagram for . The lifts are much the same as for the finite case only there will be t’s in places. What I don’t have nailed down yet is the vectors: I don’t even know where they live.

UPDATE: Two of the Scans I promised — (a) the Weyl diagram for SL_3 and (b) the Diagram for the affine Weyl group with my guess at appropriate matrix representations (once again ignoring sign issues).

(a)

(b)lsl3wm.png