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.

Probably nothing important, just a calculation I was doing last night. In the Verma module where acts by -2 , (c by 1 and d by zero). I calculated that:

So inductively, beginning with , none of these are zero.

We do have , so the outline looks like \_/, a truncated cone, not a parabola.

I want to know the shapes and weights of various representations so I can determine how paths pair up to become MV-Polytopes — more on this with pictures to come this week.

Let

Where is the killing form:, and .

To abbreviate take the convention: (similarly for ). (Don’t confuse it with similar notation used for higher rank Kac-Moody Algebras.)

Keeping in mind the previous post about Borels, split where (the large green dot) is generated by ,c, and d.

Now we are ready, given a character , to define a Verma module , where the product is taken over ( acts trivially on and action is given by ).

What we are really interested in is a quotient where
.
This will be the irreducible representation of (by left multiplication) of lowest weight .

Of course is itself a -representation and it diagonalizes under the action of , .

Let be the Verma module with a highest weight (i.e. with tensor taken over ), with a similar decomposition . With a choice of basis for each and we define a pairing such that for basis elements . The rank of gives the dimension of the -weight space of the quotient .

To familiarize myself more with the distinction between roots and coroots, I’ve been given a homework assignment.(WARNING: notation in this article differs from other notation I use, particularly the use of checks)

:

The weigth lattice (denoted, for this post only, as ) is and will be generated by the following maps.

The coweight lattice (denoted, for this post, as ) is and will be generated by

The action of on breaks down like this:

where .

So the roots are .

Circled dots are Roots, red lines are Weyl reflections numbered Lambdas are the fundamental weights: sorry for the mixed conventions.

I determine the roots also to be by solving the equations:
and

(notice in the diagram that is orthogonal to )

(The Coweight diagram is almost identical to that for the Weight space.)

Let be a Lie group. We are interested in the infinite dimensional Lie group where composition is done pointwise. One way to understand a group is by understanding its representations. In this particular case our interest is quickly narrowed to smooth, projective, positive energy representations. It turns out that a better way object of study is the semidirect product where is a particular one dimensional central extension and acts on by rotating loops (that is precomposing with a rotation).

This thing’s Lie Algebra will be (as a vectors space) . My charge, by this Sunday, is to calculate the Lie bracket. Suffice we will consider the (dense?) subalgebra

To begin with:

because rotation is commutative. The centeral extension is, well, central so we have:

What’s left are

Lets start with the first. We’ll take what I’ll call the “scenic route” doing as much explicit calculation as possible.

where and	e.g.

The second calculation is actually fixed for us by the particular type of central extension we’re using. I’ll explain tomorrow.

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

Sorry if this is boring to those who hate calculations, or a spoiler for those who need to do them (like me). But I want to post something so here’s one calculation I’ve finished and another that I’m still puzzling through. These are both from Loop groups chapter 4.

Proposition 4.3.2 says:

The adjoint action of on comes from an action of LG given by

=.

Here $latex\gamma . \xi$ denotes the adjoint action of on .

The book says verifying this is a group action is straightforward. I agree, having done it, that it is straightforward, but it is not obvious – and the formula anyway seems rather mysterious. But here we go.

Let me recall:

• G is a Lie group, is the Lie algebra of its loop group.
• We are constructing a Lie algebra = with the bracket: (notice that is central in the sense of not contributing to the bracket.)
• is given by (WLOG in some sense):
  derived from a symmetric invariant form on . (Invariant means that .)
• (We also use to denote a form on which is the average over loops of the form on .)

OK, so to my calculation: I want to verify that

On the first component this is obvious, the second component is the one where anything interesting happens.

Begining by writing out complicated bit from the left hand side we have:

which is exactly the complicated bit on the right hand side.

The calculation is straightforward, boring even — but not obvious to me until I actually write it out. This book is full of things like that. If I write it out just a little but more, that is something I can follow.

Here is another such case I found today but have not managed to untangle yet:
We are considering the map given by pulling back the evaluation map . ( is based loops.) We define to be a left invariant 3-form on G whose value at the identity is given by . When we pullback this form and evaluate it at on the vector we get, according to the book,

This calculation mystifies me more than the first.

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

The data for come in two sets of three, one set for each fundamental weight. For a fixed set of values for the elements of the affine grassmanian corresponding to that data will be the “balance towers” that lie between the “pure towers” described by those two sets.

For there’s only one set of two. We can still get two towers, but these will both be described by the same set which is self-dual.

When we move to we start getting more intermediate data. We still have the “level 1″ set in the data that describes an outer tower and a a level set in the data that describes an inner tower. But now we have additional data. I’d like to understand the additional restrictions this set (and further middle sets for higher n) will put on towers.

So far the one thing I’ve noticed is that ignoring a column of the tower (and alowing any parts leaning into that portion to “stand up”) We get a tower like those for but not necessarily balanced. There’s a subset of the that can be translated into data about this .

Let me take some notation. Let denote row vectors and denote column vectors of a representative in of an element in . The are the generators of the subspace represented by our tower. valuations of the and their exterior products for our data. What is not an official part of the data is the valuation of the determinant or the exterior product of all columns.

When we eliminate one of the columns as suggested, we will have 4 rows to generate a tower only three wide, so one of the rows will become superfluous. I argue that the valuations of the wedge products of pairs of will be unchanged despite the elimination of this row. Its because of this that I say the middle data arising in describe these related unbalanced towers’ inner parts. Clarifying exactly how that describes the original tower is one of my current goals.