Doing Without Tropical Plücker?

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.

On the “longest element”

MV cycles are the components of a variety defined as the intersection of two infinite and coinfinite subsets of the affine grassmanian. In Kamnitzer the second item in the intersection is described in terms of the “longest” element of the Weyl group. This is troublesome because the affine Weyl group doesn’t seem to have a longest element.

The papers by Anderson and Mirkovic-Vilonen however describe the two in terms of and . In retrospect the failure of the (affine)Weyl group to fully parametrize our world is not new as we saw already it fixes imaginary roots.

For , I have an inkling at this time to augment the affinte Weyl group with an articial “longest element” which acts by rotating 180 degrees around the axis.

• In terms of chamber weights this element will swap the “two” imaginary chamber weights and will create a second parabola, downward pointing and with opposite “c”-value.
• In terms of polytopes this will make our Pseudo Weyl polytopes in intersections of opposing parabolas, something like those I posted early on.
• In terms of Kamnitzer’s plucker relations this new element may categorize as a third simple reflection (though, unfortunately not fixing the real fundamental weights) possibly bringing some of his equations to bear.

That last point was one of the problems I mentioned in my last post to which I will now add detail. Kamnitzer’s inequalities restrict the values of M’s, for example, whenever we have triple satisfying ,, and either or or . If there are only two simple reflections then only two triples will be considered (e,1,2) and (e,2,1) in both cases .

Then Kamnitzer’s treatment imposes no restriction and allows all PW-polytopes to be MV-polytopes. This may be the case, indeed it is the case for . But even if this particular issue is in fact a non-issue it at least illustrates how offcolour things seem.

Rho-check for LSL2 / LPGL2?

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 ?

Calculations: Extended Loop Lie Algebra

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.

Intersection (Co)homology

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

D_\gamma for LSL_3 Part one: Lifting the Weyl group

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

Root System for Lg (Part 3)

Proof that is the semidirect product of by W, the Weyl group of G. Line by line from the book:

The lattice is a subgroup of LG, and obviously centralizes T.

Yes because are loops in T, they will commute with the constant loops in T.

On the other hand, if is the operation of rotating by u then for any we have where .

This formula, if we cancel out the on both sides says that rotating and then multiplying by a loop is the same as multiplying first by a pre-rotated loop, and then rotating: where .

If f is a homomorphism then is the constant , and so .

The first part is clear, just pull out the f(z) and cancel. The second part is a culmination of all so far: concugating an element of T by maps to T and conjugating an rotation (an element of ) by maps to : the product of a rotation and a constant loop in T.

If I understand correctly the rest of the proof aims to decompose any element of into a homomorphism and a constant element .

Conversely, it belongs to then must be a constant function of z for each u,

This follows from the formula. I would add a “constant function in T.“

which implies that is a homomorphism .

Mapping to T as I said follows from the formula. Its a bit difficult to read because now z is playing the part of u and other things play in where z was before. Homomorphism can be shown along the following lines:

since is constant in the variable formerly known as z, now 1 and x. Moving over and adding to each side we get.

i.e. the map is a homomorphism.

Furthermore must belong to the normalizer N of T in G.

When acts on an element of there is no twisting, it is exactly the pointwise action of conjugation. So if normalizes it will introduce no twist and therefore normalizes and hence each point f(z) normalizes T in G.

It follows that is in G,

So only constant loops? Don’t see this yet.

and this proves [the proposition]

This too is a mystery to me yet.

Root system for Lg (part 2)

I really want to unpack the rest of this section (Loop Groups 5.1) . But for now I only have a cartoon:

The Lie algebra for is . What is called the affine Weyl group is a semidirect product of the coweight lattice for G and the Weyl group of G (I understood the proof last weekend, but didn’t write it down fast enough.) The name “affine” comes because its action on the Lie algebra can be seen by its action on the affine plane , essentially because its does nothing in the “loop direction”.

From here the story is very similar to the “finite dimensional” case. Alcoves now play the role of chambers. Since alcoves live in the affine plane, they are in a sense cross sections of chambers in the Lie algebra, but taking every advantage to lower the dimension of our pictures we focus on the alcoves.

As they present it here, in the finite case the simple roots correspond to walls of a chosen chamber. Positive roots are those which are positive on that chamber. So for we say simple affine roots correspond to the walls of a chosen alcove (containing ?) and the positive affine roots are those positive on that alcove.

By projecting we can compare the actions of the affine Weyl group to the finite Weyl group. In the case of (and in some more general sense) All but one of the alcove walls lie on a chamber wall for the finite case. The one wall opposite the origin corresponds to a “highest root” for G but one which is translated — introducing a t factor.

In the next section (5.2) they describe as generated by these simple affine roots with relations identical to the ones for .

Root system for Lg (part 1)

Let me try to unpack chapter five of Loop groups — here is a first bit. Maybe I could make a page on this to continue to add to as I learn more.

Instead of we consider the semidirect product (I can’t find the latex for the notation I’m used to for semidirect product, a times closed on one side) where acts by loop rotation. In other words:

where

For an element to commute with we need . For such to commute with all of is to say it is a constant loop. I.e the centralizer of is . (The action of on constant loops is trivial so it is no longer semidirect product.) This roughly is how we decide to use as a maximal torus for

D_γ for SL_4

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.