GRTEALA 1: Review of the situation

I should have been posting while I was at the conference. But anyway I’ll try to post as much as I can remember, before I forget it.

Geometric Satake gives a correspondence from representation theory to subvarieties of an affine Grassmanian. MV-cycles help give a better handle on them but are still rather abstract. MV-polytopes , introduced by Jared Anderson, are more “hands on” in terms of being able to do direct computations. But you need to know what they are first, and their original definition as moment map images of MV-cycles doesn’t really help. At least if you know they are a convex hill of the torus fixed-points appear in the *closure* of an MV-cycle then you’re done — but this still requires more-or-less calculating the MV-cycles and taking their closure.

Kamnitzer’s thesis provided a few ways to get direct handle on MV-polytopes avoiding MV-cycles entirely.

• Implicit description via Plucker relations
• Inductive description (for later extended to types B and C by **FIXME**)
• Reduction to dim-2: higher dimensional MV-polytopes are all polytopes whose 2-faces are MV-polytopes.
• Construction from primitives: MV-polytopes are Minkowski sums of “primitives” and sums of primitives from the same “cluster” are MV-polytopes
• In dimension 2, clusters can be described by networks of non-overlapping cords each parallel to a sides of the Weyl polytope

A conjecture on the Uniqueness of MV-Polytopes

Fix a lattice L with and a partial order . Define


(Alternatively we could fix and define . Similarly we might fix L^+ without reference to any inner product.)

We say that a set of “characters” and a set of subsets satisfy the “tensor property” if:

1. 
2. where
   

(Here indicates convolution and includes zero.)

Suppose we also require the following about

0. 
1. 
2. 
3. (*) for any isotopy, of .

(I’m not sure if the last condition is worded correctly, but I’d like the collection to be invariant under permutations of, for example, minuscule weights.)

And, fixing a group W acting on L for which is a fundamental domain, suppose we require that:

4. 

Conjecture For there is only one (and corresponding ) satisfying the tensor property and all the above requirements.

For coming from or it seems to be true.

Zero-stability: Examples

Example zero

Suppose I consists of a single element. Then a zero stable module is pair of maps and such that is surjective, is injective and .

The existence of zero-stable modules is only possible then if the dimension of W is at least twice the dimension of V. In the case of the smallest possible example of this (when W is 2-dimensionl) we are essentially picking two non-zero orthogonal vectors in W.

Example 1

Let be a directed graph and V be chosen so that for all h. Then zero-stable modules are possible only when for all , and will correspond to a choice of zero-stable module of the type in example zero for each non-trivial .

As a more specific example consider the directed graph consisting of four vertices and four edges formed into a circle. Let and and . Then you can think of a zero-stable module as a pair of bases (each an orthogonal basis) for .

0-stable Modules Part 2

Fix a directed graph and its double (I,H).  Fix I-graded vector spaces V,W.

Let be a path in (I,H).  That is, and for .  The length of is n.  Define and .

P(H) will denote the set of all paths,  P(i,H) will denote the set of paths begining at i, and P(H,j) will denote the paths ending at j.  In practice we will actually these with finite sets depending on the dimension of V — I’ll address this in a moment.

Let (B,a,b) be a module as defined in the previous post.  refers to the component of B associated to a particular edge h and refers a composition of such maps.

Then we may alternatively define zero-stable as follows: (B,a,b) is zero-stable if for all and every

1. there exists a path such that and
2. there exists a vector such that .

The proof that this is equivelant to the previous definition is straightforward and also demonstrates that we need only consider paths that visit each vertex no more than times.

0-stable Modules

For Nakajima’s analogue of MV-cycles we need to understand what it takes for a “module” to be 0-stable. Let me review first what that means.

Fix a finite set I={1 .. r} and a directed graph (I,). Let . For two I-graded vector spaces V and W we define


.
(where s and e denote the start and end vertices of an edge.)
We have a function

which multiplies by -1 components corresponding to and fixes components corresponding to . We have a composition

which is straightforward. We have a composition

where . And we have a “moment map vanishing at the origin”

where .

From now on we fix V and W.

A point is called a “module”.

A “sub-module” of (B,a,b) is a B-invariant I-graded vector space which either contains Im(a) or is contained in Ker(b).

A module (B,a,b) is said to be “0-stable” if the only sub-modules are 0 and V.

**EDIT: I should say (B,a,b) is zero-stable if the only sub-module containing Im(a) is V and the only submodule contained in Ker(b) is 0.

Primative Polytopes and Tropical Relations

My attention was drawn this week to another part of Kamnitzer’s paper on MV-polytopes were he discusses clusters of primitive polytopes (observed by Anderson).  Primitive polytopes are a finitie set (for the finite dimensional groups anyway) of polytopes that generate all MV-polytopes under Minkowski sum.  MV-polytopes aren’t closed under Minkowski sum, but the primitive polytopes are grouped into clusters such that taking Minkowski sum within a cluster guarantees an MV-polytope.  Prehaps I should emphasize, every MV-polytope can be written as the Minkowski sum of a set of primitive polytopes found in the same cluster.

The clusters correspond to tropical choices.  When we have a tropical formula this can be rephrased as:

OR
OR

So the solution set to a tropcal formula is a union of cones.  Each of these cones (generally overlapping) correspond to a tropical choice.   If an MV polytope satisfies the tropical Plucker relations with a particular tropical choice — then it can be generated via Minkowski sum by polytopes in the cluster corresponding to that tropical choice.

I’m very interested in this because for $latex  L\mathfrak{sl}_2$ I don’t have a satisfactory sense of tropical Plucker relations, but I do have a notion of “MV-polytopes” (not yet realized as moment map images of cycles, but functioning combinatorially in the same way) and of course Minkowski sum.   So if I can observe primitives and  clusters, I may gain some insight into what sort of tropical relations to expect.

Tensoring Polytopes, Minkowski Sum Method (pictures)

Here’s a case in we should all be familiar with.

If we tensor the standard representation with the standard dual we get a nine-dimensional representation:

Say the red polytopes come from Standard and the blue come from Standard dual (black is the overlap).  In the Minkowski sum method the MV-polytopes are “added” head-to-tail style.

In this method the action on a basis vector is given by so the adjoint representation inside (recall ) looks like this:

And the trivial representation inside looks like:

The signs will be explained in the next post when I go into what I call “Anderson method”.  For now, notice that this makes and orthogonal sum with respect to the tensor basis.

Tensoring Polytopes

Fix a Kac-Moody Algebra .

For every dominant (resp. anti-dominant) weight There is a collection of MV polytopes that forms a basis for the irreducible representation, , of highest (resp. lowest) weight . Furthermore this basis respects the weight decomposition of the . ( I’d really like to get a firm grasp on how acts on this basis, but for now I only have a vague idea. )

For the tensor product of two representations, then, we can take as a basis, ordered pairs of MV-cycles — one part of the basis for the first and one part of the basis for the second. Since polytopes have a highest and lowest vertex, thinking of these like “head” and “tail” we draw these pairs in a manner analogous to summing vectors. This process basically gives the Minkowski sum.

But lately I’m beginning to think that a better method is to turn the second polytope “upside down” and draw them head to head. ( I hoped to have some pictures justifying this earlier this week, but at best I may have them up by Monday.) In words, if we associate to each polytope a path in the crystal (understanding that some paths are equivelant) then putting two polytopes head to tail is like concatenating these paths, head to head concatenating one path with the reverse of the other. The new path also corresponds to a polytope (two if you consider its reverse). I have no abstract justification for doing this other than it can be done — but the results for the handful of calculations I’ve done so far are very interesting, by which I mean indicative of symmetries.

The description by Anderson of can also be thought of as adding two polytopes (head to tail or head to head) and in terms of concatenating paths through the crystal may have some parallels (it certainly does for what might be called “balanced” paths) but I’m not to excited about those.

Let me summarize these two methods, let be the polytopes forming a basis for then a basis for can be given either by

or by
.
The second one is a theorem due to Anderson — I’m recalling it off the top of my head so I may revise it later.

Current calculation.

Currently reading through Kazhdan-Lusztig and trying to make the following calculation.
Let be representations with the same central weight. K-L define a vector space where is whose action on is not the usual action but a sort of anti-diagonal:

This pairing has a second definition which generalizes: . And satisfies the formula:

where is the tensor product that preserves central character.

Picture of Anderson Calculation

These calculations are done ala Anderson (see previous post). Up until the 2’s appear I have verified the calculations ala Fullton-Harris. To find polytopes inside specified region I don’t have to calculate the full crystal, only the paths of the crystal that lie in the polytope (I meant to mark the paths, just connect the blue dots — red dots arise from reflections.)

Besides being stunningly faster, with this method we easily see that any tensor product of irreps will have infinitely many direct summands.