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.

Tensor products of Lsl2 Reps

Adapted from Anderson (arXiv:math/0110225v1):

If and are irreducible representations with dominant weights and , and is any dominant weight, then the multiplicity of in the direct sum decomposition of into irreps is the number of MV-polytopes of weight   inside the intersection

Notice that the weights of polytopes always have zero central direction so   must have central part equal to the sum of and .  will be an up-pointing parabola and will be dow- pointing.

Pictures of this process will be up soon.

A little more on Lsl_3

Red points are the root lattice. Alpha’s are the rows of the cartan matrix. The Double bold outline around the indicates that it is displaced in one extraplanar dimension. The dashed border around the points label Lambda indicates displacement in the other extraplanar dimension. The lambda’s are the fundamental weights.

The reflection associated to each α will be reflection, except for which will be a shear-reflection.

Chamber weights, that is the image of the Lambda’s under the reflections and shear reflection will have have dashed displacement of one and double-bolded displacement forming a paraboloid shape.

Recall that in the case of we also add an imaginary fundamental weight. Pseudo-Weyl polytopes are given by associating a number to each chamber weight or imaginary weight. The number associated to the imaginary weight defines the level of the polytope, the other numbers tell how far hyperplanes are displaced to cut faces in the three dimensional polytope:

These will be unboinded, paraboloid like shapes.

We will want to cap these, as I did for , with parabolas going the opposite direction.

standard tropical Plucker relations for Lsl_3

Since the Cartan matrix for is -1 at every off diagonal we can apply the usual tropical Plucker relation. This post I begin to explore what it means for the normal (affine)Weyl group, but I suspect that eventually it will need to be interpreted for my extended Weyl group (with conjugation).

First we associate to each element of the Weyl group a simple triangle in the regular hexagonal lattice of the plane. Left multiplication by a simple reflection corresponds to reflecting over one of three universal lines, and right multiplication corresponds to reflecting over one side of the triangle.

Left Multiplication done by reflecting over colored lines, notice that the leading factor on the outside of any line is all the same (or there is a representative with that leading factor).

Right multiplication is done by flipping over colored edges.

Then the tropical plucker relation that applies states that:

For each triple such that ,; we have

Each triple corresponds to a simple hexagon. This triangle furthest from is w; the sides of that triangle interior to the hexagon correspond to i,j; and the corners of the hexagon are the subscripts of M appearing in the tropical Plucker equation.

Special case reflecting onto a vertical edge

Here is another example of a polytope construction, using the same colour scheme as last time. (There should be an arrow on the bottom row, sorry that’s missing.)

But this time we see something funny happen on the last move (into the purple box).  If you simply follow steps described in the previous post then the right side of that figure should have a red dot, a dark blue dot (reflected from the lower left corner) and two light blue dots (from the previous figure).  What happens instead is that the light blue dot from the previous diagram slides along the vertical edge.

Why does this happen?  Well returning to universal enveloping algebras, unless .  So really we shouldn’t strictly mark any points on the vertical edges but instead label those edges with partions. Then points that reflect onto a side merely specify something about the number of pieces in the partition.

I’m still unclear precisely how this works.  I’m looking into what happens further in the crystal, but calculations there take longer to verify.

Another example

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.

Lsl2 MV-Polytopes: Inductive Approach

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.

Non Parabola Verma module

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.