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.

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 .

We consider various subalgebras for (which for the purpose of this post we will refer to as ). To construct these remember that the roots of look like this:

Roots of

Example of finite positive borel

Then the roots on one side are called positive and the roots on the other side are called negative. In this case, say the blue dots are positive. Then following the standard procedure we write where the sum is taken only over in the blue region.

Now using a finite element of the (affine) Weyl group we can end up in a situation that looks like this:

This process (indeed my the picture above, if you ignore the green line) suggest the existence of four special borels which exist in the limiting case where, were we to try to construct it by putting a green line would look like:

Limit case

The blue dots are now the positive roots of the finite case with degrees of t added. A Borel may now consist of either the red or blue dots but will additionally contain some subset of the imaginary roots either those above zero or below it (or possibly some other subset ??).

As mentioned previously, the (affine) Weyl group will not transform any blue borel from the first three pictures into any red borel from because it fixes the imaginary roots. So I suggest the possibility of adding and “imaginary” reflection. There are two ways to do this — a reflection that swaps red and blue parts, and a vertical reflection — they are conjugate to each other w.r.t. the Weyl group.

Taking the limit of borels does not give us the swap, because the key is what happens to the imaginary roots. Once declared positive the (affine) Weyl group will never change that. But taking the limit of the borels does suggest that polytopes may have vertical lines.

My previous treatment of LSL2 polytopes only took into account the real roots.  Given a collection of integers associated to each real chamber weight we defined as the intersection of half-spaces facing in the direction of and displaced by .  This was some three dimensional bi-infinite shape, not similar to weights of irreps, which should be restricted to a single central character.

If you add an imaginary chamber weight though, and instead of imposing an inequality, require equality then we will get parabolic polytopes.  Furthermore each edge has integer slope which is an observed phenomenon.

Imposing equality rather than inequality may be loosely justified a few ways.  First, in some sense there are two imaginary chamber weights in opposite directions — but they are fundamentally the same so should have corresponding numbers. Then the inequalities pointing in opposite directions yield equality.  Second, given the two dimensional polytope (using one equality) and taking some non-degeneracy assumptions we can always reconstruct the three dimensional polytope (defined only by inequalities).

Unfortunately none of the Plucker relations listed in Kamnitzer apply in this situation.  And there’s still the problem of not having a “longest element” of the Weyl group.

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.)

Kamnitzer’s (BZ’s?) Tropical Plücker relations for only imply one thing: that the distance between the two “middle” sides of the polytope is the maximum of the distances between the other two pairs of opposing sides. Such a simple relation! I should be able to quickly jot down a nice large list of them.

This may not teach me anything new, but doing this will stoke my interest!