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.

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.

Guesses about LSL2 Wall-based Polytopes

Here are some conjectures which should not be difficult to prove or disprove about the 1-skeleton of these polytopes.

• Its an infinite tree (i.e. acyclic) allowing that some edges (I’ll call them “leaves”)will go to infinity and therefore have only one vertex.
• For generic polytopes every vertex has order three.
• Each edge divides the tree into finite and infinite parts, thus giving a natural orientation for each edge pointing toward the infinite part.
• With the edges so oriented every vertex will have two incoming and on outgoing edge, and there will be a bijection between cells and vertices given as: the edges of a particular cell all flow toward one of its vertices and for that vertex its two incoming edges both border on that cell.
• Starting from any edge traversing around the finite trees finite side back to that edge will take you through consecutively numbered cells (numbering the cells, as in the previous post, by the portion of the root they are perpendicular to)

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 ?

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.

More on D_\gamma

Today and tomorrow I’ll post some pictures illustrating more about the functions on my “towers”. If I remember correctly and have the same Lie algebra, at least the same chamber weights (those that are indexed over) so I think my diagrams below are good.

I was worried at first that the towers in the previous post are not “balanced” and so don’t represent elements of . If we think of them as elements of , are only defined modulo ?

To preview my analysis: for of level 1 describe a “pure” tower that covers the given one; for of level 2 describe when subtracted from the valuation of the determinant (which is zero for )describes a “pure” tower that lives inside.

UPDATE:

In this picture is a subspace of containing and . Its the image of under the matrix .

The red outlines those standard coordinates contained in the subspace — this is the “inner” tower and is calculated from the values. The blue outline is the “outer” tower and is calculated from the values.

In this picture we have the space corresponding to the matrix .

The green lines indicate the generating vector . The black line represents which is also in the space. Again the red outlines the “inner” tower and the blue the “outer” tower.

D_\gamma and a related Kamnitzer question

It hit me last night as I was reading Kamnitzer the line that says,

Fix a high weight vector in each fundamental representation of . For each chamber weight , let . Since acts on , acts on .

First, does the definition of suggest that there is no redundancy in writing chamber weights as or merely assume implicitely that your choice of will be consistent with this process or is there something more subtle going on? I think I should really figure out which one it is.

Second, The towers of dots I’ve been drawing for G(K): what does look like there? Can the function that follows be read off my dot diagrams?

For each define the function by:

The level sets of these functions are the (for which I still owe you a definition.) These , or somethings not much unlike them, were what I was using the dot towers before so I highly suspect there is a way to read from them. Getting that straightened out should advance my comprehension significantly.

Update:

(These matrices aren’t in .)

Sp_4

Kamnitzer when he’s laying out the basic MV-polytope examples. I don’t really know anything about though I can guess some about its roots from the diagrams he draws. OK, this “question” is vague; basically I’d like to look at in more detail – at least on the level of weights. I suppose I’ll be able to find material on it in Fullton & Harris.

Polytope-to-Cycle conversion

Kamnitzer specifies a formula for converting a polytope, given either by weights or by collections of integers , into subsets of the grassmanian:

Even though the formula makes sense to me I don’t have much of a sense what this means. I’m going to try to do a few examples. Even if I don’t get any answers at least I’ll have some questions which is more than I have now.