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.