A D_gamma To S^mu Relationship

Recall that are functions on the Grassmanian whose joint level sets define the GGMS strata and in polytopes correspond to the distance of each face’s hyperplane from the origin.

on the other hand is a variety in the grassmanian. It can be defined as the basin of attraction under a action defined by . In polytopes a GGMS stratum with vertices is the interesection .

So how do these two relate to each other? and how do they translate into towers?

In previous posts I outlined how the correspond to towers. At least in the case of it is straightforward: There are 6 , 3 of them (corresponding to level 1 chamber weights) tell what the “interior shell” of a tower looks like, and the other 3 (corresponding to level 2 chamber weights) tell what the “exterior shell” looks like.

What the look like as towers is more difficult to express in general. It involves making the towers stand straight by causing any leaning vectors to prefer columns in a particular order (given by w). Or rather allowing an unleaning tower with sillouette given by to lean a way that gives precedence to the columns. For its not nearly as neat as the but there is one thing to be said.

To begin with let me give a better description of in terms of matrices. The w corresponds to an ordering of the bases. Once that is set, the describes the valuation of the diagonal entries of the elements of after they have been put in upper triangular form.

Now here’s the punch line for this post: when a matrix is in upper triangular form, the valuation of the first diagonal entry is the same as the valuation of the first column. This is the relationship between and . That is, the relationship between the cone based at pointing in a direction defined by w, and the face of that cone orthogonal to .

Lets do and example with . Then describes the first column of the “inner shell”.

Now is the set of all leaning towers which, when pushed to the right until unleaning, will turn into a tower of outline given by .  Say we do this backwards, start with the unleaning tower and let it lean: replace vectors of that tower with vectors leaning left. Notice that the leftmost peak of the unleaning tower has no where to lean! So although it may have pieces from the other two towers leaning above it — it, itselft must remain “pure” and have no pure vectors above it. This is exactly the inner shell. (A picture may make this infinitely more clear … I’ll try to get on that later this week.)