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:
- Include all previous points
- Move the “moving point” in the desired direction
- Reflect any corners on the appropriate side of the reflection line and include those, and
- 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.
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.)
with highest weight 
we define new polytopes 
with highest weight 
(where 
is a fundamental coroot.) 
. If you test, you will see that only two of them will fit into the parabola for the basic representation.
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.
portion of the root they are perpendicular to)
.
, 
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 
.
, 
but what is 
?
such that
is closed,
is a manifold of dimension k, and
latex X_{n-2}$.
where 
.
is covered by open sets in 
such that each open set 
is of the form 
where L (called the “link”) is a stratified space depending only on the strata (or possibly on the component of the strata) and 
indicates the open cone ![( (0,1]\times L) / (1\times L)](http://l.wordpress.com/latex.php?latex=%28+%280%2C1%5D%5Ctimes+L%29+%2F+%281%5Ctimes+L%29&bg=ffffff&fg=000000&s=0 "( (0,1]\times L) / (1\times L)")
.
is called admissible. A chain, 
, that intersects 
will be called admissible if it can be written as the product 
where 
is an admissible chain (defined inductively) for the space 
is a chain in L with sufficiently large dimension (small co-dimension). Sufficiently large dimension isn’t mysterious; for most cases (the standard case I think) we require it to have half the dimension of the link.
, the next strata consists of three copies of 
at their intersections, and the final strata consists of two points, the origin and point of compactification.
the link is two circles, one for each hyperplane. Once again, 1-chains cannot cross the singular stratum. Also no 2-chain can touch unless it wraps around the singular part.
a bit. Let 
where w is an element of the Weyl group and 
is a fundamental weight. Before calculating 
such that 
where 
indicates the lift of w.
W is basically the set of permutation matrices only I feel like there is a trouble with signs. Ignoring signs for now think of it as generated by
.
.
when 
.)
. The lifts are much the same as for the finite case only there will be t’s in places. What I don’t have nailed down yet is the 
vectors: I don’t even know where they live.
