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Polytopes for LSL2 by Walls

By trdunlap2

The Pseudo Weyl Polytopes in the finite case are given by two descriptions: (1) by the walls one orthogonal to each chamber weight, \gamma and displaced by a distance M_\gamma, and (2) by cones one oriented by each element, w of the Weyl group and based a point \mu_w. Either of these two data \{M_\gamma\}_\gamma or \{\mu_w\}_{w\in W} is sufficient and there are relations between them.

This post will discuss the M_\cdot analogue based on the affine chamber weights from the last post.

Shadow of a polytope projected onto the TxC plane

Shadow of a polytope projected onto the TxC plane

This image represents the sillouette of any polytope where 1 = M_{(1,-1,0)}=M_{(1,0,0)}=M){(1,1,0)} when we project to the central/constant plane. These red lines are the only faces that will be perpendicular to this projection.

"Top down" view of a polytope with (1,1,1)-face shaded.

Here we see a top-down view of a polytope. The red lines are the faces orthogonal to the projection, the green lines are borders between the other faces. The teal region is the face orthogonal to (1,1,1).

Note:

  • If you imagine that we are looking top-down no face will curl “underneath”. Except in the red region the projection is one-to-one.
  • Seeing where the red walls are fixes the values M_{(1,n,0)} for n\in\{-1,0,1\}.
  • The position of the green lines will fix the relative values of the rest of M_\cdot.

Furthermore, the slope of the green line seperating the (1,n,?)-face and the (1,m,?)-face (assuming the two faces meet) will be given by the following formula

\frac{nm}{n+m} when n and m are even,
\frac{nm + 1}{n+m} when n and m are both odd, and
\frac{n^2m-nm^2+n}{m^2-n^2-1} when n is even and m is odd.

(This formula applies for -1, 0, or 1 as well, but these lines are always drawn red)

This entry was posted on October 14, 2008 at 4:54 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

3 Responses to “Polytopes for LSL2 by Walls”

  1. trdunlap2 Says:
    October 15, 2008 at 2:30 pm | Reply

    If m and n are the same sign then the higher number will be on the left of the line. If they have opposite sign then the negative number will be on the left side.

  2. trdunlap2 Says:
    October 15, 2008 at 2:32 pm | Reply

    I’m not sure what placement of planes should be possible, but I’d guess that vertices should be on the lattice, which would pose a major restriction.

  3. trdunlap2 Says:
    October 16, 2008 at 12:07 am | Reply

    Applying L’Hopital’s rule to the each formula notice that the limit as either variable goes to plus or minus infinity is the other variable.

    Also note that \phi(4,5)=\phi(3,5)=\phi(3,4)=2 (\phi here denoting the formula above.) So these three cells will never three meet. This can also be seen because their normal vectors lie on a line.

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