slackFromGalePlucker -- fill the slack matrix with Plucker coordinates of the Gale transform

Synopsis

Usage:

fillPl = slackFromGalePlucker(B, G)

fillPl = slackFromGalePlucker(B, MG)
Inputs:
- B, a list, set of hyperplane spanning set indices
- G, a list, list of Gale vectors of a polytope
- MG, a matrix, matrix of Gale vectors of a polytope
Outputs:
- fillPl, a matrix, slack matrix filled with Plucker coordinates

Description

Given a set of vectors of a Gale transform or a matrix whose columns form a Gale transform of a polytope, it fills the slack matrix of the polytope with Plucker coordinates of the Gale transform.

i1 : G = {{0, 1}, {1, 0}, {-1, -1}, {0, -1}, {-1, 0}, {1, 1}};

i2 : B = {{1, 2, 4}, {0, 2, 3}, {0, 1, 4}, {3, 4, 5}, {0, 1, 2}};

i3 : slackFromGalePlucker(B, G)

o3 = | 1 0  0 1 0  |
     | 0 -1 0 1 0  |
     | 0 0  1 1 0  |
     | 1 0  0 0 -1 |
     | 0 -1 0 0 -1 |
     | 0 0  1 0 -1 |

              6        5
o3 : Matrix ZZ  <--- ZZ

i4 : MG = matrix(RR, {{0, 1, -1, 0, -1, 1}, {1, 0, -1, -1, 0, 1}});

                2          6
o4 : Matrix RR    <--- RR
              53         53

i5 : B = {{1, 2, 4}, {0, 2, 3}, {0, 1, 4}, {3, 4, 5}, {0, 1, 2}};

i6 : slackFromGalePlucker(B, MG)

o6 = | 1 0  0 1 0  |
     | 0 -1 0 1 0  |
     | 0 0  1 1 0  |
     | 1 0  0 0 -1 |
     | 0 -1 0 0 -1 |
     | 0 0  1 0 -1 |

                6          5
o6 : Matrix RR    <--- RR
              53         53

Caveat

Does not check if B actually spans for given V, does not check orientation of simplices in B, so that the resulting slack matrix may differ by signs of each column.

Ways to use `slackFromGalePlucker` :

"slackFromGalePlucker(List,List)"
"slackFromGalePlucker(List,Matrix)"

For the programmer

The object slackFromGalePlucker is a method function.