slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plucker coordinates

Synopsis

Usage:

fillPl = slackFromPlucker(V, B)

fillPl = slackFromPlucker V

fillPl = slackFromPlucker P

fillPl = slackFromPlucker M
Inputs:
- V, a list, list of coordinates for polytope vertices, cone generators, or matroid vectors
- B, a list, set of hyperplane spanning set indices
- P, a convex polyhedron, a polytope
- M, a matroid, a matroid
Optional inputs:
- Object => ..., default value polytope, specify combinatorial object
Outputs:
- fillPl, a matrix, slack matrix filled with Plucker coordinates

Description

Given a slack matrix or a list of vertices of d-polytope or a rank d+1 matroid, or (d+1)-cone generators, it fills the corresponding slack matrix with Plucker coordinates.

i1 : V = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 0, 1}, {1, 1, 0}};

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

i3 : fillPl = slackFromPlucker(V, B)

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

              6        5
o3 : Matrix ZZ  <--- ZZ

i4 : V = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 0, 1}, {1, 1, 0}};

i5 : fillPl = slackFromPlucker V

Input has been reorderd to
{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {1, 1, 0}, {0, 0, 1}, {1, 0, 1}}

o5 = | 0 -1 0  0 1 |
     | 1 0  0  0 1 |
     | 0 -1 -1 0 0 |
     | 1 0  -1 0 0 |
     | 0 -1 0  1 0 |
     | 1 0  0  1 0 |

              6        5
o5 : Matrix QQ  <--- QQ

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 `slackFromPlucker` :

"slackFromPlucker(List)"
"slackFromPlucker(List,List)"
"slackFromPlucker(Matroid)"
"slackFromPlucker(Polyhedron)"

For the programmer

The object slackFromPlucker is a method function with options.