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# slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plücker 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 polytope
• M, , a matroid
• Optional inputs:
• Object => ..., default value "polytope", specify combinatorial object
• Outputs:
• fillPl, , slack matrix filled with Plücker 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 Plücker 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 reordered 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.