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# slackFromGalePlucker -- fill the slack matrix with Plücker 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, , matrix of Gale vectors of a polytope
• Outputs:
• fillPl, , slack matrix filled with Plücker 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 Plücker 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.