# grassmannSectionIdeal -- given a slack matrix of a polytope, a cone or a matroid, or a set of polytope vertices, cone generators, or matroid vectors, and a set of set of hyperplane spanning set indices, it computes the Grassmannian section ideal corresponding to choice B of the object with slack matrix S

## Synopsis

• Usage:
I = grassmannSectionIdeal(S, B)
I = grassmannSectionIdeal S
I = grassmannSectionIdeal(V, B)
I = grassmannSectionIdeal V
I = grassmannSectionIdeal P
I = grassmannSectionIdeal C
I = grassmannSectionIdeal M
• Inputs:
• S, , slack matrix
• B, a list, set of hyperplane spanning set indices
• V, a list, list of polytope vertex coordinates, cone generators, or matroid vectors
• P, , a polytope
• C, , a cone
• M, , a matroid
• Optional inputs:
• Outputs:
• I, an ideal, the Grassmannian section ideal corresponding to choice B of the object with slack matrix S

## Description

It computes the Grassmannian section ideal corresponding to choice B of the object with slack matrix S

 `i1 : V = {{0, 0}, {1, 0}, {2, 1}, {1, 2}, {0, 1}};` `i2 : (VV, B) = getFacetBases V;` ```i3 : I = grassmannSectionIdeal(VV, B) Order of vertices is {{0, 0}, {1, 0}, {0, 1}, {2, 1}, {1, 2}} o3 = ideal (p p - p p + p p , p p - 1,2,4 0,3,4 0,2,4 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 ------------------------------------------------------------------------ p p + p p , p p - p p + p p , 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 0,2,3 1,2,4 0,1,2 2,3,4 ------------------------------------------------------------------------ p p - p p + p p , p p - p p 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 ------------------------------------------------------------------------ + p p ) 0,1,2 0,3,4 o3 : Ideal of QQ[p , p , p , p , p , p , p , p , p , p ] 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4```
 `i4 : V = {{0, 0}, {1, 0}, {2, 1}, {1, 2}, {0, 1}};` `i5 : (VV, B) = getFacetBases V;` ```i6 : I = grassmannSectionIdeal(slackMatrix(VV), B) Order of vertices is {{0, 0}, {1, 0}, {0, 1}, {2, 1}, {1, 2}} o6 = ideal (p p - p p + p p , p p - 1,2,4 0,3,4 0,2,4 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 ------------------------------------------------------------------------ p p + p p , p p - p p + p p , 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 0,2,3 1,2,4 0,1,2 2,3,4 ------------------------------------------------------------------------ p p - p p + p p , p p - p p 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 ------------------------------------------------------------------------ + p p ) 0,1,2 0,3,4 o6 : Ideal of QQ[p , p , p , p , p , p , p , p , p , p ] 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4```

• getFacetBases -- given a slack matrix or a list of vertices of d-polytope or a rank d+1 matroid, or (d+1)-cone generators, creates a sorted list of vertices (empty if a matrix is given as input) in the order corresponding to B, and B the list of d spanning elements for each facet
• slackFromPlucker -- 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
• slackFromGalePlucker -- 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
• symbolicSlackOfPlucker -- given the number of polytope vertices, cone generators, or matroid vectors, or a set of polytope vertices, cone generators, or matroid vectors, or a slack matrix and a set of set of hyperplane spanning set indices, it fills the slack matrix with Plucker variables

## Ways to use grassmannSectionIdeal :

• grassmannSectionIdeal(Cone)
• grassmannSectionIdeal(List)
• grassmannSectionIdeal(List,List)
• grassmannSectionIdeal(Matrix)
• grassmannSectionIdeal(Matrix,List)
• grassmannSectionIdeal(Matroid)
• grassmannSectionIdeal(Polyhedron)