getFacetBases -- get a list of d-spanning elements for each facet

Synopsis

• Usage:

  (newV, B) = getFacetBases S

  (newV, B) = getFacetBases P

  (newV, B) = getFacetBases M

  (newV, B) = getFacetBases V

• Inputs:
  • S, a matrix, a slack matrix S of rank d+1
  • P, a convex polyhedron, a polytope
  • M, a matroid, a matroid
  • V, a list, a list of vertices of d-polytope, vectors of a rank d+1 matroid, or (d+1)-cone generators
• Optional inputs:
  • Object => ..., default value "polytope", specify combinatorial object
• Outputs:
  • newV, a list, a list of vertices (empty if a matrix is given as input) in the order corresponding to B
  • B, a list, list of d spanning elements for each facet

Description

i1 : V = {{0, 0}, {1, 0}, {2, 1}, {1, 2}, {0, 1}};

i2 : (newV, B) = getFacetBases V

o2 = ({{0, 0}, {1, 0}, {0, 1}, {2, 1}, {1, 2}}, {{0, 2}, {0, 1}, {1, 3}, {2,
     ------------------------------------------------------------------------
     4}, {3, 4}})

o2 : Sequence

i3 : V = {{0, 0}, {1, 0}, {1, 1}, {0, 1}};

i4 : S = slackMatrix V;

Order of vertices is 
{{0, 0}, {1, 0}, {0, 1}, {1, 1}}

              4       4
o4 : Matrix QQ  <-- QQ

i5 : (newV, B) = getFacetBases S

o5 = ({}, {{0, 2}, {1, 3}, {0, 1}, {2, 3}})

o5 : Sequence

See also

• slackFromPlucker -- fill the slack matrix of a given polytope, cone or matroid with Plücker coordinates
• symbolicSlackOfPlucker -- fill the slack matrix with Plücker variables
• grassmannSectionIdeal -- compute the Grassmannian section ideal corresponding to a slack matrix