findFlag -- computes a list of facet labels that make up a flag in a polytope

Synopsis

• Usage:

  F = findFlag P

  F = findFlag M

  F = findFlag S

  F = findFlag V

• Inputs:
  • P, a convex polyhedron, a polytope
  • M, a matroid, a matroid
  • S, a matrix, (symbolic) slack matrix
  • V, a list, a list of vertex coordinates, matroid vectors, facet labels, or cone generators
• Optional inputs:
  • FlagElement => ..., default value 0, a facet label that will be contained in a flag of facets of given polytope or matroid
  • Object => ..., default value "polytope", specify combinatorial object
• Outputs:
  • F, a list, a list of facet labels or hyperplanes or column labels or labels that make up a flag in the given polytope or matroid or matrix or object given by V

Description

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

i2 : findFlag V

Order of vertices is 
{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 0, 1}, {0, 1, 1}}
Order of facets is 
{{0, 2, 3, 5}, {0, 1, 3, 4}, {1, 2, 4, 5}, {0, 1, 2}, {3, 4, 5}}

o2 = {0, 1, 2, 4}

o2 : List

i3 : findFlag(V, Object => "matroid")

Order of hyperplanes is 
{{2, 3, 4, 5}, {0, 1, 4, 5}, {1, 3, 5}, {0, 3, 5}, {1, 2, 5}, {0, 2, 5}, {1, 3, 4}, {0, 3, 4}, {1, 2, 4}, {0, 2, 4}, {0, 1, 2, 3}}

o3 = {0, 1, 5, 7}

o3 : List

i4 : findFlag({{0, 2, 3, 5}, {0, 1, 3, 4}, {1, 2, 4, 5}, {0, 1, 2}, {3, 4, 5}}, Object => "abstractPolytope")
Order of facets is 
{{0, 2, 3, 5}, {0, 1, 3, 4}, {1, 2, 4, 5}, {0, 1, 2}, {3, 4, 5}}

o4 = {0, 1, 2, 4}

o4 : List

See also

• reconstructSlackMatrix -- a list of facet labels that make up a flag in a polytope
• reducedSlackMatrix -- a reduced slack matrix of a polytope