dualFaceLattice(ZZ,Cone) -- computes the dual face lattice of a cone

Function: dualFaceLattice
Usage:

L = dualFaceLattice C

L = dualFaceLattice(k,C)
Inputs:
- k, an integer, between 0 and the dimension of C
- C, a convex rational cone
Outputs:
- L, a list

Description

The dual face lattice of a cone C displays for eachk the faces of dimension k as a list of integers, indicating the bounding halfspaces of C that generate this face together with the hyperplanes. If no integer is given the function returns the faces of all dimensions in a list, starting with the Cone itself.

i1 : C = posOrthant 4

o1 = {ambient dimension => 4           }
      dimension of lineality space => 0
      dimension of the cone => 4
      number of facets => 4
      number of rays => 4

o1 : Cone

i2 : dualFaceLattice(2,C)

o2 = {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}}

o2 : List

Returns the faces of dimension two, where the integers give the rows in the halfspaces matrix of the cone:

i3 : R = halfspaces C

o3 = | 1 0 0 0 |
     | 0 1 0 0 |
     | 0 0 1 0 |
     | 0 0 0 1 |

              4       4
o3 : Matrix ZZ  <-- ZZ

The complete dual face lattice is returned if no integer is given:

i4 : dualFaceLattice C

o4 = {{{}}, {{0}, {1}, {2}, {3}}, {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3},
     ------------------------------------------------------------------------
     {2, 3}}, {{1, 2, 3}, {0, 2, 3}, {0, 1, 3}, {0, 1, 2}}, {{0, 1, 2, 3}}}

o4 : List

Ways to use this method:

dualFaceLattice(Cone)
dualFaceLattice(ZZ,Cone) -- computes the dual face lattice of a cone

The source of this document is in OldPolyhedra.m2:5955:0.