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parliament -- computes the parliament of polytopes to a toric vector bundle

Synopsis

Description

Given a toric vector bundle in Klyachko's description, parliament computes its parliament of polytopes as introduced in [RJS, Section 3].
i1 : E = tangentBundle(projectiveSpaceFan 2)

o1 = {dimension of the variety => 2 }
      number of affine charts => 3
      number of rays => 3
      rank of the vector bundle => 2

o1 : ToricVectorBundleKlyachko
i2 : p = parliament E

o2 = HashTable{| 0 | => Polyhedron{...1...}}
               | 1 |
               | 1 | => Polyhedron{...1...}
               | 0 |
               | 1 | => Polyhedron{...1...}
               | 1 |

o2 : HashTable
i3 : applyValues(p, vertices)

o3 = HashTable{| 0 | => | 0 0  1  |}
               | 1 |    | 0 -1 -1 |
               | 1 | => | 0 -1 -1 |
               | 0 |    | 0 0  1  |
               | 1 | => | 0 1 0 |
               | 1 |    | 0 0 1 |

o3 : HashTable
i4 : P112 = ccRefinement transpose matrix {{1,0},{0,1},{-1,-2}};
i5 : D = {1,0,0};
i6 : L = toricVectorBundle(1, P112, toList(3: matrix{{1_QQ}}), apply( D, i-> matrix{{-i}}));
i7 : details L

o7 = HashTable{| -1 | => (| 1 |, | -1 |)}
               | -2 |
               | 0 | => (| 1 |, 0)
               | 1 |
               | 1 | => (| 1 |, 0)
               | 0 |

o7 : HashTable
i8 : apply(values parliament L, vertices)

o8 = {| 0 1 0   |}
      | 0 0 1/2 |

o8 : List
If the toric variety is two-dimensional, then the result can be visualised using drawParliament2Dtikz. parliament calls internally the method groundSet.

Caveat

This method works for any toric reflexive sheaf on any toric variety.

See also

Ways to use parliament :

For the programmer

The object parliament is a method function with options.