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

## Synopsis

• Usage:
p = parliament E
• Inputs:
• g, a list, containing elements of ground set as one column matrices
• Outputs:
• p, , whose keys are elements of the ground set and the value of a key is a Polyhedron
• Consequences:
• The result of parliament will be stored as a cacheValue in E.cache#parliament. It will be used by other methods.

## 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.