# dualGrading -- The dual vertices of a polytope.

## Synopsis

• Usage:
• Inputs:
• C, , a polytope
• Outputs:
• ,

## Description

Returns the dual grading of (i.e., a matrix whose row are) the dual polytope of C. The rows are sorted according to the polytopalFacets of C.

 i1 : R=QQ[x_0..x_4] o1 = R o1 : PolynomialRing i2 : C=simplex(R) o2 = 4: x x x x x 0 1 2 3 4 o2 : complex of dim 4 embedded in dim 4 (printing facets) equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0 i3 : grading C o3 = | -1 -1 -1 -1 | | 1 0 0 0 | | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | 5 4 o3 : Matrix ZZ <--- ZZ i4 : dA=dualGrading C o4 = | -1 -1 -1 4 | | -1 -1 4 -1 | | -1 4 -1 -1 | | 4 -1 -1 -1 | | -1 -1 -1 -1 | 5 4 o4 : Matrix QQ <--- QQ i5 : dA===grading dualize C o5 = true i6 : dA===C.dualComplex.simplexRing.grading o6 = true i7 : pf=polytopalFacets C o7 = {x x x x , x x x x , x x x x , x x x x , x x x x } 0 1 2 3 0 1 2 4 0 1 3 4 0 2 3 4 1 2 3 4 o7 : List i8 : coordinates pf#0 o8 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}} o8 : List i9 : (dualGrading C)^{0} o9 = | -1 -1 -1 4 | 1 4 o9 : Matrix QQ <--- QQ

## Caveat

This just returns C.dualComplex.grading. If this data has not beed computed use verticesDualPolytope. Integrate into this verticesDualPolytope.