Macaulay2 » Documentation
Packages » SRdeformations :: dualGrading
next | previous | forward | backward | up | index | toc

# dualGrading -- The dual vertices of a polytope.

- Usage:
    `dualGrading(C)`
- Inputs:
  - C, an embedded complex, a polytope
- Outputs:
  - a matrix,

## 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 been computed use verticesDualPolytope. Integrate into this verticesDualPolytope.

## See also

- verticesDualPolytope -- The dual vertices of a polytope.
- grading -- The grading of a first order deformation or complex or polynomial ring.
- polytopalFacets -- The facets of a polytope.
- dualize -- The dual of a face or complex.
- isPolytope -- Check whether a complex is a polytope.
- facets -- The maximal faces of a complex.

## Ways to use dualGrading:

- `dualGrading(Complex)`

## For the programmer

The object dualGrading is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SRdeformations.m2:6027:0.