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dualize -- The dual of a face or complex.

Description

Returns the dual of a face or a (co)complex. This is in the sense of dual face of Polytopes, so the faces of C have to be faces of a polytope.

The dual (co)complex dC is stored in C.dualComplex=dC and dC.dualComplex=C.

Note that if C is a Stanley-Reisner subcomplex of a simplex then dualize complement C is the isomorphic geometric complex of strata.

i1 : R=QQ[x_0..x_4]

o1 = R

o1 : PolynomialRing
 
i2 : addCokerGrading R

o2 = | -1 -1 -1 -1 |
     | 1  0  0  0  |
     | 0  1  0  0  |
     | 0  0  1  0  |
     | 0  0  0  1  |

              5       4
o2 : Matrix ZZ  <-- ZZ
 
i3 : C=simplex R

o3 = 4: x x x x x  
         0 1 2 3 4

o3 : complex of dim 4 embedded in dim 4 (printing facets)
     equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0
 
i4 : bC=boundaryOfPolytope C

o4 = 3: 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

o4 : complex of dim 3 embedded in dim 4 (printing facets)
     equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1
 
i5 : F=bC.fc_2_0

o5 = x x x
      0 1 2

o5 : face with 3 vertices
 
i6 : coordinates F

o6 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}}

o6 : List
 
i7 : dualize F

o7 = v v
      0 1

o7 : face with 2 vertices
 
i8 : coordinates dualize F

o8 = {{-1, -1, -1, 4}, {-1, -1, 4, -1}}

o8 : List
 
i9 : dbC=dualize bC

o9 = 0: v  v  v  v  v  
         0  1  2  3  4

o9 : co-complex of dim 0 embedded in dim 4 (printing facets)
     equidimensional, simplicial, F-vector {0, 5, 10, 10, 5, 1}, Euler = 1
 
i10 : complement F

o10 = x x
       3 4

o10 : face with 2 vertices
 
i11 : coordinates complement F

o11 = {{0, 0, 1, 0}, {0, 0, 0, 1}}

o11 : List
 
i12 : complement bC

o12 = 0: x  x  x  x  x  
          4  3  2  1  0

o12 : co-complex of dim 0 embedded in dim 4 (printing facets)
      equidimensional, simplicial, F-vector {0, 5, 10, 10, 5, 1}, Euler = 1
 
i13 : dualize complement bC

o13 = 3: v v v v  v v v v  v v v v  v v v v  v v v v  
          1 2 3 4  0 2 3 4  0 1 3 4  0 1 2 4  0 1 2 3

o13 : complex of dim 3 embedded in dim 4 (printing facets)
      equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1
 
i14 : bC

o14 = 3: 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

o14 : complex of dim 3 embedded in dim 4 (printing facets)
      equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1
 
i15 : coordinates dualize complement F

o15 = {{-1, 4, -1, -1}, {4, -1, -1, -1}, {-1, -1, -1, -1}}

o15 : List
 
i16 : coordinates F

o16 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}}

o16 : List

See also

Ways to use dualize:

  • dualize(Complex)
  • dualize(Face)

For the programmer

The object dualize is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SRdeformations.m2:5522:0.