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
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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
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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
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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
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i5 : F=bC.fc_2_0
o5 = x x x
0 1 2
o5 : face with 3 vertices
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i6 : coordinates F
o6 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}}
o6 : List
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i7 : dualize F
o7 = v v
0 1
o7 : face with 2 vertices
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i8 : coordinates dualize F
o8 = {{-1, -1, -1, 4}, {-1, -1, 4, -1}}
o8 : List
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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
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i10 : complement F
o10 = x x
3 4
o10 : face with 2 vertices
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i11 : coordinates complement F
o11 = {{0, 0, 1, 0}, {0, 0, 0, 1}}
o11 : List
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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
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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
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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
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i15 : coordinates dualize complement F
o15 = {{-1, 4, -1, -1}, {4, -1, -1, -1}, {-1, -1, -1, -1}}
o15 : List
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i16 : coordinates F
o16 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}}
o16 : List
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