# dualize -- The dual of a face or complex.

• Usage:
dualize(F)
dualize(C)
• Inputs:
• F, a face,
• C, ,
• C, ,
• Outputs:

## 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