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colorfulPresentation -- A presentation of the Stanley-Reisner ring of the barycentric subdivision of a simplicial complex over its colorful parameter ring

Synopsis

Description

Since the barycentric subdivision of a simplicial complex D is a balanced simplicial complex B, i.e., there exists a dim(D)+1 proper vertex coloring, this colorfulPresentation takes in the barycentric subdivision of a simplicial complex, computes a colorful system of parameters for the Stanley-Reisner ring of B and then returns this quotient ring as an N-graded module over the colorful parameter ring.

i1 : S = QQ[a..e];
 
i2 : F = {a*b*c,c*d,e}

o2 = {a*b*c, c*d, e}

o2 : List
 
i3 : D = simplicialComplex F

o3 = simplicialComplex | e cd abc |

o3 : SimplicialComplex
 
i4 : colorfulPresentation D

o4 = cokernel {0} | 0 0 0 0 |
              {1} | 0 0 0 0 |
              {1} | 0 0 0 0 |
              {3} | 0 0 0 0 |
              {1} | d 0 0 0 |
              {1} | 0 d 0 c |
              {2} | 0 0 0 0 |
              {2} | 0 0 0 0 |
              {2} | 0 0 d 0 |

                                            9
o4 : QQ[b..d]-module, quotient of (QQ[b..d])
 
i5 : M = colorfulPresentation F

o5 = cokernel {0} | 0 0 0 0 |
              {1} | 0 0 0 0 |
              {1} | 0 0 0 0 |
              {3} | 0 0 0 0 |
              {1} | d 0 0 0 |
              {1} | 0 d 0 c |
              {2} | 0 0 0 0 |
              {2} | 0 0 0 0 |
              {2} | 0 0 d 0 |

                                            9
o5 : QQ[b..d]-module, quotient of (QQ[b..d])
 
i6 : degrees M

o6 = {{0}, {1}, {1}, {3}, {1}, {1}, {2}, {2}, {2}}

o6 : List

See also

Ways to use colorfulPresentation :

For the programmer

The object colorfulPresentation is a method function.