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SimplicialDecomposability :: hVector

hVector -- the h-vector of a simplicial complex

Synopsis

Description

The h-vector of the 4-simplex.
i1 : R = ZZ[a..e];
i2 : simplex = simplicialComplex{a*b*c*d*e}

o2 = | abcde |

o2 : SimplicialComplex
i3 : hVector simplex

o3 = HashTable{0 => 1}
               1 => 0
               2 => 0
               3 => 0
               4 => 0
               5 => 0

o3 : HashTable
A filled triangle with two edges attached to two vertices shows that the h-vector can have negative entries.
i4 : R = ZZ[x_1..x_5];
i5 : delta = simplicialComplex{x_1*x_2*x_3,x_2*x_4,x_3*x_5}

o5 = | x_3x_5 x_2x_4 x_1x_2x_3 |

o5 : SimplicialComplex
i6 : hVector delta

o6 = HashTable{0 => 1 }
               1 => 2
               2 => -2
               3 => 0

o6 : HashTable
The last example above can be considered in a Z^3-graded ring. Then we can compute its flag h-vector.
i7 : grading = {{1,0,0},{1,0,0},{1,0,0},{0,1,0},{0,0,1}};
i8 : R = ZZ[x_1,x_2,x_3,y,z, Degrees => grading];
i9 : gamma = simplicialComplex{x_1*y*z,x_2*y,x_3*z}

o9 = | x_1yz x_3z x_2y |

o9 : SimplicialComplex
i10 : hVector(gamma, Flag => true)

o10 = HashTable{{0, 0, 0} => 1 }
                {1, 0, 0} => 2
                {1, 0, 1} => -1
                {1, 1, 0} => -1

o10 : HashTable

Caveat

The option Flag checks if the multigrading corresponds to a properly d-coloring of D, where d is the dimension of D plus one. If it is not the case the output is an empty HashTable.

See also

Ways to use hVector :

For the programmer

The object hVector is a method function with options.