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# barycentricSubdivision(SimplicialComplex,Ring) -- create the barycentric subdivision of a simplicial complex

## Synopsis

• Function: barycentricSubdivision
• Usage:
barycentricSubdivision(D,R)
• Inputs:
• Delta, ,
• R, a ring, the ambient ring for the barycentric subdivision of $D$
• Outputs:
• , the barycentric subdivision of $D$

## Description

If $\Delta$ is an abstract simplicial complex, the barycentric subdivision of $\Delta$ is the abstract simplicial complex whose ground set (vertices) is the set of faces of $D$ and whose faces correspond to sequences $\{(F_0, F_1, \ldots, F_k)\}$ where $F_i$ is an $i$-dimensional face containing $F_{i-1}$. In order to understand how the data of the barycentric subdivision is organized, we work through a simple example.

 i1 : R = QQ[x_0..x_2]; i2 : S = QQ[y_0..y_6]; i3 : Δ = simplexComplex(2, R) o3 = simplicialComplex | x_0x_1x_2 | o3 : SimplicialComplex i4 : Γ = barycentricSubdivision(Δ, S) o4 = simplicialComplex | y_2y_5y_6 y_1y_5y_6 y_2y_4y_6 y_0y_4y_6 y_1y_3y_6 y_0y_3y_6 | o4 : SimplicialComplex i5 : ΓFacets = facets Γ o5 = {y y y , y y y , y y y , y y y , y y y , y y y } 2 5 6 1 5 6 2 4 6 0 4 6 1 3 6 0 3 6 o5 : List

To make sense of the facets of the barycentric subdivision, we order the faces of $\Delta$ as follows.

 i6 : ΔFaces = flatten for i to 1 + dim Δ list faces(i, Δ) o6 = {x , x , x , x x , x x , x x , x x x } 0 1 2 0 1 0 2 1 2 0 1 2 o6 : List

The indices of the variables appearing in each monomial (or facet) $F$ in the facets of $\Gamma$ determines a sequence of monomials (faces) in $\Delta$.

 i7 : netList for F in ΓFacets list F => ΔFaces_(indices F) +----------------------------+ o7 = |y y y => {x , x x , x x x }| | 2 5 6 2 1 2 0 1 2 | +----------------------------+ |y y y => {x , x x , x x x }| | 1 5 6 1 1 2 0 1 2 | +----------------------------+ |y y y => {x , x x , x x x }| | 2 4 6 2 0 2 0 1 2 | +----------------------------+ |y y y => {x , x x , x x x }| | 0 4 6 0 0 2 0 1 2 | +----------------------------+ |y y y => {x , x x , x x x }| | 1 3 6 1 0 1 0 1 2 | +----------------------------+ |y y y => {x , x x , x x x }| | 0 3 6 0 0 1 0 1 2 | +----------------------------+