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# matrix(SimplicialMap) -- get the underlying map of rings

## Synopsis

• Function: matrix
• Usage:
g = matrix f
• Inputs:
• Optional inputs:
• Degree => ..., default value null, unused
• Outputs:
• g, , having one row

## Description

A simplicial map is a map $f \colon \Delta \to \Gamma$ such that for any face $F \subset \Delta$, the image $f(F)$ is contained in a face of $\Gamma$. Since an abstract simplicial complex is, in this package, represented by its Stanley–Reisner ideal in a polynomial ring, the simplicial map $f$ corresponds to a ring map from the ring of $\Delta$ to the ring of $\Gamma$. The ring map is described by a matrix having one row; the entry in the $i$-th column is the image in the ring of $\Gamma$ of the $i$-th variable in the ring $\Delta$. This method returns this matrix.

For the identity map, the matrix of variables in the ambient polynomial ring.

 i1 : S = ZZ[x_0..x_5]; i2 : Δ = simplicialComplex monomialIdeal(x_0*x_5, x_1*x_4, x_2*x_3) o2 = simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | o2 : SimplicialComplex i3 : id_Δ o3 = | x_0 x_1 x_2 x_3 x_4 x_5 | o3 : SimplicialMap simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | <--- simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | i4 : matrix id_Δ o4 = | x_0 x_1 x_2 x_3 x_4 x_5 | 1 6 o4 : Matrix S <-- S i5 : assert(matrix id_Δ === vars S)

The next map projects an octahedron onto a square.

 i6 : R = ZZ[y_0..y_3]; i7 : Γ = simplicialComplex monomialIdeal(y_1*y_2) o7 = simplicialComplex | y_0y_2y_3 y_0y_1y_3 | o7 : SimplicialComplex i8 : f = map(Γ, Δ, {y_0,y_0,y_1,y_2,y_3,y_3}) o8 = | y_0 y_0 y_1 y_2 y_3 y_3 | o8 : SimplicialMap simplicialComplex | y_0y_2y_3 y_0y_1y_3 | <--- simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | i9 : matrix f o9 = | y_0 y_0 y_1 y_2 y_3 y_3 | 1 6 o9 : Matrix R <-- R

This matrix is simply extracted from the underlying map of rings.

 i10 : code(matrix, SimplicialMap) o10 = -- code for method: matrix(SimplicialMap) /usr/local/share/Macaulay2/ SimplicialComplexes/Code.m2:904:39-904:58: --source code: matrix SimplicialMap := Matrix => opts -> f -> matrix map f