arrangementSum(Arrangement,Arrangement) -- make the direct sum of two arrangements

Synopsis

Function: arrangementSum
Usage:

arrangementSum(A,B)

A ++ B
Inputs:
- A, a hyperplane arrangement,
- B, a hyperplane arrangement,
Outputs:
- a hyperplane arrangement, the sum ${\mathcal A} \oplus {\mathcal B}$

Description

Given two hyperplane arrangements ${\mathcal A}$ in $V$ and ${\mathcal B}$ in $W$, the sum ${\mathcal A} \oplus {\mathcal B}$ is the hyperplane arrangement in $V \oplus W$ with hyperplanes $\{ H \oplus W \colon H \in {\mathcal A} \} \cup \{ V \oplus H \colon H\in {\mathcal B} \}$. The ring of the direct sum is (ring A) ** (ring B) with all the generators assigned degree 1.

i1 : R = QQ[w,x];

i2 : S = QQ[y,z];

i3 : A = arrangement{w, x, w-x}

o3 = {w, x, w - x}

o3 : Hyperplane Arrangement

i4 : B = arrangement{y, z, y+z}

o4 = {y, z, y + z}

o4 : Hyperplane Arrangement

i5 : C = A ++ B

o5 = {w, x, w - x, y, z, y + z}

o5 : Hyperplane Arrangement

i6 : gens ring C

o6 = {w, x, y, z}

o6 : List

i7 : assert (degrees ring C === {{1}, {1}, {1}, {1}})

Caveat

Both hyperplane arrangements must be defined over the same coefficient ring.

Ways to use this method: