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

## Synopsis

• Function: arrangementSum
• Usage:
arrangementSum(A,B)
A ++ B
• Inputs:
• Outputs:
• , 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.