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# typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system

## Synopsis

• Function: typeD
• Usage:
typeD(n, k)
typeD(n, R)
typeD n
• Inputs:
• n, an integer, that is greater than $1$
• k, a ring, that determines the coefficient ring of the hyperplane arrangement or $R$ that determines the ambient ring
• Outputs:

## Description

Given a coefficient ring $k$, the Coxeter arrangement of type $D_n$ is the hyperplane arrangement in $k^{n}$ defined by $x_i \pm x_j$ for all $1 \leq i < j \leq n$.

 i1 : A0 = typeD(3, ZZ) o1 = {x - x , x + x , x - x , x + x , x - x , x + x } 1 2 1 2 1 3 1 3 2 3 2 3 o1 : Hyperplane Arrangement  i2 : ring A0 o2 = ZZ[x ..x ] 1 3 o2 : PolynomialRing i3 : A1 = typeD(4, QQ) o3 = {x - x , x + x , x - x , x + x , x - x , x + x , x - x , x + x , x - x , x + x , x - x , x + x } 1 2 1 2 1 3 1 3 1 4 1 4 2 3 2 3 2 4 2 4 3 4 3 4 o3 : Hyperplane Arrangement  i4 : ring A1 o4 = QQ[x ..x ] 1 4 o4 : PolynomialRing i5 : A3 = typeD(2, ZZ/2) o5 = {x + x , x + x } 1 2 1 2 o5 : Hyperplane Arrangement  i6 : trim A3 o6 = {x + x } 1 2 o6 : Hyperplane Arrangement  i7 : ring A3 ZZ o7 = --[x ..x ] 2 1 2 o7 : PolynomialRing

When the second input is a polynomial ring $R$, this ring determines the ambient ring of the Coxeter arrangement. The polynomial ring must have at least $n$ variables.

 i8 : A4 = typeD(3, ZZ[a,b,c,d]) o8 = {a - b, a + b, a - c, a + c, b - c, b + c} o8 : Hyperplane Arrangement  i9 : ring A4 o9 = ZZ[a..d] o9 : PolynomialRing i10 : A5 = typeD(2, ZZ[t][x,y,z]) o10 = {x - y, x + y} o10 : Hyperplane Arrangement  i11 : ring A5 o11 = ZZ[t][x..z] o11 : PolynomialRing

Omitting the ring (or second argument) is equivalent to setting $k := \mathbb{Q}$.

 i12 : A6 = typeD 3 o12 = {x - x , x + x , x - x , x + x , x - x , x + x } 1 2 1 2 1 3 1 3 2 3 2 3 o12 : Hyperplane Arrangement  i13 : ring A6 o13 = QQ[x ..x ] 1 3 o13 : PolynomialRing

• typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system