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

## Synopsis

• Function: typeA
• Usage:
typeA(n, k)
typeA(n, R)
typeA n
• Inputs:
• n, an integer, that is positive
• 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 $A_n$ is the hyperplane arrangement in $k^{n+1}$ defined by $x_i - x_j$ for all $1 \leq i < j \leq n+1$.

 i1 : A0 = typeA(3, ZZ) o1 = {x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 2 3 2 4 3 4 o1 : Hyperplane Arrangement  i2 : ring A0 o2 = ZZ[x ..x ] 1 4 o2 : PolynomialRing i3 : A1 = typeA(4, QQ) o3 = {x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 4 5 o3 : Hyperplane Arrangement  i4 : ring A1 o4 = QQ[x ..x ] 1 5 o4 : PolynomialRing i5 : A3 = typeA(2, ZZ/2) o5 = {x + x , x + x , x + x } 1 2 1 3 2 3 o5 : Hyperplane Arrangement  i6 : ring A3 ZZ o6 = --[x ..x ] 2 1 3 o6 : 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 $n+1$ variables.

 i7 : A4 = typeA(3, ZZ[a,b,c,d]) o7 = {a - b, a - c, a - d, b - c, b - d, c - d} o7 : Hyperplane Arrangement  i8 : ring A4 o8 = ZZ[a..d] o8 : PolynomialRing i9 : A5 = typeA(2, ZZ[t][x,y,z]) o9 = {x - y, x - z, y - z} o9 : Hyperplane Arrangement  i10 : ring A5 o10 = ZZ[t][x..z] o10 : PolynomialRing

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

 i11 : A6 = typeA 2 o11 = {x - x , x - x , x - x } 1 2 1 3 2 3 o11 : Hyperplane Arrangement  i12 : ring A6 o12 = QQ[x ..x ] 1 3 o12 : PolynomialRing

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