typeA -- Type A reflection arrangement

Synopsis

• Usage:
typeA(n) or typeA(n,R) or typeA(n,k)
• Inputs:
• n, an integer, the rank
• R, , a polynomial (coordinate) ring in n+1 variables
• k, a ring, a coefficient ring; by default, QQ
• Outputs:
• , the A_n reflection arrangement

Description

The hyperplane arrangement with hyperplanes x_i-x_j.
 i1 : A3 = typeA(3) 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 : describe A3 o2 = {x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 2 3 2 4 3 4 i3 : ring A3 o3 = QQ[x ..x ] 1 4 o3 : PolynomialRing
Alternatively, one may specify a coordinate ring,
 i4 : S = ZZ[w,x,y,z]; i5 : A3' = typeA(3,S) o5 = {w - x, w - y, w - z, x - y, x - z, y - z} o5 : Hyperplane Arrangement  i6 : describe A3' o6 = {w - x, w - y, w - z, x - y, x - z, y - z}
or a coefficient ring:
 i7 : A4 = typeA(4,ZZ/3) o7 = {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 o7 : Hyperplane Arrangement  i8 : ring A4 ZZ o8 = --[x ..x ] 3 1 5 o8 : PolynomialRing