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 a polynomial ring $R$ that determines the ambient ring
Outputs:
- a hyperplane arrangement,

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

Ways to use this method:

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

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

Synopsis

Description

See also

Ways to use this method: