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 ring, a polynomial (coordinate) ring in n+1 variables
- k, a ring, a coefficient ring; by default, QQ
Outputs:
- a hyperplane arrangement, 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 , x , x ]
         1   2   3   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 , x , x , x ]
      3  1   2   3   4   5

o8 : PolynomialRing

Ways to use `typeA` :

typeA(ZZ)
typeA(ZZ,PolynomialRing) -- A_n arrangement with specified coordinate ring
typeA(ZZ,Ring) -- A_n reflection arrangement with specified coefficient ring

typeA -- Type A reflection arrangement

Synopsis

Description

See also

Ways to use typeA :

Ways to use `typeA` :