graphic -- Make a graphic arrangement

Synopsis

• Usage:

  graphic(G) or graphic(G,k) or graphic(G,R)

• Inputs:
  • G, a list, a graph, expressed as a list of pairs of vertices
  • k, a ring, an optional coefficient ring, by default QQ
  • R, a polynomial ring, an optional coordinate ring for the arrangement
• Outputs:
  • a hyperplane arrangement, the graphic arrangement from graph G

Description

i1 : G = {{1,2},{2,3},{3,4},{4,1}}; -- a four-cycle

i2 : AG = graphic G

o2 = {- x  + x , - x  + x , - x  + x , x  - x }
         1    2     2    3     3    4   1    4

o2 : Hyperplane Arrangement 

i3 : describe AG

o3 = {- x  + x , - x  + x , - x  + x , x  - x }
         1    2     2    3     3    4   1    4

i4 : rank AG -- the number of vertices minus number of components

o4 = 3

i5 : ring AG

o5 = QQ[x ..x ]
         1   4

o5 : PolynomialRing

i6 : ring graphic(G,ZZ[x,y,z,w])

o6 = ZZ[x..z, w]

o6 : PolynomialRing

See also

• typeA -- Type A reflection arrangement