definingIdeal -- presentation of a ring of invariants as polynomial ring modulo the defining ideal

Synopsis

• Usage:

  definingIdeal S

• Inputs:
  • S, an instance of the type RingOfInvariants
• Optional inputs:
  • Variable (missing documentation) => ..., default value u, name of the varibles in the polynomial ring.
• Outputs:
  • an ideal, which defines the ring of invariants as a polynomial ring modulo the ideal.

Description

This method presents the ring of invariants as a polynomial ring modulo the defining ideal. The default variable name in the polynomial ring is u_i. You can pass the variable name you want as optional input.

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing

i2 : W = matrix{{0,1,-1,1},{1,0,-1,-1}}

o2 = | 0 1 -1 1  |
     | 1 0 -1 -1 |

              2        4
o2 : Matrix ZZ  <--- ZZ

i3 : T = diagonalAction(W, R)

             * 2
o3 = R <- (QQ )  via 

     | 0 1 -1 1  |
     | 1 0 -1 -1 |

o3 : DiagonalAction

i4 : S = R^T

o4 =             2
     QQ[x x x , x x x ]
         1 2 3   1 3 4

o4 : RingOfInvariants

i5 : definingIdeal S

o5 = ideal ()

o5 : Ideal of QQ[u ..u ]
                  1   2