groupIdeal -- ideal defining a linearly reductive group

Synopsis

• Usage:

  groupIdeal L

• Inputs:
  • L, an instance of the type LinearlyReductiveAction, of a group with coordinate ring S/I
• Outputs:
  • an ideal, I

Description

Suppose the action L consists of the linearly reductive group with coordinate ring S/I (where S is a polynomial ring) acting on a (quotient of) a polynomial ring R via the action matrix M. This function returns the ideal I.

i1 : S = QQ[z]

o1 = S

o1 : PolynomialRing

i2 : I = ideal(z^2 - 1)

            2
o2 = ideal(z  - 1)

o2 : Ideal of S

i3 : M = matrix{{(z+1)/2, (1-z)/2},{(1-z)/2, (z+1)/2}}

o3 = | 1/2z+1/2  -1/2z+1/2 |
     | -1/2z+1/2 1/2z+1/2  |

             2      2
o3 : Matrix S  <-- S

i4 : R = QQ[x,y]

o4 = R

o4 : PolynomialRing

i5 : L = linearlyReductiveAction(I, M, R)

                   2
o5 = R <- S/ideal(z  - 1) via 

     | 1/2z+1/2  -1/2z+1/2 |
     | -1/2z+1/2 1/2z+1/2  |

o5 : LinearlyReductiveAction

i6 : groupIdeal L

            2
o6 = ideal(z  - 1)

o6 : Ideal of S