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# groupIdeal -- ideal defining a linearly reductive group

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

## Description

This function is provided by the package InvariantRing.

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

## Ways to use groupIdeal:

- `groupIdeal(LinearlyReductiveAction)`

## For the programmer

The object groupIdeal is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/InvariantRing/LinearlyReductiveGroupsDoc.m2:83:0.