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# finiteAction -- the group action generated by a list of matrices

- Usage:
    `finiteAction(L, R), finiteAction(M, R)`
- Inputs:
  - L, a list, of matrices representing the generators of a finite group
  - M, a matrix, generating a finite cyclic group of matrices
  - R, a polynomial ring, on which the group elements act by linear changes of coordinates
- Outputs:
  - an instance of the type FiniteGroupAction, the action of the (finite) matrix group generated by the input matrices on the given polynomial ring

## Description

This function is provided by the package InvariantRing.

The following example defines the permutation action of a symmetric group on three elements.

    i1 : R = QQ[x_1..x_3]

    o1 = R

    o1 : PolynomialRing

    i2 : L = apply(2, i -> permutationMatrix(3, [i + 1, i + 2] ) )

    o2 = {| 0 1 0 |, | 1 0 0 |}
          | 1 0 0 |  | 0 0 1 |
          | 0 0 1 |  | 0 1 0 |

    o2 : List

    i3 : S3 = finiteAction(L, R)

    o3 = R <- {| 0 1 0 |, | 1 0 0 |}
               | 1 0 0 |  | 0 0 1 |
               | 0 0 1 |  | 0 1 0 |

    o3 : FiniteGroupAction

On the other hand, the action below corresponds to a cyclic permutation of the variables.

    i4 : P = permutationMatrix toString 231

    o4 = | 0 0 1 |
         | 1 0 0 |
         | 0 1 0 |

                  3       3
    o4 : Matrix ZZ  <-- ZZ

    i5 : C3 = finiteAction(P, R)

    o5 = R <- {| 0 0 1 |}
               | 1 0 0 |
               | 0 1 0 |

    o5 : FiniteGroupAction

## Ways to use finiteAction:

- `finiteAction(List,PolynomialRing)`
- `finiteAction(Matrix,PolynomialRing)`

## For the programmer

The object finiteAction is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/InvariantRing/FiniteGroupsDoc.m2:54:0.