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InvariantRing :: finiteAction

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

Synopsis

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 :

For the programmer

The object finiteAction is a method function.