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

Synopsis

• 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

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