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 |
The object finiteAction is a method function.