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InvariantRing :: invariants(FiniteGroupAction)

invariants(FiniteGroupAction) -- computes the generating invariants of a group action

Synopsis

Description

This function is provided by the package InvariantRing.

It implements King's algorithm to compute a minimal set of generating invariants for the action of a finite group on a polynomial ring following Algorithm 3.8.2 in:

The following example computes the invariants of the alternating group on 4 elements.

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing
i2 : L = apply({"2314","2143"},permutationMatrix);
i3 : A4 = finiteAction(L,R)

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

o3 : FiniteGroupAction
i4 : netList invariants A4

     +---------------------------------------------------------------------------------------------------------+
o4 = |x  + x  + x  + x                                                                                         |
     | 1    2    3    4                                                                                        |
     +---------------------------------------------------------------------------------------------------------+
     | 2    2    2    2                                                                                        |
     |x  + x  + x  + x                                                                                         |
     | 1    2    3    4                                                                                        |
     +---------------------------------------------------------------------------------------------------------+
     | 3    3    3    3                                                                                        |
     |x  + x  + x  + x                                                                                         |
     | 1    2    3    4                                                                                        |
     +---------------------------------------------------------------------------------------------------------+
     | 4    4    4    4                                                                                        |
     |x  + x  + x  + x                                                                                         |
     | 1    2    3    4                                                                                        |
     +---------------------------------------------------------------------------------------------------------+
     | 3 2        3 2    2   3    2 3      3 2      2 3      3   2    3   2      3 2      2 3    2   3      2 3|
     |x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x |
     | 1 2 3    1 2 3    1 2 3    1 2 4    1 3 4    2 3 4    1 2 4    2 3 4    1 3 4    1 2 4    1 3 4    2 3 4|
     +---------------------------------------------------------------------------------------------------------+

See also