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

Function: invariants
Usage:

invariants G
Inputs:
- G, an instance of the type FiniteGroupAction
Optional inputs:
- DegreeBound => ..., default value infinity, degree bound for invariants of finite groups
- DegreeLimit => ..., default value {}, GB option for invariants
- Strategy => ..., default value "Default", choose the strategy for computing invariants
- SubringLimit => ..., default value infinity, GB option for invariants
- UseCoefficientRing => ..., default value false, option to compute invariants over the given coefficient ring
- UsePolyhedra => ..., default value false, use Polyhedra package for invariants of tori
Outputs:
- L, a list, a minimal set of generating invariants for the group action

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:

Derksen, H. & Kemper, G. (2015).Computational Invariant Theory. Heidelberg: Springer.

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({[2, 3, 1, 4], [2, 1, 4, 3]}, 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|
     +---------------------------------------------------------------------------------------------------------+

Ways to use this method:

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/InvariantRing/InvariantsDoc.m2:400:0.

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

Description

See also

Ways to use this method: