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

## Synopsis

• Function: invariants
• Usage:
invariants G
• Inputs:
• Optional inputs:
• Strategy (missing documentation) => ..., default value UseNormaliz, the strategy used to compute diagonal invariants, options are UsePolyhedra or UseNormaliz.
• DegreeBound => ..., default value infinity, degree bound for invariants of finite groups
• DegreeLimit => ..., default value {}, GB option for invariants
• SubringLimit => ..., default value infinity, GB option for invariants
• UseCoefficientRing => ..., default value false, option to compute invariants over the given coefficient ring
• UseLinearAlgebra => ..., default value false, strategy for computing invariants of finite groups
• 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({"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| +---------------------------------------------------------------------------------------------------------+