invariants(FiniteGroupAction,ZZ) -- basis for graded component of invariant ring

Synopsis

Function: invariants
Usage:

invariants(G,d)
Inputs:
- G, an instance of the type FiniteGroupAction
- d, an integer, a degree or multidegree
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, an additive basis for a graded component of the ring of invariants

Description

This function is provided by the package InvariantRing

When called on a finite group action and a (multi)degree, it computes an additive basis for the invariants of the action in the given degree.

This function uses an implementation of the Linear Algebra Method described in §3.1.1 of

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

For example, consider the following action of a dihedral group.

i1 : K=toField(QQ[a]/(a^2+a+1));

i2 : R = K[x,y]

o2 = R

o2 : PolynomialRing

i3 : A=matrix{{a,0},{0,a^2}};

             2      2
o3 : Matrix K  <-- K

i4 : B=sub(matrix{{0,1},{1,0}},K);

             2      2
o4 : Matrix K  <-- K

i5 : D6=finiteAction({A,B},R)

o5 = R <- {| a 0    |, | 0 1 |}
           | 0 -a-1 |  | 1 0 |

o5 : FiniteGroupAction

i6 : invariants(D6,20)

       10 10   13 7    7 13   16 4    4 16   19       19
o6 = {x  y  , x  y  + x y  , x  y  + x y  , x  y + x*y  }

o6 : List

It is important to note that this implementation uses the group generators provided by the user, which can be recovered using generators(FiniteGroupAction). To improve efficiency the user should provide a generating set for the group that is as small as possible.

Ways to use this method: