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InvariantRing :: invariants(FiniteGroupAction,ZZ)

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

Synopsis

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 ยง3.1.1 of

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.

See also