secondaryInvariants -- computes secondary invariants for the invariant ring of a finite group

Synopsis

• Usage:

  secondaryInvariants(P,G)

• Inputs:
  • P, a list, a list of primary invariants in n variables f₁,...,f_n for the invariant ring of G defined over a field K of characteristic zero
  • G, an instance of the type FiniteGroupAction
• Optional inputs:
  • PrintDegreePolynomial => ..., default value false, an optional argument for secondaryInvariants that determines the printing of an informative polynomial
• Outputs:
  • a list, a list S of secondary invariants for the invariant ring R=K[x₁,...,x_n]^G of G that makes R into a free K[f₁,...,f_n]-module with basis S

Description

The example below computes the secondary invariants for the dihedral group with 6 elements, given a set of primary invariantsP.

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

i2 : R=K[x,y];

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 : P={x^3+y^3,-(x^3-y^3)^2};

i7 : secondaryInvariants(P,D6)

               2 2
o7 = {1, x*y, x y }

o7 : List

This function is provided by the package InvariantRing.

Caveat

Currently, a user needs to ensure that the all primary invariants are defined with coefficients in a ring that Macaulay2 recognises as a characteristic zero field (see toField for a way to do this).

Note also that the function secondaryInvariants only works when R is defined over a field of characteristic zero.