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

- Usage:
    `secondaryInvariants(P,G)`
- Inputs:
  - P, a list, a list of primary invariants in n variables f1,...,fn 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,...,xn]G of G that makes R into a free K[f1,...,fn]-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.

## Ways to use secondaryInvariants:

- `secondaryInvariants(List,FiniteGroupAction)`

## For the programmer

The object secondaryInvariants is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/InvariantRing/HawesDoc.m2:487:0.