secondaryInvariants(...,PrintDegreePolynomial=>...) -- an optional argument for secondaryInvariants that determines the printing of an informative polynomial

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
• 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

PrintDegreePolynomial takes a Boolean value and is set to false by default. If it is set to true, then secondaryInvariants will print a polynomial in the variable T. This polynomial encodes the degrees of the secondary invariants (given by the exponents of T appearing in it) and the number of secondary invariants of a given degree (the coefficient of the corresponding term in the polynomial). Specifically, if secondaryInvariants takes as input a set of primary invariants of degrees d₁,...,d_n for an invariant ring S^G of a finite group G, and H(S^G,T)denotes the Molien (Hilbert) series of S^G, then secondaryInvariants will compute the polynomial H(S^G,T)*(1-T^d₁)*...*(1-T^d_n).

The example below computes the secondary invariants for the dihedral group with 6 elements, given a set of primary invariants P. The optional argument PrintDegreePolynomial is set to true in order to see which degrees the secondary invariants should have.

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,PrintDegreePolynomial=>true)
 4    2
t  + t  + 1

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

o7 : List

Further information

• Default value: false
• Function: secondaryInvariants -- computes secondary invariants for the invariant ring of a finite group
• Option key: PrintDegreePolynomial -- an optional argument for secondaryInvariants that determines the printing of an informative polynomial

Caveat

Functions with optional argument named PrintDegreePolynomial :

• hironakaDecomposition(...,PrintDegreePolynomial=>...)
• secondaryInvariants(...,PrintDegreePolynomial=>...) -- an optional argument for secondaryInvariants that determines the printing of an informative polynomial