molienSeries -- computes the Molien (Hilbert) series of the invariant ring of a finite group

Synopsis

• Usage:

  molienSeries G

• Inputs:
  • G, a list, a finite group of invertible matrices, all defined over the same field of characteristic zero
• Outputs:
  • a divide expression, the Molien series of the invariant ring of G as a rational function in T

Description

The example below computes the Molien series for the dihedral group with 6 elements. K is the field obtained by adjoining a primitive third root of unity to QQ.

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

i2 : A=matrix{{a,0},{0,a^2}}; 

             2       2
o2 : Matrix K  <--- K

i3 : B=sub(matrix{{0,1},{1,0}},K); 

             2       2
o3 : Matrix K  <--- K

i4 : D6={A^0,A,A^2,B,A*B,A^2*B}

o4 = {| 1 0 |, | a 0    |, | -a-1 0 |, | 0 1 |, | 0    a |, | 0 -a-1 |}
      | 0 1 |  | 0 -a-1 |  | 0    a |  | 1 0 |  | -a-1 0 |  | a 0    |

o4 : List

i5 : molienSeries D6

                  1
o5 = ---------------------------
            2                 2
     (1 - T) (1 + T)(1 + T + T )

o5 : Expression of class Divide

This function is provided by the package InvariantRing.