hilbertSeries(RingOfInvariants) -- Hilbert series of the invariant ring

Synopsis

Function: hilbertSeries
Usage:

hilbertSeries S
Inputs:
- S, an instance of the type RingOfInvariants
Optional inputs:
- Order => ..., default value infinity,
- Reduce => ..., default value false,
Outputs:
- a divide expression, the Hilbert series of the invariant ring as a module over the ambient polynomial ring.

Description

This function is provided by the package InvariantRing.

This method computes the hilbert series of the ring of invariants.

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing

i2 : W = matrix{{0,1,-1,1},{1,0,-1,-1}}

o2 = | 0 1 -1 1  |
     | 1 0 -1 -1 |

              2        4
o2 : Matrix ZZ  <--- ZZ

i3 : T = diagonalAction(W, R)

             * 2
o3 = R <- (QQ )  via 

     | 0 1 -1 1  |
     | 1 0 -1 -1 |

o3 : DiagonalAction

i4 : S = R^T

o4 =     4 2 3     5   3 2   6 3 3   2
     QQ[x x x x , x x x x , x x x , x x x , x x x ]
         1 2 3 4   1 2 3 4   1 3 4   1 3 4   1 2 3

o4 : RingOfInvariants

i5 : hilbertSeries S

          10    11    12    21    22    23    33
     1 - T   - T   - T   + T   + T   + T   - T
o5 = -------------------------------------------
           12       11       10       4       3
     (1 - T  )(1 - T  )(1 - T  )(1 - T )(1 - T )

o5 : Expression of class Divide