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

• 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.

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 =             2
     QQ[x x x , x x x ]
         1 2 3   1 3 4

o4 : RingOfInvariants

i5 : hilbertSeries S

             1
o5 = ----------------
           4       3
     (1 - T )(1 - T )

o5 : Expression of class Divide