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hilbertSeries(RingOfInvariants) -- Hilbert series of the invariant 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 =     5   3 2   5 3 4     4 2 3     6 3 3   3   2     4 2 2   5 5 5 
     QQ[x x x x , x x x x , 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 2 3 4   1 3 4   1 2 3 4   1 3 4   1 2 3 
     ------------------------------------------------------------------------
              2
     x x x , x x x ]
      1 2 3   1 3 4

o4 : RingOfInvariants
i5 : hilbertSeries S

          7    8    10    11    12    13    17     18     19     20     21     22     23    24    25    28     29     30     31     32     33     34     35     36     40     41     42     43     44     45     46     47    48    51    52     53     54     55     56     57     58    59    63    64    65    66    68    69    76
     1 - T  - T  - T   - T   - T   - T   + T   + 2T   + 2T   + 2T   + 2T   + 2T   + 3T   + T   + T   - T   - 2T   - 4T   - 3T   - 3T   - 4T   - 3T   - 3T   - 2T   + 2T   + 3T   + 3T   + 4T   + 3T   + 3T   + 4T   + 2T   + T   - T   - T   - 3T   - 2T   - 2T   - 2T   - 2T   - 2T   - T   + T   + T   + T   + T   + T   + T   - T
o5 = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                     15       13       12       11       10       8       7       4       3
                                                                                                                               (1 - T  )(1 - T  )(1 - T  )(1 - T  )(1 - T  )(1 - T )(1 - T )(1 - T )(1 - T )

o5 : Expression of class Divide

Ways to use this method:


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/InvariantRing/InvariantsDoc.m2:967:0.