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definingIdeal -- presentation of a ring of invariants as polynomial ring modulo the defining ideal

Description

This function is provided by the package InvariantRing.

This method presents the ring of invariants as a polynomial ring modulo the defining ideal. The default variable name in the polynomial ring is u_i. You can pass the variable name you want as optional input.

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 : definingIdeal S

                             2        2            2        3        3      
o5 = ideal (u  - u u , u  - u , u  - u u , u  - u u , u  - u , u  - u u , u 
             5    8 9   6    9   3    8 9   1    8 9   4    9   2    8 9   7
     ------------------------------------------------------------------------
        5
     - u )
        8

o5 : Ideal of QQ[u ..u ]
                  1   9

Ways to use definingIdeal:

  • definingIdeal(RingOfInvariants)

For the programmer

The object definingIdeal is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/InvariantRing/InvariantsDoc.m2:895:0.