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invariants(...,UseLinearAlgebra=>...) -- strategy for computing invariants of finite groups

Description

This function is provided by the package InvariantRing.

This optional argument determines the strategy used to compute generating invariants of a finite group action. The default strategy uses the Reynolds operator, however this may be slow for large groups. Setting this argument to true uses the linear algebra method for computing invariants of a given degree by calling invariants(FiniteGroupAction,ZZ). This may provide a speedup at lower degrees, especially if the user-provided generating set for the group is small.

The following example computes the invariants of the symmetric group on 4 elements. Note that using different strategies may lead to different sets of generating invariants.

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing
 
i2 : L = apply({"2134","2341"},permutationMatrix);
 
i3 : S4 = finiteAction(L,R)

o3 = R <- {| 0 1 0 0 |, | 0 0 0 1 |}
           | 1 0 0 0 |  | 1 0 0 0 |
           | 0 0 1 0 |  | 0 1 0 0 |
           | 0 0 0 1 |  | 0 0 1 0 |

o3 : FiniteGroupAction
 
i4 : elapsedTime invariants S4
 -- 1.02682s elapsed

                          2    2    2    2   3    3    3    3   4    4    4  
o4 = {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    2    3  
     ------------------------------------------------------------------------
      4
     x }
      4

o4 : List
 
i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
 -- .195035s elapsed

o5 = {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    1 3    2 3    1 4    2 4    3 4   1 2 3  
     ------------------------------------------------------------------------
     x x x  + x x x  + x x x , x x x x }
      1 2 4    1 3 4    2 3 4   1 2 3 4

o5 : List

See also

Functions with optional argument named UseLinearAlgebra:

Further information

  • Default value: false
  • Function: invariants -- computes the generating invariants of a group action
  • Option key: UseLinearAlgebra -- strategy for computing invariants of finite groups

The source of this document is in InvariantRing/InvariantsDoc.m2:487:0.