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invariants(...,DegreeBound=>...) -- degree bound for invariants of finite groups

Description

This function is provided by the package InvariantRing.

This optional argument allows the user to provide an upper bound for the degree of the generating invariants of a finite group action. If no upper bound is provided, the order of the group is used as an upper bound. Providing a smaller upper bound may speed up the computation of invariants. However, if the value provided is too small the resulting list may not generate the ring of invariants.

The following example computes the invariants of the symmetric group on 4 elements.

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
 -- 1.115s 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,DegreeBound=>4)
 -- .822454s elapsed

Warning: stopping condition not met!
Output may not generate the entire ring of invariants.
Increase value of DegreeBound.


                          2    2    2    2   3    3    3    3   4    4    4  
o5 = {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

o5 : List

Caveat

If the value provided for this option is too small, then the output list does not generate the entire ring of invariants. A warning message is produced to notify the user of the issue.

See also

Functions with optional argument named DegreeBound:

Further information

  • Default value: infinity
  • Function: invariants -- computes the generating invariants of a group action
  • Option key: DegreeBound -- degree bound for invariants of finite groups

The source of this document is in InvariantRing/InvariantsDoc.m2:384:0.