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

## Synopsis

• Usage:
invariants G
• Inputs:
• Outputs:
• L, a list, a minimal set of generating invariants for the group action

## 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 -- 1.14805 seconds 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) -- 1.78056 seconds 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

## 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

## 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.