Use the InvariantRing package with non-standard coefficient fields

[Takehiko Yasuda]

I am tring to compute invariant rings for finite group actions using the package InvariantRing. It works well, when I use QQ or a finite field. But, I get wrong outputs when I use finite extensions of QQ. Doesn't this package work with such a coefficient field?

To define finite group actions, one needs to adjoin roots of unity to a prime field. I'd like to do this in characteristic zero, though I can use a finite field as an alternative to do a similar computation. Can anybody help me? 

What I tried is as follows:

+ M2 --no-readline --print-width 79
Macaulay2, version 1.22
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
               Isomorphism, LLLBases, MinimalPrimes, OnlineLookup,
               PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone

i1 : loadPackage "InvariantRing"

o1 = InvariantRing

o1 : Package

i2 : K = toField(QQ[a]/(a^3-1))

o2 = K

o2 : PolynomialRing

i3 : S = K[x,y]

o3 = S

o3 : PolynomialRing

i4 : myAction = finiteAction(matrix{{a,0},{0,a}},S)

o4 = S <- {| a 0 |}
           | 0 a |

o4 : FiniteGroupAction

i5 : invariants myAction

o5 = {0y, 0x}

o5 : List

[Federico Galetto]

Hello! I am one of the authors of InvariantRing. Thank you for using our package!

I think the unexpected result you are seeing may be related to the fact that a^3-1 is not a minimal polynomial for a over QQ.

The adjoin a cubic root of unity to the rational numbers you can do the following:

i1 : loadPackage "InvariantRing"
o1 = InvariantRing
o1 : Package

i2 : K = toField(QQ[a]/(a^2+a+1))

o2 = K
o2 : PolynomialRing

i3 : S=K[x,y]

o3 = S
o3 : PolynomialRing

i4 : myAction = finiteAction(matrix{{a,0},{0,a}},S)
o4 = S <- {| a 0 |}
                  | 0 a |
o4 : FiniteGroupAction

i5 : invariants myAction

       3     2   2    3
o5 = {y , x*y , x y, x }
o5 : List

As far as I recall, the characteristic zero methods we wrote should work just fine over finite extensions of QQ (though they may run slower).

Also, if you are interested in invariants for actions of abelian groups, I recommend setting up that kind of action since the specialized algorithms for abelian groups are much faster than those for arbitrary finite groups.

Cheers,

Fred

[Takehiko Yasuda]

Fred,  thank you so much for your prompt help! 

It was my terrible mistake. My definition of the field was wrong, as the polynomial wasn't irreducible. 

I was able to verify that the package works well for a finite extension of QQ. 

I love this package! Thank you again.

2024年5月13日月曜日 8:23:12 UTC+9 Federico Galetto: