First interface for invariant theory of finite groups #584
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Co-authored-by: Tommy Hofmann <thofma@gmail.com>
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| mutable struct InvRing{S, T, U, V, W, X} | |||
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It would be better if those one letter type names were replaced by more verbose (and self-explanatory) names, say
- S -> FldT
- T -> PolyRingT
- U -> GrpT
- V -> ActionT
- W -> SingularActionT
- X -> PolyElemT
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I would prefer that as well. @thofma, you added those, anything against this?
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Be my guest. I would have thought that the field names are self-explanatory, but we can make it more redundant.
While you are at it, you might want to change T and U so that the group type is the second parameter. This will make it a bit easier when adding methods for specific group types (i.e. InvRing{_, <: MatrixGroup} vs InvRing(_, _, <: MatrixGroup).)
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| mutable struct InvRing{S, T, U, V, W, X} | |||
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I think InvRing should be turned into an actual ring type, so in particular subtype Ring
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I agree of course that one would expect InvRing to be a Ring. However, as you already pointed out, according to the documentation, rings have to comply to a lot of things and in particular need to have elements. I know that this discussion is already going on elsewhere (and in the end I implement whatever you want), but here are my two pennies on it: I would not expect a user to want to do computations in the invariant ring as a subring of the polynomial ring the group acts on. I would think that one wants to have a presentation of the invariant ring as an affine algebra for this. So I would aim for the following interface: InvRing is just a cache and if one wants a Ring, there should be a function presentation(::InvRing) which returns the affine algebra (polynomial ring modulo ideal, I'm not sure we have a type for such a quotient ring yet?) and the map to the InvRing (or rather the polynomial ring in there?).
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There are quotient rings, see the docu under Commutative Algebra, with more functionality for affine algebras (see again the docu). Indeed, in particular for geometric invariant theory applications, a function presentation(::InvRing) is crucial. Also, as a first step towards such a presentation, we should have a function fundamental_invariants::InvRing).
Both functions have be to be developed essential from scratch (but see also the functionorbit_variety in SINGULAR).
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| primary_singular::Singular.smatrix | ||
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| function InvRing(K::S, G::U, action::Vector{V}) where {S, U, V} |
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Perhaps check that K is a field (e.g. by adding a type constraint)?
| primary_singular::Singular.smatrix | ||
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| function InvRing(K::S, G::U, action::Vector{V}) where {S, U, V} | ||
| n = degree(G) |
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What is G::U supposed to be? You call degree on it... so perhaps a permutation group? Or is degree also supposed to be the dimension of the natural module of a matrix group?
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So far we can only handle matrix groups because we pass everything to Singular. So, degree is the rank of the matrices as defined in Groups/matrices/MatGrp.jl:490.
I would think that this constructor should not restrict the type or maybe U <: AbstractAlgebra.Group if absolutely necessary. The user is not supposed to call this anyway.
| R = polynomial_ring(IR) | ||
| p = Vector{elem_type(R)}() | ||
| for i = 1:ncols(P) | ||
| push!(p, R(P[1, i])) |
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Based on the codecov messages, none of the code in this file is tested, not even with a trivial input. It'd be nice to get some simple tests.
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Sorry for not doing so earlier, but I just read DEVELOPMENT.md and realized that it might be a good idea to move the directory |
Here is a first interface to the invariant theory in the finvar library of Singular.