Meat axe

[fieker]

1st steps towards Allan's thesis: Char0 MeatAxe. Look at the TODOs in the diff

examples that I tried
G = D(8) (8 elem)
M = natural_gmodule(G, QQ)
and M \otimes M

G = 8T5
M = natural_gmodule(G, QQ)

and then
z = Oscar.GModuleFromGap.split_into_homogenous(M)
...
Oscar.GModuleFromGap.split_homogenous
resp
Oscar.GModuleFromGap.split_homogenous2

[ThomasBreuer]

Nice.
When I remove the Unbalanced flag from the call of induce_rational_reconstruction, the following examples almost work, just the second example (regular repres. of D8) runs into an error in lll_basis.

@testset begin
  examples = [
    (natural_gmodule(dihedral_group(PermGroup, 8), QQ), 2)    # 4 = 1^2 + 2
    (regular_gmodule(dihedral_group(PermGroup, 8), QQ)[1], 4) # 8 = 1^4 + 2^2
    (natural_gmodule(quaternion_group(PermGroup, 8), QQ), 4)  # 8 = 1^4 + 4
    (natural_gmodule(alternating_group(5), QQ), 4)            # 5 = 1 + 4
  ]

  @testset for (M, d) in examples
    z = Oscar.GModuleFromGap.split_into_homogenous(M)
    nonlin = z[findfirst(x -> dim(x) == d, z)]
    spl = Oscar.GModuleFromGap.split_homogenous(nonlin)
  end
end

[ThomasBreuer]

Shouldn't it be "homogeneous" instead of "homogenous"?

[fieker]

On Fri, Feb 28, 2025 at 05:50:05AM -0800, Thomas Breuer wrote: ThomasBreuer left a comment (oscar-system/Oscar.jl#4674) Nice. When I remove the `Unbalanced` flag from the call of `induce_rational_reconstruction`, the following examples almost work, just the second example (regular repres. of D8) runs into an error in

unbalanced in some version is a waiting Nemo PR... what is the problem in lll_basis?

…

`lll_basis`. ``` @testset begin examples = [ (natural_gmodule(dihedral_group(PermGroup, 8), QQ), 2) # 4 = 1^2 + 2 (regular_gmodule(dihedral_group(PermGroup, 8), QQ)[1], 4) # 8 = 1^4 + 2^2 (natural_gmodule(quaternion_group(PermGroup, 8), QQ), 4) # 8 = 1^4 + 4 (natural_gmodule(alternating_group(5), QQ), 4) # 5 = 1 + 4 ] @testset for (M, d) in examples z = Oscar.GModuleFromGap.split_into_homogenous(M) nonlin = z[findfirst(x -> dim(x) == d, z)] spl = Oscar.GModuleFromGap.split_homogenous(nonlin) end end ``` -- Reply to this email directly or view it on GitHub: #4674 (comment) You are receiving this because you authored the thread. Message ID: ***@***.***>

[ThomasBreuer]

what is the problem in lll_basis?

I get

julia> M = regular_gmodule(dihedral_group(PermGroup, 8), QQ)[1]
G-module for permutation group of degree 4 and order 8 acting on vector space of dimension 8 over QQ

julia> d = 4;

julia> z = Oscar.GModuleFromGap.split_into_homogenous(M);

julia> nonlin = z[findfirst(x -> dim(x) == d, z)]
G-module for permutation group of degree 4 and order 8 acting on subspace over QQ with 4 generators and no relations

julia> Oscar.GModuleFromGap.split_homogenous(nonlin)
s = schur_index(E) = 1
k = dim((center(E))[1]) = 1
m = div(root(div(dim(E), k), 2), s) = 2
ERROR: MethodError: no method matching lll_basis(::Hecke.AlgAssAbsOrd{ZZRing, MatAlgebra{QQFieldElem, QQMatrix}})
[...]

[fieker]

On Fri, Feb 28, 2025 at 06:42:03AM -0800, Thomas Breuer wrote: ThomasBreuer left a comment (oscar-system/Oscar.jl#4674) > what is the problem in lll_basis? I get ```julia julia> M = regular_gmodule(dihedral_group(PermGroup, 8), QQ)[1] G-module for permutation group of degree 4 and order 8 acting on vector space of dimension 8 over QQ julia> d = 4; julia> z = Oscar.GModuleFromGap.split_into_homogenous(M); julia> nonlin = z[findfirst(x -> dim(x) == d, z)] G-module for permutation group of degree 4 and order 8 acting on subspace over QQ with 4 generators and no relations julia> Oscar.GModuleFromGap.split_homogenous(nonlin) s = schur_index(E) = 1 k = dim((center(E))[1]) = 1 m = div(root(div(dim(E), k), 2), s) = 2 ERROR: MethodError: no method matching lll_basis(::Hecke.AlgAssAbsOrd{ZZRing, MatAlgebra{QQFieldElem, QQMatrix}})

Tommy changed the order of the type parameters recently - don't know which is up-to-date. The lll_basis function is in the Oscar/experimenta/GModule/src/GModule file - but poss. for the wrong order of parameters: function Oscar.lll_basis(M::Hecke.AlgAssAbsOrd{MatAlgebra{QQFieldElem, QQMatrix}, ZZRing})

…

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[ThomasBreuer]

Yes.
When I change function Oscar.lll_basis(M::Hecke.AlgAssAbsOrd{MatAlgebra{QQFieldElem, QQMatrix}, ZZRing}) to Oscar.lll_basis(M::Hecke.AlgAssAbsOrd{ZZRing, MatAlgebra{QQFieldElem, QQMatrix}}) then the @testset passes.

(The next challenge will be splitting the homogeneous 4-dim. module of Q8 over an extension field.)

[fieker]

On Fri, Feb 28, 2025 at 09:11:34AM -0800, Thomas Breuer wrote: ThomasBreuer left a comment (oscar-system/Oscar.jl#4674) Yes. When I change `function Oscar.lll_basis(M::Hecke.AlgAssAbsOrd{MatAlgebra{QQFieldElem, QQMatrix}, ZZRing})` to `Oscar.lll_basis(M::Hecke.AlgAssAbsOrd{ZZRing, MatAlgebra{QQFieldElem, QQMatrix}})` then the ***@***.***` passes. (The next challenge will be splitting the homogeneous 4-dim. module of Q8 over an extension field.)

Implemented, but due to logic error not there... minor changes: julia> Oscar.GModuleFromGap.split_homogenous2(z[5]) s = schur_index(E) = 2 k = dim((center(E))[1]) = 1 m = div(root(div(dim(E), k), 2), s) = 1 :bingo = :bingo (G-module for G acting on subspace over number field with 2 generators and no relations, Map: g-module for G acting on subspace over number field with 2 generators and no relations -> g-module for G acting on vector space of dimension 4 over number field) Which gives the split - but in general the field will not be minimal The code is there, not not linked in (yet)

…

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[ThomasBreuer]

Concerning duplicate submodules:
My understanding is that once we have found one element in basis(C) whose minimal polynomial has two different factors, M is the direct sum of kernel and image of what is called x here.
Thus we can call split_into_homogenous for kernel and image, and return the concatenation of the results.

[fieker]

The thesis is bad on images, it is using kernel only. But we can always improve it...