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Add a canonical form for lattices #6102

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48bc8a3
Add main algorithm
Kazak-11 Jun 15, 2026
22f89ef
Add test set
Kazak-11 Jun 15, 2026
a4bb47f
Add exports
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Small clear
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Added doc string
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small changes in tests
Kazak-11 Jul 6, 2026
76a763c
Merge branch 'oscar-system:master' into add_lattice_canonical_form
Kazak-11 Jul 7, 2026
7a2c44f
Update src/NumberTheory/ZLattices.jl
Kazak-11 Jul 8, 2026
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Update src/NumberTheory/ZLattices.jl
Kazak-11 Jul 8, 2026
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Update test/NumberTheory/ZLattices.jl
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Update test/NumberTheory/ZLattices.jl
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Update src/NumberTheory/ZLattices.jl
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Update src/NumberTheory/ZLattices.jl
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Update src/NumberTheory/ZLattices.jl
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Update src/NumberTheory/ZLattices.jl
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Update src/NumberTheory/ZLattices.jl
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56 changes: 56 additions & 0 deletions src/NumberTheory/ZLattices.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -211,3 +211,59 @@ function characteristic_vectors(L::ZZLat)
@hassert :Lattice 1 isone(hnf(reduce(vcat, cvL))[1:rank(L),:])
return cvL
end

function _get_canonical_form(A, char_vectors_set, canonical_ordering)

@lgoettgens lgoettgens Jul 7, 2026

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please add some types to the signature. This makes it easier to understand what the function is doing

p = length(char_vectors_set)
filter!(e->e!=p+1 && e!=p+2, canonical_ordering)
can_char_vectors_set = transpose(matrix(ZZ, reduce(vcat, char_vectors_set[canonical_ordering])))
H, U = hnf_with_transform(can_char_vectors_set)
U_inv = inv(U)
return transpose(U_inv)*A*U_inv
end

function _get_edge_labeled_graph(cv_set, gram)

@lgoettgens lgoettgens Jul 7, 2026

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here as well

p = length(cv_set)
res_graph = graph(Undirected, p+2)
max_w = QQ(0)
weightDict= Dict{Tuple{Int64, Int64}, QQFieldElem,}()
for i = 1:p
for j = i+1:p
Comment on lines +229 to +230

@simonbrandhorst simonbrandhorst Jul 7, 2026

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What about the squared length of the vector, i.e. cv_set[i]*gram*transpose(cv_set[i])? This should be relevant as well.


w = (cv_set[i]*gram*transpose(cv_set[j]))[1]
if w>max_w
max_w = w
end
Comment on lines +229 to +235

@simonbrandhorst simonbrandhorst Jul 7, 2026

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This should be considered a hot loop. Suppose that p=10000. Then the inner part is called p(p-1)/2= 49995000 times.
Some suggestions:

  • move cv_set[i]*gram to the outer loop on i. There is no dependence on j
  • use inplace operations such as transpose! and mul! with preallocated vectors
  • you can detect overflow before hand using the Cauchy-Schwarz inequality. If n= max(v*g*transpose(v)) and n^2 fits in Int, then we are fine. If this is the case (and it will be the case in almost all use cases), convert to Vector{Int} if and the matrix to Matrix{Int}. Then you can use LinearAlgebra.mul! which should be the fastest matrix vector multiplication with overflow that we have in julia. Be careful though, it does not support alialising on the first argument.
  • Is there a way to label the edge directly, instead of collecting all labels beforehand?
    Otherwise we are creating 49995000 dict entries. But I do not know enough of the internals of how labels work to judge if this makes sense.

@Kazak-11 Kazak-11 Jul 8, 2026

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Is there a way to label the edge directly, instead of collecting all labels beforehand?
Otherwise we are creating 49995000 dict entries. But I do not know enough of the internals of how labels work to judge if this makes sense.

There is option to use label!() with a newly made Dict with 1 edge. But the problem is, it looks that if you add more edges, the graph object don't merge them inside. Instead, it stores all the dictionaries as is and as Dict uses a lot of memory by default => it results in enourmous memory usage for big graphs. That's why I collect here all edges in one big dictionary.

add_edge!(res_graph, i, j)
weightDict[(i, j)] = w
end
add_edge!(res_graph, i, p+1)
end
a = 1+max_w
b = a + 1
for i = 1:p
add_edge!(res_graph, i, p+2)
merge!(weightDict, Dict((i, p+2) => a))
end
add_edge!(res_graph, p+1, p+2)
merge!(weightDict, Dict((p+1, p+2) => b))
Comment on lines +241 to +248

@simonbrandhorst simonbrandhorst Jul 7, 2026

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Can you explain what happens here?

@Kazak-11 Kazak-11 Jul 8, 2026

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Yes, here I follows the paper and it looks, that we need 2 more vertices to preserve automorphisms of graphs correctly. So, basically another step to not lose isomorphism/not get "new" isom. of resulting forms

label!(res_graph, weightDict, nothing; name=:edge)
return res_graph
end

"""
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Kazak-11 marked this conversation as resolved.
canonical_form(L::ZZLat) -> ZZMatrix
Return the canonical form of ``L``. The form is canonical in the sense, that two isomorphic latticies would have the same canonical form.

We follow ideas of Sikirić, Haensch, Voight and van Woerden [SHVW20](@cite).

!!! note
We do not give any guarantees that the canonical form stays the same
between different versions of Oscar.
"""
function canonical_form(L::ZZLat)
gram = gram_matrix(L)
char_vectors_set = characteristic_vectors(L)
graph = _get_edge_labeled_graph(char_vectors_set, gram) # transform from adjenctcy matrix A to edge-vertex weighted graph Ga, then to edge weighted graph T1(Ga)
can_order = _canonical_perm(graph; label=:edge) #_canonical_perm uses _edge_label_to_vertex_label themselfs
return _get_canonical_form(gram, char_vectors_set, can_order)
end
1 change: 1 addition & 0 deletions src/exports.jl
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Expand Up @@ -356,6 +356,7 @@ export bump!
export canonical_bundle
export canonical_divisor
export canonical_divisor_class
export canonical_form
export canonical_isomorphism
export canonical_matrix
export cartan_bilinear_form
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37 changes: 37 additions & 0 deletions test/NumberTheory/ZLattices.jl
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Expand Up @@ -8,3 +8,40 @@ end
@test 7 == length(unique!(Oscar.invariant_function_graph_hash.(L)))
@test 7 == length(unique!(Oscar.oscar_invariant_function.(L)))
end

@testset "Lattice Canonical Form" begin
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function create_gram(data) # helper function to create gram matricies of latticies
U = upper_triangular_matrix(data)
return U + transpose(U) - diagonal_matrix(diagonal(U))
end

n = 10
matrix1 = [2,1,-1,1,1,0,0,0,0,-1,2,-1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,2,1,0,0,0,0,-1,2,0,0,0,0,-1,4,-2,2,2,0,4,-2,-2,0,4,0,0,4,0,4]
matrix2 = [2,1,1,-1,-1,1,-1,1,-1,-1,2,0,-1,-1,1,-1,1,-1,0,4,-2,1,2,-2,-1,-1,0,4,1,-2,2,1,2,1,4,1,-1,0,0,1,4,-2,1,-2,-1,4,-1,2,1,4,0,-1,4,1,4]
G1 = create_gram(matrix1)
G2 = create_gram(matrix2)
U = hnf_with_transform(matrix(ZZ,n,n,rand(0:1,n^2)))[2];

@lgoettgens lgoettgens Jul 7, 2026

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the ; here and in the following lines is not necessary

L1 = integer_lattice(gram = G1);
L2 = integer_lattice(gram = G2);
L3 = lattice_in_same_ambient_space(L1,U*basis_matrix(L1));
can_form1 = canonical_form(L1)
can_form2 = canonical_form(L2)
can_form3 = canonical_form(L3)
@test can_form1 != can_form2
@test can_form1 == can_form3

n = 18
matrix1 = [4,0,0,2,1,2,2,1,0,0,0,0,0,0,0,0,0,0,4,0,2,3,1,0,2,0,0,0,0,0,0,0,0,0,0,4,2,3,3,1,1,0,0,0,0,0,0,0,0,0,0,4,4,3,2,2,0,0,0,0,0,0,0,0,0,0,6,4,2,3,0,0,0,0,0,0,0,0,0,0,4,2,2,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,2,1,2,2,1,0,0,4,0,2,3,1,0,2,0,0,4,2,3,3,1,1,0,0,4,4,3,2,2,0,0,6,4,2,3,0,0,4,2,2,0,0,2,1,0,0,2,0,0,4,0,8]
matrix2 = [2,1,1,1,0,0,0,0,1,1,-1,1,-1,0,-1,1,1,1,2,1,0,0,0,0,0,0,0,-1,1,-1,0,-1,1,0,1,2,0,0,0,0,0,1,1,-1,0,0,0,0,0,1,0,2,0,0,0,0,1,1,0,0,0,0,-1,0,0,1,2,0,0,0,-1,-1,-1,-1,1,0,0,0,0,0,2,1,-1,-1,-1,-1,1,-1,1,-1,-1,-1,1,2,-1,-1,-1,-1,1,-1,0,-1,-1,0,1,2,1,1,0,-1,1,0,0,0,1,-1,4,3,0,-1,1,0,1,1,2,0,4,1,-1,1,0,0,0,1,-1,4,0,1,-1,1,0,-1,-2,4,-3,0,-1,1,0,1,4,0,1,-1,0,-2,2,-1,-1,-1,0,4,1,1,-1,4,1,1,4,0,4]
G1 = create_gram(matrix1)
G2 = create_gram(matrix2)
U = hnf_with_transform(matrix(ZZ,n,n,rand(0:1,n^2)))[2];
L1 = integer_lattice(gram = G1);
L2 = integer_lattice(gram = G2);
L3 = lattice_in_same_ambient_space(L1,U*basis_matrix(L1));
can_form1 = canonical_form(L1)
# can_form2 = canonical_form(L2) # too long time, circa 1 hour to calculate can form
can_form3 = canonical_form(L3)
# @test can_form1 != can_form2
@test can_form1 == can_form3
end
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