Nikulin's theory on primitive embeddings

primitive_embeddings(
  G::ZZGenus,
  M::ZZLat;
  kwargs...,
) -> Bool, Vector{Tuple{ZZLat, ZZLat, ZZLat}}

Given a genus $G$ of integral $\mathbb Z$-lattice, and an integral $\mathbb Z$-lattice $M$, return a boolean T and a list $V$ of representatives of double cosets of primitive embeddings $M \hookrightarrow L$ where $L$ is a lattice in $G$, for the right and left actions of $O(M)$ and $O(L)$ respectively.

The second output $V$ consists of triples $(L, M_1, N)$ where $L$ is a lattice in $G$, $M_1$ is a primitive sublattice of $L$ isometric to $M$, and $N$ is the orthogonal complement of $M_1$ in $L$.

Alternatively, one can also choose as first input:

• a lattice $L$ representing $G$: one would therefore call primitive_embeddings(L, M; kwargs...),
• the discriminant form q and the signature pair (p, n) of $G$: one would therefore call primitive_embeddings(q, (p, n), M; kwargs...).

Here is a complete list of currently supported keyword arguments:

• right_action::MatGroup{QQFieldElem, QQMatrix} -> replace the right action of $O(M)$ by a right action of the group generated by right_action and the kernel of $O(M) \to O(D_M)$.
• right_discriminant_action::AutomorphismGroup{TorQuadModule} -> replace the right action of $O(M)$ by a right action of the largest subgroup whose discriminant representation surjects onto right_discriminant_action.
• first::Bool -> if set to true, return the first primitive embedding computed.
• exist_only::Bool -> if set to true, return only whether a primitive embedding exists together with an empty list.
• check::Bool -> is set to true and the first input is a lattice $L_0$, the function filters the output to keep only triples $(L, M_1, N)$ such that $L$ is isometric to $L_0$.

Examples

We can use such primitive embeddings algorithm to classify embedding in unimodular lattices

julia> E8 = root_lattice(:E, 8);

julia> A4 = root_lattice(:A, 4);

julia> bool, pe = primitive_embeddings(E8, A4)
(true, Tuple{ZZLat, ZZLat, ZZLat}[(Integer lattice of rank 8 and degree 8, Integer lattice of rank 4 and degree 8, Integer lattice of rank 4 and degree 8)])

julia> pe
1-element Vector{Tuple{ZZLat, ZZLat, ZZLat}}:
 (Integer lattice of rank 8 and degree 8, Integer lattice of rank 4 and degree 8, Integer lattice of rank 4 and degree 8)

julia> genus(pe[1][2]) == genus(pe[1][3])
true

To be understood: There exists a unique class of primitive embedding of the root lattice $A_4$ into the root lattice $E_8$, and the orthogonal primitive sublattice is isometric to $A_4$.

primitive_extensions(
  M::ZZLat,
  N::ZZLat;
  kwargs...,
) -> Bool, Vector{Tuple{ZZLat, ZZLat, ZZLat}}

Given two integral $\mathbb Z$-lattices $M$ and $N$, return a boolean T and a list $V$ of representatives of double cosets of primitive extensions $M \oplus N \subseteq L$, for the right and left actions of $O(M)$ and $O(N)$ respectively.

Here is a complete list of currently supported keyword arguments:

• right_action::MatGroup{QQFieldElem, QQMatrix} -> replace the right action of $O(M)$ by a right action of the group generated by right_action and the kernel of $O(M) \to O(D_M)$.
• left_action::MatGroup{QQFieldElem, QQMatrix} -> replace the left action of $O(N)$ by a left action of the group generated by left_action and the kernel of $O(N) \to O(D_N)$.
• right_discriminant_action::AutomorphismGroup{TorQuadModule} -> replace the right action of $O(M)$ by a right action of the largest subgroup whose discriminant representation surjects onto right_discriminant_action.
• left_discriminant_action::AutomorphismGroup{TorQuadModule} -> replace the left action of $O(N)$ by a left action of the largest subgroup whose discriminant representation surjects onto left_discriminant_action.
• first::Bool -> if set to true, return the first primitive extension computed.
• exist_only::Bool -> if set to true, return only whether a primitive extension exists together with an empty list.
• glue_order::AbstractVector{IntegerUnion} -> provide a list of possible orders for the glue groups associated to each primitive extension in output. If no such list is provided, then the algorithm assumes no restriction.
• form_over::Vector{TorQuadModule} -> provide a list of possible discriminant groups for the primitive extensions in output. If no such list is provided, then the algorithm assumes no restriction.
• even::Bool -> if set to true, return only even primitive extensions.

equivariant_primitive_extensions(
  M::Union{ZZLat, ZZLatWithIsom},
  N::Union{ZZLat, ZZLatWithIsom};
  kwargs...,
) -> Bool, Vector{Tuple{ZZLatWithIsom, ZZLatWithIsom, ZZLatWithIsom}}

Given two integral $\mathbb Z$-lattices with isometry $(M, f_M)$ and $(N, f_N)$, return a boolean T and a list $V$ of representatives of double cosets of equivariant primitive extensions $(M, f_M) \oplus (N, f_N) \subset (L, f_L)$, for the right and left actions of $O(M, f_M)$ and $O(N, f_N)$ respectively.

Alternatively, one can choose exactly one of the two inputs to be an integral $\mathbb Z$-lattice without isometry. In that situation, such a lattice must be definite and for each possible primitive extension, the algorithm computes representatives for the conjugacy classes of isometries which can be extended with the other fixed isometry along the primitive extension.

Here is a complete list of currently supported keyword arguments:

• right_action::MatGroup{QQFieldElem, QQMatrix} -> replace the right action of $O(M, f_M)$ by a right action of the centralizer of $f_M$ in the group generated by right_action and the kernel of $O(M) \to O(D_M)$.
• left_action::MatGroup{QQFieldElem, QQMatrix} -> replace the left action of $O(N, f_N)$ by a left action of the centralizer of $f_N$ in the group generated by left_action and the kernel of $O(N) \to O(D_N)$.
• right_discriminant_action::AutomorphismGroup{TorQuadModule} -> replace the right action of $O(M, f_M)$ by a right action of the largest subgroup whose discriminant representation surjects onto right_discriminant_action.
• left_discriminant_action::AutomorphismGroup{TorQuadModule} -> replace the left action of $O(N, f_N)$ by a left action of the largest subgroup whose discriminant representation surjects onto left_discriminant_action.
• first::Bool -> if set to true, return the first equivariant primitive extension computed.
• exist_only::Bool -> if set to true, return only whether an equivariant primitive extension exists together with an empty list.
• glue_order::AbstractVector{IntegerUnion} -> provide a list of possible orders for the glue groups associated to each primitive extension in output. If no such list is provided, then the algorithm assumes no restriction.
• form_over::Vector{TorQuadModule} -> provide a list of possible discriminant groups for the primitive extensions in output. If no such list is provided, then the algorithm assumes no restriction.
• even::Bool -> if set to true, return only even primitive extensions.
• compute_bar_Gf::Bool -> if set to true, compute the image of the map $O(L, f_L) \to O(D_L)$ for every equivariant primitive extension $(L, f_L)$ in output. This can be computed directly at the level of gluings.
• first_fitting_isometry::Bool -> if set to true and one of the lattices in input is not equipped with an isometry, the algorithm computes only one possible isometry to be extended along every primitive extensions.

admissible_equivariant_primitive_extensions(
  Afa::ZZLatWithIsom,
  Bfb::ZZLatWithIsom,
  Cfc::ZZLatWithIsom,
  p::IntegerUnion,
  q::IntegerUnion = p;
  check::Bool=true,
  test_type::Bool=true,
) -> Vector{ZZLatWithIsom}

Given a triple of lattices with isometry $(A, f_A)$, $(B, f_B)$ and $(C, f_C)$, and a prime number $p$, such that $(A, B, C)$ is $p$-admissible, return a set of representatives of the double coset $G_B\backslash S/G_A$ where:

• $G_A$ and $G_B$ are the respective images of the morphisms $O(A, f_A) \to O(D_A, D_{f_A})$ and $O(B, f_B) \to O(D_B, D_{f_B})$;
• $S$ is the set of all primitive extensions $A \oplus B \subseteq C'$ with isometry $f_C'$ where $p\cdot C' \subseteq A\oplus B$ and such that the type of $(C', (f_C')^q)$ is equal to the type of $(C, f_C)$.

If check == true the input triple is checked to a $p$-admissible triple of even lattices (with isometry) with $f_A$ and $f_B$ having relatively coprime irreducible minimal polynomials. Moreover, the function checks that $A$ and $B$ are orthogonal if $A$, $B$ and $C$ lie in the same ambient quadratic space.

Note moreover that the function computes the image of the natural map $O(C, f_C) \to O(D_C, D_{f_C})$ along the primitive extension $A\oplus B\subseteq C$ (see Algorithm 2, Line 22 of [BH23]).

If one sets test_type to false, then the function does not check if the outputs satisfy the type condition.