latwithisom.md

CurrentModule = Oscar

Lattices with isometry

We call lattice with isometry any pair $(L, f)$ consisting of an integer lattice $L$ together with an isometry $f \in O(L)$. We refer to the section about Integer Lattices of the documentation for new users.

In Oscar, such a pair is encoded in the type called ZZLatWithIsom:

ZZLatWithIsom

It is seen as a quadruple $(Vf, L, f, n)$ where $Vf = (V, f_a)$ consists of the ambient rational quadratic space $V$ of $L$, and an isometry $f_a$ of $V$ preserving $L$ and inducing $f$ on $L$. The integer $n$ is the order of $f$, which is a divisor of the order of the isometry $f_a\in O(V)$.

Given a lattice with isometry $(L, f)$, we provide the following accessors to the elements of the previously described quadruple:

ambient_isometry(::ZZLatWithIsom)
ambient_space(::ZZLatWithIsom)
isometry(::ZZLatWithIsom)
lattice(::ZZLatWithIsom)
order_of_isometry(::ZZLatWithIsom)

Note that for some computations, it is more convenient to work either with the isometry of the lattice itself, or with the fixed isometry of the ambient quadratic space inducing it on the lattice.

Constructors

We provide two ways to construct a pair $Lf = (L,f)$ consisting of an integer lattice endowed with an isometry. One way to construct an object of type ZZLatWithIsom is through the methods integer_lattice_with_isometry. These two methods do not require as input an ambient quadratic space with isometry.

integer_lattice_with_isometry(::ZZLat, ::QQMatrix)
integer_lattice_with_isometry(::ZZLat)

By default, the first constructor will always check whether the matrix defines an isometry of the lattice, or its ambient space. We recommend not to disable this parameter to avoid any further issues. Note that as in the case of quadratic spaces with isometry, both isometries of integer lattices of finite order and infinite order are supported.

Another way of constructing such lattices with isometry is by fixing an ambient quadratic space with isometry, of type QuadSpaceWithIsom, and specifying a basis for an integral lattice in that space. If this lattice is preserved by the fixed isometry of the quadratic space considered, then we endow it with the induced action.

lattice(::QuadSpaceWithIsom)
lattice(::QuadSpaceWithIsom, ::MatElem{ <:RationalUnion})
lattice_in_same_ambient_space(::ZZLatWithIsom, ::MatElem)

Attributes and first operations

Given a lattice with isometry $Lf := (L, f)$, one can have access most of the attributes of $L$ and $f$ by calling the similar function for the pair. For instance, in order to know the genus of $L$, one can simply call genus(Lf). Here is a list of what are the current accessible attributes:

basis_matrix(::ZZLatWithIsom)
characteristic_polynomial(::ZZLatWithIsom)
degree(::ZZLatWithIsom)
det(::ZZLatWithIsom)
discriminant(::ZZLatWithIsom)
genus(::ZZLatWithIsom)
gram_matrix(::ZZLatWithIsom)
is_definite(::ZZLatWithIsom)
is_even(::ZZLatWithIsom)
is_elementary(::ZZLatWithIsom, ::IntegerUnion)
is_elementary_with_prime(::ZZLatWithIsom)
is_integral(::ZZLatWithIsom)
is_positive_definite(::ZZLatWithIsom)
is_primary(::ZZLatWithIsom, ::IntegerUnion)
is_primary_with_prime(::ZZLatWithIsom)
is_negative_definite(::ZZLatWithIsom)
is_unimodular(::ZZLatWithIsom)
minimum(::ZZLatWithIsom)
minimal_polynomial(::ZZLatWithIsom)
norm(::ZZLatWithIsom)
rank(::ZZLatWithIsom)
rational_span(::ZZLatWithIsom)
scale(::ZZLatWithIsom)
signature_tuple(::ZZLatWithIsom)

Similarly, some basic operations on $\mathbb Z$-lattices and matrices are available for integer lattices with isometry.

Base.:^(::ZZLatWithIsom, ::Int)
biproduct(::Vector{ZZLatWithIsom})
direct_product(::Vector{ZZLatWithIsom})
direct_sum(::Vector{ZZLatWithIsom})
dual(::ZZLatWithIsom)
lll(::ZZLatWithIsom)
orthogonal_submodule(::ZZLatWithIsom, ::QQMatrix)
rescale(::ZZLatWithIsom, ::RationalUnion)

Type for finite order isometries

Given a lattice with isometry $Lf := (L, f)$ where $f$ is of finite order $n$, one can compute the type of $Lf$.

type(::ZZLatWithIsom)

Since determining whether two pairs of lattices with isometry are isomorphic is a challenging task, one can perform a coarser comparison by looking at the type. This set of data keeps track of some local and global invariants of the pair $(L, f)$ with respect to the action of $f$ on $L$.

is_of_type(::ZZLatWithIsom, t::Dict)
is_of_same_type(::ZZLatWithIsom, ::ZZLatWithIsom)

Finally, if the minimal polynomial of $f$ is irreducible, then we say that the pair $(L, f)$ is of hermitian type. The type of a lattice with isometry of hermitian type is called hermitian (note that the type is only defined for finite order isometries).

These names follow from the fact that, by the trace equivalence, one can associate to the pair $(L, f)$ a hermitian lattice over the equation order of $f$, if it is maximal in the associated number field $\mathbb{Q}[f]$.

is_of_hermitian_type(::ZZLatWithIsom)
is_hermitian(::Dict)

Hermitian structures and trace equivalence

As mentioned in the previous section, to a lattice with isometry $Lf := (L, f)$ such that the minimal polynomial of $f$ is irreducible, one can associate a hermitian lattice $\mathfrak{L}$ over the equation order of $f$, if it is maximal, for which $Lf$ is the associated trace lattice. Hecke provides the tools to perform the trace equivalence for lattices with isometry of hermitian type.

hermitian_structure(::ZZLatWithIsom)

Discriminant groups

Given an integral lattice with isometry $Lf := (L, f)$, if one denotes by $D_L$ the discriminant group of $L$, there exists a natural map $\pi\colon O(L) \to O(D_L)$ sending any isometry to its induced action on the discriminant group of $L$. In general, this map is neither injective nor surjective. If we denote by $D_f := \pi(f)$ then $\pi$ induces a map between centralizers $O(L, f)\to O(D_L, D_f)$. Again, this induced map is in general neither injective nor surjective, and we denote its image by $G_{L,f}$.

discriminant_group(::ZZLatWithIsom)

For simple cases as for definite lattices, $f$ being plus-or-minus the identity or if the rank of $L$ is equal to the totient of the order of $f$ (in the finite case), $G_{L,f}$ can be easily computed. For the remaining cases, we use the hermitian version of Miranda-Morrison theory as presented in BH23. The general computation of $G_{L, f}$ has been implemented in this project and it can be indirectly used through the general following method:

image_centralizer_in_Oq(::ZZLatWithIsom)

Note: hermitian Miranda-Morrison is only available for even lattices.

For an implementation of the regular Miranda-Morrison theory, we refer to the function image_in_Oq which actually computes the image of $\pi$ in both the definite and the indefinite case.

More generally, for a finitely generated subgroup $G$ of $O(L)$, we have implemented a function which computes the representation of $G$ on $D_L$:

discriminant_representation(::ZZLat, ::MatrixGroup)

We will see later in the section about enumeration of lattices with isometry that one can compute $G_{L,f}$ in some particular cases arising from equivariant primitive embeddings of lattices with isometries.

Kernel sublattices

As for single integer lattices, it is possible to compute kernel sublattices of some $\mathbb{Z}$-module homomorphisms. We provide here the possibility to compute $\ker(p(f))$ as a sublattice of $L$ equipped with the induced action of $f$, where $p$ is a polynomial with rational coefficients.

kernel_lattice(::ZZLatWithIsom, ::Union{ZZPolyRingElem, QQPolyRingElem})
kernel_lattice(::ZZLatWithIsom, ::Integer)

Note that such sublattices are by definition primitive in $L$ since $L$ is non-degenerate. As particular kernel sublattices of $L$, one can also compute the so-called invariant and coinvariant lattices of $(L, f)$:

coinvariant_lattice(::ZZLatWithIsom)
invariant_lattice(::ZZLatWithIsom)
invariant_coinvariant_pair(::ZZLatWithIsom)

Similarly, we provide the possibility to compute invariant and coinvariant sublattices given an orthogonal representation G in matrix form of a finite group on a given lattice L:

coinvariant_lattice(::ZZLat, ::MatrixGroup)
invariant_lattice(::ZZLat, ::MatrixGroup)
invariant_coinvariant_pair(::ZZLat, ::MatrixGroup)

Signatures

We conclude this introduction about standard functionalities for lattices with isometry by introducing a last invariant for lattices with finite isometry of hermitian type $(L, f)$, called the signatures. These signatures are intrinsically connected to the local archimedean invariants of the hermitian structure associated to $(L, f)$ via the trace equivalence.

signatures(::ZZLatWithIsom)

Spinor norm

Given an integer lattice with isometry $(L, f)$, one often would like to know the spinor norm of $f$ seen as an isometry of the rational quadratic space $L\times \mathbb{Q}$. See rational_spinor_norm(::QuadSpaceWithIsom) for a definition.

rational_spinor_norm(::ZZLatWithIsom)

Equality

We choose as a convention that two pairs $(L, f)$ and $(L', f')$ of integer lattices with isometries are equal if their ambient quadratic space with isometry of type QuadSpaceWithIsom are equal, and if the underlying lattices $L$ and $L'$ are equal as $\mathbb Z$-modules in the common ambient quadratic space.