spacewithisom.md

CurrentModule = Oscar

Quadratic spaces with isometry

We call quadratic space with isometry any pair $(V, f)$ consisting of a non-degenerate quadratic space $V$ together with an isometry $f\in O(V)$. We refer to the section about Spaces of the documentation for new users.

Note that currently, we support only rational quadratic forms, i.e. quadratic spaces defined over $\mathbb{Q}$.

In Oscar, such a pair is encoded by the type called QuadSpaceWithIsom:

QuadSpaceWithIsom

It is seen as a triple $(V, f, n)$ where $n$ is the order of $f$. We actually support isometries of finite and infinite order. In the case where $f$ is of infinite order, then n = PosInf. If $V$ has rank 0, then any isometry $f$ of $V$ is trivial and we set by default n = -1.

Given a quadratic space with isometry $(V, f)$, we provide the following accessors to the elements of the previously described triple:

isometry(::QuadSpaceWithIsom)
order_of_isometry(::QuadSpaceWithIsom)
space(::QuadSpaceWithIsom)

The main purpose of the definition of such objects is to define a contextual ambient space for quadratic lattices endowed with an isometry. Indeed, as we will see in the next section, lattices with isometry are attached to an ambient quadratic space with an isometry inducing the one on the lattice.

Constructors

For simplicity, we have gathered the main constructors for objects of type QuadSpaceWithIsom under the same name quadratic_space_with_isometry. The user has then the choice on the parameters depending on what they intend to do:

quadratic_space_with_isometry(::Hecke.QuadSpace, ::QQMatrix)
quadratic_space_with_isometry(::Hecke.QuadSpace)

By default, the first constructor always checks whether the matrix defines an isometry of the quadratic space. We recommend not to disable this parameter to avoid any complications. Note however that in the rank 0 case, the checks are avoided since all isometries are necessarily trivial.

Attributes and first operations

Given a quadratic space with isometry $Vf := (V, f)$, one has access to most of the attributes of $V$ and $f$ by calling the similar functions on the pair $(V, f)$ itself. For instance, in order to know the rank of $V$, one can simply call rank(Vf). Here is a list of what are the current accessible attributes:

characteristic_polynomial(::QuadSpaceWithIsom)
det(::QuadSpaceWithIsom)
diagonal(::QuadSpaceWithIsom)
dim(::QuadSpaceWithIsom)
discriminant(::QuadSpaceWithIsom)
gram_matrix(::QuadSpaceWithIsom)
is_definite(::QuadSpaceWithIsom)
is_positive_definite(::QuadSpaceWithIsom)
is_negative_definite(::QuadSpaceWithIsom)
minimal_polynomial(::QuadSpaceWithIsom)
rank(::QuadSpaceWithIsom)
signature_tuple(::QuadSpaceWithIsom)

Similarly, some basic operations on quadratic spaces and matrices are available for quadratic spaces with isometry.

Base.:^(::QuadSpaceWithIsom, ::Int)
biproduct(::Vector{QuadSpaceWithIsom})
direct_product(::Vector{QuadSpaceWithIsom})
direct_sum(::Vector{QuadSpaceWithIsom})
rescale(::QuadSpaceWithIsom, ::RationalUnion)

Spinor norm

Given a rational quadratic space $(V, \Phi)$, and given an integer $b\in\mathbb{Q}$, we define the rational spinor norm $\sigma$ on $(V, b\Phi)$ to be the group homomorphism

$\sigma\colon O(V, b\Phi) = O(V, \Phi)\to \mathbb{Q}^\ast/(\mathbb{Q}^\ast)^2$

defined as follows. For $f\in O(V, b\Phi)$, there exist elements $v_1,\ldots, v_r\in V$ where $1\leq r\leq \text{rank}(V)$ such that $f = \tau_{v_1}\circ\cdots\circ \tau_{v_r}$ is equal to the product of the associated reflections. We define

$\sigma(f) := (-\frac{b\Phi(v_1, v_1)}{2})\cdots(-\frac{b\Phi(v_r,v_r)}{2}) \mod (\mathbb{Q}^{\ast})^2.$

rational_spinor_norm(::QuadSpaceWithIsom)

Equality

We choose as a convention that two pairs $(V, f)$ and $(V', f')$ of quadratic spaces with isometries are equal if $V$ and $V'$ are the same space, and $f$ and $f'$ are represented by the same matrix with respect to the standard basis of $V = V'$.