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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 — Type
QuadSpaceWithIsom

A container type for pairs $(V, f)$ consisting of a rational quadratic space $V$ of type QuadSpace and an isometry $f$ given as a QQMatrix representing the action on the standard basis of $V$.

We store the order of $f$ too, which can finite or infinite.

To construct an object of type QuadSpaceWithIsom, see the set of functions called quadratic_space_with_isometry

Examples

julia> V = quadratic_space(QQ, 4);

julia> quadratic_space_with_isometry(V, neg=true)
Quadratic space of dimension 4
  with isometry of finite order 2
  given by
  [-1    0    0    0]
  [ 0   -1    0    0]
  [ 0    0   -1    0]
  [ 0    0    0   -1]

julia> L = root_lattice(:E, 6);

julia> V = ambient_space(L);

julia> f = matrix(QQ, 6, 6, [ 1  2  3  2  1  1;
                             -1 -2 -2 -2 -1 -1;
                              0  1  0  0  0  0;
                              1  0  0  0  0  0;
                             -1 -1 -1  0  0 -1;
                              0  0  1  1  0  1]);

julia> Vf = quadratic_space_with_isometry(V, f)
Quadratic space of dimension 6
  with isometry of finite order 8
  given by
  [ 1    2    3    2    1    1]
  [-1   -2   -2   -2   -1   -1]
  [ 0    1    0    0    0    0]
  [ 1    0    0    0    0    0]
  [-1   -1   -1    0    0   -1]
  [ 0    0    1    1    0    1]
source

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 — Method
isometry(Vf::QuadSpaceWithIsom) -> QQMatrix

Given a quadratic space with isometry $(V, f)$, return the underlying isometry $f$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> isometry(Vf)
[-1    0]
[ 0   -1]
source
order_of_isometry — Method
order_of_isometry(Vf::QuadSpaceWithIsom) -> IntExt

Given a quadratic space with isometry $(V, f)$, return the order of the underlying isometry $f$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> order_of_isometry(Vf) == 2
true
source
space — Method
space(Vf::QuadSpaceWithIsom) -> QuadSpace

Given a quadratic space with isometry $(V, f)$, return the underlying space $V$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> space(Vf) === V
true
source

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 — Method
quadratic_space_with_isometry(V:QuadSpace, f::QQMatrix; check::Bool = false)
                                                        -> QuadSpaceWithIsom

Given a quadratic space $V$ and a matrix $f$, if $f$ defines an isometry of $V$ of order $n$ (possibly infinite), return the corresponding quadratic space with isometry pair $(V, f)$.

Examples

julia> V = quadratic_space(QQ, QQ[ 2 -1;
                                  -1  2])
Quadratic space of dimension 2
  over rational field
with gram matrix
[ 2   -1]
[-1    2]

julia> f = matrix(QQ, 2, 2, [1  1;
                             0 -1])
[1    1]
[0   -1]

julia> Vf = quadratic_space_with_isometry(V, f)
Quadratic space of dimension 2
  with isometry of finite order 2
  given by
  [1    1]
  [0   -1]
source
quadratic_space_with_isometry — Method
quadratic_space_with_isometry(V::QuadSpace; neg::Bool = false) -> QuadSpaceWithIsom

Given a quadratic space $V$, return the quadratic space with isometry pair $(V, f)$ where $f$ is represented by the identity matrix.

If neg is set to true, then the isometry $f$ is negative the identity on $V$.

Examples

julia> V = quadratic_space(QQ, QQ[ 2 -1;
                                  -1  2])
Quadratic space of dimension 2
  over rational field
with gram matrix
[ 2   -1]
[-1    2]

julia> Vf = quadratic_space_with_isometry(V)
Quadratic space of dimension 2
  with isometry of finite order 1
  given by
  [1   0]
  [0   1]
source

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 — Method
characteristic_polynomial(Vf::QuadSpaceWithIsom) -> QQPolyRingElem

Given a quadratic space with isometry $(V, f)$, return the characteristic polynomial of the underlying isometry $f$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> characteristic_polynomial(Vf)
x^2 + 2*x + 1
source
det — Method
det(Vf::QuadSpaceWithIsom) -> QQFieldElem

Given a quadratic space with isometry $(V, f)$, return the determinant of the underlying space $V$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> is_one(det(Vf))
true
source
diagonal — Method
diagonal(Vf::QuadSpaceWithIsom) -> Vector{QQFieldElem}

Given a quadratic space with isometry $(V, f)$, return the diagonal of the underlying space $V$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> diagonal(Vf)
2-element Vector{QQFieldElem}:
 1
 1
source
dim — Method
dim(Vf::QuadSpaceWithIsom) -> Integer

Given a quadratic space with isometry $(V, f)$, return the dimension of the underlying space of $V$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> dim(Vf) == 2
true
source
discriminant — Method
discriminant(Vf::QuadSpaceWithIsom) -> QQFieldElem

Given a quadratic space with isometry $(V, f)$, return the discriminant of the underlying space $V$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> discriminant(Vf)
-1
source
gram_matrix — Method
gram_matrix(Vf::QuadSpaceWithIsom) -> QQMatrix

Given a quadratic space with isometry $(V, f)$, return the Gram matrix of the underlying space $V$ with respect to its standard basis.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> is_one(gram_matrix(Vf))
true
source
is_definite — Method
is_definite(Vf::QuadSpaceWithIsom) -> Bool

Given a quadratic space with isometry $(V, f)$, return whether the underlying space $V$ is definite.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> is_definite(Vf)
true
source
is_positive_definite — Method
is_positive_definite(Vf::QuadSpaceWithIsom) -> Bool

Given a quadratic space with isometry $(V, f)$, return whether the underlying space $V$ is positive definite.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> is_positive_definite(Vf)
true
source
is_negative_definite — Method
is_negative_definite(Vf::QuadSpaceWithIsom) -> Bool

Given a quadratic space with isometry $(V, f)$, return whether the underlying space $V$ is negative definite.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> is_negative_definite(Vf)
false
source
minimal_polynomial — Method
minimal_polynomial(Vf::QuadSpaceWithIsom) -> QQPolyRingElem

Given a quadratic space with isometry $(V, f)$, return the minimal polynomial of the underlying isometry $f$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> minimal_polynomial(Vf)
x + 1
source
rank — Method
rank(Vf::QuadSpaceWithIsom) -> Integer

Given a quadratic space with isometry $(V, f)$, return the rank of the underlying space $V$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> rank(Vf) == 2
true
source
signature_tuple — Method
signature_tuple(Vf::QuadSpaceWithIsom) -> Tuple{Int, Int, Int}

Given a quadratic space with isometry $(V, f)$, return the signature tuple of the underlying space $V$.

Examples

julia> V = quadratic_space(QQ, 2);

julia> Vf = quadratic_space_with_isometry(V; neg = true);

julia> signature_tuple(Vf)
(2, 0, 0)
source

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

^ — Method
^(Vf::QuadSpaceWithIsom, n::Int) -> QuadSpaceWithIsom

Given a quadratic space with isometry $(V, f)$ and an integer $n$, return the pair $(V, f^n)$.

Examples

julia> V = quadratic_space(QQ, QQ[ 2 -1;
                                  -1  2])
Quadratic space of dimension 2
  over rational field
with gram matrix
[ 2   -1]
[-1    2]

julia> f = matrix(QQ, 2, 2, [1  1;
                             0 -1])
[1    1]
[0   -1]

julia> Vf = quadratic_space_with_isometry(V, f)
Quadratic space of dimension 2
  with isometry of finite order 2
  given by
  [1    1]
  [0   -1]

julia> Vf^2
Quadratic space of dimension 2
  with isometry of finite order 1
  given by
  [1   0]
  [0   1]
source
biproduct — Method
biproduct(x::Vector{QuadSpaceWithIsom}) -> QuadSpaceWithIsom, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
biproduct(x::Vararg{QuadSpaceWithIsom}) -> QuadSpaceWithIsom, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}

Given a collection of quadratic spaces with isometries $(V_1, f_1), \ldots, (V_n, f_n)$, return the quadratic space with isometry $(V, f)$ together with the injections $V_i \to V$ and the projections $V \to V_i$, where $V$ is the biproduct $V := V_1 \oplus \ldots \oplus V_n$ and $f$ is the isometry of $V$ induced by the diagonal actions of the $f_i$'s.

For objects of type QuadSpaceWithIsom, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain $(V, f)$ as a direct sum with the injections $V_i \to V$, one should call direct_sum(x). If one wants to obtain $(V, f)$ as a direct product with the projections $V \to V_i$, one should call direct_product(x).

Examples

julia> V1 = quadratic_space(QQ, QQ[2 5;
                                   5 6])
Quadratic space of dimension 2
  over rational field
with gram matrix
[2   5]
[5   6]

julia> Vf1 = quadratic_space_with_isometry(V1, neg=true)
Quadratic space of dimension 2
  with isometry of finite order 2
  given by
  [-1    0]
  [ 0   -1]

julia> V2 = quadratic_space(QQ, QQ[ 2 -1;
                                   -1  2])
Quadratic space of dimension 2
  over rational field
with gram matrix
[ 2   -1]
[-1    2]

julia> f = matrix(QQ, 2, 2, [1  1;
                             0 -1])
[1    1]
[0   -1]

julia> Vf2 = quadratic_space_with_isometry(V2, f)
Quadratic space of dimension 2
  with isometry of finite order 2
  given by
  [1    1]
  [0   -1]

julia> Vf3, inj, proj = biproduct(Vf1, Vf2)
(Quadratic space with isometry of finite order 2, AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space], AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space])

julia> Vf3
Quadratic space of dimension 4
  with isometry of finite order 2
  given by
  [-1    0   0    0]
  [ 0   -1   0    0]
  [ 0    0   1    1]
  [ 0    0   0   -1]

julia> space(Vf3)
Quadratic space of dimension 4
  over rational field
with gram matrix
[2   5    0    0]
[5   6    0    0]
[0   0    2   -1]
[0   0   -1    2]

julia> matrix(compose(inj[1], proj[1]))
[1   0]
[0   1]

julia> matrix(compose(inj[1], proj[2]))
[0   0]
[0   0]
source
direct_product — Method
direct_product(algebras::StructureConstantAlgebra...; task::Symbol = :sum)
  -> StructureConstantAlgebra, Vector{AbsAlgAssMor}, Vector{AbsAlgAssMor}
direct_product(algebras::Vector{StructureConstantAlgebra}; task::Symbol = :sum)
  -> StructureConstantAlgebra, Vector{AbsAlgAssMor}, Vector{AbsAlgAssMor}

Returns the algebra $A = A_1 \times \cdots \times A_k$. task can be ":sum", ":prod", ":both" or ":none" and determines which canonical maps are computed as well: ":sum" for the injections, ":prod" for the projections.

source
direct_product(F::FreeMod{T}...; task::Symbol = :prod) where T

Given free modules $F_1\dots F_n$, say, return the direct product $\prod_{i=1}^n F_i$.

Additionally, return

  • a vector containing the canonical projections $\prod_{i=1}^n F_i\to F_i$ if task = :prod (default),
  • a vector containing the canonical injections $F_i\to\prod_{i=1}^n F_i$ if task = :sum,
  • two vectors containing the canonical projections and injections, respectively, if task = :both,
  • none of the above maps if task = :none.
source
direct_product(M::ModuleFP{T}...; task::Symbol = :prod) where T

Given modules $M_1\dots M_n$, say, return the direct product $\prod_{i=1}^n M_i$.

Additionally, return

  • a vector containing the canonical projections $\prod_{i=1}^n M_i\to M_i$ if task = :prod (default),
  • a vector containing the canonical injections $M_i\to\prod_{i=1}^n M_i$ if task = :sum,
  • two vectors containing the canonical projections and injections, respectively, if task = :both,
  • none of the above maps if task = :none.
source
direct_product(x::Vector{QuadSpaceWithIsom}) -> QuadSpaceWithIsom, Vector{AbstractSpaceMor}
direct_product(x::Vararg{QuadSpaceWithIsom}) -> QuadSpaceWithIsom, Vector{AbstractSpaceMor}

Given a collection of quadratic spaces with isometries $(V_1, f_1), \ldots, (V_n, f_n)$, return the quadratic space with isometry $(V, f)$ together with the projections $V \to V_i$, where $V$ is the direct product $V := V_1 \times \ldots \times V_n$ and $f$ is the isometry of $V$ induced by the diagonal actions of the $f_i$'s.

For objects of type QuadSpaceWithIsom, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain $(V, f)$ as a direct sum with the injections $V_i \to V$, one should call direct_sum(x). If one wants to obtain $(V, f)$ as a biproduct with the injections $V_i \to V$ and the projections $V \to V_i$, one should call biproduct(x).

Examples

julia> V1 = quadratic_space(QQ, QQ[2 5;
                                   5 6])
Quadratic space of dimension 2
  over rational field
with gram matrix
[2   5]
[5   6]

julia> Vf1 = quadratic_space_with_isometry(V1, neg=true)
Quadratic space of dimension 2
  with isometry of finite order 2
  given by
  [-1    0]
  [ 0   -1]

julia> V2 = quadratic_space(QQ, QQ[ 2 -1;
                                   -1  2])
Quadratic space of dimension 2
  over rational field
with gram matrix
[ 2   -1]
[-1    2]

julia> f = matrix(QQ, 2, 2, [1  1;
                             0 -1])
[1    1]
[0   -1]

julia> Vf2 = quadratic_space_with_isometry(V2, f)
Quadratic space of dimension 2
  with isometry of finite order 2
  given by
  [1    1]
  [0   -1]

julia> Vf3, proj = direct_product(Vf1, Vf2)
(Quadratic space with isometry of finite order 2, AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space])

julia> Vf3
Quadratic space of dimension 4
  with isometry of finite order 2
  given by
  [-1    0   0    0]
  [ 0   -1   0    0]
  [ 0    0   1    1]
  [ 0    0   0   -1]

julia> space(Vf3)
Quadratic space of dimension 4
  over rational field
with gram matrix
[2   5    0    0]
[5   6    0    0]
[0   0    2   -1]
[0   0   -1    2]
source
direct_sum — Method
direct_sum(g1::QuadSpaceCls, g2::QuadSpaceCls) -> QuadSpaceCls

Return the isometry class of the direct sum of two representatives.

source
direct_sum(M::ModuleFP{T}...; task::Symbol = :sum) where T

Given modules $M_1\dots M_n$, say, return the direct sum $\bigoplus_{i=1}^n M_i$.

Additionally, return

  • a vector containing the canonical injections $M_i\to\bigoplus_{i=1}^n M_i$ if task = :sum (default),
  • a vector containing the canonical projections $\bigoplus_{i=1}^n M_i\to M_i$ if task = :prod,
  • two vectors containing the canonical injections and projections, respectively, if task = :both,
  • none of the above maps if task = :none.
source
direct_sum(x::Vector{QuadSpaceWithIsom}) -> QuadSpaceWithIsom, Vector{AbstractSpaceMor}
direct_sum(x::Vararg{QuadSpaceWithIsom}) -> QuadSpaceWithIsom, Vector{AbstractSpaceMor}

Given a collection of quadratic spaces with isometries $(V_1, f_1), \ldots, (V_n, f_n)$, return the quadratic space with isometry $(V, f)$ together with the injections $V_i \to V$, where $V$ is the direct sum $V := V_1 \oplus \ldots \oplus V_n$ and $f$ is the isometry of $V$ induced by the diagonal actions of the $f_i$'s.

For objects of type QuadSpaceWithIsom, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain $(V, f)$ as a direct product with the projections $V \to V_i$, one should call direct_product(x). If one wants to obtain $(V, f)$ as a biproduct with the injections $V_i \to V$ and the projections $V \to V_i$, one should call biproduct(x).

Examples

julia> V1 = quadratic_space(QQ, QQ[2 5;
                                   5 6])
Quadratic space of dimension 2
  over rational field
with gram matrix
[2   5]
[5   6]

julia> Vf1 = quadratic_space_with_isometry(V1, neg=true)
Quadratic space of dimension 2
  with isometry of finite order 2
  given by
  [-1    0]
  [ 0   -1]

julia> V2 = quadratic_space(QQ, QQ[ 2 -1;
                                   -1  2])
Quadratic space of dimension 2
  over rational field
with gram matrix
[ 2   -1]
[-1    2]

julia> f = matrix(QQ, 2, 2, [1  1;
                             0 -1])
[1    1]
[0   -1]

julia> Vf2 = quadratic_space_with_isometry(V2, f)
Quadratic space of dimension 2
  with isometry of finite order 2
  given by
  [1    1]
  [0   -1]

julia> Vf3, inj = direct_sum(Vf1, Vf2)
(Quadratic space with isometry of finite order 2, AbstractSpaceMor[Map: quadratic space -> quadratic space, Map: quadratic space -> quadratic space])

julia> Vf3
Quadratic space of dimension 4
  with isometry of finite order 2
  given by
  [-1    0   0    0]
  [ 0   -1   0    0]
  [ 0    0   1    1]
  [ 0    0   0   -1]

julia> space(Vf3)
Quadratic space of dimension 4
  over rational field
with gram matrix
[2   5    0    0]
[5   6    0    0]
[0   0    2   -1]
[0   0   -1    2]
source
rescale — Method
rescale(Vf::QuadSpaceWithIsom, a::RationalUnion)

Given a quadratic space with isometry $(V, f)$, return the pair $(V^a, f$) where $V^a$ is the same space as $V$ with the associated quadratic form rescaled by $a$.

Examples

julia> V = quadratic_space(QQ, QQ[ 2 -1;
                                  -1  2])
Quadratic space of dimension 2
  over rational field
with gram matrix
[ 2   -1]
[-1    2]

julia> Vf = quadratic_space_with_isometry(V)
Quadratic space of dimension 2
  with isometry of finite order 1
  given by
  [1   0]
  [0   1]

julia> Vf2 = rescale(Vf, 1//2)
Quadratic space of dimension 2
  with isometry of finite order 1
  given by
  [1   0]
  [0   1]

julia> space(Vf2)
Quadratic space of dimension 2
  over rational field
with gram matrix
[    1   -1//2]
[-1//2       1]
source

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 — Method
rational_spinor_norm(Vf::QuadSpaceWithIsom; b::Int = -1) -> QQFieldElem

Given a rational quadratic space with isometry $(V, b, f)$, return the rational spinor norm of $f$.

If $\Phi$ is the form on $V$, then the spinor norm is computed with respect to $b\Phi$.

source

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 $V = V'$.