Vector spaces

AbstractSpace{F}

An abstract type representing a vector space. The type F represents the number field (or number type) over which the vector space is defined.

basis(::AbstractSpace)

Returns a basis of the given vector space.

dim(::AbstractSpace) -> Int

Returns the dimension of the given vector space.

PolySpace <: AbstractSpace

A structure representing a vector space generated by polynomials.

AbstractDirectSum{F} <: AbstractSpace{F}

An abstract type representing a direct sum of vector spaces over the field (or number type) F.

summands(::AbstractDirectSum)

Returns the summands of the given direct sum of vector spaces.

nsummands(::AbstractDirectSum) -> Int

Returns the number of summands in the given direct sum of vector spaces.

AbstractSymmetricPower{F} <: AbstractSpace{F}

An abstract type representing a symmetric power of a vector space over the field (or number type) F.

base_space(::AbstractSymmetricPower) -> AbstractSpace

Returns the base vector space of the given symmetric power. I.e. for $\mathrm{Sym}^n(V)$, this function returns $V$.

power(::AbstractSymmetricPower) -> Int

Returns the power of the given symmetric power. I.e. for $\mathrm{Sym}^n(V)$, this function returns $n$.

FixedDegreePolynomials <: AbstractSymmetricPower

A type representing a space of polynomials of a fixed degree. I.e., for the variables $x_1, \dots, x_n$ and degree $d$ gives $\mathbb{F}[x_1, \dots, x_n]_{d}$, the space of polynomials in $n$ variables of degree $d$.

Constructors

FixedDegreePolynomials(vars::Vector{<:Variable}, degree::Int)

Examples

julia> @polyvar x[1:3];

julia> V = FixedDegreePolynomials(x, 2)
FixedDegreePolynomials space of dimension 6
 variables: x₁, x₂, x₃
 degree: 2

julia> base_space(V)
VariableSpace with 3 variables
 number type (or field): ComplexF64
 variables: x₁, x₂, x₃

julia> power(V)
2

AbstractTensorProduct{F} <: AbstractSpace{F}

An abstract type representing a tensor product of vector spaces over the field F.

factors(::AbstractTensorProduct)

Returns an iterator through the factors of the given tensor product.

FixedMultidegreePolynomials <: AbstractTensorProduct

A type representing a space of polynomials of a fixed multidegree in certain groups of variables. For example, for 2 groups of variables $x_1,\dots,x_m$ and $y_1,\dots,y_n$, and degrees $d_1$ and $d_2$, this gives

\[\mathbb{F}[x_1, \dots, x_m, y_1, \dots, y_n]_{(d_1, d_2)} \cong \mathbb{F}[x_1, \dots, x_m]_{d_1} \otimes \mathbb{F}[y_1, \dots, y_n]_{d_2},\]

the space of multi-homogeneous polynomials in $m+n$ variables of degree $d_1$ in the first group and $d_2$ in the second group.

Constructors

FixedMultidegreePolynomials(var_groups::Vector{Vector{<:Variable}}, degrees::Vector{Int})

Examples

julia> @polyvar x[1:3] y[1:2];

julia> V = FixedMultidegreePolynomials([x, y], [1, 1])
FixedMultidegreePolynomials space of dimension 6
 variable groups: [x₁, x₂, x₃], [y₁, y₂]
 multidegree: [1, 1]

julia> fs = factors(V)
2-element Vector{FixedDegreePolynomials{ComplexF64, Commutative{CreationOrder}, Graded{LexOrder}}}:
 FixedDegreePolynomials space of dimension 3
 FixedDegreePolynomials space of dimension 2

julia> fs[2]
FixedDegreePolynomials space of dimension 2
 variables: y₁, y₂
 degree: 1

HighestWeightModule <: AbstractSpace

Represents a highest weight module in the context of Lie group representations.

Examples

julia> @polyvar x y z;

julia> vars = [x, y, z];

julia> SO3 = LieGroup("SO", 3);

julia> a = MatrixGroupAction(SO3, [vars]);

julia> V = FixedDegreePolynomials(vars, 2);

julia> ρ = GroupRepresentation(a, V);

julia> irr_decomp = irreducibles(ρ)
IrreducibleDecomposition of SO(3)-action on 6-dimensional vector space
 number of irreducibles: 2
 dimensions of irreducibles: 1, 5

julia> H = space(irr_decomp[1])
HighestWeightModule of dimension 1
 Lie group: SO(3)
 highest weight: [0]

julia> basis(H)
1-element Vector{Polynomial{Commutative{CreationOrder}, Graded{LexOrder}, ComplexF64}}:
 z² + y² + x²