GroupsWithPropertyT/SemiDirectProduct.jl

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2017-03-13 16:20:40 +01:00
module SemiDirectProduct
import Base: convert, show, isequal, ==, size, inv
import Base: +, -, *, //
export SemiDirectProductElement, matrix_repr
"""
Implements elements of a semidirect product of groups H and N, where N is normal in the product. Usually written as H N.
The multiplication inside semidirect product is defined as
(h₁, n₁) (h₂, n₂) = (h₁h₂, n₁φ(h₁)(n₂)),
where φ:H Aut(N) is a homomorphism.
In the case below we implement H = GL(n,K) and N = Kⁿ, the Affine Group (i.e. GL(n,K) Kⁿ) where elements of GL(n,K) act on vectors in Kⁿ via matrix multiplication.
# Arguments:
* `h::Array{T,2}` : square invertible matrix (element of GL(n,K))
* `n::Vector{T,1}` : vector in Kⁿ
* `φ = φ(h,n) = φ(h)(n)` :2-argument function which defines the action of GL(n,K) on Kⁿ; matrix-vector multiplication by default.
"""
immutable SemiDirectProductElement{T<:Number}
h::Array{T,2}
n::Vector{T}
φ::Function
function SemiDirectProductElement(h::Array{T,2},n::Vector{T},φ::Function)
# size(h,1) == size(h,2)|| throw(ArgumentError("h has to be square matrix"))
det(h) 0 || throw(ArgumentError("h has to be invertible!"))
new(h,n,φ)
end
end
SemiDirectProductElement{T}(h::Array{T,2}, n::Vector{T}, φ) =
SemiDirectProductElement{T}(h,n,φ)
SemiDirectProductElement{T}(h::Array{T,2}, n::Vector{T}) =
SemiDirectProductElement(h,n,*)
SemiDirectProductElement{T}(h::Array{T,2}) =
SemiDirectProductElement(h,zeros(h[:,1]))
SemiDirectProductElement{T}(n::Vector{T}) =
SemiDirectProductElement(eye(eltype(n), n))
convert{T<:Number}(::Type{T}, X::SemiDirectProductElement) =
SemiDirectProductElement(convert(Array{T,2},X.h),
convert(Vector{T},X.n),
X.φ)
size(X::SemiDirectProductElement) = (size(X.h), size(X.n))
matrix_repr{T}(X::SemiDirectProductElement{T}) =
[X.h X.n; zeros(T, 1, size(X.h,2)) [1]]
show{T}(io::IO, X::SemiDirectProductElement{T}) = print(io,
"Element of SemiDirectProduct over $T of size $(size(X)):\n",
matrix_repr(X))
function isequal{T}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{T})
X.h == Y.h || return false
X.n == Y.n || return false
X.φ == Y.φ || return false
return true
end
function isequal{T,S}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{S})
W = promote_type(T,S)
warn("Comparing elements with different coefficients! trying to promoting to $W")
X = convert(W, X)
Y = convert(W, Y)
return isequal(X,Y)
end
(==)(X::SemiDirectProductElement, Y::SemiDirectProductElement) = isequal(X, Y)
function semidirect_multiplication{T}(X::SemiDirectProductElement{T},
Y::SemiDirectProductElement{T})
size(X) == size(Y) || throw(ArgumentError("trying to multiply elements from different groups!"))
return SemiDirectProductElement(X.h*Y.h, X.n + X.φ(X.h, Y.n))
end
(*){T}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{T}) =
semidirect_multiplication(X,Y)
inv{T}(X::SemiDirectProductElement{T}) =
SemiDirectProductElement(inv(X.h), X.φ(inv(X.h), -X.n))
end