89 lines
3.1 KiB
Julia
89 lines
3.1 KiB
Julia
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module SemiDirectProduct
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import Base: convert, show, isequal, ==, size, inv
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import Base: +, -, *, //
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export SemiDirectProductElement, matrix_repr
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"""
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Implements elements of a semidirect product of groups H and N, where N is normal in the product. Usually written as H ⋉ N.
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The multiplication inside semidirect product is defined as
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(h₁, n₁) ⋅ (h₂, n₂) = (h₁h₂, n₁φ(h₁)(n₂)),
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where φ:H → Aut(N) is a homomorphism.
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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.
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# Arguments:
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* `h::Array{T,2}` : square invertible matrix (element of GL(n,K))
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* `n::Vector{T,1}` : vector in Kⁿ
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* `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.
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"""
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immutable SemiDirectProductElement{T<:Number}
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h::Array{T,2}
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n::Vector{T}
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φ::Function
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function SemiDirectProductElement(h::Array{T,2},n::Vector{T},φ::Function)
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# size(h,1) == size(h,2)|| throw(ArgumentError("h has to be square matrix"))
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det(h) ≠ 0 || throw(ArgumentError("h has to be invertible!"))
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new(h,n,φ)
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end
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end
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SemiDirectProductElement{T}(h::Array{T,2}, n::Vector{T}, φ) =
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SemiDirectProductElement{T}(h,n,φ)
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SemiDirectProductElement{T}(h::Array{T,2}, n::Vector{T}) =
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SemiDirectProductElement(h,n,*)
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SemiDirectProductElement{T}(h::Array{T,2}) =
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SemiDirectProductElement(h,zeros(h[:,1]))
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SemiDirectProductElement{T}(n::Vector{T}) =
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SemiDirectProductElement(eye(eltype(n), n))
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convert{T<:Number}(::Type{T}, X::SemiDirectProductElement) =
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SemiDirectProductElement(convert(Array{T,2},X.h),
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convert(Vector{T},X.n),
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X.φ)
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size(X::SemiDirectProductElement) = (size(X.h), size(X.n))
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matrix_repr{T}(X::SemiDirectProductElement{T}) =
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[X.h X.n; zeros(T, 1, size(X.h,2)) [1]]
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show{T}(io::IO, X::SemiDirectProductElement{T}) = print(io,
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"Element of SemiDirectProduct over $T of size $(size(X)):\n",
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matrix_repr(X))
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function isequal{T}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{T})
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X.h == Y.h || return false
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X.n == Y.n || return false
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X.φ == Y.φ || return false
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return true
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end
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function isequal{T,S}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{S})
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W = promote_type(T,S)
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warn("Comparing elements with different coefficients! trying to promoting to $W")
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X = convert(W, X)
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Y = convert(W, Y)
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return isequal(X,Y)
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end
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(==)(X::SemiDirectProductElement, Y::SemiDirectProductElement) = isequal(X, Y)
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function semidirect_multiplication{T}(X::SemiDirectProductElement{T},
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Y::SemiDirectProductElement{T})
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size(X) == size(Y) || throw(ArgumentError("trying to multiply elements from different groups!"))
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return SemiDirectProductElement(X.h*Y.h, X.n + X.φ(X.h, Y.n))
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end
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(*){T}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{T}) =
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semidirect_multiplication(X,Y)
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inv{T}(X::SemiDirectProductElement{T}) =
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SemiDirectProductElement(inv(X.h), X.φ(inv(X.h), -X.n))
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end
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