From fb2fa1c3acb1bcf7d9b23529760a0d867c377ee9 Mon Sep 17 00:00:00 2001
From: kalmar <kalmar@amu.edu.pl>
Date: Fri, 13 Jan 2017 18:42:07 +0100
Subject: [PATCH] SemiDirect Products of matrix groups

---
 SemiDirectProduct.jl | 88 ++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 88 insertions(+)
 create mode 100644 SemiDirectProduct.jl

diff --git a/SemiDirectProduct.jl b/SemiDirectProduct.jl
new file mode 100644
index 0000000..52b7d0b
--- /dev/null
+++ b/SemiDirectProduct.jl
@@ -0,0 +1,88 @@
+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