2017-03-13 18:03:48 +01:00
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module GroupAlgebras
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2017-05-16 18:27:32 +02:00
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using Nemo
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import Nemo: Group, GroupElem, Ring
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2017-03-13 19:44:26 +01:00
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import Base: convert, show, isequal, ==
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import Base: +, -, *, //
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import Base: size, length, norm, rationalize
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2017-03-13 18:03:48 +01:00
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2017-03-13 19:44:26 +01:00
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2017-05-16 18:28:32 +02:00
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type GroupRing <: Ring
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group::Group
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pm::Array{Int,2}
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basis::Vector{GroupElem}
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basis_dict::Dict{GroupElem, Int}
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2017-03-13 19:44:26 +01:00
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2017-05-16 18:28:32 +02:00
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GroupRing(G::Group) = new(G)
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end
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2017-03-13 19:44:26 +01:00
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2017-05-16 18:29:14 +02:00
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type GroupRingElem{T<:Number}
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coeffs::AbstractVector{T}
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parent::GroupRing
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2017-03-13 19:44:26 +01:00
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2017-05-16 18:29:14 +02:00
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function GroupRingElem(coeffs::AbstractVector)
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return new(coeffs)
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end
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2017-03-13 19:44:26 +01:00
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end
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2017-05-16 18:31:45 +02:00
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export GroupRing, GroupRingElem
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2017-03-13 19:44:26 +01:00
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2017-05-16 18:31:26 +02:00
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GroupRingElem{T}(c::AbstractVector{T}, A::GroupRing) = GroupRingElem{T}(c,A)
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convert{T<:Number}(::Type{T}, X::GroupRingElem) =
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GroupRingElem(parent(X), convert(AbstractVector{T}, X.coeffs))
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2017-05-16 18:32:12 +02:00
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function GroupRing(G::Group, pm::Array{Int,2})
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size(pm,1) == size(pm,2) || throw("pm must be of size (n,n), got
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$(size(pm))")
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return GroupRing(Group, pm)
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end
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function GroupRing(G::Group, pm::Array{Int,2}, basis::Vector)
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size(pm,1) == size(pm,2) || throw("pm must be of size (n,n), got
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$(size(pm))")
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eltype(basis) == elem_type(G) || throw("basis must consist of elements of $G")
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basis_dict = Dict(g => i for (i,g) in enumerate(basis))
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return GroupRing(Group, pm, basis, basis_dict)
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end
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function GroupRing(G::Group; complete=false)
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A = GroupRing(Group)
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if complete
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complete(A)
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end
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return A
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end
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2017-05-16 18:34:21 +02:00
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function show(io::IO, A::GroupRing)
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print(io, "GroupRing of $(A.group)")
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end
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function show(io::IO, X::GroupRingElem)
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T = eltype(X.coeffs)
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print(io, "Element of Group Algebra of $(parent(X)) over $T:\n $(X.coeffs)")
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end
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2017-03-13 19:44:26 +01:00
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end
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(==)(X::GroupAlgebraElement, Y::GroupAlgebraElement) = isequal(X,Y)
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function add{T<:Number}(X::GroupAlgebraElement{T}, Y::GroupAlgebraElement{T})
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X.product_matrix == Y.product_matrix || throw(ArgumentError(
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"Elements don't seem to belong to the same Group Algebra!"))
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return GroupAlgebraElement(X.coefficients+Y.coefficients, X.product_matrix)
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end
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function add{T<:Number, S<:Number}(X::GroupAlgebraElement{T},
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Y::GroupAlgebraElement{S})
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warn("Adding elements with different base rings!")
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return GroupAlgebraElement(+(promote(X.coefficients, Y.coefficients)...),
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X.product_matrix)
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end
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(+)(X::GroupAlgebraElement, Y::GroupAlgebraElement) = add(X,Y)
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(-)(X::GroupAlgebraElement) = GroupAlgebraElement(-X.coefficients, X.product_matrix)
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(-)(X::GroupAlgebraElement, Y::GroupAlgebraElement) = add(X,-Y)
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function algebra_multiplication{T<:Number}(X::AbstractVector{T}, Y::AbstractVector{T}, pm::Array{Int,2})
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result = zeros(X)
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for (j,y) in enumerate(Y)
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if y != zero(T)
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for (i, index) in enumerate(pm[:,j])
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if X[i] != zero(T)
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index == 0 && throw(ArgumentError("The product don't seem to belong to the span of basis!"))
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result[index] += X[i]*y
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end
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end
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end
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end
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return result
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end
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function group_star_multiplication{T<:Number}(X::GroupAlgebraElement{T},
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Y::GroupAlgebraElement{T})
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X.product_matrix == Y.product_matrix || ArgumentError(
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"Elements don't seem to belong to the same Group Algebra!")
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result = algebra_multiplication(X.coefficients, Y.coefficients, X.product_matrix)
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return GroupAlgebraElement(result, X.product_matrix)
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end
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function group_star_multiplication{T<:Number, S<:Number}(
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X::GroupAlgebraElement{T},
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Y::GroupAlgebraElement{S})
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S == T || warn("Multiplying elements with different base rings!")
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return group_star_multiplication(promote(X,Y)...)
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end
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(*){T<:Number, S<:Number}(X::GroupAlgebraElement{T},
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Y::GroupAlgebraElement{S}) = group_star_multiplication(X,Y);
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(*){T<:Number}(a::T, X::GroupAlgebraElement{T}) = GroupAlgebraElement(
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a*X.coefficients, X.product_matrix)
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function scalar_multiplication{T<:Number, S<:Number}(a::T,
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X::GroupAlgebraElement{S})
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promote_type(T,S) == S || warn("Scalar and coefficients are in different rings! Promoting result to $(promote_type(T,S))")
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return GroupAlgebraElement(a*X.coefficients, X.product_matrix)
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end
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(*){T<:Number}(a::T,X::GroupAlgebraElement) = scalar_multiplication(a, X)
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//{T<:Rational, S<:Rational}(X::GroupAlgebraElement{T}, a::S) =
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GroupAlgebraElement(X.coefficients//a, X.product_matrix)
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//{T<:Rational, S<:Integer}(X::GroupAlgebraElement{T}, a::S) =
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X//convert(T,a)
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length(X::GroupAlgebraElement) = length(X.coefficients)
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size(X::GroupAlgebraElement) = size(X.coefficients)
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function norm(X::GroupAlgebraElement, p=2)
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if p == 1
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return sum(abs(X.coefficients))
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elseif p == Inf
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return max(abs(X.coefficients))
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else
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return norm(X.coefficients, p)
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end
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end
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ɛ(X::GroupAlgebraElement) = sum(X.coefficients)
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function rationalize{T<:Integer, S<:Number}(
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::Type{T}, X::GroupAlgebraElement{S}; tol=eps(S))
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v = rationalize(T, X.coefficients, tol=tol)
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return GroupAlgebraElement(v, X.product_matrix)
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end
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end
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