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remove DirectProducts and WreathProducts

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These are part of the Groups.jl package
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kalmar 2017-06-22 15:19:08 +02:00
parent c6a56fcadb
commit 77c3ba1c61
2 changed files with 0 additions and 484 deletions
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248
DirectProducts.jl
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module DirectProducts
using Nemo
import Base: show, ==, hash, deepcopy_internal
import Base: ×, *, inv
import Nemo: parent, parent_type, elem_type
import Nemo: elements, order, Group, GroupElem, Ring
export DirectProductGroup, DirectProductGroupElem
###############################################################################
#
# DirectProductGroup / DirectProductGroupElem
#
###############################################################################
doc"""
DirectProductGroup(factors::Vector{Group}) <: Group
Implements direct product of groups as vector factors. The group operation is
`*` distributed component-wise, with component-wise identity as neutral element.
"""
type DirectProductGroup <: Group
factors::Vector{Group}
operations::Vector{Function}
end
type DirectProductGroupElem <: GroupElem
elts::Vector{GroupElem}
parent::DirectProductGroup
DirectProductGroupElem{T<:GroupElem}(a::Vector{T}) = new(a)
end
###############################################################################
#
# Type and parent object methods
#
###############################################################################
elem_type(G::DirectProductGroup) = DirectProductGroupElem
parent_type(::Type{DirectProductGroupElem}) = DirectProductGroup
parent(g::DirectProductGroupElem) = g.parent
###############################################################################
#
# DirectProductGroup / DirectProductGroupElem constructors
#
###############################################################################
DirectProductGroup(G::Group, H::Group) = DirectProductGroup([G, H], Function[(*),(*)])
DirectProductGroup(G::Group, H::Ring) = DirectProductGroup([G, H], Function[(*),(+)])
DirectProductGroup(G::Ring, H::Group) = DirectProductGroup([G, H], Function[(+),(*)])
DirectProductGroup(G::Ring, H::Ring) = DirectProductGroup([G, H], Function[(+),(+)])
DirectProductGroup{T<:Ring}(X::Vector{T}) = DirectProductGroup(Group[X...], Function[(+) for _ in X])
×(G::Group, H::Group) = DirectProductGroup(G,H)
function DirectProductGroup{T<:Group, S<:Group}(G::Tuple{T, Function}, H::Tuple{S, Function})
return DirectProductGroup([G[1], H[1]], Function[G[2],H[2]])
end
function DirectProductGroup(groups::Vector)
for G in groups
typeof(G) <: Group || throw("$G is not a group!")
end
ops = Function[typeof(G) <: Ring ? (+) : (*) for G in groups]
return DirectProductGroup(groups, ops)
end
###############################################################################
#
# Parent object call overloads
#
###############################################################################
(G::DirectProductGroup)() = G([H() for H in G.factors]; checked=false)
function (G::DirectProductGroup)(g::DirectProductGroupElem; checked=true)
if checked
return G(g.elts)
else
g.parent = G
return g
end
end
doc"""
(G::DirectProductGroup)(a::Vector; checked=true)
> Constructs element of the direct product group `G` by coercing each element
> of vector `a` to the corresponding factor of `G`. If `checked` flag is set to
> `false` no checks on the correctness are performed.
"""
function (G::DirectProductGroup)(a::Vector; checked=true)
length(a) == length(G.factors) || throw("Cannot coerce $a to $G: they have
different number of factors")
if checked
for (F,g) in zip(G.factors, a)
try
F(g)
catch
throw("Cannot coerce to $G: $g cannot be coerced to $F.")
end
end
end
elt = DirectProductGroupElem([F(g) for (F,g) in zip(G.factors, a)])
elt.parent = G
return elt
end
###############################################################################
#
# Basic manipulation
#
###############################################################################
function deepcopy_internal(g::DirectProductGroupElem, dict::ObjectIdDict)
G = parent(g)
return G(deepcopy(g.elts))
end
function hash(G::DirectProductGroup, h::UInt)
return hash(G.factors, hash(G.operations, hash(DirectProductGroup,h)))
end
function hash(g::DirectProductGroupElem, h::UInt)
return hash(g.elts, hash(g.parent, hash(DirectProductGroupElem, h)))
end
doc"""
eye(G::DirectProductGroup)
> Return the identity element for the given direct product of groups.
"""
eye(G::DirectProductGroup) = G()
###############################################################################
#
# String I/O
#
###############################################################################
function show(io::IO, G::DirectProductGroup)
println(io, "Direct product of groups")
join(io, G.factors, ", ", " and ")
end
function show(io::IO, g::DirectProductGroupElem)
print(io, "("*join(g.elts,",")*")")
end
###############################################################################
#
# Comparison
#
###############################################################################
function (==)(G::DirectProductGroup, H::DirectProductGroup)
G.factors == H.factors || return false
G.operations == H.operations || return false
return true
end
doc"""
==(g::DirectProductGroupElem, h::DirectProductGroupElem)
> Return `true` if the given elements of direct products are equal, otherwise return `false`.
"""
function (==)(g::DirectProductGroupElem, h::DirectProductGroupElem)
parent(g) == parent(h) || return false
g.elts == h.elts || return false
return true
end
###############################################################################
#
# Binary operators
#
###############################################################################
function direct_mult(g::DirectProductGroupElem, h::DirectProductGroupElem)
parent(g) == parent(h) || throw("Can't multiply elements from different groups: $g, $h")
G = parent(g)
return G([op(a,b) for (op,a,b) in zip(G.operations, g.elts, h.elts)])
end
doc"""
*(g::DirectProductGroupElem, h::DirectProductGroupElem)
> Return the direct-product group operation of elements, i.e. component-wise
> operation as defined by `operations` field of the parent object.
"""
(*)(g::DirectProductGroupElem, h::DirectProductGroupElem) = direct_mult(g,h)
###############################################################################
#
# Inversion
#
###############################################################################
doc"""
inv(g::DirectProductGroupElem)
> Return the inverse of the given element in the direct product group.
"""
# TODO: dirty hack around `+` operation
function inv(g::DirectProductGroupElem)
G = parent(g)
return G([(op == (*) ? inv(elt): -elt) for (op,elt) in zip(G.operations, g.elts)])
end
###############################################################################
#
# Misc
#
###############################################################################
doc"""
elements(G::DirectProductGroup)
> Returns `Task` that produces all elements of group `G` (provided that factors
> implement the elements function).
"""
# TODO: can Base.product handle generators?
# now it returns nothing's so we have to collect ellements...
function elements(G::DirectProductGroup)
cartesian_prod = Base.product([collect(elements(H)) for H in G.factors]...)
return (G(collect(elt)) for elt in cartesian_prod)
end
doc"""
order(G::DirectProductGroup)
> Returns the order (number of elements) in the group.
"""
order(G::DirectProductGroup) = prod([order(H) for H in G.factors])
end # of module DirectProduct

236
WreathProducts.jl
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module WreathProducts
using Nemo
using DirectProducts
import Base: convert, deepcopy_internal, show, isequal, ==, hash, size, inv
import Base: +, -, *, //
import Nemo: Group, GroupElem, elem_type, parent_type, parent, elements, order
###############################################################################
#
# WreathProduct / WreathProductElem
#
###############################################################################
doc"""
WreathProduct <: Group
> Implements Wreath product of a group N by permutation (sub)group P < Sₖ,
> usually written as $N \wr P$.
> The multiplication inside wreath product is defined as
> (n, σ) * (m, τ) = (n*ψ(σ)(m), σ*τ),
> where ψ:P → Aut(Nᵏ) is the permutation representation of Sₖ restricted to P.
# Arguments:
* `::Group` : the single factor of group N
* `::PermutationGroup` : full PermutationGroup
"""
type WreathProduct <: Group
N::DirectProductGroup
P::PermutationGroup
function WreathProduct(G::Group, P::PermutationGroup)
N = DirectProductGroup(typeof(G)[G for _ in 1:P.n])
return new(N, P)
end
end
type WreathProductElem <: GroupElem
n::DirectProductGroupElem
p::perm
parent::WreathProduct
function WreathProductElem(n::DirectProductGroupElem, p::perm)
length(n.elts) == parent(p).n
return new(n, p)
end
end
export WreathProduct, WreathProductElem
###############################################################################
#
# Type and parent object methods
#
###############################################################################
elem_type(::WreathProduct) = WreathProductElem
parent_type(::WreathProductElem) = WreathProduct
parent(g::WreathProductElem) = g.parent
###############################################################################
#
# WreathProduct / WreathProductElem constructors
#
###############################################################################
# converts???
###############################################################################
#
# Parent object call overloads
#
###############################################################################
function (G::WreathProduct)(g::WreathProductElem)
try
G.N(g.n)
catch
throw("Can't coerce $(g.n) to $(G.N) factor of $G")
end
try
G.P(g.p)
catch
throw("Can't coerce $(g.p) to $(G.P) factor of $G")
end
elt = WreathProductElem(G.N(g.n), G.P(g.p))
elt.parent = G
return elt
end
doc"""
(G::WreathProduct)(n::DirectProductGroupElem, p::perm)
> Creates an element of wreath product `G` by coercing `n` and `p` to `G.N` and
> `G.P`, respectively.
"""
function (G::WreathProduct)(n::DirectProductGroupElem, p::perm)
result = WreathProductElem(n,p)
result.parent = G
return result
end
(G::WreathProduct)() = G(G.N(), G.P())
doc"""
(G::WreathProduct)(p::perm)
> Returns the image of permutation `p` in `G` via embedding `p -> (id,p)`.
"""
(G::WreathProduct)(p::perm) = G(G.N(), p)
doc"""
(G::WreathProduct)(n::DirectProductGroupElem)
> Returns the image of `n` in `G` via embedding `n -> (n,())`. This is the
> embedding that makes sequence `1 -> N -> G -> P -> 1` exact.
"""
(G::WreathProduct)(n::DirectProductGroupElem) = G(n, G.P())
###############################################################################
#
# Basic manipulation
#
###############################################################################
function deepcopy_internal(g::WreathProductElem, dict::ObjectIdDict)
G = parent(g)
return G(deepcopy(g.n), deepcopy(g.p))
end
function hash(G::WreathProduct, h::UInt)
return hash(G.N, hash(G.P, hash(WreathProduct, h)))
end
function hash(g::WreathProductElem, h::UInt)
return hash(g.n, hash(g.p, hash(parent(g), h)))
end
###############################################################################
#
# String I/O
#
###############################################################################
function show(io::IO, G::WreathProduct)
print(io, "Wreath Product of $(G.N.factors[1]) and $(G.P)")
end
function show(io::IO, g::WreathProductElem)
# println(io, "Element of WreathProduct over $T of size $(size(X)):")
# show(io, "text/plain", matrix_repr(X))
print(io, "($(g.n)≀$(g.p))")
end
###############################################################################
#
# Comparison
#
###############################################################################
function (==)(G::WreathProduct, H::WreathProduct)
G.N == H.N || return false
G.P == H.P || return false
return true
end
function (==)(g::WreathProductElem, h::WreathProductElem)
parent(g) == parent(h) || return false
g.n == h.n || return false
g.p == h.p || return false
return true
end
###############################################################################
#
# Binary operators
#
###############################################################################
function wreath_multiplication(g::WreathProductElem, h::WreathProductElem)
parent(g) == parent(h) || throw("Can not multiply elements from different
groups!")
G = parent(g)
w=G.N((h.n).elts[inv(g.p).d])
return G(g.n*w, g.p*h.p)
end
doc"""
*(g::WreathProductElem, h::WreathProductElem)
> Return the wreath product group operation of elements, i.e.
>
> g*h = (g.n*g.p(h.n), g.p*h.p),
>
> where g.p(h.n) denotes the action of `g.p::perm` on
> `h.n::DirectProductGroupElem` via standard permutation of coordinates.
"""
(*)(g::WreathProductElem, h::WreathProductElem) = wreath_multiplication(g,h)
###############################################################################
#
# Inversion
#
###############################################################################
doc"""
inv(g::WreathProductElem)
> Returns the inverse of element of a wreath product, according to the formula
> g^-1 = (g.n, g.p)^-1 = (g.p^-1(g.n^-1), g.p^-1).
"""
function inv(g::WreathProductElem)
G = parent(g)
w = G.N(inv(g.n).elts[g.p.d])
return G(w, inv(g.p))
end
###############################################################################
#
# Misc
#
###############################################################################
matrix_repr(g::WreathProductElem) = Any[matrix_repr(g.p) g.n]
function elements(G::WreathProduct)
iter = Base.product(collect(elements(G.N)), collect(elements(G.P)))
return (G(n)*G(p) for (n,p) in iter)
end
order(G::WreathProduct) = order(G.P)*order(G.N)
end # of module WreatProduct