From 00f5fc1f239daec1c8a5377ac8ff948d23ccce55 Mon Sep 17 00:00:00 2001
From: kalmar <kalmar@amu.edu.pl>
Date: Thu, 22 Jun 2017 14:21:25 +0200
Subject: [PATCH] add DirectProducts and WreathProducts

---
 src/DirectProducts.jl | 248 ++++++++++++++++++++++++++++++++++++++++++
 src/WreathProducts.jl | 236 ++++++++++++++++++++++++++++++++++++++++
 2 files changed, 484 insertions(+)
 create mode 100644 src/DirectProducts.jl
 create mode 100644 src/WreathProducts.jl

diff --git a/src/DirectProducts.jl b/src/DirectProducts.jl
new file mode 100644
index 0000000..c0332bf
--- /dev/null
+++ b/src/DirectProducts.jl
@@ -0,0 +1,248 @@
+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
diff --git a/src/WreathProducts.jl b/src/WreathProducts.jl
new file mode 100644
index 0000000..24aa589
--- /dev/null
+++ b/src/WreathProducts.jl
@@ -0,0 +1,236 @@
+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