add WreathProducts

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