Added Automorpshim_groups (of free groups)

src/Groups.jl

@@ -174,5 +174,6 @@ end
 (^){T<:GSymbol}(x::T, n::Integer) = GWord(x)^n
 
 include("free_groups.jl")
+include("automorphism_groups.jl")
 
 end # of module Groups

src/automorphism_groups.jl

@@ -0,0 +1,101 @@
+using Permutations
+
+export AutSymbol, AutWord, rmul_AutSymbol, lmul_AutSymbol, flip_AutSymbol, symmetric_AutSymbol
+
+immutable AutSymbol <: GSymbol
+    gen::String
+    pow::Int
+    ex::Expr
+end
+
+(==)(s::AutSymbol, t::AutSymbol) = s.gen == t.gen && s.pow == t.pow
+hash(s::AutSymbol, h::UInt) = hash(s.gen, hash(s.pow, hash(:AutSymbol, h)))
+
+IdSymbol(::Type{AutSymbol}) = AutSymbol("(id)", 0, :(IdAutomorphism(N)))
+
+function change_pow(s::AutSymbol, n::Int)
+
+    if n == 0
+        return one(s)
+    end
+
+    symbol = s.ex.args[1]
+    if symbol == :ɛ
+        return flip_AutSymbol(s.ex.args[2], pow=n)
+    elseif symbol == :σ
+        return symmetric_AutSymbol(s.ex.args[2], pow=n)
+    elseif symbol == :ϱ
+        return rmul_AutSymbol(s.ex.args[2], s.ex.args[3], pow=n)
+    elseif symbol == :λ
+        return lmul_AutSymbol(s.ex.args[2], s.ex.args[3], pow=n)
+    elseif symbol == :IdAutomorphism
+        return s
+    else
+        warn("Changing an unknown type of symbol! $s")
+        return AutSymbol(s.gen, n, s.ex)
+    end
+end
+
+inv(f::AutSymbol) = change_pow(f, -1*f.pow)
+(^)(s::AutSymbol, n::Integer) = change_pow(s, s.pow*n)
+
+function rmul_AutSymbol(i,j; pow::Int=1)
+    gen = string('ϱ',Char(8320+i), Char(8320+j)...)
+    return AutSymbol(gen, pow, :(ϱ($i,$j)))
+end
+
+function lmul_AutSymbol(i,j; pow::Int=1)
+    gen = string('λ',Char(8320+i), Char(8320+j)...)
+    return AutSymbol(gen, pow, :(λ($i,$j)))
+end
+
+function flip_AutSymbol(j; pow::Int=1)
+    gen = string('ɛ', Char(8320 + j))
+    return AutSymbol(gen, (2+ pow%2)%2, :(ɛ($j)))
+end
+
+function symmetric_AutSymbol(perm::Vector{Int}; pow::Int=1)
+    # if perm == collect(1:length(perm))
+    #     return one(AutSymbol)
+    # end
+    perm = Permutation(perm)
+    ord = order(perm)
+    pow = pow % ord
+    perm = perm^pow
+    gen = string('σ', [Char(8320 + i) for i in array(perm)]...)
+    return AutSymbol(gen, 1, :(σ($(array(perm)))))
+end
+
+function getperm(s::AutSymbol)
+    if s.ex.args[1] == :σ
+        return s.ex.args[2]
+    else
+        throw(ArgumentError("$s is not a permutation automorphism!"))
+    end
+end
+
+typealias AutWord GWord{AutSymbol}
+
+function simplify_perms!(W::AutWord)
+    reduced = true
+    for i in 1:length(W.symbols) - 1
+        current = W.symbols[i]
+        if current.ex.args[1] == :σ
+            if current.pow != 1
+                current = symmetric_AutSymbol(perm(current), pow=current.pow)
+            end
+            next_s = W.symbols[i+1]
+            if  next_s.ex.args[1] == :σ
+                reduced = false
+                if next_s.pow != 1
+                    next_s = symmetric_AutSymbol(perm(next_s), pow=next_s.pow)
+                end
+                p1 = Permutation(getperm(current))
+                p2 = Permutation(getperm(next_s))
+                W.symbols[i] = one(AutSymbol)
+                W.symbols[i+1] = symmetric_AutSymbol(array(p1*p2))
+            end
+        end
+    end
+    return reduced
+end