GroupsWithPropertyT/Projections.jl

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###############################################################################
#
# Characters of PermutationGroup
#
###############################################################################
function chars(G::PermutationGroup)
permtype_unsorted(σ::Nemo.perm) = [length(c) for c in cycles(σ)]
permtype(σ::Nemo.perm) = sort(permtype_unsorted(σ))
χ_id(σ::Nemo.perm) = 1
χ_sgn(σ::Nemo.perm) = (-1)^parity(σ)
function χ_reg(σ::Nemo.perm)
fixed_points = countnz([(x == y? 1 : 0) for (x,y) in enumerate(σ.d)])
return fixed_points - 1
end
χ_regsgn(σ::Nemo.perm) = (-1)^parity(σ)*χ_reg(σ)
function χ_regviaS3(σ::Nemo.perm)
@assert parent(σ).n == 4
t = permtype(σ)
if t == [1,1,1,1]
result = 2
elseif t == [2,2]
result = 2
elseif t == [1,3]
result = -1
else
result = 0
end
return result
end
chars = [χ_id, χ_sgn, χ_regviaS3, χ_reg, χ_regsgn]
if G.n == 1
return chars[1:1]
elseif G.n == 2
return chars[1:2]
elseif G.n == 3
return [chars[1:2]..., chars[4]]
elseif G.n == 4
return chars[1:5]
else
throw("Characters for $G unknown!")
end
end
###############################################################################
#
# Character of DirectProducts
#
###############################################################################
function epsilon(i, g::DirectProducts.DirectProductGroupElem)
return reduce(*, 1, ((-1)^isone(g.elts[j]) for j in 1:i))
end
###############################################################################
#
# Projections
#
###############################################################################
function central_projection(RG::GroupRing, char::Function, T::Type=Rational{Int})
result = RG(T)
for g in RG.basis
result[g] = char(g)
end
return convert(T, char(RG.group())//Int(order(RG.group))*result)
end
function rankOne_projections(G::PermutationGroup, T::Type=Rational{Int})
RG = GroupRing(G)
projections = [central_projection(RG, χ, T) for χ in chars(G)]
if G.n == 1 || G.n == 2
return projections
elseif G.n == 3
rankone_projs = [
projections[1],
projections[2],
1//2*(one(RG) - RG(RG.group([2,1,3])))*projections[3]
]
return rankone_projs
elseif G.n == 4
rankone_projs = [
projections[1],
projections[2],
1//2*(one(RG) - RG(RG.group([2,1,3,4])))*projections[3],
1//2*(one(RG) - RG(RG.group([2,1,3,4])))*projections[4],
1//2*(one(RG) + RG(RG.group([2,1,3,4])))*projections[5]]
return rankone_projs
else
throw("Rank-one projections for $G unknown!")
end
end
function rankOne_projections(BN::WreathProducts.WreathProduct, T::Type=Rational{Int})
N = BN.P.n
# projections as elements of the group rings RSₙ
SNprojs_nc = [rankOne_projections(PermutationGroup(i), T) for i in 1:N]
# embedding into group ring of BN
RBN = GroupRing(BN)
RFFFF_projs = [central_projection(GroupRing(BN.N), g->epsilon(i,g), T)
for i in 0:BN.P.n]
Qs = [RBN(q, g -> BN(g)) for q in RFFFF_projs]
function incl(k::Int, g::perm, WP::WreathProduct=BN)
@assert length(g.d) + k <= WP.P.n
arr = [1:k; g.d .+ k; (length(g.d)+k+1):WP.P.n]
return WP(WP.P(arr))
end
all_projs=[Qs[1]*RBN(p, g-> incl(0,g)) for p in SNprojs_nc[N]]
for i in 1:N-1
Sk_first = [RBN(p, g->incl(0,g)) for p in SNprojs_nc[i]]
Sk_last = [RBN(p, g->incl(i,g)) for p in SNprojs_nc[N-i]]
append!(all_projs, [Qs[i+1]*p1*p2
for (p1,p2) in Base.product(Sk_first,Sk_last)])
end
append!(all_projs, [Qs[N+1]*RBN(p, g-> incl(0,g)) for p in SNprojs_nc[N]])
return all_projs
end