using DirectProducts using WreathProducts ############################################################################### # # 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