commit tmp.jl as an example of what works

tmp.jl

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+using Pkg
+Pkg.activate(@__DIR__)
+
+using Revise
+using StarAlgebras
+import StarAlgebras as SA
+using PermutationGroups
+import PermutationGroups.AP as AP
+
+SA.star(x::Number) = x'
+SA.star(g::AP.AbstractPermutation) = inv(g)
+
+G = PermGroup(perm"(1,2,3,4,5,6)", perm"(1,2)")
+g, h = rand(G, 2)
+
+db = SA.DiracBasis{UInt32}(G)
+RG = SA.StarAlgebra(G, db)
+
+ad = SA.AugmentedBasis(db)
+
+ad[SA.AugmentedDirac(h)]
+
+IG = SA.StarAlgebra(G, ad)
+
+xcfs = SA.SparseCoefficients([one(G), g], [1, -1])
+x = SA.AlgebraElement(xcfs, RG)
+
+ycfs = SA.SparseCoefficients([one(G), inv(g)], [1, -1])
+y = SA.AlgebraElement(ycfs, RG)
+
+zcfs = SA.SparseCoefficients([one(G), h], [1, -1])
+z = SA.AlgebraElement(zcfs, RG)
+
+xycfs = SA.SparseCoefficients([one(G), g, inv(g)], [2, -1, -1])
+xy = SA.AlgebraElement(xycfs, RG)
+
+xzcfs = SA.SparseCoefficients([one(G), g, h, g * h], [1, -1, -1, 1])
+xz = SA.AlgebraElement(xzcfs, RG)
+
+@assert x != y
+@assert x' == y
+@assert SA.mstructure(basis(RG))(g, h) == SA.Dirac(g * h)
+@assert x * y == xy
+@assert x * z == xz
+
+axcfs = SA.coeffs(coeffs(x), basis(RG), basis(IG))
+aycfs = SA.coeffs(coeffs(y), basis(RG), basis(IG))
+azcfs = SA.coeffs(coeffs(z), basis(RG), basis(IG))
+ax = SA.AlgebraElement(axcfs, IG)
+ay = SA.AlgebraElement(aycfs, IG)
+az = SA.AlgebraElement(azcfs, IG)
+
+@assert coeffs(ax * ay) == SA.coeffs(coeffs(x * y), basis(RG), basis(IG))
+@assert coeffs(ax * az) == SA.coeffs(coeffs(x * z), basis(RG), basis(IG))
+
+rcfs = SA.SparseCoefficients(rand(G, 100), rand(-2:2, 100))
+r = SA.AlgebraElement(rcfs, RG)
+scfs = SA.SparseCoefficients(rand(G, 100), rand(-2:2, 100))
+s = SA.AlgebraElement(scfs, RG)
+
+@time r * s
+
+aug(r)
+aug(s)
+@assert aug(r * s) == aug(r) * aug(s)
+
+m = PermutationGroups.order(UInt16, G)
+fb = SA.FixedBasis(collect(G), SA.DiracMStructure(*), (m, m))
+fRG = SA.StarAlgebra(G, fb)
+
+fr = SA.AlgebraElement(coeffs(rcfs, basis(RG), basis(fRG)), fRG)
+fs = SA.AlgebraElement(coeffs(scfs, basis(RG), basis(fRG)), fRG)
+
+@assert aug(fr) == aug(r)
+@assert aug(fs) == aug(s)
+
+@assert aug(fr * fs) == aug(fr) * aug(fs)
+
+r * s
+fr * fs
+frs = SA.AlgebraElement(coeffs(coeffs(r * s), basis(RG), basis(fRG)), fRG)
+@assert fr * fs == frs
+
+fr * fs
+frs
+
+let mt = SA.mstructure(basis(fRG)).table
+    count(i -> isassigned(mt, i), eachindex(mt)), length(mt)
+end
+using BenchmarkTools
+@btime $(star(r)) * $s
+@btime $(star(fr)) * $fs
+
+star(star(fr))
+
+@btime star($fr)
+@btime star($r)
+@edit star(fr)
+
+mulN(a, b, N) = [a * b for i in 1:N]
+
+# using Profile
+
+@profview mulN(fr, fs, 1000)