add numerical simple_projection + try_harder kwarg

src/direct_summands.jl

@@ -26,3 +26,102 @@ function SparseArrays.droptol!(
     droptol!(image_basis(ds), tol)
     return ds
 end
+
+function simple_projection(
+    ds::SymbolicWedderburn.DirectSummand,
+    G::Group,
+    hom::SymbolicWedderburn.InducedActionHomomorphism;
+    atol=eps(eltype(ds))*max(size(ds)...)
+)
+    issimple(ds) && return ds
+    @assert degree(ds) > 1
+    @assert rem(size(ds, 1), degree(ds)) == 0
+    @assert multiplicity(ds) == size(ds, 1)÷degree(ds)
+
+    # ds is a projection onto subspace of dimension m·d
+    # where m=multiplicity(ds), d = degreee(ds)
+    # here we separate the image into m orthogonal (simple) subspaces
+    # each of dimension d
+
+    Uπ = SymbolicWedderburn.image_basis(ds)
+    T = eltype(ds)
+    @assert T == Float64
+    subspaces = Vector{typeof(Uπ)}()
+    qrs = LinearAlgebra.QRCompactWY{T, Matrix{T}}[]
+    sizehint!(subspaces, multiplicity(ds))
+    sizehint!(qrs, multiplicity(ds))
+
+    @assert isapprox(Uπ*Uπ', I, atol=atol)
+
+    for (i, b) in enumerate(eachrow(Uπ))
+        b_orth = if isempty(subspaces)
+            b
+        else
+            b_orth = b - sum(_orth_proj(V, b, QR) for (V, QR) in zip(subspaces, qrs))
+        end
+        norm(b_orth, 2) < atol && continue
+
+        # we found a new subspace!
+        # zero_small!(b_orth, atol=atol)
+        b_orth ./= norm(b_orth, 2)
+        # seach subspace is of dimension degree(ds)
+        orbit_subspace = Matrix{T}(undef, length(b_orth), degree(ds))
+        orbit_subspace[:, 1] .= b_orth
+
+        dim = 1
+        # cached qrfact to be reused when b_orth is already in the orbit_subspace
+        qrfact = qr(@view orbit_subspace[:, 1:dim])
+        for g in G
+            isone(g) && continue
+            k = action(hom, g, b_orth)
+            @assert isapprox(norm(k, 2), 1.0, atol=atol) "Norm of translated vector should be ≈ 1.0, got $(norm(k, 2))"
+
+            # check if we've seen this one
+            for c in eachcol(orbit_subspace)
+                isapprox(k, c, atol=atol) && continue
+            end
+            # or maybe it's already in the span of others?
+            residual = k - _orth_proj(@view(orbit_subspace[:, 1:dim]), k, qrfact)
+            (rnorm = norm(residual, 2)) < atol && continue
+            if !isempty(subspaces)
+                before_norm = norm(residual, 2)
+                residual = residual - sum(_orth_proj(V, residual, QR) for (V, QR) in zip(subspaces, qrs))
+                @info "norms" before_norm after_norm=norm(residual, 2)
+                (rnorm = norm(residual, 2)) < atol && continue
+            end
+
+            # ok, it's not so we found a new vector in the subspace!
+            dim += 1
+            orbit_subspace[:, dim] .= residual ./ norm(residual, 2)
+            qrfact = qr(@view orbit_subspace[:, 1:dim])
+
+            if dim == degree(ds)
+                @info orbit_subspace' * orbit_subspace
+                @assert isapprox(orbit_subspace' * orbit_subspace, I, atol=atol)
+                push!(subspaces, orbit_subspace)
+                push!(qrs, qrfact)
+                break
+            end
+        end
+        # check that we exited via break, i.e. found all vectors in the subspace
+        @assert dim == degree(ds)
+    end
+    # and that we found exactly the right number of subspaces
+    @assert length(subspaces) == multiplicity(ds) "Expected $(multiplicity(ds)) subspaces, found $(length(subspaces))"
+
+    # as the projection we take the average of all vectors in the subspace
+    # new_Uπ = typeof(Uπ)(transpose(reduce(hcat, sum.(subspaces, dims=2))))
+    # new_Uπ ./= degree(ds)
+    # TODO: math says the the first one would be equally fine
+    new_Uπ = typeof(Uπ)(transpose(reduce(
+        hcat,
+        (@view(s[:, 1]) for s in subspaces)
+    )))
+
+    return SymbolicWedderburn.DirectSummand(
+        new_Uπ,
+        degree(ds),
+        multiplicity(ds),
+        true
+    )
+end

src/wedderburn_decomposition.jl

@@ -13,6 +13,7 @@ function WedderburnDecomposition(
     basis_half,
     S = Rational{Int};
     semisimple = false,
+    try_harder = false,
 )
     basis = StarAlgebras.Basis{UInt32}(basis_full)
     invariants = invariant_vectors(G, action, basis)
@@ -24,6 +25,10 @@ function WedderburnDecomposition(
         sa_basis = symmetry_adapted_basis(T, tbl, ehom; semisimple = semisimple)
     end
 
+    if try_harder
+        Uπs .= simple_projection.(Uπs, Ref(G), Ref(ehom))
+    end
+
     return WedderburnDecomposition(basis, invariants, Uπs, ehom)
 end