image_basis fails over finite fields

[laurentbartholdi]

I tried the following, based on the example in action_permutation.jl:

julia> G = PermGroup(perm"(1,2,3)", perm"(1,2)")
julia> F = SymbolicWedderburn.Characters.FiniteFields.GF{42013}
julia> struct Word{T}
           alphabet::Vector{T}
           letters::Vector{Int}

           function Word(a::AbstractVector{T}, l::AbstractVector{<:Integer}) where {T}
               all(i -> 1 <= i <= length(a), l) ||
                   throw(ArgumentError("Invalid word over alphabet $a: $w"))
               return new{T}(a, l)
           end
       end

julia> Base.show(io::IO, w::Word) = join(io, w.alphabet[w.letters], "·")

julia> function Base.:(==)(w::Word, v::Word)
           return w.alphabet == v.alphabet && w.letters == v.letters
       end

julia> function Base.hash(w::Word, h::UInt = UInt(0))
           return hash(w.alphabet, hash(w.letters, hash(Word, h)))
       end

julia> struct OnLetters <: SymbolicWedderburn.ByPermutations end

julia> function SymbolicWedderburn.action(
           ::OnLetters,
           p::PermutationGroups.AbstractPermutation,
           w::Word,
       )
           return Word(w.alphabet, [w.letters[i]^p for i in eachindex(w.letters)])
       end

julia> action = OnLetters()

julia> function allwords(A, radius)
           words = [Word(A, [i]) for i in 1:length(A)]
           for r in 2:radius
               append!(
                   words,
                   [Word(A, collect(w)) for w in Iterators.product(fill(1:3, r)...)],
               )
           end
           return words
       end
allwords (generic function with 1 method)

julia> words = let A = [:a, :b, :c], radius = 4
               w = Word(A, [1, 2, 3, 2, 1])

               # (a·b·c·b·a)^(2,3) == a·c·b·c·a
               @test SymbolicWedderburn.action(OnLetters(), perm"(2,3)", w) ==
                     Word(A, [1, 3, 2, 3, 1])

               words = allwords(A, radius)
           end

julia> tbl = SymbolicWedderburn.CharacterTable(F, G)

julia> ehom = SymbolicWedderburn.CachedExtensionHomomorphism(
               G,
               action,
               words;
               precompute = true,
           )

julia> all(g ∈ keys(ehom.cache) for g in G)

julia> ψ = SymbolicWedderburn.action_character(ehom, tbl)

julia> SymbolicWedderburn.image_basis(ehom,SymbolicWedderburn.irreducible_characters(tbl)[1])
ERROR: promotion of types Rational{Int64} and SymbolicWedderburn.Characters.FiniteFields.GF{42013} failed to change any arguments
Stacktrace:
  [1] error(::String, ::String, ::String)
    @ Base ./error.jl:44
  [2] sametype_error(input::Tuple{Rational{Int64}, SymbolicWedderburn.Characters.FiniteFields.GF{42013}})
    @ Base ./promotion.jl:420
  [3] not_sametype(x::Tuple{…}, y::Tuple{…})
    @ Base ./promotion.jl:414
  [4] promote
    @ ./promotion.jl:397 [inlined]
  [5] *(x::Rational{Int64}, y::SymbolicWedderburn.Characters.FiniteFields.GF{42013})
    @ Base ./promotion.jl:426
  [6] matrix_projection_irr_acc!(result::SparseArrays.SparseMatrixCSC{…}, hom::SymbolicWedderburn.CachedExtensionHomomorphism{…}, class_values::Vector{…}, conjugacy_cls::Vector{…}, weight::Rational{…})
    @ SymbolicWedderburn ~/.julia/packages/SymbolicWedderburn/hBhTJ/src/matrix_projections.jl:234
  [7] matrix_projection_irr_acc!(result::SparseArrays.SparseMatrixCSC{…}, hom::SymbolicWedderburn.CachedExtensionHomomorphism{…}, χ::SymbolicWedderburn.Characters.Character{…}, weight::Int64)
    @ SymbolicWedderburn ~/.julia/packages/SymbolicWedderburn/hBhTJ/src/matrix_projections.jl:217
  [8] matrix_projection_irr
    @ ~/.julia/packages/SymbolicWedderburn/hBhTJ/src/matrix_projections.jl:145 [inlined]
  [9] matrix_representation(hom::SymbolicWedderburn.CachedExtensionHomomorphism{…}, χ::SymbolicWedderburn.Characters.Character{…})
    @ SymbolicWedderburn ~/.julia/packages/SymbolicWedderburn/hBhTJ/src/matrix_projections.jl:360
 [10] image_basis
    @ SymbolicWedderburn ~/.julia/packages/SymbolicWedderburn/hBhTJ/src/matrix_projections.jl:419 [inlined]
 [11] image_basis(hom::SymbolicWedderburn.CachedExtensionHomomorphism{…}, α::SymbolicWedderburn.Characters.Character{…})
    @ SymbolicWedderburn ~/.julia/packages/SymbolicWedderburn/hBhTJ/src/matrix_projections.jl:419
 [12] top-level scope
    @ REPL[41]:1
Some type information was truncated. Use `show(err)` to see complete types.

[kalmarek]

as already observed the immediate error could be fixed by defining

Base.promote_rule(::Type{<:GF{p}}, ::Type{<:Rational}) where p = GF{p}
GF{p}(q::Rational, check=false) where p = GF{p}(numerator(q), check)/
numerator(q)

but runs into problems deeper down the chain

[kalmarek]

as of today master defining the following two methods:

import SymbolicWedderburn.Characters.FiniteFields as FF
Base.promote_rule(::Type{<:FF.GF{p}}, ::Type{<:Rational}) where p = FF.GF{p}
FF.GF{p}(q::Rational, check=false) where p = FF.GF{p}(numerator(q), check)/numerator(q)

makes the call to image_basis return what it should (well, hopefully, I didn't check the values):

julia> SymbolicWedderburn.image_basis(ehom,SymbolicWedderburn.irreducible_characters(tbl)[1])

22×120 SparseArrays.SparseMatrixCSC{SymbolicWedderburn.Characters.FiniteFields.GF{42013}, Int64} with 42 stored entries:
⎡⠉⠁⠀⠀⠐⠒⠀⠀⠀⠒⠂⠀⠀⠀⠒⠀⠀⠤⢀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠠⠀⠄⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠤⠀⠀⎤
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠁⠀⠁⠊⠀⠁⠀⠀⠀⠀⠀⠈⠀⠓⠈⠀⠁⠀⠐⠉⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎦

I'm not convinced that I want to support here the GF{p} & Rational promotion, but if there are more requests for the feature the path is open ;)