double the quaternionic characters in affordable_real

[kneshat]

Adding example that contains tall three possibilities(real, complex, and quaternion-type)
Distinguishing the constructions of characters for real-type and quaternion-type irreps

[kalmarek]

Thanks!

It'd be better indeed to have this example as a test case (so in ./test directory), where instead of printing and eye-balling the result we write @tests. In this case one could

1. Check that the quaternionic character is really doubled by affordable real
2. compute the dimension of the image of the corresponding projection and check it is as expected.

you could end up as a separate @testset in e.g. ./test/projections.jl

[kneshat]

Don't you want that test to be in ./test/sa_basis.jl? I can use SL(2,3) instead of $C_2 \oplus C_5$ since SL(2,3) has all three possibilities. What do you think?
How can I use SmallPermGroups for SL(2,3)? What number should I put instead of the question mark in SmallPermGroups[24][?]

[kneshat]

There is an issue I want to address. I cannot use charsR, _ = SymbolicWedderburn.affordable_real(chars) and I get the error: InexactError: Int64(0.5). This is because I divide the multiplicity by 2 for the quaternionic irreps here: push!(mls_real, multiplicities[i]/2). If we have a real representation, the number of times that a complex irrep with FS indicator -1 occurs is an even number. It cannot be an odd number. This concept is very similar to say when we have a real representation, if an complex irrep with FS indicator 0 occurs, the conjugate of this irrep must occur.

Let $V$ be a REAL representation. If $W$ is a complex irrep with FS indicator -1 that occurs $2k$ times in $V$, then $W^{\mathbb{R}}$, where $W^{\mathbb{R}} \otimes_{\mathbb{R}} \mathbb{C} = W \oplus W$, is a quaternion-type real representation that occurs $k$ times in $V$. Now, would you tell me how we need to proceed?

[kalmarek]

maybe the solution is to remove the default value of multiplicities from affordable_real and assert that the the entry corresponding to a irrep of quaternionic type iseven?
After the assertion we can safely use div (the integer division) to compute the resulting multiplicity.

[kneshat]

Do you mean

function affordable_real(
    irreducible_characters,
    multiplicities,
)
    irr_real = similar(irreducible_characters, 0)
    mls_real = similar(multiplicities, 0)
    for (i, χ) in pairs(irreducible_characters)
        ι = Characters.frobenius_schur(χ)
        if ι == 1 # real
            @debug "real" χ
            push!(irr_real, χ)
            push!(mls_real, multiplicities[i])
        elseif ι == -1 # quaternionic
            @debug "quaternionic" χ
            @assert iseven(multiplicities[i]) "Multiplicities of quaternionic character must be even."
            push!(irr_real, 2 * χ)
            push!(mls_real, div(multiplicities[i], 2))
        else # complex
            cχ = conj(χ)
            k = findfirst(==(cχ), irreducible_characters)
            @assert k !== nothing "Conjugate character not found."
            @debug "complex" χ "conj(χ) =" irreducible_characters[k]
            if k > i # we haven't already observed a conjugate
                @assert multiplicities[i] == multiplicities[k] "Multiplicities of complex conjugates must be equal."
                push!(irr_real, χ + cχ)
                push!(mls_real, multiplicities[i])
            end
        end
    end

    return irr_real, mls_real
end

??
We access characters as follows

G = SmallPermGroups[10][2] # C₂⊕C₅
tbl = SymbolicWedderburn.CharacterTable(Rational{Int}, G)
chars = SymbolicWedderburn.irreducible_characters(tbl)

Now, how can I access multiplicities of irreps so that I can run SymbolicWedderburn.affordable_real(chars,multiplicities) ?

[kalmarek]

Yes, something like this. It would be easier to discuss, if you directly commit here, as then the review comments could be attached directly to code lines.

as for the use of affordable_real, search this repository or grep the test folder ;)

For the multiplicities basically construct your own representation that you want to realify by passing to affordable_real. Then its character decomposes into irreducibles with some multiplicities which you need to pass here.
E.g. if the character of the representation is 2χ₁ + 3χ₂ + 1χ₄, then the args will be [χ₁, χ₂, χ₄] and [2, 3, 1].

[kneshat]

I see. In case you were wondering why I asked this question, I had a stupid misunderstanding. I thought the multiplicity of each character was stored in the character table in addition to its value, and if there is no representation, then the predefined multiplicity is 1. That is why I thought I could access the multiplicities using the character table. The source of this wrong preassumption is that I thought there were two numbers in each entry of the character table:a\ \ b. But now, I know a\ \ b is just the representation of 1 rational number but not two different numbers.

Anyhow, I am thinking in the cases that our representation is not realizable over real numbers, Julia returns something like "This representation is not realizable over real numbers either because there is an irrep with Schur index 0 without its conjugate or because there is an odd number of irrep with Schur index -1" instead of arbitrary error. Would you suggest how can I do that?

[kalmarek]

yes, sounds good to me, just modify the corresponding lines with @asserts, or replace them with throwing those errors

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 87.94%. Comparing base (8f64019) to head (5b25731).
Report is 35 commits behind head on master.

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==========================================
- Coverage   90.74%   87.94%   -2.80%     
==========================================
  Files          23       22       -1     
  Lines        1329     1228     -101     
==========================================
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- Misses        123      148      +25     

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