Changing affordable_real and related tests

[kneshat]

Now, runtest.jl works without any error.
I made some changes which are as follows:
1- Modifying affordable_real function to cover quaternionic characters.
2- Modifying @testset "affordable real degrees/dot" to accommodate affordable_real.
3- Modifying @testset "Orthogonality of projections" to accommodate affordable_real.
4- Removing ex_SL(2,3).jl

[kneshat]

I also added a test in projections.jl based on the group SL(2,3) for doubling the quaternionic characters.

[kalmarek]

@kneshat thanks, tests look good!

I'll squash and merge once the tests finish.

[kneshat]

I am wondering why tests do not work here, while "runtest.jl" works fine.
Can it relate to the warning Warning: Possibly wrong rank estimate? (numerical) 4 ≠ 2 (expected) @ SymbolicWedderburn C:\Users\raven\.julia\packages\SymbolicWedderburn\lbiby\src\image_basis.jl:124 that I get?

Would you also run the branch kneshat/correcting-issimple? How can I consider merging this new branch in this pull request?

[kalmarek]

here I need to approve tests to run, since CI services have been abused to perform computations unrelated to the package.

[kalmarek]

we should check the Frobenius Schur indicator here and test exactly that issimple really does what we want

[kalmarek]

replying here to your comment (#82 (comment)) to keep the discussion related to PR in one place:

using frobenius_schur(b.character) for b in sa_basisR does not work since not all b.character are irreducible in the sense that you defined with the function isirreducible. My two first questions are: did you define isirreducible only for complex characters? What is the difference between issimple and isirreducible?

Now, I am wondering if I need to modify this function so that we can define frobenius_schur for irreducible real characters or not. In theory, frobenius_schur has a meaning for any ground field K that makes K G semisimple. This indicator relates to the possibility that the underlying irrep (not necessarily complex) accepts invariant symmetric bilinear forms and invariant skew-symmetric bilinear forms.

Do you have any other suggestion instead of checking frobenius_schur(b.character)? So that we can proceed without modifying isirreducible and frobenius_schur

issimple refers to DirectSummands (I'm no longer sure this is a good name, but this renaming is a separate issue #77), whereas isirreducible is defined for Characters only.

---

fair point with the irreducibility of characters! In the suggestion above I forgot that we already have realified the character. Taking a step back: what I want is a set of tests that will test precisely what we mean by issimple, now that we're changing its definition.

---

we have essentially three cases with three different meanings of issimple that need to be checked.

• the standard case -- isirreducible(b.character)
• sum of two different conjugates --

   mult = Characters.multiplicities(b.character)
   !isirreducible(b.character) && count(!iszero, mult) == 2 && all(m->m in (0,1), mult)

  Alternatively

   k = findfirst(!iszero, mult)
   χ = Character(table(b.character), k)
   χc = conj(χ) 
   χ ≠ χc && b.character == χ + χc

• a double of self-conjugate character

   mult = Characters.multiplicities(b.character)
   k = findfirst!(!iszero, mult)
   b.character == 2Character(table(b.character), k)

We could wrap those checks in small functions within the testset and use them there e.g.

if isirreducible(b.character)
    @test multiplicity(b) == size(b, 1)
elseif _is_irr_plus_conj(b.character)
    @test 2*multiplicity(b) == size(b, 1)
elseif _is_quaternionic_double(b.character)
    @test 4*multiplicity(b) == size(b, 1)
else
    throw("Unexpected character associated to a direct summand")
end

or something along those lines

[kalmarek]

@kneshat do you intend to work on this further, or should I carry those changes over the finish line?

[kneshat]

@kneshat do you intend to work on this further, or should I carry those changes over the finish line?

Sorry, I was busy with another project. Now, I come back and intend to work. Is there any major change since then that I should be aware of?

[kalmarek]

@kneshat sure, me too ;) no problem. I think there were hardly any significant changes in the package. I'd need to get back up to speed with the proposed changes though ;)