Add character_degrees for a given group or character table, or for given invariants of a finite abelian group and finite field

[ThomasBreuer]

add character_degrees:

• for a given group or character table, delegate to GAP,
• for given invariants of a finite abelian group and finite field, use Julia code.

and improve _irred_abelian:

• add Oscar.DiscLog.element_of_given_order, as suggested by Claus, and use this function in _irred_abelian(G::Oscar.GAPGroup, F::FinField),
• work with a smaller field L (somehow this idea had been lost in the previous version)

[ThomasBreuer]

The failures happen with Julia nightly, they are not related to the proposed changes.

[fingolfin]

Since this adds a new function, I think it should appear in the manual, so I adjusted the labels and title accordingly.

I'll leave it to @fieker to review and approve, as this functionality he requested, I think?

[ThomasBreuer]

One question to @fieker is whether MSet is the appropriate type for the result of character_degrees.
Logically, it fits.
Technically, I find it a bit complicated to check equality of MSets (see the tests), and sometimes it is useful to have a sorted version. (The underlying GAP function returns a sorted list of degree/multiplicity pairs.)

[fieker]

isn't == defined for MSet s? If not that is the problem... This seems to work as expected:

julia>  ==(MSet([1]), MSet([2]))
false

julia>  ==(MSet([1]), MSet([1]))
true

julia>  ==(MSet([1]), MSet([1,1]))
false

julia>  ==(MSet([1]), MSet([1,1]))
false

julia>  ==(MSet([1,1]), MSet([1,1]))
true

The doc tests are tricky...

[ThomasBreuer]

Concerning the comparisons:
What we can do is this.

D = character_degrees([3, 3, 5], 2)
@test D == MSet(ZZRingElem[1, 2, 4], [1, 4, 9])

Concerning the doctests:
If the ordering of the entries in the show result for MSets is not safe then it is better not to show these MSets at all in doctests. (Would it then make sense to introduce a sorted variant for the case that the keys can be sorted?)

[fieker]

On Mon, Oct 20, 2025 at 08:39:00AM -0700, Thomas Breuer wrote: @ThomasBreuer commented on this pull request. > ***@***.*** "character degrees" begin + # for character tables + D = character_degrees(character_table("S5")) + @test sort(collect(D)) == [1, 1, 4, 4, 5, 5, 6] + @test D isa MSet{ZZRingElem} + D2 = character_degrees(Int, character_table("S5")) + @test sort(collect(D)) == sort(collect(D2)) + @test D2 isa MSet{Int} + @test sort(collect(character_degrees(character_table("S5", 2)))) == [1, 4, 4] + + # for groups + @test sort(collect(character_degrees(symmetric_group(5)))) == sort(collect(D)) + + # for invariant lists of abelian groups, and order of a finite field + D = character_degrees([3, 3, 5], 2) + @test sort(collect(D)) == sort(collect(MSet([1, 2, 4], [1, 4, 9]))) Concerning the comparisons: What we can do is this. ``` D = character_degrees([3, 3, 5], 2) @test D == MSet(ZZRingElem[1, 2, 4], [1, 4, 9]) ``` Concerning the doctests: If the ordering of the entries in the `show` result for `MSet`s is not safe then it is better not to `show` these `MSet`s at all in doctests. (Would it then make sense to introduce a sorted variant for the case that the keys can be sorted?)

I'd rather either - not compare the output (can be done) - show, by comparison that the result is correct (not using show) But I don't like adding unnecessary complexity to the code just to please the doc tests. My opinion.

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[fingolfin]

@ThomasBreuer tests still fail