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oscar-system / Oscar.jl

A comprehensive open source computer algebra system for computations in algebra, geometry, and number theory.
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Request: all absolutely irreducible representations/G-modules over minimal number fields #995

Open thofma opened 2 years ago

thofma commented 2 years ago

So that I don't forget about it, here is a small feature request relating to some topic in the book: Given a finite group G, I would like to have for every irreducible complex character (or a set of non-Galois conjugate characters) a G-module realizing this character with the field having minimal degree. We may assume that G is solvable and that all rational Schur indices are trivial. The one thing I am not sure about is how the function should be called. Maybe something like

(absolutely)_irreducible_gmodules(QQ, G, minimal = true)

(although I am not sure what the QQ is doing, maybe extensions of QQ). Or maybe

absolutely_irreducible_gmodule(NumField, G, minimal = true)

to say that I want number fields?!

In Magma I would iterate over the characters and given such an irreducible character chi I would call GModule(chi : Minimal := true) (or something like that). But as far as I understand, we already have code to determine G-modules realizing all irreducible complex characters (thanks @fieker!), so it makes more sense to ask for them all at once.

CC: @fieker

ThomasBreuer commented 2 years ago

If the representations are independent in the sense that the entries of the matrices of two different representations lie in different number fields then it is not possible to form direct sums or tensor products of such representations. Don't we need the possibility to prescribe a common number field over which all matrices are expected to live?

thofma commented 2 years ago

If the representations are independent in the sense that the entries of the matrices of two different representations lie in different number fields then it is not possible to form direct sums or tensor products of such representations.

Sure, they will be G-modules over different number fields. I do not intend to combine them in any way.

Don't we need the possibility to prescribe a common number field over which all matrices are expected to live?

This functionality should probably exist but I think it is not relevant when one is asking for minimal fields of realization (common field and minimality of the individual fields is incompatible).