Extend meataxe to be able to detect invariant forms of reducible modules (just one possible form is returned)

[fingolfin]

No description provided.

[hulpke]

A main change seems a change from IsomorphismIrred to IsomorphismModules. What I want to make sure is that there will not be a chance of running in infinite recursion.

I have not looked again into the code of these two functions, but the former is a straightforward spinning algorithm, while the second is a rather elaborate calculation that uses a lot of other MeatAxe routines.
This means that with this change, InvariantXXXForm cannot be used within routines that might get called from IsomorphismModules. This means that the tests need to cover a variety of module structures (direct sums of multiple isomorphic and non-isomorphic indecomposable components).

[fingolfin]

Calls to MTX.Invariant*Form in GAP:

$ git grep -P -l '\.Invariant.*Form' grp lib
lib/meataxe.gi

So nothing outside of lib/meataxe.gi calls this.
Looking closer at those calls in lib/meataxe.gi:

• SMTX.InvariantQuadraticForm calls InvariantBilinearForm as a first step
• SMTX.OrthogonalSign calls MTX.InvariantBilinearForm and MTX.InvariantQuadraticForm
• however nothing in the library calls OrthogonalSign (only tests)

So all in all I don't see how this could cause a problem?

[fingolfin]

@ThomasBreuer @hulpke @jandebeule any ideas what the commented code above (inside SMTX.InvariantSesquilinearForm) is meant to be good for?

It looks to me like an attempt to "normalize" the isomorphism (and hence invariant form) we compute here). But the form can in fact be scaled by an arbitrary scalar (not just by a norm), so I don't understand why it we choose this specific one.

The reason I stumbled over this, and subsequently commented it out (to be deleted) is that it utterly fails if the given module is e.g. the sum of two isomorphic modules - then isot above in general won't be diagonal at all (test cases for that added below). On the other hand I can't really think of any reason why it would be useful. But I am sure whoever added this had a good reason, so I wonder what I am missing.

Any suggestions would be highly welcome!