[GAP Forum] Action of general linear group on polynomials

[Steven R. Costenoble]

Alexander,

Thanks, that works. I don’t need anything more efficient than that at the moment – I was able to use it to verify something I thought was happening based on some other computations. It beats trying to compute by hand an orbit under the action of a group with 20,160 elements!

--Steve


From: Hulpke,Alexander <Alexander.Hulpke at colostate.edu>
Date: Monday, December 28, 2020 at 6:04 PM
To: Steven R. Costenoble <Steven.R.Costenoble at hofstra.edu>, forum at gap-system.org <forum at gap-system.org>
Subject: Re: [GAP Forum] Action of general linear group on polynomials
EXTERNAL MESSAGE

Dear Steven Costenoble, Dear Forum,

> Apologies if this is elementary: Related to a research project, I’d like to do some computations involving the action of GL(n,2) on the ring of polynomials over GF(2) in n variables. (Specifically n=4 at the moment.) I know how to compute with the action of SymmetricGroup(n), using OnIndeterminates, but is there an easy/standard way in Gap to let all of GL(n,2) act on polynomials?

At this point, this action is not implemented -- the issue being that it *should* be done better than just using `Value', but doing so requires a bit more thought. (And so far nobody had asked for it.)

I append a basic kludgy implementation of such an action that *does* use `Value' after all. It is not as good/fast as I would like, and does no test of parameters, but might be better than nothing.

All the best,

Alexander Hulpke

# basic action on polynomials. No check of characteristic etc.
OnPolynomials:=function(pol,mat)
local n,v,r;
  r:=DefaultRing(OneImmutable(CoefficientsFamily(FamilyObj(pol))));
  n:=Length(mat);
  v:=List([1..n],x->X(r,x));
  return Value(pol,v,List([1..n],i->Sum([1..n],j->v[j]*mat[i][j])));
end;


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