[GAP Forum] Questions on defining an action from GL(n,p)

[Arturo Magidin]

Dear Forum,

I am pretty new to GAP. I have a problem I want to take a look at, and 
wanted to ask whether there is already a way to do it with pre-defined 
utilities, before trying to go ahead and program them myself.

Let p be a prime, n a positive integer. Let U be the vector space of 
dimension n over GF(p); let V be isomorphic to U/\U (the alternating 
product, of dimension Binomial(n,2)).

1) Is there a way to handle U/\U directly? Right now, I have a vector
    space of dimension Binomial(n,2), and then I constructed an n x n
    table, with entries satisfy v[i][j]=-v[j][i]. I then place elements
    of the canonical basis for GF(p)^{Binomial(n,2)} in the table and
    refer to them that way.

2) Given an element of GL(n,p), this gives an automorphism (call it f)
    of U, and this in turn induces an automorphism on U/\U by mapping
    v/\w to f(v)/\f(w). We can then use this automorphism to define an
    action on Subspaces(U/\U).

    The condition I want to check is invariant under that action. Thus,
    I would like to define the corresponding action, pick an element
    from each orbit in Subspaces(U/\U), and check the property for that
    element (I already have a short function to do the checking).


What would be the simplest way of doing this? If there is no way of 
doing this directly, should I take some generating set for GL(n,p), 
and describe the action explicitly on V in some way? Or can one use 
the natural action of GL(n,p) to make this faster/simpler?

Thanks for any advice you might have.

Arturo Magidin
magidin at member.ams.org