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
[EMAIL PROTECTED]
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