I'm writing some gap code to work with representations of simple Lie algebras.
Although I'm already very pleased with the performance, (the gap version is
more than 10 times faster than a Mathematica version!); I am finding that no
less than half of the time (according to "Profile") is spent looking up the
position of elements in lists -- using "Position" or "PositionProperty",... .
Now the next set of calculations involves even more of this type of operation.
So I would like any pointers on how to improve the efficiency of locating
elemets in list.

The key general idea is to use Sets rather than lists.

Suppose you have a = List[ <stuff> ] and you will want to do
Position(a, <thing>) lots of times, it is better to do:

b := ShallowCopy(a);
c := [1..Length(a)];
SortParallel(b,c);

then Position(a,<thing>) = c[Position(b,<thing>)]

and the Position calculation in b can be done by binary chop rather
than linear search.

Steve