[GAP Forum] Difficulty Understanding how to Generate Group and Compute Molien Series

[saad khalid]

Hello everyone!

I'm coming from using Macaulay2 (and not knowing much of anything about
representation theory) to generate groups and their corresponding Molien
series. In Macaulay2, I would create a field with some qth root of unity.
For example, if I wanted q = 8, I would use:
K = toField(QQ[zet]/(zet^4 + 1))

Then, I would pick some matrix I wanted to use to generate the group:

A = matrix{{zet^(a1),0,0},{0,zet^(a2),0},{0,0,zet^(a3)}} #a1, a2, a3 are
what I would pick.

Notice that this is diagonal(all the matrices I'm dealing with are
diagonal).

Then, I would generate the group for A in the field K, and compute its
molienseries:
B = generateGroup({A},K)
molienSeries B

I would like to do this process in GAP, but I'm having trouble figuring out
how to generate the group. In actuality, I don't need the group at all, I'm
only interested in the final molienseries representation. However, to
generate the molien series using GAP, I think I need to know the character
of the group. I believe I have sort of an understanding of what that means.
At least, in my case since they're just diagonal matrices, the character of
each matrix in the group is just the trace of that matrix. Even if that is
the case, I still don't know how to put that in GAP. Say I had q = 4 and
picked a1 = 1, a2 = 2, a3 = 3; what should I input into GAP as the
character of this group for it to generate the Molien series?

If there isn't some easy way to just figure out and  type in the character
of the group, I assume that I'll have to create the group in GAP and then
use GAP to get its character. I followed the examples given in the GAP
documentation, but I couldn't figure out how to apply that to my situation.
I just don't know how to translate what I did in Macaulay2 to GAP. I would
like to learn representation theory in the future, and I hope that I will
learn some by working with GAP. Until that happens though, I would like to
apologize for how novice my questions here are. Thank you for taking the
time to read my post! I would really appreciate any help you can give.

-Saad