Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would appreciate advice into what specific areas/theorems I should learn that apply to this particular question.
Here is the problem: I have a group $G$ acting on a very big vector space $V$, which is over $\mathbb{F}_2$. I want to compute the number of orbits of the action of $G$ on $V$. I could use Cauchy-Frobenius and sum the fixed points, but $V$ is too big for that to be feasible. My hope is to be able to use the basis of $V$, which is small, instead. If I could factor the image of each basis vector $e_i$ under the action by $g \in G$ into a basis representation $$e_i^g = f_1e_1 + f_2e_2 + \cdots + f_ne_n$$ then I could calculate the matrix form for each $g$ and use that to find the fixed points. But this task is not easy—I can compute the image of a vector under a transformation by $g$, but I do not know of any way to write this new vector as a linear combination of the basis vectors.
I figure there might be a very general way of understanding the orbits of group representations—that is, of elements of the vector space that are "distinct $\mod G$"—but I do not know enough about character theory, homology, etc. to see the direction to go. If anyone could recommend to me some specific readings that are relevant to problems like this, I would very much appreciate it. Thanks.
Edit: I will also note that I am able to determine the orbits of the action of $G$ on each basis vector (just not in terms of the other basis vectors), so much hope is to exploit this.
According to your request, here are some wonderful sources on representation theory; Standford group actions, Bowdoin powerpoint on representation theory and orbital varieties, hope this helps!