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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 find the orbits (specifically, representatives) of the action of $G$ on $V$. I could just go through each element and then sort for redundancies, 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. I can compute the image of each basis vector under each linear transformation induced by $G$. More specifically, I can determine the coefficients of each image too. Is there a way to understand and calculate specific orbits of the entire vector space using information about the basis specifically?

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.

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  • $\begingroup$ Usually you know that a subset is a basis because you can prove that any vector can be written as a linear combination of your subset in a unique way. It seems that your situation is different, isn’t it? $\endgroup$ Dec 29, 2021 at 6:26
  • $\begingroup$ @FernandoMuro, I think that, when the question says "If I could factor [sic]", it means not "if it were possible"—for, as you say, it certainly is—but "if I could compute the coefficients efficiently", or perhaps just "if I could compute the coefficients at all (as opposed to just proving their existence)". $\endgroup$
    – LSpice
    Dec 29, 2021 at 6:37
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    $\begingroup$ This is a difficult question, even for specialists. For example, while there is usually an orbit of length $|G|$ on vectors when the action of $G$ on $V$ is faithful, but there are exceptions, and they are difficult to classify. $\endgroup$ Dec 29, 2021 at 12:22
  • $\begingroup$ @LSpice New development: I actually do now know how to compute the coefficients of the image of each basis vector under each linear transformation. The question now consists of understanding the orbits specifically based on the images of the basis vectors. $\endgroup$
    – healynr
    Dec 30, 2021 at 0:58
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    $\begingroup$ Since invariant polynomials can separate orbits, you could try to find all the invariant polynomials of degree 0, 1, 2, etc and maybe this will be a graded algebra that you can understand. (The low-degree stuff is likely to be interesting to you even if the approach doesn't pan out!) If the algebra is understandable, then probably any question about the orbits can be answered. $\endgroup$ Dec 30, 2021 at 1:26

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