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I came across an extremely useful Python software library for the Monster group: https://github.com/Martin-Seysen/mmgroup which allows for all sorts of manipulations involving the sporadic finite groups.

Is there a similar fast and straightforward software library for $\operatorname{PSL}_n(q)$ where $q$ is a prime power? In particular I'd like to construct the smallest dimensional ($N$) complex faithful representations of this group for various choices of $n$ and $q$, meaning that the generators should be given explicitly in terms of $N \times N$ matrices. I know generally it's a difficult problem but for given $n$ and $q$ are there algorithmic constructions which are already in a software library?

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    $\begingroup$ Someone more familiar with GAP or Magma could speak to possible capabilities of them, but it's probably worth mentioning the galois package here: github.com/mhostetter/galois, which is an extension of numpy to finite fields. If GAP or Magma are lacking the functionality you need this would give you a decent starting point for implementing things yourself. $\endgroup$
    – multi_porpoise
    Commented Nov 28 at 23:44

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All representations of quasi-simple groups up to degree 250 were constructed by Allan Steel, as described here. Unfortunately this webpage contains links to only a small selection of them.

The files defining generating matrices for the database up to degree about $60$ are included in the distributed version of Magma in the Magma library directory $\mathtt{c9lattices}$, with versions over a minimal degree number field over which the representation can be defined, and over the integers. Unfortunately there does not appear to be a convenient user interface. If there is some particular representation you are looking for, then I might be able to help you define it, or you could write to Allan Steel and ask whether the complete database is available somehow.

You can easily compute the degrees of the representations of a specific group using $\mathtt{CharacterDegrees}$.

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  • $\begingroup$ Allan Steel's PhD thesis is very useful, exactly the kind of thing I was looking for! magma.maths.usyd.edu.au/users/allan/reps/AllanSteelPhD.pdf Now what I'm still a bit uncertain is how do I deduce which representations are complex and which of them are real, starting on page 159, which gives a very neat list. I'm guessing it has to do with the column C, but I'm not sure. $\endgroup$
    – Fetchinson0234
    Commented Nov 30 at 18:20
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    $\begingroup$ The column $C$ is just the minimal degree over the rationals of a field over which the representation can be written, so that does not tell you whether it is real (unless it is $1$). For that you need to know the Frobenius-Schur indicator, which can be $-1$, $0$, or $1$, and is $1$ for real representations. That is is not given in the tables in Steel's thesis, but it is given, for example, in the tables in the Hiss-Malle table which is being referenced. $\endgroup$
    – Derek Holt
    Commented Dec 1 at 9:11
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While it is not a software library, the site ATLAS of Finite Group Representations might be what you are looking for. It lists several smallest degree representations for groups $\text{PSL}_n(q)$ (called $\text{L}_n(q)$ there) for many pairs $n,q$.

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    $\begingroup$ Going through the ATLAS tables it seems there are only a couple of cases where the representation is complex (and not over Z or GF(integer)). Is there an intuitive reason why the vast majority of the representations of the vast majority of the simple finite groups are not complex? $\endgroup$
    – Fetchinson0234
    Commented Nov 29 at 14:01
  • $\begingroup$ Oh, maybe you need to "unaccept" my answer. It seems that this site omits some low degree representations if they are not defined over the rationals. $\endgroup$
    – Peter Mueller
    Commented Nov 29 at 15:08
  • $\begingroup$ But in some cases (generally, not only $PSL_n(q)$ but general simple finite groups) they do include complex representations. Or you say it's not systematic, sometimes they include them, sometimes not? $\endgroup$
    – Fetchinson0234
    Commented Nov 29 at 15:31
  • $\begingroup$ I do not know by which criteria the reps were chosen. It does not seem to be documented. $\endgroup$
    – Peter Mueller
    Commented Nov 29 at 15:38
  • $\begingroup$ And you'd say there isn't a similar other collection of reps for simple finite groups? This seemed really neat I just wish it were more comprehensive for the particular case of complex reps. $\endgroup$
    – Fetchinson0234
    Commented Nov 29 at 19:33

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