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I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated.

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My original question was ambiguous. Let me explain and give a few further details.

I want to understand some combinatorial properties (expansion of some type) of the group $GL_{\mathbb{F}}(n)$, where the $\mathbb{F}$ is a finite field. One possible approach for doing that (that was successful in, e.g. understanding similar aspects of the permutation group) is through unitary representations of that group. As far as I can tell, most of the texts cover $GL(n)$ over fields of characteristic zero, which are not what I'm interested in. So I'm asking for sources for unitary representations of the linear group over finite fields..

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To clarify further, by unitary representations I mean homomorphisms of GL(n) of a finite fields, into the group of finite-dimensional unitary matrices over $\mathbb C$. As you might guess, I'm a cs/combinatorics person, and far from expert on representation theory -- please excuse my lack of verbal skills in this area and otherwise..

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  • $\begingroup$ Try "Representations of Finite Groups of Lie Type" by Digne and Michel: books.google.co.uk/books?id=x6CtmIlf6TYC $\endgroup$
    – Neil Strickland
    Commented May 28, 2017 at 11:17
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    $\begingroup$ What do you intend to mean by unitary representation over a finite field? $\endgroup$
    – Geoff Robinson
    Commented May 28, 2017 at 11:28
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    $\begingroup$ @GeoffRobinson I had assumed OP meant homomorphisms of the form $GL(n, \mathbb{F}_q) \to U(k)$ where the codomain consists of unitary transformations on $\mathbb{C}^k$. $\endgroup$
    – Todd Trimble
    Commented May 28, 2017 at 13:16
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    $\begingroup$ @Todd Trimble : If that is indeed what the OP intended, there is no real need to specify unitary representations since all finite dimensional complex representations of finite groups are equivalent to unitary ones. I now see there is an ambiguity in the question which I read differently to you $\endgroup$
    – Geoff Robinson
    Commented May 28, 2017 at 13:33
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    $\begingroup$ @user1258240, you still haven't specified representations on what (not of what). As @ GeoffRobinson points out, if you mean representations on a complex vector space, then there is no need to say 'unitary'; every representation of a finite group is unitarisable. If you mean representations on a finite vector space (not just of GL of a finite vector space), then it is important clearly to say so (and probably also to say what you mean by 'unitary'). $\endgroup$
    – LSpice
    Commented May 28, 2017 at 14:05

5 Answers 5

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All finite dimensional complex representations of finite groups are equivalent to unitary representations, so the requirement that the representations be unitary is not really a restriction.

The 1955 work of J.A. Green gives the definitive description of the complex characters of the groups ${\rm GL}(n,q)$ for q a prime power: see

http://www.ams.org/journals/tran/1955-080-02/S0002-9947-1955-0072878-2/S0002-9947-1955-0072878-2.pdf

Later work of Deligne-Lusztig studied the complex characters of other finite classical groups (and finite groups of Lie type), and the book of Digne and Michel mentioned by Neil Strickland is a good source of information.

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  • $\begingroup$ The work of Green which mentioned in Geoff Robinson's answer has also been discussed in more detail in I. G. Macdonald's book "Symmetric Functions and Hall Polynomials", see specially Chapter IV: The Characters of $GL_n(q)$ Over A Finite Field, which discuss the Green's paper. $\endgroup$
    – Farrokh Shirjian
    Commented Jun 1, 2017 at 22:49
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Especially for combinatorialists, I found the book "Representations of finite classical groups. A Hopf algebra approach. Lecture Notes in Mathematics, 869" by Andrei Zelevinsky useful. It devolops a theory which covers both $S_n$ and $GL(n,\mathbb F_q)$. The point of view is that representations of these groups should be studied simultaneously for all $n$.

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I think the standard reference for representations over finite fields still is

I you want a much briefer introduction, the last chapters of Serre's book might be enough.

In fact, even if you want to study Alperin, Serre might be a good place to start.

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I would highly recommend "Complex Representations of GL(2,K) for Finite Fields K" by Piatetski-Shapiro. (I know you're interested in GL(n), but this book is a great place to start.)

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On-line and a good starting point with GL(2, F_q):

Paul Garret's notes "Representations of GL2 and SL2 over finite fields" http://www-users.math.umn.edu/~garrett/m/repns/notes_2014-15/04_finite_GL2.pdf

And also:

Amritanshu Prasad "Representations of GL2(Fq) and SL2(Fq), and some remarks about GLn(Fq)" http://www.imsc.res.in/~amri/html_notes/notes.html

or similar in arxiv:

https://arxiv.org/abs/0712.4051

PS

The bonus is that both authors are at MO so you might get answer if something is unclear.

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