Does anyone know please of some/few references, where people explain and construct explicitly some (finite-dimensional irreducible) complex representations of finite groups of Lie type? I am particularly interested in groups consisting of two by two matrices with entries in some finite field (and similar related finite groups), as well as their complex representations.

I know that there is a whole theory of such representations, but I am not familiar with algebraic groups, only with Lie groups (over $\mathbb{R}$ and over $\mathbb{C}$), and would like for the time being to study some "baby" examples.

Edit 1: as an update, I found online the following notes by Gerhard Hiss http://www.math.rwth-aachen.de/~Gerhard.Hiss/Preprints/StAndrewsBath09.pdf. I think this may be a good start for me.

  • $\begingroup$ Actually, the algebraic group theory is not quite that relevant to this, since that mainly helps with reps in the defining characteristic (since those are the reps that make sense for the algebraic group). That said, how much theory are you comfortable with? For example, does it look like arxiv.org/abs/1705.05083 uses too much about algebraic groups for your taste? $\endgroup$ – Tobias Kildetoft May 23 '18 at 19:30
  • $\begingroup$ @TobiasKildetoft, thank you for that reference. I purchased a book by M. Geck, but had no time to read it yet. I mean, I can understand the article probably, if I sat down and focused, but do you know of some more concrete constructions, at least for groups of two by two matrices? I do realize things could be tricky, because the group itself could have characteristic $p$, while the representation I would like to have is over $\mathbb{C}$, which has characteristic $0$. I kind of have no idea what to expect. Do you happen to know of maybe more explicit examples, say at the expense of generality? $\endgroup$ – Malkoun May 23 '18 at 19:40

There is a lot of sophisticated theory on the classification of these representations and their characters (Deligne-Lusztig and many others, the notes mentioned above give a short introduction).

But there seems to be not much published about concrete representations. In "practice" only baby examples occur because the smallest dimensions of non-trivial representations (complex or other non-defining characteristic) grow quickly with the group (e.g., over the complex numbers for $SL_2(q)$ with odd $q$ this is $(q-1)/2$ or for $E_8(q)$ it is $q(q^4-q^2+1)(q^8-q^6+q^4-q^2+1)(q^4+1)(q^8-q^4+1)(q^2+1)^2$).

One reference with explicit representations over the smallest possible field is Böge, S., Realisierung (p−1)-dimensionaler Darstellungen von PSL(2,p), Arch. Math. (Basel) 60 (1993), no. 2, 121–127.

  • $\begingroup$ Thank you. My German is not so good though unfortunately. By the way what is the dimension of the smallest representation of $SU_2(q^2)$ please? $\endgroup$ – Malkoun May 24 '18 at 15:03
  • 1
    $\begingroup$ Same as $SL_2(q)$, that is $(q-1)/2$ for odd $q$ and $q-1$ for even $q$. $\endgroup$ – Frank Lübeck May 25 '18 at 9:09

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