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Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and $\operatorname{SL}_n(\mathbb{F}_q)$ the special linear group in $n$ variables.

What is the minimum dimension of nontrivial real representations of $\operatorname{SL}_n(\mathbb{F}_q)$? What about for the general liner group $\operatorname{GL}_n(\mathbb{F}_q)$?

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    $\begingroup$ The general linear group admits the determinant homomorphism to ${\mathbb F} _q^{*}\cong {\mathbb Z}/(q-1)$. The latter admits in turn a non-trivial real representation of dimension $1$ if $q$ is odd, dimension $2$ if $q=2^m$ with $m>1$. $\endgroup$
    – user43326
    Nov 6, 2019 at 20:43
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    $\begingroup$ There is a $\frac{q^n-q}{q-1}$ dimensional representation of $SL_n(\mathbb{F})$ over any field $k$ given by the action of $SL_n(\mathbb{F}_q)$ on the space of $k$-value functions on $\mathbb{P}(\mathbb{F}_q^n)$ of total sum zero. If $k$ is of characteristic zero this is known to be the smallest dimensional non-trivial representation for all but finitely many (explicitly known) pairs $(n,q)$. $\endgroup$
    – Nate
    Nov 6, 2019 at 23:19
  • $\begingroup$ @Nate Can you give me the link that gives the proof of your statement? $\endgroup$
    – user148117
    Nov 7, 2019 at 15:12
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    $\begingroup$ Take a look at "Minimal characters of the finite classical groups" by Tiep and Zalesskii. $\endgroup$
    – Nate
    Nov 7, 2019 at 20:07

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