# Finding the minimum dimension of $\operatorname{SL}_n(\mathbb{F}_q)$'s nontrivial real representations

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 $$\operatorname{GL}_n(\mathbb{F}_q)$$?

• 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 $m>1$. – user43326 Nov 6 '19 at 20:43
• 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 $(n,q)$. – Nate Nov 6 '19 at 23:19
• @Nate Can you give me the link that gives the proof of your statement? – Nguyễn Văn Thế Nov 7 '19 at 15:12
• Take a look at "Minimal characters of the finite classical groups" by Tiep and Zalesskii. – Nate Nov 7 '19 at 20:07