$\DeclareMathOperator\SL{SL}$ Let $V$ be the adjoint representation of $\SL_n(\mathbb{F}_2)$, so $V$ is the space of trace zero $n \times n$ matrices over $\mathbb{F}_2$. For a paper she is writing, one$\DeclareMathOperator\SL{SL} \newcommand{\gl}{\mathfrak{gl}} \newcommand{\sl}{\mathfrak{sl}}$One of my graduate students asked me for a reference for the following fact:. Let $k$ be a general field (she's particularly interested in $k = \mathbb{F}_2$) and let $n \geq 2$. Consider the representation $\gl_n(k)$ of $\SL_n(k)$. Let $V$ be a nonzero proper subrepresentation of $\gl_n(k)$. Then $V$ is either
- if $n$ is odd, thenthe $V$ is irreducibe; and$1$-dimensional line of scalar matrices; or
- if $n$ is even, then the only nontrivial subrepresentation of $V$ is the subspace $W \cong \mathbb{F}_2$$\sl_n(k)$ of scalartrace-$0$ matrices.
I didn't know one off the top of my head. Can anyone provide such a reference?
(thanks to YCor for cleaning up my original statement)
