TITLE: Low-dimensional irreducible 2-modular representations of the symmetric group
QUESTION [10 upvotes]: I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even though I suspect it's easily known by the experts.
What are the lowest-dimensional irreducible representations over $\mathbb{F}_2$ of the full symmetric group $S_n$ and in particular whether there exist any of dimension $\leq n - 2$ (resp. $\leq n - 3$) if $n$ is odd (resp. even)?
I know that for each $n \geq 3$ we have the $(n - 1)$-dimensional standard representation over any field. Moreover, if $n$ is even, we get an $(n - 2$)-dimensional representation as a quotient of this over characteristic $2$ (because then the vector with all $1$'s lies in the standard representation but is fixed by all of $S_n$). It's easy to see that these representations are irreducible. What I would specifically like to know is whether or not they are the lowest-dimensional irreducible representations of $S_n$ over $\mathbb{F}_{2}$.
(I'm curious about slightly more general questions as well, like whether or not the analogous representations to the ones I described above are the lowest-dimensional over any given positive characteristic.)

REPLY [10 votes]: An earlier reference for the minimality of these degrees over any field is L.E.Dickson, Representations of the general symmetric group as linear groups in finite and infinite fields, Trans. Amer. Math. Soc. 9 (1908), 121-148. This can be found in Dickson's Collected Works.