Low-dimensional irreducible 2-modular representations of the symmetric group

Question

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.)

Answer 1

I believe that the minimality of these degrees for representations over the field of order $2$ was proved originally in the paper

A. Wagner. The faithful linear representations of least degree of $S_n$ and $A_n$ over a field of characteristic $2$. Math. Zeit, 151:127–137, 1976; DOI: 10.1007/BF01241824.

For $S_n$, the minimal nontrivial irreducible representation has dimension $n-1$ or $n-2$ for all $n$, as you conjectured. For $A_n$, there are a couple of exceptions: $A_7$ and $A_8$ have representations of degree $4$.

Answer 2

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.

Answer 3

If you fix a prime $p$ then for $n$ sufficiently large these $n-1$ or $n-2$ dimensional representations are indeed of minimal dimension. For this I believe the right reference is "On the minimal dimensions of irreducible representations of symmetric groups" by Gordon James, although I don't have access to it at the moment.

I certainly would not be surprised if there are counterexamples for small $n$ and $p$ though.