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.)



Low-dimensional irreducible 2-modular representations of the symmetric group
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 $2$](https://eudml.org/doc/172514). Math. Zeit, 151:127–137, 1976; [DOI: 10.1007/BF01241824](https://doi.org/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 $4$.


