Let $V$ be the $(n-1)$-dimensional standard representation of the symmetric group $S_n$ over the finite field of order $p.$
For which triples $(n,p,k)$ is $\bigwedge^k V$ irreducible? Or for a weaker version, for which pairs $(n,p)$ is $\bigwedge^k V$ irreducible for all $0\leq k \leq n-1$?
I know that Fulton-Harris answers this in characteristic 0, but I haven't found a reference for other fields.
Best Answer
Well a first observation is that if $p \vert n$ then the $n-1$ dimensional standard representation itself is not irreducible, and this will propagate to its exterior powers.
It turns out this is the only obstruction and that if $p \not\vert n$ then indeed these are indeed irreducible. One can probably prove this directly in this case, but I'll just appeal to a general theorem:
Carter's criterion (formulated in full by James and Mathas, with the last piece proved by Fayers) gives a combinatorial condition for which Specht modules $S^\lambda$ remain irreducible in characteristic $p$. I won't state it in full, but it involves looking at the $p$-valuations of the hook lengths. In the case of hook shape partitions the condition degenerates into just looking at whether $p$-divides the corner hook length or not.