TITLE: Irreducible $S_n$-modules and $S_n$-actions on projective spaces
QUESTION [8 upvotes]: Let $V$ be an $(N+1)$-dimensional vector space with an action of the symmetric group $S_n$, such that $V$ is an irreducible $S_n$ -module. 
Let $\{p_1,...,p_h\}\in \mathbb{P}(V)$ be $h\geq N+2$ points such that $S_n$ acts transitively on $\{p_1,...,p_h\}$. 
Is it true then that there exist $N+2$ points $p_{i_1},...,p_{i_{N+2}}\in \{p_1,...,p_h\}$ that are in linear general position in $\mathbb{P}(V)$ ?

REPLY [5 votes]: EDIT:  Actually, I made this way too difficult.  
Points in projective space are in general linear position if the corresponding lines are the multiples of a basis of $V$.  Thus, the question is just if the lines corresponding to $p_1,\dots, p_h$ span the vector space $V$ (since in this case, some subset will correspond to a basis).  The span of these lines is invariant and non-zero, and so must be all of $V$ by its irreducibility.