= Invariant vector criterion for a Specht module
{title2=$(S^\lambda)^{S_n}$}
In positive <characteristic> $p$, $(S^\lambda)^{S_n}$ is nonzero exactly when $\lambda_i\equiv-1\pmod{p^{t_i}}$ for every adjacent pair, where $t_i$ is least with $\lambda_{i+1}<p^{t_i}$. The only invariant line in $M^\lambda$ is generated by the sum $z_\lambda$ of all <tabloids>. The map $\psi_{i,v}$ sends it to $\binom{\lambda_i+\lambda_{i+1}-v}{\lambda_i}z_{\lambda(i,v)}$. Thus the <kernel intersection theorem for Specht modules> gives vanishing of $\binom{\lambda_i+j}{j}$ modulo $p$ for $1\le j\le\lambda_{i+1}$. By <Lucas's theorem>, all these <binomial coefficients> vanish exactly when the last $t_i$ base-$p$ digits of $\lambda_i$ are $p-1$.
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