James submodule theorem
= James submodule theorem
{c}
Let $U$ be a submodule of the Young permutation module $M^\lambda$ over any field. Then either
$$
S^\lambda\subseteq U
\qquad\text{or}\qquad
U\subseteq(S^\lambda)^\perp
$$
for the <tabloid bilinear form>. The key identity is $b_tu=\langle u,e(t)\rangle e(t)$: if one pairing is nonzero, the cyclic generator $e(t)$ and hence the whole Specht module lies in $U$.