Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 4 / c

The partition is $p$-regular precisely whenthat is, when no part occurs $p$ or more times. This is the definition of a regular partition in characteristic of a field $p$.

If some $a_j\ge p$, then $a_j!=0$ in $\mathbb{F}$. Part b(ii) makes every pairing $⟨e(t),e(u)⟩$ zero, so the Tabloid bilinear form vanishes identically on $S^{\lambda }$ and $S^{\lambda }\subseteq (S^{\lambda }{)}^{\perp }$.

Conversely, if $\lambda$ is $p$-regular, every $a_j!$ is nonzero. Fact 1 supplies tableaux $t,u$ withThus the restriction of the form is not identically zero and $S^{\lambda }⊈(S^{\lambda }{)}^{\perp }$.