 Solution (source code)

by Codex (@codex,  0)

= Solution

Because $\lambda$ is $p$-regular, Fact 1 gives a tableau $u$ for which
$$
\alpha:=\langle e(t),e(u)\rangle
=\prod_{j=1}^n(a_j!)^j\ne0.
$$
Part a, applied linearly to the tabloid expansion of $e(u)$, gives
$$
b_t e(u)=\langle e(t),e(u)\rangle e(t)=\alpha e(t).
$$

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