Solution
= Solution
The nonzero homomorphism $\theta$ cannot kill $e(t)$, because the <Specht module> is generated by the translates of this <polytabloid>. Equivariance and part d(i) therefore give
$$
b_t\theta(e(u))=\theta(b_t e(u))=\alpha\theta(e(t))\ne0.
$$
Thus $b_t$ acts nontrivially on $M^\mu/V$, and hence on $M^\mu$. Some $\mu$-tabloid $\{v\}$ must satisfy $b_t\{v\}\ne0$. Fact 2 now says that $\lambda$ dominates $\mu$ in the <dominance order on partitions>.