For $\lambda=(6,5,3)$ use the three-bead <beta set> $\{8,6,3\}$. Its runners give the 3-<quotient of a partition>
$$
Q_3(\lambda)=\bigl((1,1),\varnothing,(2)\bigr).
$$
The hook lengths divisible by three and their images are
$$
\begin{aligned}
f(H_{1,3}(\lambda))&=H_{1,1}(\lambda^{(2)}),&6&=3\cdot2,\\
f(H_{1,5}(\lambda))&=H_{1,2}(\lambda^{(2)}),&3&=3\cdot1,\\
f(H_{2,1}(\lambda))&=H_{1,1}(\lambda^{(0)}),&6&=3\cdot2,\\
f(H_{3,1}(\lambda))&=H_{2,1}(\lambda^{(0)}),&3&=3\cdot1.
\end{aligned}
$$
There are no others, as the complete hook-length rows are $(8,7,6,4,3,1)$, $(6,5,4,2,1)$, and $(3,2,1)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 3 / a / ii / Solution

Apply the <Murnaghan–Nakayama rule> successively to the $w$ disjoint $e$-cycles. A complete term requires a sequence of $w$ removable $e$-hooks. If $w>w_e(\lambda)$, no such sequence exists after the <Weight of a partition> is exhausted, so the character value is zero.

Suppose $w=w_e(\lambda)$. Every complete sequence ends at the <Core of a partition> $C_e(\lambda)$. Under the <abacus divisible-hook correspondence>, a removal chooses one cell from one component $\lambda^{(i)}$ of the <quotient of a partition>. The choices of which runner is used occur in
$$
\binom{w}{|\lambda^{(0)}|,\ldots,|\lambda^{(e-1)}|}
$$
orders. Within runner $i$, the signed complete removal sum is the degree $\chi^{\lambda^{(i)}}(1)$, and all inter-runner removal orders have the common <Sign of an abacus hook-removal sequence> $\varepsilon\in\{1,-1\}$. The remaining permutation $\gamma$ acts on the core, giving
$$
\chi^\lambda(\rho\gamma)
=\varepsilon
\binom{w}{|\lambda^{(0)}|,\ldots,|\lambda^{(e-1)}|}
\chi^{C_e(\lambda)}(\gamma)
\prod_{i=0}^{e-1}\chi^{\lambda^{(i)}}(1).
$$

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 3 / c / Solution

{title2=$Q_e(\lambda)$}

The $e$-quotient $Q_e(\lambda)=(\lambda^{(0)},\ldots,\lambda^{(e-1)})$ is the tuple of partitions represented by the individual runners after their positions are divided by $e$.


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 Incoming links: Quotient of a partition