The e-quotient controls the hooks of e-divisible length

Write a hook $H=H_{i,j}(\lambda )$ of $e$-divisible length in its bead-gap form $(b,g)$ with $b\in X$, $g\mathrm{\notin }X$, $0\le g<b$ and $h:=\mathrm{\mid }H\mathrm{\mid }=b-g$. By $e\mid h$ we have $b\equiv g(\mathrm{m}\mathrm{o}\mathrm{d}e)$; write the common residue as $s\in \{0,\dots ,e-1\}$ and $b=b^{\mathrm{\prime }}e+s,g=g^{\mathrm{\prime }}e+s,b^{\mathrm{\prime }}=\frac{b-s}{e}>g^{\mathrm{\prime }}=\frac{g-s}{e}\ge 0.$ The bead $b$ is a bead of $X$ on runner $s$ and the gap $g$ is a gap of $X$ on runner $s$.

Applying step 1.1 conversely, a bead $b^{\mathrm{\prime }}\in X_s^{(e)}$ and a gap $g^{\mathrm{\prime }}\mathrm{\notin }{X}_s^{(e)}$ with $0\le g^{\mathrm{\prime }}<b^{\mathrm{\prime }}$ give the bead-gap pair $(b,g)=(b^{\mathrm{\prime }}e+s,g^{\mathrm{\prime }}e+s)$ of $X$ with $e\mid (b-g)$ and $b-g=e(b^{\mathrm{\prime }}-g^{\mathrm{\prime }})$. Hence $f(H):=$ the hook of ${\lambda }^{(s)}$ attached to the bead-gap pair $(b^{\mathrm{\prime }},g^{\mathrm{\prime }})$ of the $\beta$-set $X_s^{(e)}$ is well defined, and $f$ is bijective onto the hooks of ${⨆}_i{\lambda }^{(i)}$ because both sides are in bijection with the bead-gap pairs of $X$ whose two positions share a runner, respectively with the union over $s$ of the bead-gap pairs of $X_s^{(e)}$; the length identity is $\mathrm{\mid }H\mathrm{\mid }=b-g=e(b^{\mathrm{\prime }}-g^{\mathrm{\prime }})=e\mathrm{\mid }f(H)\mathrm{\mid }$.

For the removal identity, let $X^{\mathrm{\prime }}=(X\setminus \{b\})\cup \{g\}$ be the $\beta$-set of $\lambda \setminus H$ from [F1], again of size $m$ and hence normalised by [F3]. Because $b$ and $g$ lie on runner $s$, the beads of $X^{\mathrm{\prime }}$ on a runner $t\mathrm{\ne }s$ are exactly the beads of $X$ on runner $t$, and the beads on runner $s$ have row labels $(X_s^{(e)}\setminus \{b^{\mathrm{\prime }}\})\cup \{g^{\mathrm{\prime }}\}$. Consequently, computing the quotient from the normalised sets by [F2], ${\lambda }^{\mathrm{\prime }(t)}=P((X^{\mathrm{\prime }}{)}_t^{(e)})={\lambda }^{(t)}(t\mathrm{\ne }s),{\lambda }^{\mathrm{\prime }(s)}=P((X_s^{(e)}\setminus \{b^{\mathrm{\prime }}\})\cup \{g^{\mathrm{\prime }}\}),$ and the last partition is ${\lambda }^{(s)}$ with its hook $f(H)$ removed, by [F1] applied to the $\beta$-set $X_s^{(e)}$ of ${\lambda }^{(s)}$. This is exactly the assertion $Q_e(\lambda \setminus H)=Q_e(\lambda )\setminus f(H)$.

Steps 2.1 and 3.1 together exhibit the required bijection $f$ with the length and removal identities, and the description of $f$ on the abacus is the rescaling of step 1.1 followed by the bead-gap dictionary of the runner in step 2.1.