The QG Fock-crystal arrows are the modular good-node operators

For every partition $\mu$ and every color $i$, the selection operations of the two readings correspond under transposition. By [F2] the top-to-bottom $i$-node word of $\mu$ equals the increasing-content $(-i)$-node word of ${\mu }^{\mathrm{\prime }}$ position by position, with the same signs, so the reduced survivor words coincide; consequently ${\epsilon }_i(\mu )={\epsilon }_{-i}({\mu }^{\mathrm{\prime }})$ and ${\phi }_i(\mu )={\phi }_{-i}({\mu }^{\mathrm{\prime }})$, and if ${\epsilon }_i(\mu )>0$ then the good $i$-node of $\mu$ is the transposed node of the good $(-i)$-node of ${\mu }^{\mathrm{\prime }}$, while if ${\phi }_i(\mu )>0$ then the cogood $i$-node of $\mu$ is the transposed node of the cogood $(-i)$-node of ${\mu }^{\mathrm{\prime }}$. In particular ${\epsilon }_i(\mu )=0$ if and only if ${\epsilon }_{-i}({\mu }^{\mathrm{\prime }})=0$.

There is no naming-polarity swap between the two conventions: in the modular reading the good node is by definition removable and the cogood node addable ([F1]), in the $p$-restricted operator definition the good node is removable and the cogood node addable ([F3]), and in the Fock-space reading the good node is the surviving $-$, hence removable, while the cogood node is the surviving $+$, hence addable ([F4]); the normalized Fock operators remove the good node and add the cogood node ([F5]). Hence both dictionaries compare a deletion with a deletion and an addition with an addition, and the only change is the color reflection $i\mapsto -i$.

The transported crystal operators on all partition residues are given by the modular signature rule. The field operators $J^{-1}{\tilde{e}}_{-i}^{\mathrm{Q}\mathrm{G}}J$ and $J^{-1}{\tilde{f}}_{-i}^{\mathrm{Q}\mathrm{G}}J$ preserve $(F_0{)}_{\mathrm{s}\mathrm{y}\mathrm{m}}$ by [F5]; let $b_{\mu }:=M_{\mu }+q(F_0{)}_{\mathrm{s}\mathrm{y}\mathrm{m}}$. The induced map of residue spaces sends $b_{\mu }$ to $\mathrm{\mid }{\mu }^{\mathrm{\prime }}⟩+q{F}_0$. By [F4]-[F5] its raising image is the residue of the partition obtained by deleting the good $(-i)$-node of ${\mu }^{\mathrm{\prime }}$, or zero when no such node exists. Step 1.1 identifies that node with the transpose of the modular good $i$-node $A$ of $\mu$, so ${\tilde{e}}_i^{\mathrm{s}\mathrm{y}\mathrm{m}}{b}_{\mu }=b_{\mu \setminus A}$ when ${\epsilon }_i(\mu )>0$, and zero otherwise. The same argument gives ${\tilde{f}}_i^{\mathrm{s}\mathrm{y}\mathrm{m}}{b}_{\mu }=b_{\mu \cup B}$ for the modular cogood node $B$, and zero when ${\phi }_i(\mu )=0$. These are equalities of crystal residues: on raw standard vectors they assert congruences modulo $q(F_0{)}_{\mathrm{s}\mathrm{y}\mathrm{m}}$, including at zero arrows, rather than exact field-vector equalities.

Weights transform by the reflection. By [F1] the transported weight satisfies $\mathrm{wt}(M_{\mu })={\mathrm{\Lambda }}_0-{\sum }_i{m}_i(\mu ){\alpha }_i$ and $m_i(\mu )=m_{-i}({\mu }^{\mathrm{\prime }})$; hence $\mathrm{wt}(\mathrm{\mid }{\mu }^{\mathrm{\prime }}⟩)={\mathrm{\Lambda }}_0-{\sum }_j{m}_j({\mu }^{\mathrm{\prime }}){\alpha }_j={\mathrm{\Lambda }}_0-{\sum }_j{m}_{-j}(\mu ){\alpha }_j=\tau (\mathrm{wt}(M_{\mu }))$, and the weight of a crystal arrow is preserved by the node moves of step 2.1 because those moves change one residue count by one on each side consistently.

Restricted case. Let $\lambda$ be $e$-restricted, so that ${\lambda }^{\mathrm{\prime }}$ is $e$-regular by [F1] and hence a vertex of the empty component of the full Fock crystal by [F5], on which the operators ${\tilde{e}}^{\mathrm{Q}\mathrm{G}}$, ${\tilde{f}}^{\mathrm{Q}\mathrm{G}}$ of [F5] act. By step 2.1 applied at $\mu =\lambda$, the modular operator ${\tilde{e}}_i$ on $\lambda$ (delete the good $i$-node, zero if ${\epsilon }_i(\lambda )=0$) is intertwined with ${\tilde{e}}_{-i}^{\mathrm{Q}\mathrm{G}}$ on ${\lambda }^{\mathrm{\prime }}$: $C{\tilde{e}}_i\lambda ={\tilde{e}}_{-i}^{\mathrm{Q}\mathrm{G}}C\lambda$, and likewise $C{\tilde{f}}_i\lambda ={\tilde{f}}_{-i}^{\mathrm{Q}\mathrm{G}}C\lambda$, both sides being zero together by the count agreement of step 1.1. Since transposition is a bijection from the $e$-restricted onto the $e$-regular labels and [F5] identifies the latter precisely as the vertex set of the empty component, the conjugation of the modular restricted graph is exactly that empty component, with colors negated; for $e=p$ prime the modular side is the operator system of [F3], and in general it is the reading fixed in [F1].

Finally, the arrows are not read off from ordinary Chevalley coefficients. The selection is the signature reduction of [F4], and [F4]-[F5] record that a pole or a unique Chevalley term does not determine the crystal arrow; the transported operator ${\tilde{e}}_i^{\mathrm{s}\mathrm{y}\mathrm{m}}$ of step 2.1 is the normalization inherited from ${\tilde{e}}_{-i}^{\mathrm{Q}\mathrm{G}}$, not the transported Chevalley operator $E_i^{\mathrm{s}\mathrm{y}\mathrm{m}}=J^{-1}{E}_{-i}^{\mathrm{Q}\mathrm{G}}J$, whose coefficients (such as $q^{-1}$ from the formulas of Leclerc (38)-(39)) differ from the arrow data.