The regular-partition label convention and the Mullineux crystal involution

Well-definedness and involution. Let $v_0$ be the normalized highest vector. Twist $V({\mathrm{\Lambda }}_0)$ by the diagram automorphism $\tau$; the defining relations are preserved and the twisted highest weight is again ${\mathrm{\Lambda }}_0$, so the uniqueness in [F2] gives a $\mathbb{Q}(q)$-linear map $J_{\tau }$ with $J_{\tau }{v}_0=v_0$ and $J_{\tau }(xv)=\tau (x)J_{\tau }(v)$. Its square is a module map fixing the cyclic generator, hence is the identity. The unique $i$-string decomposition is carried to the $-i$-string decomposition, so $J_{\tau }{\tilde{f}}_i={\tilde{f}}_{-i}{J}_{\tau }$ and likewise for ${\tilde{e}}_i$. The canonical $A_0$-lattice is the span of the iterated Kashiwara lowering words on $v_0$, not the span of ordinary divided-generator words (The canonical crystal $B(\lambda )$ is a highest weight crystal of highest weight $\lambda$); therefore $J_{\tau }$ preserves that lattice and its crystal residue set. Its induced involution is $M$. Applying this fixed residue map to any lowering path proves path independence.

Weight compatibility. Since $\tau ({\mathrm{\Lambda }}_0)={\mathrm{\Lambda }}_0$, $\tau (\delta )=\delta$ and $\tau ({\alpha }_i)={\alpha }_{-i}$, the involution satisfies $\mathrm{w}\mathrm{t}M=\tau (\mathrm{w}\mathrm{t})$, and it is an involution of the regular label set; no affine longest element formula is used.

Conjugation versus Mullineux. By [F3] conjugation sends residue $i$ to $-i$ and compares the regular and restricted models, so it is a model dictionary rather than a self-map of the regular set: for $e\ge 3$ the conjugate of a regular partition is generally restricted and not regular, and for $e=2$ the reflection acts trivially while conjugation changes partitions visibly, so the two maps are distinct. The restricted sign involution $CM{C}^{-1}$ is the transport of $M$, and dominance on equal-size partitions is the published prefix-sum order, reversed by conjugation; horizontal-strip inclusion is not dominance.

Conclusion. Regular means part multiplicities $<e$, restricted means adjacent differences $<e$, conjugation is the regular/restricted comparison sending $i$ to $-i$, and the Mullineux involution $M$ is the well-defined same-model involution obtained from the datum reflection with weight action $\tau$, distinct from both conjugation and the adjoint transpose.