The symmetric-group level-one residue and partition-label convention

For every $a\ge 1$ the multiplicity of the part $a$ in ${\lambda }^{\mathrm{\prime }}$ equals ${\lambda }_a-{\lambda }_{a+1}$. Indeed the columns of $[\lambda ]$ of height at least $a$ are exactly the first ${\lambda }_a$ columns, because column $j$ has height $\mathrm{\#}\{r:{\lambda }_r\ge j\}\ge a$ if and only if $j\le {\lambda }_a$; hence the columns of height exactly $a$ number ${\lambda }_a-{\lambda }_{a+1}$, and these heights are precisely the parts of ${\lambda }^{\mathrm{\prime }}$ by [F1].

Transposition of nodes: $(r,c)\in [\lambda ]$ if and only if $(c,r)\in [{\lambda }^{\mathrm{\prime }}]$, so $(r,c)\mapsto (c,r)$ is a bijection from the nodes of $\lambda$ onto the nodes of ${\lambda }^{\mathrm{\prime }}$. It preserves addability and removability: deleting $(r,c)$ from $[\lambda ]$ leaves the diagram of a partition $\mu$ if and only if deleting $(c,r)$ from $[{\lambda }^{\mathrm{\prime }}]$ leaves the diagram of ${\mu }^{\mathrm{\prime }}$, and the same argument with diagrams enlarged by one node gives addability. It negates the residue, $\mathrm{res}(c,r)=r-c=-(c-r)=-\mathrm{res}(r,c)$ in $I=\mathbb{Z}\mathrm{/}e\mathbb{Z}$.

Distinct corners of a partition have distinct contents. Let $x=(r,c)\mathrm{\ne }y=(s,d)$ be addable or removable nodes with equal content $c-r=d-s$; by [F2] each is of the form $(a,{\lambda }_a)$ (removable) or $(a,{\lambda }_a+1)$ (addable, the new bottom row included via ${\lambda }_{k+1}=0$). If $r=s$, two distinct corners must be the removable and addable endpoints of that row, whose contents differ by $1$, so they cannot have equal content. Interchange the nodes if needed to take $r<s$. If $x$ and $y$ are both removable with $r<s$, then ${\lambda }_r\ge {\lambda }_s$ while ${\lambda }_r-r={\lambda }_s-s$ forces ${\lambda }_r-{\lambda }_s=r-s<0$, a contradiction. If both are addable with $r<s$, then ${\lambda }_r\ge {\lambda }_s$ while equal contents force ${\lambda }_r-{\lambda }_s=r-s<0$, again a contradiction. If $x$ is removable and $y$ addable with $r<s$, equal contents give ${\lambda }_r-{\lambda }_s=r+1-s$; for $s\ge r+2$ the right side is at most $-1$, so ${\lambda }_r<{\lambda }_s$, contradicting ${\lambda }_r\ge {\lambda }_s$, and for $s=r+1$ it gives ${\lambda }_r={\lambda }_s$, which contradicts the addability of $(s,{\lambda }_s+1)$, that is ${\lambda }_{s-1}={\lambda }_r>{\lambda }_s$ (here $s\ge 2$ because $r\ge 1$). If $x$ is addable with smaller row $r$ than the removable $y=(s,{\lambda }_s)$, then ${\lambda }_r\ge {\lambda }_s$ while ${\lambda }_r+1-r={\lambda }_s-s$ forces ${\lambda }_s-{\lambda }_r=s+1-r\ge 2$, contradicting ${\lambda }_r\ge {\lambda }_s$. All possible pairs are covered, so equal contents force $x=y$.

Consequently $\lambda$ is $e$-restricted, i.e. ${\lambda }_a-{\lambda }_{a+1}<e$ for all $a$, if and only if the part $a$ occurs in ${\lambda }^{\mathrm{\prime }}$ fewer than $e$ times, i.e. ${\lambda }^{\mathrm{\prime }}$ is $e$-regular: step 1.1 identifies the two lists of integers compared with $e$. For $\lambda =\mathrm{\varnothing }$, where ${\lambda }_a=0$ for all $a$, both conditions are the empty family of inequalities and both hold; for $a>k$ one has ${\lambda }_a-{\lambda }_{a+1}=0<e$, so the restrictedness condition is finite, and $m_i(\mathrm{\varnothing })=0$ for all $i$.

Step 1.2 also gives the weight bookkeeping: the transposition bijects the residue-$i$ nodes of $\lambda$ onto the residue-$(-i)$ nodes of ${\lambda }^{\mathrm{\prime }}$, so $m_i(\lambda )=m_{-i}({\lambda }^{\mathrm{\prime }})$ for every $i$. Hence $\mathrm{wt}(\mathrm{\mid }{\lambda }^{\mathrm{\prime }}⟩)={\mathrm{\Lambda }}_0-{\sum }_j{m}_j({\lambda }^{\mathrm{\prime }}){\alpha }_j={\mathrm{\Lambda }}_0-{\sum }_j{m}_{-j}(\lambda ){\alpha }_j=\tau (\mathrm{wt}(M_{\lambda }))$: applying the transposition of labels to the weight is the reflection $i\mapsto -i$ on the coefficients. In particular the relabelled basis vector $M_{\lambda }$ is a weight vector of weight $\mathrm{wt}(M_{\lambda })$ for the transported toral action, since $K_h^{\mathrm{s}\mathrm{y}\mathrm{m}}{M}_{\lambda }=J^{-1}{K}_{\tau h}^{\mathrm{Q}\mathrm{G}}J{M}_{\lambda }=q^{⟨\tau h,\mathrm{wt}(\mathrm{\mid }{\lambda }^{\mathrm{\prime }}⟩)⟩}{M}_{\lambda }=q^{⟨h,\tau \mathrm{wt}(\mathrm{\mid }{\lambda }^{\mathrm{\prime }}⟩)⟩}{M}_{\lambda }=q^{⟨h,\mathrm{wt}(M_{\lambda })⟩}{M}_{\lambda }$, using that $\tau$ preserves the pairing and exchanges $h_j$ with $h_{-j}$ (The cyclic affine type A Cartan datum and its residues).

The transported operators act by node moves with reflected colors. For any partition $\mu$, the QG operator $E_{-i}^{\mathrm{Q}\mathrm{G}}$ sends $\mathrm{\mid }\mu ⟩$ to a linear combination of $\mathrm{\mid }\mu \setminus \gamma ⟩$ over the removable nodes $\gamma$ of $\mu$ of residue $-i$, and $F_{-i}^{\mathrm{Q}\mathrm{G}}$ sends $\mathrm{\mid }\mu ⟩$ to a combination of $\mathrm{\mid }\mu \cup \beta ⟩$ over the addable nodes $\beta$ of $\mu$ of residue $-i$ (The level-one partition Fock space and node residues); conjugating by $J$ and using step 1.2, $E_i^{\mathrm{s}\mathrm{y}\mathrm{m}}$ sends $M_{\lambda }$ ($J{M}_{\lambda }=\mathrm{\mid }{\lambda }^{\mathrm{\prime }}⟩$) to the $J^{-1}$-images of the results of removing residue-$(-i)$ removable nodes of ${\lambda }^{\mathrm{\prime }}$, that is, to combinations of $M_{\lambda \setminus \gamma }$ over the removable nodes $\gamma$ of $\lambda$ of residue $i$, and likewise $F_i^{\mathrm{s}\mathrm{y}\mathrm{m}}$ adds residue-$i$ nodes of $\lambda$. Thus at the level of nodes the symmetric-group model uses the same residue labels as the modular signature of the Definition, not their negatives.

At quantum characteristic $e=0$ the colors form the set $I=\mathbb{Z}$ and, by [F2] and step 1.3, each residue class contains at most one node of $\lambda$: an $i$-node is a corner of content exactly $i$, and there is at most one such corner. Hence the residue word of every color has length at most one, no cancellation can occur, and for every nonempty $\lambda$ the removable corner $(k,{\lambda }_k)$ of its last row is the good node of its own residue, so repeated good deletions carry $\lambda$ to $\mathrm{\varnothing }$; consult the $e=0$ case of the survey's level-one example, under which every level-one partition is a crystal label. No finite affine datum, charge or weight formula is asserted here.

The conjugation $C(\lambda )={\lambda }^{\mathrm{\prime }}$ is a bijection from the $e$-restricted partition class onto the $e$-regular class by step 2.1, and it is not an involution of the $e$-restricted class: at $e=3$ it sends the restricted shape $(1,1,1)$ to the non-restricted shape $(3)$, and at $e=2$ it sends the restricted shape $(1,1,1,1)$ to the non-restricted shape $(4)$; the same-model sign (Mullineux) involution of the survey instead preserves the restricted label set and is computed by negating the residues of a good-node path inside it. In the group regime the published dictionary for conjugate-sign duality of simple labels is exactly the stated identification $D(\lambda )\cong D^{{\lambda }^{\mathrm{\prime }}}\otimes \mathrm{sgn}$ (p-regular and p-restricted labels under transpose and sign).