The crystal identification does not elementarily identify the cellular simple labels

At level one, distinguish the cellular labels $D(\mu )$, the nonzero Murphy cell-form quotients (the dual-Specht heads at $q=1$), from the inductive labels $\dot{D}(\mu )$ defined by $\dot{D}(\mathrm{\varnothing })=F,\dot{D}(\mu )=\mathrm{hd}(F_i\dot{D}({\tilde{e}}_i\mu ))$ for any defined good deletion ${\tilde{e}}_i\mu$. Here $F$ is any field, the Hecke parameter is invertible with finite quantum characteristic $e\ge 2$, and $\mu$ is $e$-restricted. At $q=1$, $e=p$ and the functors are those of Generalized Jucys-Murphy primary idempotents and the residue-summand functors $E_i$, $F_i$; the general Hecke functors are constructed in General Hecke residue branching endpoints from normalized Schur cells. The group label convention is fixed by p-regular and p-restricted labels under transpose and sign.

The equality $\dot{D}(\mu )\cong D(\mu )$ and independence of the chosen deletion follow at level one from the proved cellular branching endpoints and induction on rank, as explained below. They do not follow from the combinatorial inverse property alone (Good-node deletion and cogood-node addition operators). This local comparison uses no Fock-space classification.

The historical warning in Kleshchev's survey §11.2, equation (11.1), concerns the broader cyclotomic labelling theorem for restricted multipartitions. The survey attributes its proof to Ariki and a later graded argument, and describes both as relying on the wider theory. That warning does not preclude the finite level-one comparison here. Identifying the resulting graph with the canonical crystal $B({\mathrm{\Lambda }}_0)$ is a separate result of General Hecke cellular simple labels match the basic crystal ↗, using the independent Fock-component suppliers and their declared Axiom of Choice. Neither that identification nor Fock reachability is a premise of the cellular branching proofs on this page.

Every nonempty $e$-restricted shape has a good deletion. Its bottom removable corner has residue ${\mu }_h-h$, while the only boundary corner below it is the addable $(h+1,1)$ of residue $-h$. Since $0<{\mu }_h<e$, that bottom minus cannot cancel, so a surviving minus exists. The finite signature argument in General Hecke residue branching endpoints from normalized Schur cells proves that good deletion remains restricted and that its inverse is the prescribed cogood addition. For the group specialization, reachability is also supplied by Every $p$-restricted partition is reached from the empty partition by cogood additions.

Now induct on rank. At rank zero both modules are $F$. For each allowed predecessor $\nu ={\tilde{e}}_i\mu$, induction gives $\dot{D}(\nu )\cong D(\nu )$. The cellular induction endpoint is $\mathrm{hd}(F_{i}D(\nu ))\cong D({\tilde{f}}_i\nu )=D(\mu )$. Thus every allowed deletion yields the same isomorphism class, and the bottom-corner argument ensures that a deletion exists. The argument is finite and choice-free; in the group specialization its induction endpoint is also The cogood node gives the simple socle and head of the residue induction.