Ariki's canonical basis theorem in characteristic zero

Projectives to the lower basis (1). By [F1] the transported isomorphism ${\varphi }^{\mathrm{H}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{e}}$ is the QG-8 transport of the KLR-side identification, under which the normalized self-dual indecomposable projectives correspond bijectively to the lower global basis; the transport preserves the classes of the underlying modules and internal shifts, so the class of $Y(\lambda )$ is sent to $G(\lambda )$ for every restricted label $\lambda$ [F1, F2].

Simples to the upper basis (2). The same clause identifies the transported normalized self-dual simples with the upper global basis, and the upper family is form-dual to the lower family under the contravariant form by [F2]; hence ${\mathrm{\Psi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}^{\mathrm{H}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{e}}([D(\lambda )])=G^{\ast }(\lambda )$ with $B(G(\lambda ),G^{\ast }({\lambda }^{\mathrm{\prime }}))={\delta }_{\lambda {\lambda }^{\mathrm{\prime }}}$ [F1, F2].

Labels and non-assertion (3). On the KLR side the crystals are constructed from the one-step socle and head operators and identified with $B({\mathrm{\Lambda }}_0)$; the transport preserves the crystal structure, so each transported class carries its intrinsic label in $RP$ and no permutation of labels is left open [F1]. The identification of these crystal labels with the cellular or Dipper--James head labels is deliberately not asserted here; only the transported identifications of clauses (1) and (2) are claimed [F1, F4].

The ungraded specialization (4). The forgetful maps satisfy $\zeta (qm)=\zeta (m)$ and specialize $q$ to $1$ [F3]; a graded composition series of a finite-dimensional graded module specializes to an ungraded composition series with the same multiplicities, and the transported identifications therefore specialize to the ungraded statement that the self-dual projective classes are the canonical basis and the simple classes the dual-canonical basis of the basic module, which is Ariki's ungraded canonical-basis statement [F1, F3]. Combining this specialization with the graded decomposition theorem of this page gives the ungraded decomposition-number statement; that combination is recorded as the pointer in clause (4) and is not used in the present proofs [F3, F4].

Choice. The transport functor, the label comparison and the specialization are explicit finite constructions on finite-dimensional blocks; the only choice is inherited through the global-basis suppliers, as recorded in [C1].