On the computation of the canonical basis for irreducible highest weight $U_q({\mathfrak{g}\mathfrak{l}}_{\mathrm{\infty }})$-module

We study canonical basis elements in higher-level Fock spaces associated with the quantum group $U_q({\mathfrak{g}\mathfrak{l}}_{\mathrm{\infty }})$, which are conjecturally related to Calogero-Moser theory for complex reflection groups. We generalize the Leclerc-Miyachi formula to arbitrary levels by introducing new explicit constructions based on symbols, including a column removal theorem and closed formulas in several cases. These results provide explicit descriptions of canonical basis elements with applications to Calogero-Moser cellular characters and to the decomposition matrices of Ariki-Koike algebras.