On the computation of the canonical basis for irreducible highest weight $U_q({\mathfrak{gl}}_{\infty })$-modules

Abstract

We study canonical basis elements in higher-level Fock spaces associated with the quantum group $U_q({\mathfrak{gl}}_{\infty })$, 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.