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

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.

v2: 24 pages, comments welcome. Theorem 4.1 from v1 has been removed due to an irreparable error in the proof