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

TL;DR

This paper develops explicit formulas and constructions for canonical basis elements in higher-level Fock spaces of $U_q(rak{gl}_inite)$, linking them to Calogero-Moser theory and algebraic decompositions.

Contribution

It generalizes the Leclerc-Miyachi formula to arbitrary levels and introduces new explicit constructions based on symbols, including a column removal theorem.

Findings

Provides explicit descriptions of canonical basis elements.

Establishes connections to Calogero-Moser cellular characters.

Offers formulas for decomposition matrices of Ariki-Koike algebras.

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.