Canonical bases for Fock spaces and tensor products

TL;DR

This paper explores the relationship between canonical bases in Fock space representations of quantum affine algebras and their restrictions, providing new insights into decomposition numbers and branching coefficients in Schur algebras.

Contribution

It establishes a connection between canonical bases of quantum affine algebra representations and their restrictions, extending to parabolic subalgebras and generalizing existing results.

Findings

Results on decomposition numbers in positive characteristic

Branching coefficients for Schur algebras

Generalization of Kleshchev, Tan, and Teo's work

Abstract

We relate the canonical basis of the Fock space representation of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$, as defined by Leclerc and Thibon, to the canonical basis of its restriction to $U_q(\mathfrak{sl}_{n})$, regarded as a based module in the sense of Lusztig. More generally we consider the restriction to any parabolic subalgebra. We deduce results on decomposition numbers and branching coefficients of Schur algebras over fields of positive characteristic, generalising those of Kleshchev and of Tan and Teo.