Product formulas for the cyclotomic v-Schur algebra and for the canonical bases of the Fock space

TL;DR

This paper proves a product formula for the canonical basis of the Fock space related to cyclotomic v-Schur algebras, supporting Yvonne's conjecture linking decomposition numbers to canonical bases.

Contribution

It establishes a new product formula for the canonical basis of the Fock space, complementing previous results on decomposition numbers of cyclotomic v-Schur algebras.

Findings

Proved a product formula for the canonical basis of the Fock space.

Provided evidence supporting Yvonne's conjecture on decomposition numbers.

Linked algebraic structures to combinatorial bases in representation theory.

Abstract

In our earlier work, we have proved a product formula for certain decomposition numbers of the cyclotomic v-Schur algebra associated to the Ariki-Koike algebra. It is conjectured by Yvonne that the decomposition numbers of this algebra can be described in terms of the canonical basis of the higher level Fock space studied by Uglov. In this paper we prove a product formula related to the canonical basis of the Fock space. In view of Yvonne's conjecture, this formula is regarded as a counter-part for the Fock space of our previous formula.