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

Let Fq[s] be the q-deformed Fock space of level l with multi-charge s=(s1,…,sl). Uglov defined canonical bases G±(λ,s) of Fq[s] for l-partitions λ, and the polynomials are defined as the coefficients of the transition matrix between the canonical bases and the standard basis of Fq[s]. In this paper, we prove a product formula for for certain l-partitions λ,μ, which expresses as a product of such polynomials related to various smaller Fock spaces Fq[s[i]]. Yvonne conjectures that are related to the q-decomposition numbers dλμ(q) of the cyclotomic v-Schur algebra S(Λ), where the parameters v,Q1,…,Ql are roots of unity in C determined from s. In our earlier work, we have proved a product formula for dλμ(q) of S(Λ), and the product formula for is regarded as a counter-part of that formula for the case of the Fock space.