Some decomposition numbers and v-decomposition numbers for q-Schur algebras

Abstract

We show that the closed formula for the $v$-decomposition numbers arising from the canonical basis of the Fock space representation of $U_v(\widehat{\mathfrak{sl}}_n)$ achieved by Chuang and Tan, and independently by Leclerc and Miyachi, work for a larger class of partitions than they had originally worked on. We also show that the results for the decomposition numbers of the $q$-Schur algebra, by Tan, also work in this larger class of partitions.