On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras

TL;DR

This paper establishes a product formula for v-decomposition numbers of cyclotomic q-Schur algebras, extending known results and providing new insights into their structure via Jantzen filtrations and Schur functors.

Contribution

It introduces a product formula for v-decomposition numbers of cyclotomic q-Schur algebras, generalizing previous results and utilizing Schur functors for proof.

Findings

Proved a product formula for v-decomposition numbers of cyclotomic q-Schur algebras.

Extended Shoji-Wada's results to a v-analogue setting.

Derived similar formulas for Ariki-Koike algebras using Schur functors.

Abstract

Let $\Sc(\vL)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\He_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $\Sc(\vL)$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product formula for $v$-decomposition numbers of $\Sc(\vL)$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $\He_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula for $v$-decomposition numbers of $\He_{n,r}$ by using a Schur functor.