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

Wada, Kentaro

doi:10.1090/S1088-4165-2010-00376-3

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

Represent. Theory 14 (2010), 417-434

DOI: https://doi.org/10.1090/S1088-4165-2010-00376-3

Published electronically: May 18, 2010

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Abstract:

Let $\mathscr {S}(\varLambda )$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\mathscr {H}_{n,r}$, introduced by Dipper, James and Mathas. In this paper, we consider $v$-decomposition numbers of $\mathscr {S}(\varLambda )$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji and Wada, a product formula for $v$-decomposition numbers of $\mathscr {S}(\varLambda )$, 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 $\mathscr {H}_{n_i,r_i}$ of smaller rank. Moreover, we prove a similar formula for $v$-decomposition numbers of $\mathscr {H}_{n,r}$ by using a Schur functor.

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