Graded $q$-Schur algebras

TL;DR

This paper introduces a grading on $q$-Schur algebras and proves a graded version of a conjecture related to their decomposition numbers, extending recent work in the field.

Contribution

It provides the first graded analogue of the decomposition number conjecture for $q$-Schur algebras, generalizing previous ungraded results.

Findings

Established a grading on Dipper-James' $q$-Schur algebra.

Proved a graded version of Leclerc and Thibon's conjecture.

Extended the understanding of decomposition numbers in graded $q$-Schur algebras.

Abstract

Generalizing recent work of Brundan and Kleshchev, we introduce grading on Dipper-James' $q$-Schur algebra, and prove a graded analogue of the Leclerc and Thibon's conjecture on the decomposition numbers of the $q$-Schur algebra when $q^2\neq1$ and $q^3\neq1$.