GRADED q-SCHUR ALGEBRAS
(g), the nega-tive half of the quantized enveloping algebra associated with a simply-lacedquiver. The algebra is called the Khovanov-Lauda algebra. They also pro-posed the study of cyclotomic Khovanov-Lauda algebras. Soon after that,Brundan and Kleshchev proved in [4] that the cyclotomic Khovanov-Laudaalgebras associated with a cyclic quiver are nothing but block algebras ofthe cyclotomic Hecke algebras of type G(m,1,n) and, more recently, theyproved the graded analogue of an old result of the author of this note [3] in[5]. The aim of this note is to introduce grading on the q-Schur algebra andobtain the graded analogue of the decomposition number conjecture for theq-Schur algebra considered in [23].The author is grateful to Professor Khovanov and Dr. Lauda for somecommunication about the content of [15], and to Professor Kleshchev for hiscomment that their proof in [6] works for Specht modules but it does notapply to the permutation modules. This motivated the author to write thisnote. He also thanks Dr. Fayers for some communication. The researchwas carried out during the author’s visit to the Isaac Newton Institute inCambridge for attending the program Algebraic Lie Theory. He appreciatesnice research environment he enjoyed there.2. Preliminaries I; the Hecke algebraLet F be a ﬁeld, q ∈ F