On the Decomposition Numbers of the Finite General Linear Groups. II

Abstract

Let $q$ be a prime power, $G = {\operatorname {GL} _n}(q)$ and let $r$ be a prime not dividing $q$. Using representations of Hecke algebras associated with symmetric groups over arbitrary fields, the $r$-modular irreducible $G$-modules are classified. The decomposition matrix $D$ of $G$ (with respect to $r$) is partly described in terms of decomposition matrices of Hecke algebras, and it is shown that $D$ is lower unitriangular, provided the irreducible characters and irreducible Brauer characters of $G$ are suitably ordered.