On the decomposition numbers of the finite general linear groups. II

Abstract

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