On the Decomposition Numbers of the Finite General Linear Groups

Abstract

Let $G = {\text {GL}_n}(q)$, $q$ a prime power, and let $r$ be an odd prime not dividing $q$. Let $s$ be a semisimple element of $G$ of order prime to $r$ and assume that $r$ divides. ${q^{\deg (\Lambda )}} - 1$ for all elementary divisors $\Lambda$ of $s$. Relating representations of certain Hecke algebras over symmetric groups with those of $G$, we derive a full classification of all modular irreducible modules in the $r$-block ${B_s}$ of $G$ with semisimple part $s$. The decomposition matrix $D$ of ${B_s}$ may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the Hecke algebras above. Moreover $D$ is lower unitriangular. This applies in particular to all $r$-blocks of $G$ if $r$ divides $q - 1$. Thus, in this case, the $r$-decomposition matrix of $G$ is lower unitriangular.