The vertices of the components of the permutation module induced from parabolic groups

Abstract

We consider the permutation module $k_P{\uparrow^{{\rm GL}_n(p^f)}}$, where $P$ is a parabolic group in the general linear group ${\rm GL}_n(p^f)$ and $k$ is an algebraically closed field of prime characteristic $p$. The vertices of the components of these modules have been calculated in [9] by Tinberg, who studied these modules for all groups with split BN-pairs in characteristic $p$. In this paper we show that the idea of suitability is strong enough to find all $p$-groups that are vertex of some component of $k_P{\uparrow^{{\rm GL}_n(p^f)}}$. Furthermore using a result of Burry and Carlson we show that all components have a different vertex.