THE VERTICES OF THE COMPONENTS OF THE PERMUTATION MODULE INDUCED FROM PARABOLIC GROUPS

Abstract

We consider the permutation module kP↑GLn(p f ), where P is a parabolic group in the general linear group GLn(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 kP↑GLn(p f ). Furthermore using a result of Burry and Carlson we show that all components have a different vertex.