The Permutation Modules for GL(n+1, Fq) Acting on Pn(Fq) and Formula

The paper studies the permutation representations of a finitegeneral linear group, first on finite projective space and thenon the set of vectors of its standard module. In both casesthe submodule lattices of the permutation modules are determined.In the case of projective space, the result leads to the solutionof certain incidence problems in finite projective geometry,generalizing the rank formula of Hamada. In the other case,the results yield as a corollary the submodule structure ofcertain symmetric powers of the standard module for the finitegeneral linear group, from which one obtains the submodule structureof all symmetric powers of the standard module of the ambientalgebraic group.