NOTE: This replaces the earlier paper with a simialr title




The Permutation Module for 
$\text{GL}(n+1,\Bbb F_q)$ Acting on $\Bbb P^n(\Bbb F_q)$ and ${\Bbb F_q}^{n+1}$.

Authors: Matthew Bardoe, no affilialition.

         Peter Sin
                        Department of Mathematics
                        University of Florida 
                        358 Little Hall
                        PO Box 118105
                        Gainesville, FL 32611-8105 


Abstract. This paper studies the permutation representations of
a finite general linear group, first on finite projective space and
then on the set of vectors of its standard module. In both cases
the submodule lattices of the permutation modules are determined.
In the case of projective space, the result leads to the solution
of 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 of certain symmetric powers 
of the standard module for the finite general linear group, from which
one obtains the submodule structure of all symmetric powers of
the standard module of the ambient algebraic group.


Status: Preprint.