The permutation action of finite symplectic groups of odd characteristic on their standard modules

Abstract

Motivated by the incidence problems between points and flats of a symplectic polar space, we study a large class of submodules of the space of functions on the standard module of a finite symplectic group of odd characteristic. Our structure results on this class of submodules allow us to determine the $p$-ranks of the incidence matrices between points and flats of the symplectic polar space. In particular, we give an explicit formula for the $p$-rank of the generalized quadrangle $\mathrm{W}(3,q)$, where $q$ is an odd prime power. Combined with the earlier results of Sastry and Sin on the 2-rank of $\mathrm{W}(3,2^t)$, it completes the determination of the $p$-ranks of $\mathrm{W}(3,q)$.

arXiv comment22 pages