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

David B. Chandler·Peter Sin·Qing Xiang
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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 ${\rm W}(3,q)$, where $q$ is an odd prime power. Combined with the earlier results of Sastry and Sin on the 2-rank of ${\rm W}(3,2^t)$, it completes the determination of the $p$-ranks of ${\rm W}(3,q)$.

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David B. Chandler, Peter Sin, Qing Xiang. 2006-03-03. The permutation action of finite symplectic groups of odd characteristic on their standard modules. https://arxiv.org/abs/math/0603100

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