Abstract
Let ASG(2ν + l, ν;F q ) be the (2ν + l)-dimensional affine-singular symplectic space over the finite field F q and ASp2ν+l,ν (F q ) be the affine-singular symplectic group of degree 2ν + l over F q . Let O be any orbit of flats under ASp2ν+l,ν (F q ). Denote by LJ the set of all flats which are joins of flats in O such that O ⊆ LJ and assume the join of the empty set of flats in ASG(2ν + l, ν;F q ) is ∅. Ordering LJ by ordinary or reverse inclusion, then two lattices are obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice LJ, when the lattices form geometric lattice, lastly gives the characteristic polynomial of LJ.
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Supported by the National Natural Science Foundation of China under Grant No.61179026 and No.11701558.
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Gao, Y., Xue, Yy., Xiao, Yt. et al. Lattices generated by joins of the flats in orbits under finite affine-singular symplectic group and its characteristic polynomials. Acta Math. Appl. Sin. Engl. Ser. 33, 919–932 (2017). https://doi.org/10.1007/s10255-017-0707-9
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DOI: https://doi.org/10.1007/s10255-017-0707-9