Abstract Let ASG(2ν + l, ν;F_{q}) be the (2ν + l)-dimensional affine-singular symplectic space over the finite field F q and ASp_{2ν+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 ASp_{2ν+l,ν}(F_{q}). Denote by L^{J} the set of all flats which are joins of flats in O such that O ⊆ L^{J} and assume the join of the empty set of flats in ASG(2ν + l, ν;F_{q}) is Ø. Ordering L^{J} 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 L^{J}, when the lattices form geometric lattice, lastly gives the characteristic polynomial of L^{J}.

You GAO,Yan-yan XUE,Yu-ting XIAO等. Lattices Generated by Joins of the Flats in Orbits under Finite Affine-singular Symplectic Group and its Characteristic Polynomials[J]. Acta Mathematicae Applicatae Sinica, English Serie, 2017, 33(4): 919-932.

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