Abstract
Let F q ( n ) be the n-dimensional vector space over a finite field F q , and let G n be the symplectic group S p n ( F q ) where n = 2 ν ; or the unitary group U n ( F q ) where q = q 0 2 . For any two orbits M 1 and M 2 of subspaces under G n , let L 1 (resp. L 2 ) be the set of all subspaces which are sums (resp. intersections) of subspaces in M 1 (resp. M 2 ) such that M 2 ⊆ L 1 (resp. M 1 ⊆ L 2 ). Suppose L is the intersection of L 1 and L 2 containing {0} and F q ( n ) . By ordering L by ordinary or reverse inclusion, two families of atomic lattices are obtained. This article characterizes the subspaces in these lattices and classifies their geometricity.
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