Lattices Generated by Two Orbits of Subspaces Under Finite Singular Classical Groups

Abstract

Let be the n + l-dimensional vector space over a finite field 𝔽 _q , and let G _{n+l, n} be the singular symplectic group Sp _{n+l, n} (𝔽 _q ) where n = 2ν; or the singular unitary group U _{n+l, n} (𝔽 _q ) where . For any two orbits M ₁ and M ₂ of subspaces under G _{n+l, 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 . By ordering ℒ by ordinary or reverse inclusion, two families of atomic lattices are obtained. This article characterizes the subspaces in the two lattices and classifies geometricity of these lattices.

Key Words:

• Geometric lattice
• Orbit
• Singular symplectic group
• Singular unitary group

2000 Mathematics Subject Classification:

• 20G40
• 51D25

ACKNOWLEDGMENT

This research is supported by NSF of China (10771023), NSF of Hebei Province (A2008000128), and Educational Committee of Hebei Province (2008142).

Notes

Communicated by L. Small.