Normalized Matching Property of Posets Generated by Orbits of Subspaces Under Finite Symplectic Groups

Abstract

Let be the 2ν-dimensional symplectic space over a finite field 𝔽 q , and let ℳ be a given nontrivial orbit of subspaces in under the symplectic group Sp 2ν(𝔽 q ). Denote by 𝒫 the set of subspaces which are intersections of subspaces in ℳ. By ordering 𝒫 by inclusion, 𝒫 is a finite graded poset. In this article we show that 𝒫 has the normalized matching (NM) property, which implies that 𝒫 has the strong Sperner property and the Lubell-Yamamoto-Meschalkin (LYM) property.