## Journal of Applied Mathematics

### Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II

#### Abstract

Let ${\mathrm{\Bbb F}}_{q}^{(2\nu +\delta +l)}$ be a $(2\nu +\delta +l)$-dimensional vector space over the finite field ${\mathrm{\Bbb F}}_{q}$. In this paper we assume that ${\mathrm{\Bbb F}}_{q}$ is a finite field of odd characteristic, and ${O}_{2\nu +\delta +l,\mathrm{ }\mathrm{\Delta }}({\mathrm{\Bbb F}}_{q})$ the singular orthogonal groups of degree $2\nu +\delta +l$ over ${\mathrm{\Bbb F}}_{q}$. Let $\scr M$ be any orbit of subspaces under ${O}_{2\nu +\delta +l,\mathrm{ }\mathrm{\Delta }}({\mathrm{\Bbb F}}_{q})$. Denote by $\scr L$ the set of subspaces which are intersections of subspaces in $\scr M$, where we make the convention that the intersection of an empty set of subspaces of ${\mathrm{\Bbb F}}_{q}^{(2\nu +\delta +l)}$ is assumed to be ${\mathrm{\Bbb F}}_{q}^{(2\nu +\delta +l)}$. By ordering $\scr L$ by ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these lattices $\scr L$ are geometric lattices.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 387132, 16 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.jam/1355495251

Digital Object Identifier
doi:10.1155/2012/387132

Mathematical Reviews number (MathSciNet)
MR2959992

Zentralblatt MATH identifier
1255.51002

#### Citation

Gao, You; Fu, XinZhi. Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II. J. Appl. Math. 2012 (2012), Article ID 387132, 16 pages. doi:10.1155/2012/387132. https://projecteuclid.org/euclid.jam/1355495251