Research Article | Open Access

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

**Academic Editor:**Ch Tsitouras

#### Abstract

Let be a -dimensional vector space over the finite field . In this paper we assume that is a finite field of odd characteristic, and the singular orthogonal groups of degree over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in , where we make the convention that the intersection of an empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these lattices are geometric lattices.

#### 1. Introduction

Let be a finite field with elements, where is an odd prime power. We choose a fixed nonsquare element in . Let be a -dimensional row vector space over the finite field , and let be one of the singular orthogonal groups of degree over . There is an action of on defined as follows:
Let be an -dimensional subspace of , and be a basis of . Then, the matrix:
is called a *matrix representation* of . We usually denote a matrix representation of the -dimensional subspace still by . The above action induces an action on the set of subspaces of , that is, a subspace is carried by into the subspace . The set of subspaces of is partitioned into orbits under . Clearly, and are two trivial orbits. Let be any orbit of subspaces under . Denote the set of subspaces which are intersections of subspaces in by and call the set of subspaces generated by . We agree that the intersection of an empty set of subspaces is . Then, . Partially ordering by ordinary or reverse inclusion, we get two posets and denote them by and , respectively. Clearly, for any two elements ,
where is a subspace generated by and . Therefore, is a finite lattice.

Similarly, for any two elements , so is also a finite lattice. Both and are called the lattices generated by .

The results on the geometricity of lattices generated by subspaces in -bounded distance-regular graphs can be found in Guo et al. [1]; on the geometricity and the characteristic polynomial of lattices generated by orbits of flats under finite affine-classical groups can be found in Wang and Feng [2], Wang and Guo [3]; on inclusion relations, the geometricity and the characteristic polynomial of lattices generated by orbits of subspaces under finite nonsingular classical groups and a characterization of subspaces contained in lattices can be found in Huo [4–6], Huo and Wan [7, 8]; on inclusion relations, the geometricity and the characteristic polynomial of lattices generated by orbits of subspaces under finite singular symplectic groups, singular unitary groups, and singular pseudosymplectic groups and a characterization of subspaces contained in lattices can be found in Gao and You [9–12]. In [13], the authors studied the various lattices and generated by different orbits of subspaces under singular orthogonal group . The study contents include the inclusion relations between different lattices, the characterization of subspaces contained in a given lattice (resp., ), and the characteristic polynomial of . The purpose of this paper is to study the questions when (resp., ) are geometric lattices.

#### 2. Preliminaries

In the following, we recall some definitions and facts on ordered sets and lattices (see [8, 14]).

Let be a partially ordered set, and . We say that *covers * and write , if and there exists no such that . An element is called the *minimal element* if there exists no elements such that . If has unique minimal element, denote it by and we say that is a poset with .

Let be a poset with and . If all maximal ascending chains starting from with endpoint have the same finite length, this common length is called the *rank * of . If rank is defined for every , is said to have the rank function , where is the set consisting of all positive integers and .

A poset is said to satisfy the *Jordan-Dedekind (JD) condition* if any two maximal chains between the same pair of elements of have the same finite length.

Proposition 2.1 ([14, Proposition 2.1]). *Let be a poset with 0. If satisfies the JD condition then has the rank function which satisfies*(i),(ii).*Conversely, if admits a function satisfying and , then satisfies the JD condition with as its rank function.**Let be a poset with 0. An element is called an atom of if . A lattice with 0 is called an atomic lattice or a if every element is a supremum of atoms, that is, .*

*Definition 2.2 ([14, page 46]). *A lattice is called a *semimodular lattice* if for all ,

Proposition 2.3 ([14, Theorem 2.27]). *Let be a lattice with 0. Then, is a semimodular lattice if and only if possesses a rank function such that for all *

*Definition 2.4 ([14, page 52]). *A lattice is called a geometric lattice if it isan atomic lattice,a semimodular lattice,without infinite chains in .

According to Definition 2.2, Proposition 2.3, and Definition 2.4, we can obtain the following proposition.

Proposition 2.5. *Let be a lattice with 0. Then, is a geometric lattice if and only if*for every element , , possesses a rank function and for all , (2.2) holds,without infinite chains in .

Let
where , , or , and
The set of all nonsingular matrices over satisfying
forms a group which will be called the *singular orthogonal group* of degree , rank, and with definite part over and denoted by Clearly, consists of all nonsingular matrices of the form:
where , and is nonsingular.

Two matrices and are called to be *cogredient* if there exists a nonsingular matrix such that .

An -dimensional subspace is said to be a *subspace of type *, if is cogredient to , where the matrix , respectively, is as follows
or

Let be a basis of , where
1 is in the th position. Denote by the -dimensional subspace of generated by . An -dimensional subspace is called a *subspace of type *if(i) is a subspace of type ,(ii).

Denote the set of all subspaces of type in by. By [15, Theorem 6.28], we know that is nonempty if and only if or

Moreover, if is nonempty, then it forms an orbit of subspaces under . Let denote the set of subspaces which are intersections of subspaces in , where we make the convention that the intersection of an empty set of subspaces of is assumed to be . Partially ordering by ordinary or reverse inclusion, we get two finite lattices and denote them by and , respectively.

The case has been discussed in [8]. So, we only discuss the case in this paper.

By [13], we have the following results.

Theorem 2.6. *Let , , assume that satisfies conditions (2.10) and (2.11). Then,
**
if and only if
*

Theorem 2.7. *Let , . Assume that satisfies condition (2.10), then consists of and all the subspaces of type , where satisfies condition (2.13).*

Theorem 2.8. * Let , , and satisfy
**
For any , define
**
then is a rank function of the lattice .*

Theorem 2.9. *Let , , and satisfy (2.14). For any , define
**
then is a rank function of the lattice .*

#### 3. The Geometricity of Lattices

Theorem 3.1. *Let , assume that satisfies conditions (2.10) and (2.11). Then*(i)*each of and is a finite geometric lattice, when , and is a finite atomic lattice, but not a geometric lattice when ;*(ii)*when , is a finite atomic lattice, but not a geometric lattice.*

*Proof. *By Theorem 2.8, the rank function of is defined by formula (2.15), we will show the condition of Proposition 2.5 holds for . and it is the minimal element, so all 1-dim subspaces in are atoms of .

Let , by Theorem 2.7, is a subspace of type and satisfies condition (2.13). If , then is an atom of . Assume , then
where , or .

Let be an th row vector of , then is a subspace of type , or , and . By Theorem 2.7, we know , so is an atom of , and , hence, is a union of atoms in . Since , there exist , such that . are unions of atoms in , hence, is a union of atoms in , therefore, holds.

In the following, we prove (i) and (ii).*The Proof of (i)*. We only prove the formula (2.2) holds for . The other can be obtained in the similar way. We consider two cases:

(a) . consists of and subspaces of type . Let , if are , respectively, then , so . If is or , the other is a subspace of type , then is or subspace of type , is a subspace of type or , so . If and are subspaces of type , then , so .

Hence, (2.2) holds and is a finite geometric lattice when .

(b) . Let , , where , then . When or , then is a nonsquare element or a square element, respectively. Thus, is cogredient to either or , and is a subspace of type , where , or a subspace of type . So , and we have , . By the definition of rank function, , , we have .

Hence, is a finite atomic lattice, but not a geometric lattice when .*The Proof of (ii)*. We will show there exist such that the formula (2.2) does not hold. As to , or , we only show the proof of , others can be obtained in the similar way. We distinguish the following three cases.

(a) , or , . Then, the formula (2.10) is changed into . Let , we distinguish the following two subcases.

(a.1) . From and , we have . Let
where , then is a subspace of type , is a subspace of type . When or , then is a nonsquare element or a square element, respectively, thus is cogredient to either or , and is a subspace of type or type . Consequently, , . Thus, we have , , , , Then,

(a.2) . From , we have , . Let
then is a subspace of type , is a subspace of type , is a subspace of type . Consequently, , . Thus, we have , , , , , Then,
Therefore, there exist such that formula (2.2) does not hold.

(b) . Then, the formula (2.10) is changed into . Let , we distinguish the following two subcases.

(b.1) . From , and , we have . Let
where , then is a subspace of type , is a subspace of type . When or , similar to the proof of the case (a.1), is a subspace of type or . Consequently, , , and the formula (2.2) does not hold.

(b.2) . From , we have . Let
then is a subspace of type , is a subspace of type , when or , is subspace of type or . Similar to the proof of the case (a.1), the formula (2.2) does not hold for and .

(c) . Then, the formula (2.10) is changed into . Let , we distinguish the following two subcases.

(c.1) . From , and , we have . Let
where and , then is a subspace of type , is a subspace of type . But when or , similar to the proof of the case (a.1), is a subspace of type or . Consequently, , , and the formula (2.2) does not hold.

(c.2) . From , we have and . We choose and being two linearly independent solutions of the equation . Let
then is a subspace of type , is a subspace of type . Let
because , hence, is cogredient to . Then,
is cogredient to
Therefore, is a subspace of type . Similar to the proof of the case (a.2), the formula (2.2) does not hold for and .

#### 4. The Geometricity of Lattices

Theorem 4.1. *Let , , assume that satisfies conditions (2.10) and (2.11). Then,*(i)*each of , and is a finite geometric lattice when , and is a finite atomic lattice, but not a geometric lattice when ;*(ii)*when , is a finite atomic lattice, but not a geometric lattice.*

*Proof . *By Theorem 2.9, the rank function of is defined by formula (2.16), is the minimal element of , all subspaces of type in are atoms of .*The Proof of (i)*. By [8], , , and are finite geometric lattices when ; in the following, we will show that they are finite atomic lattices, but not geometric lattices when .

(a) Let
Then, both and are subspaces of type , and , is a subspace of type . Consequently, , , , . Thus,
That is, (2.2) does not hold for and . Hence, are not geometric lattices when .

(b) Let
Then, both and are subspaces of type , and , is a subspace of type or when or . Consequently, , , , . Thus,
That is, (2.2) does not hold for and . Hence, are not geometric lattices when .

(c) From the condition (2.10), the following hold.(i)If , then , that is, , hence , and it is a contradiction.(ii)If , then , that is, , hence , or . When , from , we obtain , and it is a contradiction. When , we have . That is, in this situation, holds.(iii)If , then , that is, , hence , or . When , we have , then . When , we have , then .

From the discussion above, we know that

(c.1) If , then , and we have , ; , , and , three possible cases. For , here we just give the proof of the case , , others can be obtained in the similar way. We choose and being two linearly independent solutions of the equation . Let
then is a subspace of type , is a subspace of type , and is a subspace of type . Consequently, , . Thus, we have , , , , , . Then,
That is, (2.2) does not hold for and . Hence, are not geometric lattices when .

(c.2) If , then we have ; or , . As to , we consider , , and three cases. Here we just give the proof of the case , and we also discuss the following three subcases:

(c.2.1) , . For , let
then is a subspace of type , is a subspace of type , and is a subspace of type