Journal of Applied Mathematics

Volume 2013 (2013), Article ID 732808, 7 pages

http://dx.doi.org/10.1155/2013/732808

Research Article

Lattices Generated by Two Orbits of Subspaces under Finite Singular Symplectic Groups

Xuemei Liu

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 29 September 2012; Revised 18 November 2012; Accepted 21 November 2012

Academic Editor: P. G. L. Leach

Copyright © 2013 Xuemei Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In the paper titled “Lattices generated by two orbits of subspaces under finite classical group” by Wang and Guo. The subspaces in the lattices are characterized and the geometricity is classified. In this paper, the result above is generalized to singular symplectic space. This paper characterizes the subspaces in these lattices, classifies their geometricity, and computes their characteristic polynomials.

1. Introduction

In the following we recall some definitions and facts on ordered sets and lattices (see [1]).

Let denote a finite set. A partial order on is a binary relation on such that

(1) for any .(2) and implies .(3) and implies .

By a partial ordered set (or poset for short), we mean a pair , where is a finite set and is a partial order on . As usual, we write whenever and . By abusing notation, we will suppress reference to , and just write instead of .

Let be a poset and let be a commutative ring with the identical element. A binary function on with values in is said to be the Möbius function of if

For any two elements , we say covers , denoted by , if and there exists no such that . An element of is said to be minimal, (resp., maximal) whenever there is no element such that , (resp., ). If has a unique minimal, (resp., maximal) element, then we denote it by , (resp., ) and say that is a poset with , (resp., ). Let be a finite poset with . By a rank function on , we mean a function from to the set of all the nonnegative integers such that

(1),(2), whenever .

Let be a finite poset with 0 and 1. The polynomial

is called the characteristic polynomial of , where is the rank function of .

A poset is said to be a lattice if both and exist for any two elements . Let be a finite lattice with 0. By an atom in , we mean an element in covering 0. We say is atomic lattice if any element in is a union of atoms. A finite atomic lattice is said to be a geometric lattice if admits a rank function satisfying

In this section we will introduce the concepts of subspaces of type in singular symplectic spaces. Notation and terminologies will be adopted from Wan’s book [2].

We always assume that

Let be a finite field with elements, where is a prime power, and let denote the subspace of generated by , where is the row vector in whose th coordinate is and all other coordinates are .

The singular symplectic group of degree over , denoted by , consists of all nonsingular matrices over satisfying . The row vector space together with the right multiplication action of is called the -dimensional singular symplectic space over . An -dimensional subspace in the -dimensional singular symplectic space is said to be of type , if is of rank and dim. In particular, subspaces of type are called -dimensional totally isotropic subspaces. Clearly, singular symplectic group is transitive on the set of all subspaces of the same type in , see [2, Theorems 3.22].

The results on the lattices generated by one orbit of subspaces under finite classical groups may be found in Gao and You [3], Huo et al. [4–6], Huo and Wan [7], Orlik and Solomon [8], Wang and Feng [9], Wang and Guo [10], and Wang and Li [11].

For , , let , (resp., ) denote the set of all subspaces which are sums (resp. intersections) of subspaces in , (resp., ) such that , (resp., ). Suppose denotes the intersection of and containing 0 and . By ordering by ordinary or reverse inclusion, two families of atomic lattices are obtained, denoted by or , respectively. Wang and Guo [12] discussed the geometricity of the two lattices when . In this paper, we generalized their result to general case, characterizes the subspaces in these lattices in Section 2, classify their geometricity in Section 3, and computes their characteristic polynomials in Section 4.

2. Characterization of Subspaces Contained in

Lemma 1. Let , , and . For any subspace of type , there are two subspaces and of type such that .

Proof. Assume that

Write , where . Then and are subspace of type , such that .

Lemma 2. Let , , and . For any subspace of type , there are two subspaces and of type such that .

Proof. Assume that =, write where and are the -th and -th row vectors of , respectively.

and are the -th and -th row vectors of , respectively.

Then

are subspace of type , such that .

Theorem 3. Let , assume that , satisfy , , , . For any subspace of type , there are subspace of type such that if and only if

Proof. Suppose that and satisfy condition (7). Let , since , , , , By Lemma 1 the desired result follows. Suppose and satisfy condition (8). Let , . By Lemma 1, any subspace of type is the sum some subspaces of type . By Lemma 2, any subspace of type is the sum some subspaces of type . Hence the desired result follows.

Suppose and satisfy condition (9). Let , , , By Lemma 1, we have any subspace of type is the sum some subspace of type . By Lemma 2, we have any subspace of type is the sum some subspace of type . By Lemma 1, we have any subspace of type is the sum some subspace of type . Hence the desired result follows.

Conversely, If , then , . Let , there exists , such that , hence , dim. Therefore, , , condition (7) hold.

If , let , , such that , then , and dim dim. If , then and , Assume where rank , rank and . Since , there exists a matrix with rank , such that is a matrix representation of subspace . Thus we can assume , where is a matrix with rank  . Because is a subspaces of type , we can assume . Then can be considered as a subspace of type in singular symplectic space . Hence , condition (8) hold. Similarly we also can prove condition (9) hold.

Theorem 4. Let , assume that and satisfy , , , . For any subspace of type , there are subspace of type such that if and only if

(i),(ii), and ,(iii), and .

Proof. By [3], it is directed.

Theorem 5. For , consist of , and all subspaces of type in such that

(i), ,(ii) and , ,(iii), and , .

Proof. By Theorems 3 and 4, it is directed.

3. The Geometricity of Lattices + and +

Lemma 6 (see [3]). If , then

(i), ,(ii), .

Lemma 7 (see [3]). If and , then

(i), ,(ii), .

Theorem 8. Let . Assume that , satisfies , , , and . Then

(i) is a finite geometric lattice if and only if ,(ii) is a finite geometric lattice if and only if , ,(iii) is not a geometric lattice when .

Proof. For any , define

Then is the rank function of .

(i) For lattice .

If , by Lemma 7, , by [12], is a finite geometric lattice.

If , by Lemma 7, , by [12], is a finite geometric lattice.

If < . Let = and = , then and both are of type , is of type , , is of type , hence , , . We have

That is is not a geometric lattice when .

(ii) For lattice , by Lemma 6.

If , , and is a geometric lattice.

If , let be a basis of . Since , we can take , . Hence , , , , . We have

That is is not a geometric lattice when .

(iii) For lattice , when .

Case (a). .

(a₁), Let

Then is a of type (), is a of type (), is a of type (), , , , ,, . We have

Hence, is not a geometric lattice when (a₁).

(a₂) , or .

When , we have , when , , we have , .

Let

Then is a of type , is a of type , is a of type , clearly, we have

Hence, is not a geometric lattice when (a₂).

Case (b). .

In this case , is not a geometric lattice.

Hence is not a geometric lattice when .

Theorem 9. Let . Assume that , satisfies , , , and . Then

(i) is a finite geometric lattice if and only if ,(ii) is a finite geometric lattice if and only if , ,(iii) is not a geometric lattice when .

Proof. (i) For , define

For lattice .

If , by Lemma 7, , by [12], is a finite geometric lattice.

If , by Lemma 7, , by [12], is a finite geometric lattice.

If < < , let = . Then , both are of type , is of type , is of type . We have

Hence, is not a geometric lattice when .

(ii) if , , by Lemma 7   when , , is a geometric lattice.

If , by [7], is not a geometric lattice.

(iii) Case (a). if , then .

Let

Then are of type , is of type , is of type , , .

So , , . We have

Hence is not a geometric lattice when (a).

Case (b). , from , and , we have , .

Let

Then , are of type , is of type , and is of type .

So , , .

Hence is not a geometric lattice when (b).

Hence is not a geometric lattice when .

4. Characteristic Polynomial of Lattice

In this section we compute the characteristic polynomial of the lattice .

Theorem 10. Let , Assume that satisfies , and . Then

where .

Proof. Define

Then is the rank function on . Let , , we have

For any , define

Clearly, , , when ,

By inversion to Möbius we have

by Theorem 5 we have

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 61179026 and Supported by the Fundamental Research Funds for the Central Universities (ZXH2012 K003).

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