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

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 𝑃 denote a finite set. A partial order on 𝑃 is a binary relation ≤ on 𝑃 such that(1)𝑎≤𝑎 for any 𝑎∈𝑃.(2)𝑎≤𝑏 and 𝑏≤𝑎 implies 𝑎=𝑏.(3)𝑎≤𝑏 and 𝑏≤𝑐 implies 𝑎≤𝑐.

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

Let 𝑃 be a poset and let 𝑅 be a commutative ring with the identical element. A binary function 𝜇(𝑎,𝑏) on 𝑃 with values in 𝑅 is said to be the Möbius function of 𝑃 if _{𝑎≤𝑐≤𝑏}𝜇(𝑎,𝑐)=1,if𝑎=𝑏,0,otherwise. (1)

For any two elements 𝑎,𝑏∈𝑃, we say 𝑎covers 𝑏, denoted by 𝑏<⋅𝑎, if 𝑏<𝑎 and there exists no 𝑐∈𝑃 such that 𝑏<𝑐<𝑎. An element 𝑚 of 𝑃 is said to be minimal, (resp., maximal) whenever there is no element 𝑎∈𝑃 such that 𝑎<𝑚, (resp., 𝑎>𝑚). If 𝑃 has a unique minimal, (resp., maximal) element, then we denote it by 0, (resp., 1) and say that 𝑃 is a poset with 0, (resp., 1). Let 𝑃 be a finite poset with 0. By a rank function on 𝑃, we mean a function 𝑟 from 𝑃 to the set of all the nonnegative integers such that(1)𝑟(0)=0,(2)𝑟(𝑎)=𝑟(𝑏)+1, whenever 𝑏<⋅𝑎.

Let 𝑃 be a finite poset with 0 and 1. The polynomial 𝜒(𝑃,𝑥)=_𝑎∈𝑃𝜇(0,𝑎)𝑥^{𝑟(1)−𝑟(𝑎)}, (2) is called the characteristic polynomial of 𝑃, where 𝑟 is the rank function of 𝑃.

A poset 𝑃 is said to be a lattice if both 𝑎∨𝑏∶=sup{𝑎,𝑏} and 𝑎∧𝑏∶=inf{𝑎,𝑏} exist for any two elements 𝑎,𝑏∈𝑃. Let 𝑃 be a finite lattice with 0. By an atom in 𝑃, we mean an element in 𝑃 covering 0. We say 𝑃 is atomic lattice if any element in 𝑃⧵{0} is a union of atoms. A finite atomic lattice 𝑃 is said to be a geometric lattice if 𝑃 admits a rank function 𝑟 satisfying 𝑟(𝑎∧𝑏)+𝑟(𝑎∨𝑏)≤𝑟(𝑎)+𝑟(𝑏),∀𝑎,𝑏∈𝑃. (3)

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

We always assume that 𝐾_𝜈,𝑙=0𝐼^(𝜈)−𝐼^(𝜈)00^(𝑙). (4)

Let 𝐹_𝑞 be a finite field with 𝑞 elements, where 𝑞 is a prime power, and let 𝐸 denote the subspace of 𝐹_𝑞^(2𝜈+𝑙) generated by 𝑒_2𝜈+1,𝑒_2𝜈+2,…,𝑒_2𝜈+𝑙, where 𝑒_𝑖 is the row vector in 𝐹_𝑞^(2𝜈+𝑙) whose 𝑖th coordinate is 1 and all other coordinates are 0.

The singular symplectic group of degree 2𝜈+𝑙 over 𝐹_𝑞, denoted by 𝑆𝑝_{2𝜈+𝑙,2𝜈}(𝐹_𝑞), consists of all (2𝜈+𝑙)×(2𝜈+𝑙) nonsingular matrices 𝑇 over 𝐹_𝑞 satisfying 𝑇𝐾_𝑙𝑇^𝑡=𝐾_𝑙. The row vector space 𝐹_𝑞^(2𝜈+𝑙) together with the right multiplication action of 𝑆𝑝_{2𝜈+𝑙,2𝜈}(𝐹_𝑞) is called the (2𝜈+𝑙)-dimensional singular symplectic space over 𝐹_𝑞. An 𝑚-dimensional subspace 𝑃 in the (2𝜈+𝑙)-dimensional singular symplectic space is said to be of type (𝑚,𝑠,𝑘), if 𝑃𝐾_𝑙𝑃^𝑡 is of rank 2𝑠 and dim(𝑃∩𝐸)=𝑘. In particular, subspaces of type (𝑚,0,0) are called 𝑚-dimensional totally isotropic subspaces. Clearly, singular symplectic group 𝑆𝑝_{2𝜈+𝑙,2𝜈}(𝐹_𝑞) is transitive on the set of all subspaces of the same type in 𝐹_𝑞^(2𝜈+𝑙), 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 1≤𝑚₁≤𝑚₂≤2𝜈−1, 0≤𝑘₁≤𝑘₂≤𝑙, let 𝐿₁(𝑚₁,𝑠₁,𝑘₁;2𝜈+𝑙,𝜈), (resp., 𝐿₂(𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈)) denote the set of all subspaces which are sums (resp. intersections) of subspaces in 𝑀(𝑚₁,𝑠₁,𝑘₁;2𝜈+𝑙,𝜈), (resp., 𝑀(𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈)) such that 𝑀(𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈)⊆𝐿₁(𝑚₁,𝑠₁,𝑘₁;2𝜈+𝑙,𝜈), (resp., 𝑀(𝑚₁,𝑠₁,𝑘₁;2𝜈+𝑙,𝜈)⊆𝐿₂(𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈)). Suppose 𝐿(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) denotes the intersection of 𝐿₁(𝑚₁,𝑠₁,𝑘₁;2𝜈+𝑙,𝜈) and 𝐿₂(𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) containing 0 and 𝐹_𝑞^(2𝜈+𝑙). By ordering 𝐿(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) by ordinary or reverse inclusion, two families of atomic lattices are obtained, denoted by 𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) or 𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈), respectively. Wang and Guo [12] discussed the geometricity of the two lattices when 𝑙=0. 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 𝐿(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈)

Lemma 1. Let 2𝜈+𝑙>0, 0≤𝑘≤𝑙, 2𝑠≤𝑚−𝑘≤𝜈+𝑠 and 𝑚−𝑘≥1. For any subspace 𝑃 of type (𝑚+1,𝑠,𝑘+1), there are two subspaces 𝑃₁ and 𝑃₂ of type (𝑚,𝑠,𝑘) such that 𝑃=𝑃₁+𝑃₂.

Proof. Assume that 𝑃𝐾_𝜈𝑃^𝑇=𝑘_𝑠00()2𝑠𝑚+1−𝑘+1−2𝑠𝑘+1. (5) Write 𝑃=^𝑃11𝑃12_{0𝑃222𝜈𝑙}^{𝑚+1−(𝑘+1)}_𝑘+1, where 𝑃₂₂=^𝑃′22_𝑧1𝑧2¹₁^𝑘−1. Then 𝑃₁=^𝑃11𝑃12_0𝑃^′_220𝑧1 and 𝑃₂=^𝑃11𝑃12_0𝑃^′_220𝑧21^{𝑚+1−(𝑘+1)}_𝑘−1 are subspace of type (𝑚,𝑠,𝑘), such that 𝑃=𝑃₁+𝑃₂.

Lemma 2. Let 2𝜈+𝑙>0, 0≤𝑘≤𝑙, 2𝑠≤𝑚−𝑘≤𝜈+𝑠 and 𝑚≠2𝜈+𝑙. For any subspace 𝑃 of type (𝑚+1,𝑠+1,𝑘), there are two subspaces 𝑃₁ and 𝑃₂ of type (𝑚,𝑠,𝑘) such that 𝑃=𝑃₁+𝑃₂.

Proof. Assume that 𝑃𝐾_𝜈𝑃^𝑇=^𝑃0𝑥_1𝑦1𝐾_𝜈^𝑃0𝑥_1𝑦1^𝑇=𝑘_𝑠0𝑘₁02𝑘2𝑠𝑚+1−2𝑠−𝑘−2, write 𝑃^{=𝑃11𝑃12}_0𝑃^′_{222𝜈𝑙𝑘}^{,𝑚+1−𝑘} where 𝑃^=𝑃₁₁^0𝑥_1𝑦1¹₁^{,𝑚+1−𝑘−2}𝑥₁ and 𝑦₁ are the (𝑚+1−(𝑘+1))-th and (𝑚−𝑘+1)-th row vectors of 𝑃₁₁, respectively.
𝑃^=𝑃₁₂^′0_𝑥^′_1𝑦′1¹₁^{,𝑚+1−𝑘−2}𝑥^′₁ and 𝑦^′₁ are the (𝑚+1−(𝑘+1))-th and (𝑚−𝑘+1)-th row vectors of 𝑃₁₂, respectively.
Then 𝑃₁=𝑃₀𝑃^′₀𝑥₁𝑥^′₁0𝑃₂₂1𝑘,𝑃𝑚+1−𝑘−2₂=𝑃₀𝑃^′₀𝑦₁𝑦^′₁0𝑃₂₂1𝑘𝑚+1−𝑘−2 (6) are subspace of type (𝑚,𝑠,𝑘), such that 𝑃=𝑃₁+𝑃₂.

Theorem 3. Let 2𝜈+𝑙>0, assume that (𝑚,𝑠,𝑘), (𝑚₁,𝑠₁,𝑘₁) satisfy 0≤𝑘≤𝑙, 2𝑠≤𝑚−𝑘≤𝜈+𝑠, 0≤𝑘₁≤𝑙, 2𝑠≤𝑚₁−𝑘₁≤𝜈+𝑠. For any subspace 𝑃 of type (𝑚,𝑠,𝑘), there are subspace 𝑃₁,𝑃₂,…,𝑃_𝑖 of type (𝑚₁,𝑠₁,𝑘₁) such that 𝑃=𝑃₁+𝑃₂+⋯+𝑃_𝑖 if and only if (i)𝑠=𝑠₁=𝜈,0≤𝑘₁()≤𝑘≤𝑙,ii𝑠<𝜈,𝑘₁=𝑘=𝑙,𝑚−𝑚₁≥𝑠−𝑠₁()≥0,iii𝑠<𝜈,0≤𝑘₁()−𝑚≤𝑘≤𝑙,𝑚−𝑘₁−𝑘₁≥𝑠−𝑠₁≥0. (7)(8)(9)

Proof. Suppose that (𝑚,𝑠,𝑘) and (𝑚₁,𝑠₁,𝑘₁) satisfy condition (7). Let 𝑘−𝑘₁=ℎ(ℎ≥0), since 𝑠=𝑠₁=𝜈, 𝑚=2𝜈+𝑘, 𝑚₁=2𝜈+𝑘₁, 𝑚−𝑚₁=𝑘−𝑘₁=ℎ, By Lemma 1 the desired result follows. Suppose (𝑚,𝑠,𝑘) and (𝑚₁,𝑠₁,𝑘₁) satisfy condition (8). Let 𝑠−𝑠₁=𝑡, 𝑚−𝑚₁=𝑡+𝑡^′(𝑡,𝑡^′≥0). By Lemma 1, any subspace of type (𝑚,𝑠,𝑙) is the sum some subspaces of type (𝑚−𝑡,𝑠,𝑙). By Lemma 2, any subspace of type (𝑚−𝑡,𝑠,𝑙) is the sum some subspaces of type (𝑚₁,𝑠₁,𝑙). Hence the desired result follows.
Suppose (𝑚,𝑠,𝑘) and (𝑚₁,𝑠₁,𝑘₁) satisfy condition (9). Let 𝑠−𝑠₁=𝑡, 𝑘−𝑘₁=ℎ, 𝑚−𝑚₁=𝑡+ℎ+𝑡^′(𝑡,𝑡^′,ℎ≥0), By Lemma 1, we have any subspace of type (𝑚,𝑠,𝑘) is the sum some subspace of type (𝑚−𝑡^′,𝑠,𝑘). By Lemma 2, we have any subspace of type (𝑚−𝑡^′,𝑠,𝑘) is the sum some subspace of type (𝑚−𝑡^′−𝑡,𝑠₁,𝑘). By Lemma 1, we have any subspace of type (𝑚−𝑡^′−𝑡,𝑠₁,𝑘) is the sum some subspace of type (𝑚₁,𝑠₁,𝑘₁). Hence the desired result follows.
Conversely, If 𝑠₁=𝜈, then 𝑠=𝜈, 𝑘₁≤𝑙. Let 𝑄∈𝑀(𝑚,𝜈,𝑘;2𝜈+𝑙,𝜈), there exists 𝑃∈𝑀(𝑚₁,𝜈,𝑘₁;2𝜈+𝑙,𝜈), such that 𝑃⊂𝑄, hence 𝑃∩𝐸⊂𝑄∩𝐸, 𝑘₁=dim(𝑃∩𝐸)≤ dim(𝑄∩𝐸)=𝑘. Therefore, 𝑠=𝑠₁=𝜈, 0≤𝑘₁≤𝑘≤𝑙, condition (7) hold.
If 𝑠₁<𝜈, let 𝑃∈𝑀(𝑚₁,𝜈,𝑘₁;2𝜈+𝑙,𝜈), 𝑄∈𝑀(𝑚,𝜈,𝑘∶2𝜈+𝑙,𝜈), such that 𝑃⊂𝑄, then 𝑃∩𝐸⊂𝑄∩𝐸, and 𝑘₁= dim(𝑃∩𝐸)≤ dim(𝑄∩𝐸)=𝑘. If 𝑘₁=𝑙, then 𝑘=𝑙 and 𝑚₁≥𝑙, Assume 𝑄=^𝑄11𝑄12_{0𝑄22𝑙2𝜈𝑙}^𝑚−1 where rank 𝑄₁₁=𝑚−𝑙, rank 𝑄₂₂=𝑙 and 𝑄₁₁𝐾𝑄^𝑡=𝑀(𝑚−𝑙,𝑠). Since 𝑃∩𝐸⊂𝑄∩𝐸, there exists a 𝑙×𝑙 matrix 𝑃₂₂ with rank 𝑃₂₂=𝑙, such that 𝑃₂₂𝑄₂₂ is a matrix representation of subspace 𝑃∩𝐸. Thus we can assume 𝑃=^𝑃11𝑃12_0𝑃22^𝑚_2𝜈𝑙¹_𝑙⁻¹, where 𝑃₁₁ is a matrix with rank  𝑚₁−𝑙. Because 𝑃 is a subspaces of type (𝑚₁,𝑠₁,𝑙), we can assume 𝑃₁₁𝑀(𝑚−𝑙,𝑠)𝑃^𝑡₁₁=𝑀(𝑚₁−𝑙,𝑠). Then 𝑄₁₁ can be considered as a subspace of type (𝑚₁−𝑙,𝑠₁) in singular symplectic space 𝐹_𝑞^{(2𝑠+(𝑚−2𝑠−𝑙))}. Hence 𝑚₁−𝑙−𝑠−𝑠₁≤𝑚−2𝑠−𝑙, condition (8) hold. Similarly we also can prove condition (9) hold.

Theorem 4. Let 2𝜈+𝑙>0, assume that (𝑚,𝑠,𝑘) and (𝑚₂,𝑠₂,𝑘₂) satisfy 0≤𝑘≤𝑙, 2𝑠≤𝑚−𝑘≤𝜈+𝑠, 0≤𝑘₂≤𝑙, 2𝑠₂≤𝑚₂−𝑘₂≤𝜈+𝑠. For any subspace 𝑃 of type (𝑚,𝑠,𝑘), there are subspace 𝑃₁,𝑃₂,…,𝑃_𝑗 of type (𝑚₂,𝑠₂,𝑘₂) such that 𝑃=𝑃₁∩𝑃₂∩⋯∩𝑃_𝑗 if and only if(i)𝑠=𝑠₂=𝜈,0≤𝑘₂≤𝑘≤𝑙,(ii)𝑠<𝜈, 𝑘₂=𝑘=𝑙 and 𝑚₂−𝑚≥𝑠₂−𝑠≥0,(iii)𝑠<𝜈, 𝑘≤𝑘₂≤𝑙 and (𝑚₂−𝑘₂)−(𝑚−𝑘)≥𝑠₂−𝑠≥0.

Proof. By [3], it is directed.

Theorem 5. For 1≤𝑚₁≤𝑚₂≤2𝜈+𝑙, 𝐿(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) consist of {0}, 𝐹_𝑞^(2𝜈+𝑙) and all subspaces of type (𝑚,𝑠,𝑘) in 𝐹_𝑞^(2𝜈+𝑙) such that(i)𝑠=𝑠₁=𝑠₂=𝜈, 0≤𝑘₁≤𝑘₂≤𝑘≤𝑙,(ii)𝑠<𝜈,𝑘₁=𝑘=𝑘₂=𝑙 and 𝑚−𝑚₁≥𝑠−𝑠₁≥0, 𝑚₂−𝑚≥𝑠₂−𝑠≥0,(iii)𝑠<𝜈, 𝑘₁≤𝑘≤𝑘₂≤𝑙 and (𝑚₂−𝑘₂)−(𝑚−𝑘)≥𝑠₂−𝑠≥0, (𝑚−𝑘)−(𝑚₁−𝑘₁)≥𝑠−𝑠₁≥0.

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

3. The Geometricity of Lattices 𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈 + 𝑙,𝜈) and 𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈 + 𝑙,𝜈)

Lemma 6 (see [3]). If 0<𝑘<𝑙, then(i)𝐿_𝑅(2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈)≃𝐿_𝑅(𝑘,𝑙), 𝐿_𝑂(2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈)≃𝐿_𝑂(𝑘,𝑙),(ii)𝐿_𝑅(𝑘,0,𝑘;2𝜈+𝑙,𝜈)≃𝐿_𝑅(𝑘,𝑙), 𝐿_𝑂(𝑘,0,𝑘;2𝜈+𝑙,𝜈)≃𝐿_𝑂(𝑘,𝑙).

Lemma 7 (see [3]). If 0≤𝑠<𝜈 and 2𝑠≤𝑚−𝑙≤𝜈+𝑠, then(i)𝐿_𝑅(𝑚,𝑠,𝑙;2𝜈+𝑙,𝜈)≃𝐿_𝑅(𝑚−𝑙,𝑠;2𝜈), 𝐿_𝑂(𝑚,𝑠,𝑙;2𝜈+𝑙,𝜈)≃𝐿_𝑂(𝑚−𝑙,𝑠;2𝜈),(ii)𝐿_𝑅(𝑚,𝑠,0;2𝜈+𝑙,𝜈)≃𝐿_𝑅(𝑚,𝑠;2𝜈), 𝐿_𝑂(𝑚,𝑠,0;2𝜈+𝑙,𝜈)≃𝐿_𝑂(𝑚,𝑠;2𝜈).

Theorem 8. Let 2𝜈+𝑙>0. Assume that (𝑚₁,𝑠₁,𝑘₁), (𝑚₂,𝑠₂,𝑘₂) satisfies 0≤𝑘₁≤𝑙, 2𝑠₁≤𝑚₁−𝑘₁≤𝜈+𝑠₁, 0≤𝑘₂≤𝑙, 2𝑠₂≤𝑚₂−𝑘₂≤𝜈+𝑠₂ and 1≤𝑚₁≤𝑚₂<2𝜈+𝑙. Then(i)𝐿_𝑂(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈) is a finite geometric lattice if and only if 𝑘=0,𝑙,(ii)𝐿_𝑂(𝑘,0,𝑘;2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈) is a finite geometric lattice if and only if 𝑘=1, 𝑙−1,(iii)𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) is not a geometric lattice when 2≤𝑚₁−𝑘₁≤𝑚₂−𝑘₂≤2𝜈−2.

Proof. For any 𝑋∈𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈), define 𝑟_𝑂(𝑋)=𝑚0,if𝑋=0,₂−𝑚₁+2,if𝑋=𝐹_𝑞^(2𝜈+𝑙)dim𝑋−𝑚₁+1,otherwise., (10) Then 𝑟_𝑂 is the rank function of 𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈).
(i) For lattice 𝐿_𝑂(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈).
If 𝑘=0, by Lemma 7, 𝐿_𝑂(1,0,0;2𝜈−𝑙,𝜈−1,0;2𝜈+𝑙,𝜈)≃𝐿_𝑂(1,0,2𝜈−1,𝜈−1;2𝜈), by [12], 𝐿_𝑂(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈) is a finite geometric lattice.
If 𝑘=𝑙, by Lemma 7, 𝐿_𝑂(𝑙+1,0,𝑙;2𝜈−𝑙+𝑙,𝜈−1,𝑙;2𝜈+𝑙,𝜈)≃𝐿_𝑂(1,0,2𝜈−1,𝜈−1;2𝜈), by [12], 𝐿_𝑂(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈) is a finite geometric lattice.
If 0 < 𝑘<𝑙. Let 𝑈 = ⟨𝑒₁,𝑒_2𝜈+1,…,𝑒_2𝜈+𝑘⟩ and 𝑊 = ⟨𝑒_𝜈+1,𝑒_2𝜈+2,…,𝑒_2𝜈+𝑘+1⟩, then 𝑈 and 𝑊 both are of type (𝑘+1,0,𝑘), ⟨𝑈,𝑊⟩ is of type (𝑘+3,1,𝑘+1), ⟨𝑈,𝑊⟩∈𝐿_𝑂(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈), 𝑈∩𝑊 is of type (𝑘−1,0,𝑘−1), hence 𝑟_𝑂(𝑈∧𝑊)=0, 𝑟_𝑂(𝑈∨𝑊)=3, 𝑟_𝑂(𝑈)=𝑟_𝑂(𝑊)=1. We have 𝑟_𝑂(𝑈∧𝑊)+𝑟_𝑂(𝑈∨𝑊)=3>𝑟_𝑂(𝑈)+𝑟_𝑂(𝑊)=2. (11) That is 𝐿_𝑂(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈) is not a geometric lattice when 0<𝑘<𝑙.
(ii) For lattice 𝐿_𝑂(𝑘,0,𝑘;2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈), by Lemma 6𝐿_𝑂(𝑘,0,𝑘;2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈)≃𝐿_𝑂(𝑘,𝑙).
If 𝑘=1, 𝑙−1, 𝐿_𝑂(1,𝑙) and 𝐿_𝑂(𝑙−1,𝑙) is a geometric lattice.
If 2≤𝑘≤𝑙−2, let 𝑣₁,𝑣₂,…,𝑣_𝑙 be a basis of 𝐹_𝑞^(𝑙). Since 2≤𝑘≤𝑙−2, we can take 𝑈=⟨𝑣₁,𝑣₂,…,𝑣_𝑘⟩, 𝑊=⟨𝑣₃,𝑣₄,…,𝑣_𝑘+2⟩∈𝐿_𝑂(𝑘,𝑙). Hence 𝑈∧𝑊={0}, 𝑈∨𝑊=⟨𝑈,𝑊⟩, 𝑟_𝑂(𝑈∧𝑊)=0, 𝑟_𝑂(𝑈∨𝑊)=3, 𝑟_𝑂(𝑈)=𝑟_𝑂(𝑊)=1. We have 𝑟_𝑂(𝑈∧𝑊)+𝑟_𝑂(𝑈∨𝑊)=3>𝑟_𝑂(𝑈)+𝑟_𝑂(𝑊)=2. (12) That is 𝐿_𝑂(𝑘,0,𝑘;2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈) is not a geometric lattice when 2≤𝑘≤𝑙−2.
(iii) For lattice 𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈), when 2≤𝑚₁−𝑘₁≤𝑚₂−𝑘₂≤2𝜈−2.
Case (a). 𝑘₁≤𝑘₂≤𝑙.
(a₁)𝑠₂>0, Let𝐼𝑈=^(𝑠2−1)000𝑠₂−10000000000𝐼^(𝑠2)0000𝐼^{(𝑚2−𝑘2−2𝑠2)}00000000000𝐼^(𝑘2)1𝑚₂−𝑘₂−2𝑠₂𝜈+𝑠₂−𝑚₂+𝑘₂𝑠₂𝑚₂−𝑘₂−2𝑠₂𝜈+𝑠₂−𝑚₂+𝑘₂𝑘₂0000𝑙−𝑘₂,00𝑠𝑊=₂−1001000000000000𝐼^(𝑘2)1𝑚₂−𝑘₂−2𝑠₂1𝜈+𝑠₂−𝑚₂+𝑘₂𝑠₂𝑚₂−𝑘₂−2𝑠₂𝜈+𝑠₂−𝑚₂+𝑘₂𝑘₂00𝑙−𝑘₂. (13)
Then 𝑈 is a of type (𝑚₂−1,𝑠₂−1,𝑘₂), 𝑊 is a of type (𝑘₂+1,0,𝑘₂), ⟨𝑈,𝑊⟩ is a of type (𝑚₂,𝑠₂−1,𝑘₂), 𝑟_𝑂(𝑈∨𝑊)=𝑟_𝑂(𝐹_𝑞^(2𝜈+𝑙))=𝑚₂−𝑚₁+2, 𝑟_𝑂(𝑈∧𝑊)=𝑘₂−𝑚₁+1, 𝑟_𝑂(𝑈)=𝑚₂−1−𝑚₁+1=𝑚₂−𝑚₁, 𝑟_𝑂(𝑊)=𝑘₂+1−𝑚₁+1=𝑘₂−𝑚₁+2,𝑟_𝑂(𝑈∨𝑊)+𝑟_𝑂(𝑈∧𝑊)=𝑚₂−2𝑚₁+𝑘₂+3, 𝑟_𝑂(𝑈)+𝑟_𝑂(𝑊)=𝑚₂−2𝑚₁+𝑘₂+2. We have 𝑟_𝑂(𝑈∨𝑊)+𝑟_𝑂(𝑈∧𝑊)>𝑟_𝑂(𝑈)+𝑟_𝑂(𝑊). (14) Hence, 𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) is not a geometric lattice when (a₁).
(a₂) 𝑠₂>0, 𝑚₂−𝑘₂=𝜈+𝑠₂ or 𝑠₂=0.
When 𝑠₂=0, we have 𝑚₂−𝑘₂−2𝑠₂≥1, when 𝑠₂>0, 𝑚₂−𝑘₂=𝜈+𝑠₂, we have 𝜈−2≥𝑠₂, 𝑚₂−𝑘₂−2𝑠₂−1=𝜈−𝑠₂−1≥1.
Let𝐼𝑈=^(𝑠2)000𝑠₂00000000000𝐼^(𝑠2)𝐼0000_{(𝑚2−𝑘2−2𝑠2)}00000000000000𝐼^(𝑘2)𝑚₂−𝑘₂−2𝑠₂−11𝜈+𝑠₂−𝑚₂+𝑘₂𝑠₂𝑚₂−𝑘₂−2𝑠₂−11𝜈+𝑠₂−𝑚₂+𝑘₂𝑘₂0000𝑙−𝑘₂,00𝑠𝑊=₂00100000000𝐼^(𝑘2)1𝜈−𝑠₂−1𝑠₂1𝜈−𝑠₂−1𝑘₂00𝑙−𝑘₂. (15)
Then 𝑈 is a of type (𝑚₂−1,𝑠₂,𝑘₂), 𝑊 is a of type (𝑘₂+1,0,𝑘₂), ⟨𝑈,𝑊⟩ is a of type (𝑚₂,𝑠₂+1,𝑘₂), clearly, we have 𝑟_𝑂(𝑈∨𝑊)+𝑟_𝑂(𝑈∧𝑊)>𝑟_𝑂(𝑈)+𝑟_𝑂(𝑊). (16)
Hence, 𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) is not a geometric lattice when (a₂).
Case (b). 𝑘₁=𝑘₂=𝑙.
In this case 𝐿_𝑂(𝑚₁,𝑠₁,𝑙;𝑚₂,𝑠₂,𝑙;2𝜈+𝑙,𝜈)≃𝐿_𝑂(𝑚₁−𝑙,𝑠₁;𝑚₂−𝑙,𝑠₂;2𝜈), 𝐿_𝑂(𝑚₁,𝑠₁,𝑙;𝑚₂,𝑠₂,𝑙;2𝜈+𝑙,𝜈) is not a geometric lattice.
Hence 𝐿_𝑂(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) is not a geometric lattice when 2≤𝑚₁−𝑘₁≤𝑚₂−𝑘₂≤2𝜈−2.

Theorem 9. Let 2𝜈+𝑙>0. Assume that (𝑚₁,𝑠₁,𝑘₁), (𝑚₂,𝑠₂,𝑘₂) satisfies 0≤𝑘₁≤𝑙, 2𝑠₁≤𝑚₁−𝑘₁≤𝜈+𝑠₁, 0≤𝑘₂≤𝑙, 2𝑠₂≤𝑚₂−𝑘₂≤𝜈+𝑠₂ and 1≤𝑚₁≤𝑚₂<2𝜈+𝑙. Then(i)𝐿_𝑅(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈) is a finite geometric lattice if and only if 𝑘=0,𝑙,(ii)𝐿_𝑅(𝑘,0,𝑘;2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈) is a finite geometric lattice if and only if 𝑘=1, 𝑙−1,(iii)𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) is not a geometric lattice when 2≤𝑚₁−𝑘₁≤𝑚₂−𝑘₂≤2𝜈−2.

Proof. (i) For 𝑋∈𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈), define 𝑟_𝑅(𝑋)=0,if𝑋=𝐹_𝑞^{((2𝜈+𝑙))},𝑚₂−𝑚₁+2,if𝑚𝑋=0,₂+1−dim𝑋,otherwise. (17)
For lattice 𝐿_𝑅(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈).
If 𝑘=0, by Lemma 7, 𝐿_𝑅(1,0,0;2𝜈−𝑙,𝜈−1,0;2𝜈+𝑙,𝜈)≃𝐿_𝑅(1,0,2𝜈−1,𝜈−1;2𝜈), by [12], 𝐿_𝑅(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈) is a finite geometric lattice.
If 𝑘=𝑙, by Lemma 7, 𝐿_𝑅(𝑙+1,0,𝑙;2𝜈−𝑙+𝑙,𝜈−1,𝑙;2𝜈+𝑙,𝜈)≃𝐿_𝑅(1,0,2𝜈−1,𝜈−1;2𝜈), by [12], 𝐿_𝑅(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈) is a finite geometric lattice.
If 0 < 𝑘< 𝑙, let 𝑈 = ⟨𝑒₁,…,𝑒_𝜈−1,𝑒_𝜈+1,…,𝑒_2𝜈,𝑒_2𝜈+1,…,𝑒_2𝜈+𝑘⟩,𝑊=⟨𝑒₂,…,𝑒_𝜈,𝑒_𝜈+2,…,𝑒_2𝜈+1,𝑒_2𝜈+2,…,𝑒_2𝜈+𝑘⟩. Then 𝑈, 𝑊 both are of type (2𝜈−1+𝑘,𝜈−1,𝑘), 𝑈∧𝑊=⟨𝑈,𝑊⟩ is of type (2𝜈+1+𝑘,𝜈,𝑘+1), 𝑈∨𝑊=𝑈∩𝑊 is of type (2𝜈−3+𝑘,𝜈,𝑘). We have 𝑟(𝑈∧𝑊)+𝑟(𝑈∨𝑊)=3>𝑟(𝑈)+𝑟(𝑊)=2. (18)
Hence, 𝐿_𝑅(𝑘+1,0,𝑘;2𝜈−1+𝑘,𝜈−1,𝑘;2𝜈+𝑙,𝜈) is not a geometric lattice when 0<𝑘<𝑙.
(ii) if 𝑘=0, 𝑙−1, by Lemma 7  𝐿_𝑅(𝑘,0,𝑘;2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈)≃𝐿_𝑅(𝑘,𝑙) when 𝑘=1, 𝑙−1, 𝐿_𝑅(𝑘,0,𝑘;2𝜈+𝑘,𝜈,𝑘;2𝜈+𝑙,𝜈) is a geometric lattice.
If 2≤𝑘≤𝑙−2, by [7], 𝐿_𝑅(𝑘,𝑙) is not a geometric lattice.
(iii) Case (a). if 𝑚₁−𝑘₁<𝜈+𝑠₁, then 𝜈+𝑠₁−𝑚₁+𝑘₁−1≥0.
Let𝐼𝑈=^(𝑠1)0000𝑠₁00000000000𝐼^(𝑠1)0000100000000𝐼^{(𝑚1−𝑘1−2𝑠1)}0000000000000𝐼^(𝑘1)1𝑚₁−𝑘₁−2𝑠₁𝜈+𝑠₁−𝑚₁+𝑘₁−1𝑠₁1𝑚₁−𝑘₁−2𝑠₁𝜈+𝑠₁−𝑚₁+𝑘₁𝑘₁00000𝑙−𝑘₁,000001𝐼𝑊=^(𝑠1)00000000000𝐼^(𝑠1)000000100000𝐼^{(𝑚1−𝑘1−2𝑠1)}0000000000000𝐼^(𝑘1)𝑠₁𝑚₁−𝑘₁−2𝑠₁𝜈+𝑠₁−𝑚₁+𝑘₁−11𝑠₁𝑚₁−𝑘₁−2𝑠₁𝜈+𝑠₁−𝑚₁+𝑘₁−1𝑘₁00000𝑙−𝑘₁. (19)
Then 𝑈,𝑊 are of type (𝑚₁+1,𝑠₁,𝑘₁), 𝑈∩𝑊 is of type (𝑚₁,𝑠₁−1,𝑘₁), ⟨𝑈,𝑊⟩ is of type (𝑚₁+1,𝑠₁,𝑘₁), ⟨𝑈,𝑊⟩∈𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈), 𝑈∩𝑊∉𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈).
So 𝑟_𝑅(𝑈∨𝑊)=𝑚₂−𝑚₁+2, 𝑟_𝑅(𝑈∧𝑊)=𝑚₂−𝑚₁−1, 𝑟_𝑅(𝑈)=𝑚₂+1−(𝑚₁+1)=𝑚₂−𝑚₁=𝑟_𝑅(𝑊). We have 𝑟_𝑅(𝑈∧𝑊)+𝑟_𝑅(𝑈∨𝑊)=2𝑚₂−2𝑚₁+1>𝑟_𝑅(𝑈)+𝑟_𝑅(𝑊)=2𝑚₂−2𝑚₁. (20) Hence 𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) is not a geometric lattice when (a).
Case (b). 𝑚₁−𝑘₁=𝜈+𝑠₁, from 𝑚₁−𝑘₁=𝜈+𝑠₁, and 2≤𝑚₁−𝑘₁≤2𝜈−2, we have 𝜈−2≥𝑠₁, 𝑚₁−𝑘₁−2𝑠₁−2=𝜈−𝑠₁−2≥0.
Let𝐼𝑈=^(𝑠1)000000𝑠₁00000000000000𝐼^(𝑠1)000001000000000010000000000𝐼^{(𝑚1−𝑘1−2𝑠1)}00000000000001000000000000𝐼^(𝑘1)11𝑚₁−𝑘₁−2𝑠₁𝜈+𝑠₁−𝑚₁+𝑘₁𝑠₁11𝑚₁−𝑘₁−2𝑠₁𝜈+𝑠₁−𝑚₁+𝑘₁𝑘₁0000000𝑙−𝑘₁,𝐼𝑊=^(𝑠1)000000𝑠₁00000000000000𝐼^(𝑠1)000000000010000010000000000𝐼^{(𝑚1−𝑘1−2𝑠1)}00000000000001000000000000𝐼^(𝑘1)11𝑚₁−𝑘₁−2𝑠₁𝜈+𝑠₁−𝑚₁+𝑘₁𝑠₁11𝑚₁−𝑘₁−2𝑠₁𝜈+𝑠₁−𝑚₁+𝑘₁𝑘₁0000000𝑙−𝑘₁. (21)
Then 𝑈, 𝑊 are of type (𝑚₁+1,𝑠₁+1,𝑘₁), 𝑈∩𝑊 is of type (𝑚₁,𝑠₁+1,𝑘₁), and ⟨𝑈,𝑊⟩ is of type (𝑚₁+2,𝑠₁+2,𝑘₁).
So 𝑟_𝑅(𝑈∨𝑊)=𝑚₂−𝑚₁, 𝑟_𝑅(𝑈∧𝑊)=𝑚₂+1−(𝑚₁+2)=𝑚₂−𝑚₁−1, 𝑟_𝑅(𝑈∧𝑊)+𝑟_𝑅(𝑈∨𝑊)=2𝑚₂−2𝑚₁+1>𝑟_𝑅(𝑈)+𝑟_𝑅(𝑊)=2𝑚₂−2𝑚₁.
Hence 𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) is not a geometric lattice when (b).
Hence 𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) is not a geometric lattice when 2≤𝑚₁−𝑘₁≤𝑚₂−𝑘₂≤2𝜈−2.

4. Characteristic Polynomial of Lattice 𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈)

In this section we compute the characteristic polynomial of the lattice 𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈).

Theorem 10. Let 2𝜈+𝑙>0, Assume that (𝑚,𝑠,𝑘) satisfies 0≤𝑘≤𝑙, 2𝑠≤𝑚−𝑘≤𝜈+𝑠 and 0<𝑚<2𝜈+𝑙. Then 𝜒𝐿_𝑅𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈,𝑡=𝑡^{𝑚2−𝑚1+2}−^𝑘_𝑘=𝑘1^𝑠2_𝑠=𝑠1^{𝑚1−(𝑘−𝑘1)+(𝑠−𝑠1)}_{𝑚=𝑚2−𝑘2+𝑠−𝑘2−𝑠+1}𝑁(𝑚,𝑠,𝑘;2𝜈+𝑙,𝜈)𝑔_𝑚(𝑡)+^𝑘1𝑠_𝑘=0¹⁻¹_𝑠=0^{𝜈+𝑠+𝑘}_{𝑚=2𝑠+𝑘}𝑁(𝑚,𝑠,𝑘;2𝜈+𝑙,𝜈)𝑔_𝑚(𝑡)+^𝑙_𝑘=𝑘2^𝜈_𝑠=𝑠2+1^{𝜈+𝑠+𝑘}_{𝑚=2𝑠+𝑘}𝑁(𝑚,𝑠,𝑘;2𝜈+𝑙,𝜈)𝑔_𝑚(𝑡), (22) where 𝑔_𝑚(𝑡)=∏^𝑚−1_𝑖=0(𝑡−𝑞^𝑖).

Proof. Define 𝑟_𝑅(𝑋)=0,if𝑋=𝐹_𝑞^(2𝜈+𝑙),𝑚₂−𝑚₁+2,if{0},𝑚𝑋=₂+1−dim(𝑋),otherwise. (23)
Then 𝑟_𝑅 is the rank function on 𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈). Let 𝑉=𝐹_𝑞^(2𝜈+𝑙), 𝐿_𝑂=𝐿_𝑅(2𝜈+𝑙,𝜈), 𝐿=𝐿_𝑅(𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈) we have 𝜒𝐿_𝑂,𝑡=^{2𝜈+𝑙−1}_𝑖=0𝑡−𝑞^𝑖. (24) For any 𝑃∈𝐿, define 𝐿^𝑃={}={},𝐿𝑄∈𝐿∣𝑄⊂𝑃𝑄∈𝐿∣𝑄≥𝑃_𝑃𝑂=𝑄∈𝐿_𝑂∣𝑄⊂𝑃=𝑄∈𝐿_𝑂∣𝑄≥𝑃. (25) Clearly, 𝐿^𝑉=𝐿, 𝐿^𝑃=𝐿^𝑃_𝑂, when 𝑃≠{0}, 𝑃≠𝑉𝜒𝐿^𝑉,𝑡=𝜒(𝐿,𝑡)=_𝑃∈𝐿𝜇(0,𝑃)𝑡^{𝑟(1)−𝑟(𝑃)}=_𝑃∈𝐿𝜇(0,𝑃)𝑡^{𝑟(0)−𝑟(𝑃)}. (26) By inversion to Möbius we have 𝜒(𝐿,𝑡)=𝜒𝐿^𝑉,𝑡=𝑡^{𝑚2−𝑚1+2}−_{{𝑉}𝑃∈𝐿⧵}𝜒𝐿^𝑃,𝑡, (27) by Theorem 5 we have 𝜒𝐿_𝑅𝑚₁,𝑠₁,𝑘₁;𝑚₂,𝑠₂,𝑘₂;2𝜈+𝑙,𝜈,𝑡=𝑡^{𝑚2−𝑚1+2}−^𝑘2_𝑘=𝑘1^𝑠2_𝑠=𝑠1^{𝑚1−(𝑘−𝑘1)+(𝑠−𝑠1)}_{𝑚=𝑚2−𝑘2+𝑠−𝑘2−𝑠+1}𝑁(𝑚,𝑠,𝑘;2𝜈+𝑙,𝜈)𝑔_𝑚(𝑡)+^𝑘1𝑠_𝑘=0¹⁻¹_𝑠=0^{𝜈+𝑠+𝑘}_{𝑚=2𝑠+𝑘}𝑁(𝑚,𝑠,𝑘;2𝜈+𝑙,𝜈)𝑔_𝑚(𝑡)+^𝑙_𝑘=𝑘2^𝜈_𝑠=𝑠2+1^{𝜈+𝑠+𝑘}_{𝑚=2𝑠+𝑘}𝑁(𝑚,𝑠,𝑘;2𝜈+𝑙,𝜈)𝑔_𝑚(𝑡). (28)

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).