Linear Algebra and its Applications

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1 Linear Algebra and its Applications 431 (29) Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: Lattices associated with totally isotropic subspaces in classical spaces Jun Guo a, Zengti Li a, Kaishun Wang b, a Math. and Inf. College, Langfang Teachers College, Langfang 65, China b Sch. Math. Sci. and Lab. Math. Com. Sys., Beijing Normal University, Beijing 1875, China A R T I C L E I N F O A B S T R A C T Article history: Received 11 April 28 Accepted 1 April 29 Available online 12 May 29 Submitted by R.A. Brualdi AMS classification: 2G4 51D25 Keywords: Classical space Totally isotropic subspace Lattice Characteristic polynomial Let F (2ν+δ) be one of (2ν + δ)-dimensional classical spaces over the finite field F, where δ =, 1, or 2. For i <ν,letp denote a maximal totally isotropic subspace of F (2ν+δ), and let Q denote an (i + ω, ω)-totally isotropic subspace of F (2ν+δ) contained in P, here ω = or 1. Suppose L(Q, P, ω; 2ν + δ) denotes the set of all the totally isotropic subspaces U such that U P = Q including F (2ν+δ). Partially ordered by ordinary or reverse inclusion, two families of finite lattices are obtained. This paper discusses their atomic property, geometricity, and compute their characteristic polynomials. 29 Elsevier Inc. All rights reserved. 1. Introduction Now we recall some definitions and terminologies about finite poset and lattices. The reader is referred to [1,9] for details. Let P be a poset with partial order. As usual, we write a < b whenever a b and a /= b. For any two elements a, b P, wesaya covers b, denoted by b < a, ifb < a and there exists no c P such that b < c < a. IfP has the minimum (resp. maximum) element, then we denote it by (resp. 1). In this case we say that P is a poset with (resp. 1). Let P be a finite poset with. By a rank function on P, we mean a function r from P to the set of all the nonnegative integers such that Corresponding author. address: wangks@bnu.edu.cn (K. Wang) /$ - see front matter 29 Elsevier Inc. All rights reserved. doi:1.116/j.laa

2 J. Guo et al. / Linear Algebra and its Applications 431 (29) (i) r() =. (ii) r(a) = r(b) + 1 whenever b < a. Observe the rank function of P is uniue if it exists. Let P be a finite poset with and 1. The polynomial χ(p, t) = a P μ(, a)t r(1) r(a) is called the characteristic polynomial of P, where r is the rank function on P. A poset L 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 L.LetL be a finite lattice with. By an atom in L, we mean an element in L covering. We say L is atomic if any element in L \{} is a union of atoms. A finite atomic lattice L is said to be geometric if L admits a rank function r satisfying r(a b) + r(a b) r(a) + r(b) for any two distinct elements a, b L. The results on the lattices generated by orbits of subspaces under finite classical groups may be found in Huo, Liu and Wan [4 6], Huo and Wan [7 9], Gao and You [2,3], Wang and Feng [11], Wang and Guo [12,13], Wang and Li [14]. In this paper, we construct two families of finite lattices from totally isotropic subspaces in classical spaces, discuss their atomic property, geometricity, and compute their characteristic polynomials. 2. Classical spaces In this section we shall first introduce the concepts of totally isotropic subspaces in classical spaces, and then introduce our main results. Notation and terminology will be adopted from Wan s book [1]. Suppose ( I K = (ν) ) ( I I (ν), S s, = (s) ) I (s), ( ) s, Ss, S s,1 =, S 1 s,2 = S The symplectic group of degree 2ν over F, denoted by Sp 2ν (F ), consists of all 2ν 2ν matrix T over F satisfying TKT t = K. The row vector space F (2ν) together with the right multiplication action of Sp 2ν (F ) is called the 2ν-dimensional symplectic space over F.Anm-dimensional subspace P in 2νdimensional symplectic space is said to be of type (m, s),ifpkp t is of rank 2s. In particular, subspaces of type (m, ) are called (m, )-totally isotropic subspaces, and (ν, )-totally isotropic subspaces are called maximal totally isotropic subspaces. Let = 2, where is a prime power. Then F has an involutive automorphism a ā = a. The unitary group of degree 2ν + δ (δ = or1)overf, denoted by U 2ν+δ (F ), consists of all (2ν + δ) (2ν + δ) matrix T over F satisfying TS ν,δ T t = S ν,δ. The row vector space F (2ν+δ) together with the right multiplication action of U 2ν+δ (F ) is called the (2ν + δ)-dimensional unitary space over F.An m-dimensional subspace P in (2ν + δ)-dimensional unitary space is said to be of type (m, r),ifps ν,δ P t is of rank r. In particular, subspaces of type (m, ) are called (m, )-totally isotropic subspaces, and (ν,)-totally isotropic subspaces are called maximal totally isotropic subspaces. Suppose is even. The pseudo-symplectic group of degree 2ν + δ (δ = 1or2) over F, denoted by Ps 2ν+δ (F ), consists of all (2ν + δ) (2ν + δ) matrix T over F satisfying TS ν,δ T t = S ν,δ. The row vector space F (2ν+δ) together with the right multiplication action of Ps 2ν+δ (F ) is called the (2ν + δ)- dimensional pseudo-symplectic space over F.Anm-dimensional subspace P in (2ν + δ)-dimensional

3 19 J. Guo et al. / Linear Algebra and its Applications 431 (29) pseudo-symplectic space is a subspace of type (m,2s + τ, s, ε), where τ =, 1, or 2 and ε = or1, if (i) PS ν,δ P t is cogredient to diag (S s,τ, (m 2s τ) ), and (ii) e 2ν+1 P or e 2ν+1 P accoding to ε = or 1, respectively. In particular, subspaces of type (m + ω,,, ω) are called (m + ω, ω)-totally isotropic subspaces, and (ν + ω, ω)-totally isotropic subspaces are called maximal totally isotropic subspaces, where ω = or 1 according to δ = 1 or 2, respectively. Denote by K 2ν+δ (δ =, 1 or 2) the set of all (2ν + δ) (2ν + δ) alternate matrices over F.Two (2ν + δ) (2ν + δ)matrices A and B over F are said to be congruent mod K 2ν+δ, denoted A B (mod K 2ν+δ ), if A B K 2ν+δ. Clearly, is an euivalence relation on the set of all (2ν + δ) (2ν + δ) matrices. Let [A] denote the euivalence class containing A. Two matrix classes [A] and [B] are said to be cogredient if there is a nonsingular (2ν + δ) (2ν + δ) matrix Q over F such that [QAQ t ] [B]. For being odd, let (, if δ =, Ss, S 2s+δ, Δ =, where Δ = (1) or (z), if δ = 1, Δ) diag(1, z), if δ = 2, where z is a fixed non-suare element of F.Forbeing even, let S 2s+δ, Δ = I(s), if δ =,, where Δ = ( (1), if δ = 1, α 1 Δ, if δ = 2, α) where α is a fixed element of F such that α/ {x 2 + x x F }. The orthogonal group of degree 2ν + δ over F with respect to S 2ν+δ, Δ, denoted by O 2ν+δ, Δ (F ), consists of all (2ν + δ) (2ν + δ) matrices T over F satisfying [TS 2ν+δ, Δ T t ] [S 2ν+δ, Δ ]. The row vector space F (2ν+δ) together with the right multiplication action of O 2ν+δ, Δ (F ) is called the (2ν + δ)-dimensional orthogonal space over F.Anm-dimensional subspace P in (2ν + δ)-dimensional orthogonal space is a subspace of type (m,2s + γ, s, Ɣ) if PS 2ν+δ, Δ P t is cogredient to diag (S 2s+γ, Ɣ, (m 2s γ) ). In particular, subspaces of type (m,, ) are called (m, )-totally isotropic subspaces, and (ν, )-totally isotropic subspaces are called maximal totally isotropic subspaces. LetF (2ν+δ) denote one of(2ν + δ)-dimensional classical spaces, and let G 2ν+δ be the corresponding classical group. For each i, we assume that e i denotes the (2ν + δ)-dimensional row vector whose i-th component is 1 and other components are s. For i <ν and ω = or1,letp denote a maximal totally isotropic subspace of F (2ν+δ), and let Q P denote an (i + ω, ω)-totally isotropic subspace of F (2ν+δ). Since G 2ν+δ acts transitively on the set of such pairs (P, Q ), we may assume that P = e 1, e 2,..., e ν, ωe 2ν+1, Q = e 1, e 2,..., e i, ωe 2ν+1. Let M(Q, P, m + ω, ω; 2ν + δ) denote the set of all (m + ω, ω)-totally isotropic subspaces Q of F (2ν+δ) L(Q, P, ω; 2ν + δ) = ν m=i satisfying P Q = Q, and let M(Q, P, m + ω, ω; 2ν + δ) {F (2ν+δ) }. Partially ordered by ordinary or reverse inclusion, L(Q, P, ω; 2ν + δ) is a poset, denoted by L O (Q, P, ω; 2ν + δ) or L R (Q, P, ω; 2ν + δ), respectively. For any two subspaces U, W L O (Q, P, ω; 2ν + δ), U W = U W, U W = {P L O (Q, P, ω; 2ν + δ) P U + W}.

4 J. Guo et al. / Linear Algebra and its Applications 431 (29) Similarly, for any two subspaces U, W L R (Q, P, ω; 2ν + δ), U W = {P L R (Q, P, ω; 2ν + δ) P U + W}, U W = U W. Therefore, both L O (Q, P, ω; 2ν + δ) and L R (Q, P, ω; 2ν + δ) are finite lattices. In this paper, we obtain the following results. Theorem 2.1. Given a (2ν + δ)-dimensional classical space. Let i <νand ω = or1. Then (i) L O (Q, P, ω; 2ν + δ) (resp. L R (Q, P, ω; 2ν + δ) except the orthogonal space F (2ν) ) is atomic. (ii) L O (Q, P, ω; 2ν + δ)(resp. L R (Q, P, ω; 2ν + δ)except the orthogonal space F (2ν) ) is geometric if and only if i = ν 1. Theorem 2.2. With the assumption of Theorem 2.1, the characteristic polynomial of L R (Q, P, ω; 2ν + δ) is ν χ(l R (Q, P, ω; 2ν + δ), t) = t ν+1 i M(Q, P, m + ω, ω; 2ν + δ) g m i (t), where g m i (t) = m i 1 l= (t l ), and M(Q, P, m + ω, ω; 2ν + δ) is given by Lemma 3.1. m=i 3. Some lemmas In this section we give some lemmas which are needed in the proof of Theorem 2.1. Lemma 3.1. Let i ν, i m ν, and ω = or 1. Then where M(Q, P, m + ω, ω; 2ν + δ) [ ] = (m i)(ν m) ν i m i (m i)(m i+1)/2, (m i)2 /2+(m i)δ, (m i)(m i 1)/2+(m i)δ, (m i)(m i+1)/2, [ ] ν i m i = ν i l=ν m+1 (l 1)/ m i l=1 (l 1). the symplectic case, the unitary case, the orthogonal case, the pseudo symplectic case, Proof. We only prove the theorem in the symplectic space, the proofs in other spaces are similar and will be omitted. Each Q M(Q, P, m,; 2ν) has a matrix representation of the form ( I (i) (i) ), (1) A 2 A 4 where both A 2 and A 4 are (m i) (ν i) matrices, rank A 4 = m i and A 2 A t 4 A 4A t 2 =. Clearly, A 4 is an (m i)-dimensional subspace of F (ν i). For any (m i)-dimensional subspace A 4 of F(ν i), since the transitivity of the symplectic group Sp 2ν (F ) on the set of all (m,)-totally isotropic subspaces, the number of subspaces of form (1) is eual to the number of subspaces with the form ( I (i) (i) ) A 2 A. 4

5 192 J. Guo et al. / Linear Algebra and its Applications 431 (29) Without loss of generality, we may pick A 4 = (I (m i), (m i, ν m) ). Then ( I Q = (i) (m i) (i) ) A 21 A 22 I (m i) (m i, ν m), (2) where A t = 21 A 21. Since the matrix representation of Q as the form (2) is uniue, [ ] ν M(Q, P, m,; 2ν) = (m i)(2ν m i+1)/2 i m. i Lemma 3.2. Let i ν, i m ν, and ω = or 1. ForanytwoQ, Q M(Q, P, m + ω, ω; 2ν + δ), there exists T G 2ν+δ such that P = P T and Q = QT. Proof. We only prove the theorem in the symplectic space, the proofs in other spaces are similar and will be omitted. Since P is a maximal totally isotropic subspace, both P + Q and P + Q are subspaces of type (ν + m i, m i). By the transitivity of Sp 2ν (F ) on the set of subspaces with the same type, there exists T Sp 2ν (F ) such that P = P T and Q = QT. Lemma 3.3. Let i ν, i m ν, and ω = or 1. Then for any Q M(Q, P, m + ω, ω; 2ν + δ), the number of subspaces containing Q in M(Q, P, ν + ω, ω; 2ν + δ) is (ν m)(ν m+1)/2, the symplectic case, ϑ = (ν m)(ν m+2δ)/2, the unitary case, (ν m)(ν m 1+2δ)/2, the orthogonal case, the pseudo symplectic case. (ν m)(ν m+1)/2, Proof. By Lemma 3.2, for any Q M(Q, P, m + ω, ω; 2ν + δ), the number of elements in M(Q, P, ν + ω, ω; 2ν + δ) containing Q is a constant. Let M be a binary matrix with row-indexed (resp. column-indexed) by M(Q, P, m + ω, ω; 2ν + δ) (resp. M(Q, P, ν + ω, ω; 2ν + δ)) such that M(A, B) = 1 if and only if A B. If we count the number of 1 s in the matrix by rows, by Lemma 3.1 we obtain M(Q, P, m + ω, ω; 2ν + δ) ϑ. If we count the number of 1 s in the matrix by columns, by Lemmas 3.1 and [1, Corollary 1.9], we obtain [ ] ν i M(Q, P, ν + ω, ω; 2ν + δ) m. i Therefore, [ ] ν i M(Q, P, ν + ω, ω; 2ν + δ) m i ϑ =. M(Q, P, m + ω, ω; 2ν + δ) The desired results follow by Lemma 3.1. Lemma 3.4. Let i + 2 ν and ω = or 1. For any Q M(Q, P, i ω, ω; 2ν + δ), the number of subspaces containing Q in M(Q, P, i ω, ω; 2ν + δ), denoted by ζ, is eual to, the symplectic case, [ ] ν ζ = ν i 2 i 1 1/2+δ, the unitary case, 1 δ, the orthogonal case,, the pseudo-symplectic case. Proof. The proof is similar to that of Lemma 3.3, and will be omitted.

6 J. Guo et al. / Linear Algebra and its Applications 431 (29) Proof of main results In this section we shall prove our main results. Proof of Theorem 2.1. For any U L(Q, P, ω; 2ν + δ), define { ν + 1 i, if U = F (2ν+δ) r O (U) =, dim U i ω, otherwise; {, if U = F r R (U) = (2ν+δ), ν ω dim U, otherwise. Then r O (resp. r R ) is the rank function on L O (Q, P, ω; 2ν + δ) (resp. L R (Q, P, ω; 2ν + δ)). (i) Note that Q is the minimum element and M(Q, P, i ω, ω; 2ν + δ) is the set of atoms of L O (Q, P, ω; 2ν + δ). Given any (m + ω, ω)-totally isotropic subspace U = e 1,..., e i, α 1,, α m i, ωe 2ν+1 L O (Q, P, ω; 2ν + δ), U j = e 1,..., e i, α j, ωe 2ν+1 is an atom for each j = 1,..., m i. Since U = U 1 U m i, the lattice L O (Q, P, ω; 2ν + δ) is atomic. Note that F (2ν+δ) is the minimum element and M(Q, P, ν + ω, ω; 2ν + δ) is the set of atoms of L R (Q, P, ω; 2ν + δ). For any U M(Q, P, ν 1 + ω, ω; 2ν + δ), by Lemma 3.3, there exist two distinct subspaces U 1, U 2 M(Q, P, ν + ω, ω; 2ν + δ) except the orthogonal space F (2ν) such that U = U 1 U 2. Now suppose that each element of M(Q, P, m ω, ω; 2ν + δ) is a union of some atoms, where m ν 2. For any U M O (Q, P, m + ω, ω; 2ν + δ), by Lemma 3.3 there [ exists ] a subspace U 1 M(Q, P, ν + ω, ω; 2ν + δ) containing U. By[1, Corollary 1.9], there exist ν m 1 2 elements of M(Q, P, m ω, ω; 2ν + δ) containing U and contained in U 1. For two distinct such elements V 1 and V 2, U = V 1 V 2 ; and so the lattice L R (Q, P, ω; 2ν + δ) is atomic. (ii) It is routine to show that L O (Q, P, ω; 2ν + δ) (resp. L R (Q, P, ω; 2ν + δ) except the orthogonal space F (2ν) ) is geometric when i = ν 1. Now suppose that i ν 2. For a given U M(Q, P, i ω, ω; 2ν + δ), by Lemma 3.4, the number of subspaces in M(Q, P, i ω, ω; 2ν + δ) containing U is eual to ζ.by[1, Corollary 1.9], for each U 1 M(Q, P, i ω, ω; 2ν + δ) containing U, the number of subspaces of U 1 contained in M(Q, P, i ω, ω; 2ν + δ) is eual to + 1. By Lemma 3.1, M(Q, P, i ω, ω; 2ν + δ) ζ 2. It follows that there exists V M(Q, P, i ω, ω; 2ν + δ) \{U} such that U + V / M(Q, P, i ω, ω; 2ν + δ); and so U V = F (2ν+δ). Therefore, r O (U V) + r O (U V) = ν i + 1 > 2 = r O (U) + r O (V). Hence L O (Q, P, ω; 2ν + δ) is not geometric whenever i ν 2. ForagivenU M(Q, P, ν + ω, ω; 2ν + δ), by[1, Corollary 1.9], the number of subspaces of U in M(Q, P, ν 1 + ω, ω; 2ν + δ) is eual to ( ν i 1)/( 1). For each U 1 M(Q, P, ν 1 + ω, ω; 2ν + δ) contained in U, by Lemma 3.3 the number of elements in M(Q, P, ν + ω, ω; 2ν + δ) containing U 1 is eual to, the symplectic case, α = 1/2+δ, the unitary case, δ, the orthogonal case,, the pseudo symplectic case. By Lemma 3.1, M(Q, P, ν + ω, ω; 2ν + δ) (α 1)( ν i 1)/( 1) 2. It follows that there exists V M(Q, P, ν + ω, ω; 2ν + δ) \{U} such that dim(u V) ν + ω 2. So r R (U V) + r R (U V) 3 > 2 = r R (U) + r R (V). Hence L R (Q, P, ω; 2ν + δ) is not geometric whenever i ν 2.

7 194 J. Guo et al. / Linear Algebra and its Applications 431 (29) Proof of Theorem 2.2. For convenience, we write L = L R (Q, P, ω; 2ν + δ). ForU L, let L U ={V L V U}. Note that L F(2ν+δ) = L. Since Q is the maximum element and F (2ν+δ) is the minimum element in L, the characteristic polynomial of L is χ(l, t) = U L By the Möbius inversion formula t ν+1 i = U L χ(l U, t). μ(f (2ν+δ), U)t ν+1 i r R(U). Hence χ(l, t) = t ν+1 i = t ν+1 i U L\{F (2ν+δ) } ν m=i χ(l U, t) M(Q, P, m + ω, ω; 2ν + δ) g m i (t), where g m i (t) = m i 1 l= (t l ), g (t) = 1. Remark. For i <νand ω = or1.let L(i + ω, P, ω; 2ν + δ) ν = Q 1 M(i+ω,ω;P ) m=i+1 M(Q 1, P, m + ω, ω; 2ν + δ) {{}, F (2ν+δ) }, where M(i + ω, ω; P ) is the set of all (i + ω, ω)-totally isotropic subspaces of F (2ν+δ) contained in P. Partially ordered by ordinary or reverse inclusion, L(i + ω, P, ω; 2ν + δ) is a finite lattice, respectively. We may obtain the results similar to Theorem 2.1 and Theorem 2.2. Acknowledgements This research is supported by NCET-8-52, NSF of China (187127), NSF of Hebei Province (A28128) and Educational Committee of Hebei Province (28142). References [1] M. Aigner, Combinatorial Theory, Springer-Verlag, Berlin, [2] Y. Gao, Lattices generated by orbits of subspaces under finite singular unitary group and its characteristic polynomials, Linear Algebra Appl. 368 (23) [3] Y. Gao, H. You, Lattices generated by orbits of subspaces under finite singular classical groups and its characteristic polynomials, Comm. Algebra 31 (23) [4] Y. Huo, Y. Liu, Z. Wan, Lattices generated by transitive sets of subspaces under finite classical groups I, Comm. Algebra 2 (1992) [5] Y. Huo, Y. Liu, Z. Wan, Lattices generated by transitive sets of subspaces under finite classical groups II, the orthogonal case of odd characteristic, Comm. Algebra 2 (1993) [6] Y. Huo, Y. Liu, Z. Wan, Lattices generated by transitive sets of subspaces under finite classical groups III, the orthogonal case of even characteristic, Comm. Algebra 21 (1993) [7] Y. Huo, Z. Wan, Lattices generated by transitive sets of subspaces under finite pseudo-symplectic groups, Comm. Algebra 23 (1995) [8] Y. Huo, Z. Wan, On the geomericity of lattices generated by orbits of subspaces under finite classical groups, J. Algebra 243 (21)

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