Rank-generating functions and Poincaré polynomials of lattices in finite orthogonal space of even characteristic

Abstract

The geometry of classical groups over finite fields is widely used in many fields. In this paper, we study the rank-generating function, the characteristic polynomial, and the Poincaré polynomial of lattices generated by the orbits of subspaces under finite orthogonal groups of even characteristic. We also determine their expressions.

Keywords

Lattices / orthogonal space of even characteristic / rank-generating function / characteristic polynomial / Poincaré polynomial

1 Introduction

Earlier literature has introduced and studied the rank-generating functions and characteristic polynomials in partially ordered sets and lattices, as seen in references [1, 2, 7]. Typical group geometry over finite fields displays sound properties, making it widely used in many fields. For example, reference [5] studied the critical problem of non-singular subspaces in orthogonal spaces of even characteristic. References [3, 4] studied the rank function and characteristic polynomial of lattices generated by orbits of subspaces under finite classical groups, and provided corresponding definitions and expressions. Reference [9] studied the rank generating functions and characteristic polynomials generated by orbits of subspaces of finite linear groups, finite symplectic groups, and finite unitary groups under {\mathbb{F}}_q^n,{\mathbb{F}}_q^{2\nu },{\mathbb{F}}_{q^2}^n, respectively. Reference [6] studied the rank-generating functions and characteristic polynomials in finite orthogonal spaces of even characteristic.

This paper draws on some research methods from reference [6], supplements the rank-generating function and Poincaré polynomial of lattices generated by the orbits of subspaces when the finite orthogonal group of even characteristic O_{2\nu +\delta }\left({\mathbb{F}}_q\right) is under {\mathbb{F}}_q^{\left(2\nu +\delta \right)} in reference [9], and presents its characteristic polynomial again. Moreover, the calculation method is simpler than that in reference [9]. This paper adopts the terms, notations, and some results from references [1, 6, 8, 9].

2 Preliminaries

As seen from references [5, 8, 9], assume q is the power of 2, and {\mathbb{F}}_q is a finite field with q elements. Let n=2\nu +\delta, where \nu is a non-negative integer while \delta =0,1,2. Introduce the notation

\mathrm{\Delta }=\{\begin{array}{l}\varnothing ,\mathrm{i}\mathrm{f}\delta =0;\\ 1,\mathrm{i}\mathrm{f}\delta =1;\\ \left(\begin{array}{cc}\alpha & 1\\ & \alpha \end{array}\right),\mathrm{i}\mathrm{f}\delta =2.\end{array}

Assume G_2\nu +\delta =\left[\left(\begin{array}{cc}0& I^{\left(\nu \right)}\\ 0& 0\end{array}\right),\mathrm{\Delta }\right], then \nu is said to be the exponent of G_2\nu +\delta, while \mathrm{\Delta } is its definite part [5, 8, 9]. The group constructed through multiplication of the complete \left(2\nu +\delta \right)\times \left(2\nu +\delta \right) matrix T on {\mathbb{F}}_q satisfying T{G}_{2\nu +\delta }{T}^{\mathrm{T}}\equiv G_{2\nu +\delta } is called an orthogonal group of 2\nu +\delta order with respect to G_{2\nu +\delta } on {\mathbb{F}}_q, denoted by O_{2\nu +\delta }\left({\mathbb{F}}_q\right). The effect of row vector space {\mathbb{F}}_q^{\left(2\nu +\delta \right)} of 2\nu +\delta dimension and orthogonal group O_{2\nu +\delta }\left({\mathbb{F}}_q\right) on it is jointly called finite orthogonal space of 2\nu +\delta dimension of even characteristic on {\mathbb{F}}_q, denoted by {\mathbb{F}}_q^{\left(2\nu +\delta \right)}.

Assume P is a subspace of m dimension of {\mathbb{F}}_q^{\left(2\nu +\delta \right)}. An m\times \left(2\nu +\delta \right) matrix of m rank is also expressed as P. Its row vector generates a subspace P, then we say P is the matrix representation of the subspace P. One subspace P of m dimension in the orthogonal space {\mathbb{F}}_q^{\left(2\nu +\delta \right)} of 2\nu +\delta dimension is said to be \left(m,2s+\gamma ,s,\Gamma \right)-type subspace with respect to {\mathit{G}}_{2\nu +\delta } if P{\mathit{G}}_{2\nu +\delta }{P}^{\mathrm{T}} displays congruence at [5, 8, 9].

\mathit{M}\left(m,2s+\gamma ,s\right)=\{\begin{array}{ll}\left[{\mathit{G}}_{2s},0^{\left(m-2s\right)}\right],& \gamma =0;\\ \left[{\mathit{G}}_{2s},0^{\left(m-2s\right)},1\right],& \gamma =1;\\ \left[{\mathit{G}}_{2s},\left(\begin{array}{cc}\alpha & 1\\ & \alpha \end{array}\right),0^{\left(m-2s\right)}\right],& \gamma =2.\end{array}

\begin{array}{rl}& G_{2s}=\left(\begin{array}{cc}0& I^{\left(s\right)}\\ & 0\end{array}\right).\\ & \Gamma =\{\begin{array}{ll}0\mathrm{o}\mathrm{r}1,\mathrm{i}\mathrm{f}{e}_{2\nu +1}\notin P\mathrm{o}\mathrm{r}{e}_{2\nu +1}\in P,& \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\delta =\gamma =1;\\ \varphi \left(\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{b}\mathrm{e}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{d}\right),& \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\delta \ne 1\mathrm{o}\mathrm{r}\gamma \ne 1.\end{array}\end{array}

Here, e_{2v+1} is the row vector of 2\nu +\delta dimension with the (2\nu +1)th component as 1 and the remaining components as 0. If \delta \ne 1 or \gamma \ne 1, then the \left(m,2s+\gamma ,s,\Gamma \right)-type subspace is denoted by \left(m,2s+\gamma ,s\right), where s is the exponent of \mathit{M}\left(m,2s+\gamma ,s\right)[9], 0⩽s⩽\left[\frac{m-\gamma }{2}\right], \gamma =0,1,2.

Proposition 2.1 [5, 8, 9]　 The \left(m,2s+\gamma ,s,\Gamma \right)-type subspace with respect to {\mathit{G}}_{2v+\delta } in {\mathbb{F}}_q^{\left(2\nu +\delta \right)} exists if and only if

(2.1)2s+\gamma ⩽m⩽\{\begin{array}{ll}\nu +s+\mathrm{m}\mathrm{i}\mathrm{n}\{\gamma ,\delta \},& if\delta \ne 1or\gamma \ne 1,or\gamma =\delta =1and\Gamma =1;\\ \nu +s,& if\gamma =\delta =1and\Gamma =0.\end{array}

\mathcal{M}=\mathcal{M}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) is used to represent the set of complete (m,2s+\gamma ,s,\Gamma )-type subspaces with respect to {\mathit{G}}_{2v+\delta } in {\mathbb{F}}_q^{\left(2\nu +\delta \right)} and |\mathcal{M}(m,2s+\gamma ,s,\Gamma ;2\nu +\delta )|=N\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right). N\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) is given by [8, Theorem 7.28]. When \delta \ne 1or \gamma \ne 1, it is denoted by \mathcal{M}\left(m,2s+\gamma ,s,2\nu +\delta \right). \mathcal{M} is an orbit generated by a set of subspaces in {\mathbb{F}}_q^{\left(2\nu +\delta \right)} under orthogonal group O_{2\nu +\delta }\left({\mathbb{F}}_q\right) [8, 9].

Based on similar literature [6, 9], let \mathcal{L}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) be the set generated by the intersection of all non-empty subspaces in \mathcal{M} and assume {\mathbb{F}}_q^{\left(2\nu +\delta \right)} is the intersection of zero subspaces in \mathcal{M}. If the partial order of \mathcal{L}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) is determined to be ⩾ according to the revere inclusion of subspaces, then \mathcal{L}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) is a finite lattice, called lattice generated by subspace orbit \mathcal{M}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) under orthogonal group O_{2v+\delta }\left({\mathbb{F}}_q\right), denoted by \mathcal{L}\left(\mathcal{M}\right)=\mathcal{L}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right). From [9, Corollary 6.18], it can be seen that the minimum element of \mathcal{L}\left(\mathcal{M}\right) is {\mathbb{F}}_q^{\left(2\nu +\delta \right)} while the maximum element is {\bigcap }_X\in \mathcal{M}X=\left\{0\right\} [8, 9].

Proposition 2.2 [9]　 Assume 2\nu +\delta >m⩾1, and \left(m,2s+\gamma ,s,\Gamma \right) satisfies (2.1). If

(2.2)

holds, then \mathcal{L}\left(\mathcal{M}\right) consists of {\mathbb{F}}_q^{\left(2\nu +\delta \right)} and all \left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1\right)-type subspaces satisfying

(2.3)2m-2m_1⩾\left(2s+\gamma \right)-\left(2s_1+{\gamma }_1\right)+\left|\gamma -{\gamma }_1\right|⩾2\left|\gamma -{\gamma }_1\right|

and {\Gamma }_1\ne1.

In the text below, for the sake of convenience, it is denoted by

f\left(x,y;{\gamma }_1\right)=\sum _{m_1=2s_1+{\gamma }_1}^{m-s+s_1-x}\sum _{s_1=0}^{\mathrm{m}\mathrm{i}\mathrm{n}\{\left[\left(m_1-{\gamma }_1\right)/2\right],s-y\}}N\left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1;2\nu +\delta \right),

where \left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1\right) satisfies (2.2) and (2.3) in Proposition 2.2 but {\Gamma }_1\ne 1,{\gamma }_1=0,1,2.

Lemma 2.1　 Let 1⩽m_1⩽m⩽2\nu +\delta -1. Subspaces \left(m,2s+\gamma ,s,\Gamma \right) and (m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1) on {\mathbb{F}}_q^{\left(2\nu +\delta \right)} satisfy (2.2) and (2.3). Assume P\in \mathcal{L}\left(\mathcal{M}\right) and P\ne {\mathbb{F}}_q^{\left(2\nu +\delta \right)}, \mathrm{d}\mathrm{i}\mathrm{m}P=m_1, then the number N_{\gamma }\left(P\right) of P in \mathcal{L}\left(\mathcal{M}\right) satisfies

(1) if \gamma =0, then N_0\left(P\right)=f\left(0,0;0\right)+f\left(0,1;1\right)+f\left(0,2;2\right);

(2) if \gamma =1, then N_1\left(P\right)=f\left(1,0;0\right)+f\left(0,0;1\right)+f\left(0,1;2\right);

(3) if \gamma =2, thenN_2\left(P\right)=f\left(2,0;0\right)+f\left(1,0;1\right)+f\left(0,0;2\right).

Proof　Prove (3) only, as it is similar to cases (1) and (2). Computations for the three cases of {\gamma }_1=0,{\gamma }_1=1,{\gamma }_1=2 are conducted below.

(a) \gamma =2,{\gamma }_1=0. Since subspaces\left(m,2s+\gamma ,s,\Gamma \right) and \left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1\right) satisfy (2.2) and (2.3) holds, (2.2) becomes

(2.4)

(2.3) becomes

(2.5)m-m_1\ge s-s_1+2⩾2.

Given m_1, then 0⩽s_1⩽\mathrm{m}\mathrm{i}\mathrm{n}\{\left[\frac{m_1}{2}\right],s\} can be obtained through (2.4) and (2.5). So, when \left(m_1,2s_1,s_1\right) satisfies (2.2) and (2.3), the number of \left(m_1,2s_1,s_1\right)-type subspaces in \mathcal{L}\left(\mathcal{M}\right) is

\sum _{s_1=0}^{\mathrm{m}\mathrm{i}\mathrm{n}\{\left[m_1/2\right],s\}}N\left(m_1,2s_1,s_1;2\nu +\delta \right).

From (2.5), there is 0⩽m_1⩽m-s+s_1-2, and , and further through (2.4), it can be obtained that 2s_1⩽m_1⩽m-s+s_1-2. Therefore, when \left(m_1,2s_1,s_1\right) satisfies (2.2) and (2.3), the number of P\ne {\mathbb{F}}_q^{\left(2\nu +\delta \right)} in \mathcal{L}\left(\mathcal{M}\right) is

(2.6){\sum }_{m_1=2s_1}^{m-s+s_1-2}{\sum }_{s_1=0}^{\mathrm{m}\mathrm{i}\mathrm{n}\{\left[m_1/2\right],s\}}N\left(m_1,2s_1,s_1;2\nu +\delta \right)=f\left(2,0;0\right).

(b) \gamma =2, {\gamma }_1=1. Here, (2.2) becomes

(2.3) becomes m-m_1⩾s-s_1+1⩾1.

As (a) is proved, when (m_1,2s_1+1,s_1,{\Gamma }_1) satisfies (2.2) and (2.3), the number of P\ne {\mathbb{F}}_q^{\left(2\nu +\delta \right)} in \mathcal{L}\left(\mathcal{M}\right) is

(2.7){\sum }_{m_1=2s_1+1}^{m-s+s_1-1}{\sum }_{s_1=0}^{\mathrm{m}\mathrm{i}\mathrm{n}\{\left[\left(m_1-1\right)/2\right],s\}}N\left(m_1,2s_1+1,s_1,{\Gamma }_1;2\nu +\delta \right)=f\left(1,0;1\right).

(c) \gamma =2,{\gamma }_1=2. Here, (2.2) becomes , and (2.3) becomes m-m_1⩾s-s_1⩾0.

As (a) is proved, when \left(m_1,2s_1+2,s_1\right) satisfies (2.2) and (2.3), the number of P\ne {\mathbb{F}}_q^{\left(2\nu +\delta \right)} in \mathcal{L}\left(\mathcal{M}\right) is

(2.8){\sum }_{m_1=2s_1+2}^{m-s+s_1}{\sum }_{s_1=0}^{\mathrm{m}\mathrm{i}\mathrm{n}\{\left[\left(m_1-2\right)/2\right],s\}}N\left(m_1,2s_1+2,s_1;2\nu +\delta \right)=f\left(0,0;2\right).

Lemma 2.1 (3) is proved based on (2.6)‒(2.8).

3 Rank-generating function of \mathcal{L}\left(\mathit{m}\mathbf{,}\mathbf{2}\mathit{s}\mathbf{+}\mathit{\gamma }\mathbf{,}\mathit{s}\mathbf{,}\mathit{\Gamma }\mathbf{;}\mathbf{2}\mathit{\nu }\mathbf{+}\mathit{\delta }\right)

Definition 3.1 [6, 9]　If P is a graded partially ordered set of rank n, and there are p_i elements whose rank is i, then we say polynomial

F\left(P,q\right)=\sum _{i=0}^n{p}_i{q}^i

is the rank-generating function of P. When P is a lattice L,F\left(P,q\right) is said to be the rank-generating function of lattice L.

Lemma 3.1 [6, 9]　 Assume n=2\nu +\delta >m⩾1, and \left(m,2s+\gamma ,s,\Gamma \right) satisfies (2.2). Define X\in \mathcal{L}\left(\mathcal{M}\right)= \mathcal{L}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) as

(3.1)r\left(X\right)=\{\begin{array}{ll}m+1-\mathrm{d}\mathrm{i}\mathrm{m}X,& ifX\ne {\mathbb{F}}_q^{\left(2\nu +\delta \right)};\\ 0,& ifX={\mathbb{F}}_q^{\left(2\nu +\delta \right)}.\end{array}

Then r:\mathcal{L}\left(\mathcal{M}\right)\to \mathbb{N} is the rank function of \mathcal{L}\left(\mathcal{M}\right).

Theorem 3.1　 Let 1⩽m_1⩽m⩽2\nu +\delta -1, \mathcal{L}=\mathcal{L}\left(\mathcal{M}\right) and P\in \mathcal{L},Pis an \left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1\right)-type subspace on {\mathbb{F}}_q^{\left(2\nu +\delta \right)}. If subspaces \left(m,2s+\gamma ,s,\Gamma \right) and \left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1\right) satisfy (2.2), (2.3), then

(1) when \gamma =0, the rank-generating function of lattice \mathcal{L}\left(m,2s,s;2\nu +\delta \right) is

F\left(\mathcal{L},q\right)=f\left(0,0;0\right)q^{m+1-m_1}+f\left(0,1;1\right)q^{m+1-m_1}+f\left(0,2;2\right)q^{m+1-m_1}+1;

(2) when \gamma =1, the rank-generating function of lattice \mathcal{L}(m,2s+1,s,\Gamma ;2\nu +\delta ) is

F\left(\mathcal{L},q\right)=f\left(1,0;0\right)q^{m+1-m_1}+f\left(0,0;1\right)q^{m+1-m_1}+f\left(0,1;2\right)q^{m+1-m_1}+1;

(3) when \gamma =2, the rank-generating function of lattice \mathcal{L}\left(m,2s+2,s;2\nu +\delta \right) is

F\left(\mathcal{L},q\right)=f\left(2,0;0\right)q^{m+1-m_1}+f\left(1,0;1\right)q^{m+1-m_1}+f\left(0,0;2\right)q^{m+1-m_1}+1.

Proof　Let \mathcal{L}\left(\mathcal{M}\right)=\mathcal{L}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right). From Lemma 3.1, it follows that X\in \mathcal{L}\left(\mathcal{M}\right),r:\mathcal{L}\left(\mathcal{M}\right)\to \mathbb{N} is the rank function of the lattice. Assume P\in \mathcal{L}\left(\mathcal{M}\right), and \mathcal{L}\left(\mathcal{M}\right) comprises subspaces. If P={\mathbb{F}}_q^{\left(2\nu +\delta \right)}, r\left(P\right)=0, and the number of subspaces with a rank of 0 in \mathcal{L}\left(\mathcal{M}\right) is 1. If P\ne {\mathbb{F}}_q^{\left(2\nu +\delta \right)}, let \mathrm{d}\mathrm{i}\mathrm{m}P=m_1 and \left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1\right) satisfies (2.2), (2.3). It can be seen from [8] that N\left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1;2\nu +\delta \right)=\left|\mathcal{M}\left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1;2\nu +\delta \right)\right| is the number of \left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1\right)-type subspaces in {\mathbb{F}}_q^{\left(2\nu +\delta \right)} that satisfy (2.2). Then the number of subspaces with a rank of i= m+1-m_1 in \mathcal{L}\left(\mathcal{M}\right) is determined by that of \left(m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1\right)\left({\Gamma }_1\ne 1\right).

In this way, the number of subspaces with a rank of i=m+1-m_1\ne 0 in \mathcal{L}\left(\mathcal{M}\right) is given by the number of P\ne {\mathbb{F}}_q^{\left(2\nu +\delta \right)}. It follows through Lemma 2.1 that N_{\gamma }\left(P\right) \left(\gamma =0,1,2\right), and the number of subspaces with a rank of 0 in \mathcal{L}\left(\mathcal{M}\right) is 1, so the theorem is established according to Definition 3.1. □

4 Characteristic polynomial of \mathcal{L}\left(\mathit{m}\mathbf{,}\mathbf{2}\mathit{s}\mathbf{+}\mathit{\gamma }\mathbf{,}\mathit{s}\mathbf{,}\mathit{\Gamma }\mathbf{;}\mathbf{2}\mathit{\nu }\mathbf{+}\mathit{\delta }\right)

Definition 4.1 [6, 9]　Assume P is a finite partially ordered set with a minimum element 0 and a maximum element 1. There are rank function r and Möbius function \mu on P. Then, polynomial

\chi \left(P,x\right)=\sum _{a\in P}\mu \left(0,a\right)x^{r\left(1\right)-r\left(P\right)}

is said to be a characteristic polynomial on P.

Reference [9] presented the characteristic polynomial of \mathcal{L}\left(\mathcal{M}\right). Here, we will present it in a more simplified manner.

Theorem 4.1　 Assume n=2\nu +\delta >m⩾m_1⩾1, \left(m,2s+\gamma ,s,\Gamma \right) and (m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1) satisfy (2.2), (2.3). Then

(1) when \gamma =0, the characteristic polynomial \chi \left(\mathcal{L}\left(\mathcal{M}\right),x\right) on lattice \mathcal{L}\left(m,2s,s;2\nu +\delta \right) is

\begin{array}{rl}{g}_{m+1}\left(x\right)& +\sum _{m_1=0}^m{\left[\begin{array}{c}m+1\\ m_1\end{array}\right]}_q{g}_{m_1}\left(x\right)\\ & -f\left(0,0;0\right)g_{m_1}\left(x\right)-f\left(0,1;1\right)g_{m_1}\left(x\right)-f\left(0,2;2\right)g_{m_1}\left(x\right);\end{array}

(2) when \gamma =1, the characteristic polynomial \chi \left(\mathcal{L}\left(\mathcal{M}\right),x\right) on lattice \mathcal{L}(m,2s+1,s,\Gamma ;2\nu +\delta ) is

\begin{array}{rl}{g}_{m+1}\left(x\right)+& \sum _{m_1=0}^m{\left[\begin{array}{c}m+1\\ m_1\end{array}\right]}_q{g}_{m_1}\left(x\right)\\ & -f\left(1,0;0\right)g_{m_1}\left(x\right)-f\left(0,0;1\right)g_{m_1}\left(x\right)-f\left(0,1;2\right)g_{m_1}\left(x\right);\end{array}

(3) when \gamma =2, the characteristic polynomial \chi \left(\mathcal{L}\left(\mathcal{M}\right),x\right) on lattice \mathcal{L}(m,2s+2,s;2\nu +\delta ) is

\begin{array}{rl}{g}_{m+1}\left(x\right)& +\sum _{m_1=0}^m{\left[\begin{array}{c}m+1\\ m_1\end{array}\right]}_q{g}_{m_1}\left(x\right)\\ & -f\left(2,0;0\right)g_{m_1}\left(x\right)-f\left(1,0;1\right)g_{m_1}\left(x\right)-f\left(0,0;2\right)g_{m_1}\left(x\right),\end{array}

where g_{m+1}\left(x\right),g_{m_1}\left(x\right) is a Gauss polynomial, and x is unknown (see [9]).

Proof　Similar to the proof of [6, Theorem 3.2], it can be readily proved by citing Lemma 2.1. □

The following is a special case of Theorem 4.1.

Corollary 4.1　 Let n=2\nu +\delta >1. Then

\begin{array}{rl}& x\left(\mathcal{L}\left(2\nu +\delta -1,2\nu +\delta -1,\nu -1+\mathrm{m}\mathrm{i}\mathrm{n}\left\{\delta ,1\right\},\Gamma ,2\nu +\delta \right),x\right)\\ & ={\sum }_{s_1=0}^{\nu }N\left(\nu +{\mathrm{s}}_1+\delta ,{2s}_1+\delta ,s_1,\Gamma ;2\nu +\delta \right)g_{\mathrm{v}}\left(x\right)\gamma \left(x\right),\end{array}

where g_{\nu }\left(x\right) is a Gauss polynomial, and \gamma \left(x\right)\in \mathbb{Z}\left[x\right] is a monic polynomial.

5 Poincaré polynomial of \mathcal{L}\left(\mathit{m}\mathbf{,}\mathbf{2}\mathit{s}\mathbf{+}\mathit{\gamma }\mathbf{,}\mathit{s}\mathbf{,}\mathit{\Gamma }\mathbf{;}\mathbf{2}\mathit{\nu }\mathbf{+}\mathit{\delta }\right)

In the following, some results of Poincaré polynomial of lattices generated by the orbit of subspaces under a finite orthogonal group of even characteristic are discussed.

Assume 1⩽m⩽2\nu +\delta -1,\mathcal{M}=\mathcal{M}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) is the orbit of {\mathbb{F}}_q^{\left(2\nu +\delta \right)} under O_2\nu +\delta \left({\mathbb{F}}_q\right), and \mathcal{L}=\mathcal{L}\left(\mathcal{M}\right)is the lattice generated by \mathcal{M}. From [9, Proposition 1.3],\mathcal{L}\left(\mathcal{M}\right) has an Möbius function, and from [9, Corollary 6.18], {\bigcap }_{X\in \mathcal{M}}X=\left\{0\right\} (subspace with a dimension of 0), then \text{Poincaré} polynomial of the lattice \mathcal{L}\left(\mathcal{M}\right) can be defined.

Definition 5.1　For X\in \mathcal{L}\left(\mathcal{M}\right), the rank function r of \mathcal{L}\left(\mathcal{M}\right) is given by (3.1), and x is an indeterminate, then Poincaré polynomial of \mathcal{L}\left(\mathcal{M}\right) is

(5.1)\pi \left(\mathcal{L}\left(\mathcal{M}\right),x\right)={\sum }_{X\in \mathcal{L}\left(\mathcal{M}\right)}\mu \left({\mathbb{F}}_q^{\left(2\nu +\delta \right)},X\right){\left(-x\right)}^{r\left(X\right)}.

Theorem 5.1　 Assume 1⩽m⩽2\nu +\delta -1,\mathcal{M}=\mathcal{M}\left(m,2s+\gamma ,s,\Gamma ;2\nu +\delta \right) is the orbit of {\mathbb{F}}_q^{\left(2\nu +\delta \right)} under O_2\nu +\delta \left({\mathbb{F}}_q\right),\mathcal{L}\left(\mathcal{M}\right) is the lattice generated by \mathcal{M}. For X\in \mathcal{L}\left(\mathcal{M}\right), the rank function \tau of \mathcal{L}\left(\mathcal{M}\right) is given by (3.1), and x is an indeterminate, then

\pi \left(\mathcal{L}\left(\mathcal{M}\right),x\right)={\left(-x\right)}^{m+1}\chi \left(\mathcal{L}\left(\mathcal{M}\right),-x^{-1}\right).

Proof　The minimum element of the lattice \mathcal{L}\left(\mathcal{M}\right) is {\mathbb{F}}_q^{\left(2\nu +\delta \right)}, the maximum element is {\bigcap }_X\in \mathcal{M}X=\left\{0\right\}, the rank function r of \mathcal{L}\left(\mathcal{M}\right) is given by (3.1), and r\left(\left\{0\right\}\right)=m+1. Based on Definition 4.1, the characteristic polynomial of \mathcal{L}\left(\mathcal{M}\right) is

\begin{array}{rl}\chi \left(\mathcal{L}\left(\mathcal{M}\right),x\right)& =\sum _{X\in \mathcal{L}\left(\mathcal{M}\right)}\mu \left({\mathbb{F}}_q^{\left(2\nu +\delta \right)},X\right)x^{r\left(\left\{0\right\}\right)-r\left(X\right)}\\ & =\sum _{X\in \mathcal{L}\left(\mathcal{M}\right)}\mu \left({\mathbb{F}}_q^{\left(2\nu +\delta \right)},X\right)x^{m+1-r\left(X\right)}.\end{array}

Hence,

\chi \left(\mathcal{L}\left(\mathcal{M}\right),-x^{-1}\right)=\sum _{X\in \mathcal{L}\left(\mathcal{M}\right)}\mu \left({\mathbb{F}}_q^{\left(2\nu +\delta \right)},X\right){\left(-x\right)}^{-\left(m+1\right)+r\left(X\right)},

or

(5.2){\left(-x\right)}^{m+1}\chi \left(\mathcal{L}\left(\mathcal{M}\right),-x^{-1}\right)=\sum _{X\in \mathcal{L}\left(\mathcal{M}\right)}\mu \left({\mathbb{F}}_q^{\left(2\nu +\delta \right)},X\right){\left(-x\right)}^{r\left(X\right)}.

After comparison of (5.1) and (5.2), the Poincaré polynomial can be obtained through the available characteristic polynomial of the lattice \mathcal{L}\left(\mathcal{M}\right). □

If n⩽m+1, let

{\psi }_n\left(x\right)={\left(-x\right)}^{m+1}{g}_n\left(-x^{-1}\right)={\left(-x\right)}^{m-n+1}\left(1+x\right)\left(1+qx\right)\cdots \left(1+q^{n-1}x\right),

then with 2.1, Theorem 4.1 and Theorem 5.1, we have the following theorem.

Theorem 5.2　 Assume 1⩽m_1⩽m⩽2\nu +\delta -1,\left(m,2s+\gamma ,s,\Gamma \right) and (m_1,2s_1+{\gamma }_1,s_1,{\Gamma }_1) satisfy (2.2), (2.3), then

(1) when \gamma =0, the Poincaré polynomial of \mathcal{L}\left(m,2s,s;2\nu +\delta \right) is

\begin{array}{rl}{\psi }_{m+1}\left(x\right)& +\sum _{m_1=0}^m{\left[\begin{array}{c}m+1\\ m_1\end{array}\right]}_q{\psi }_{m_1}\left(x\right)-f\left(0,0;0\right){\psi }_{m_1}\left(x\right)\\ & -f\left(0,1;1\right){\psi }_{m_1}\left(x\right)-f\left(0,2;2\right){\psi }_{m_1}\left(x\right);\end{array}

(2) when \gamma =1, the Poincaré polynomial of \mathcal{L}\left(m,2s+1,s,\Gamma ;2\nu +\delta \right) is

\begin{array}{rl}{\psi }_{m+1}\left(x\right)& +\sum _{m_1=0}^m{\left(\begin{array}{c}m+1\\ m_1\end{array}\right)}_q{\psi }_{m_1}\left(x\right)-f\left(1,0;0\right){\psi }_{m_1}\left(x\right)\\ & -f\left(0,0;1\right){\psi }_{m_1}\left(x\right)-f\left(0,1;2\right){\psi }_{m_1}\left(x\right);\end{array}

(3) when \gamma =2, the Poincaré polynomial of \mathcal{L}\left(m,2s+2,s;2\nu +\delta \right) is

\begin{array}{rl}{\psi }_{m+1}\left(x\right)& +\sum _{m_1=0}^m{\left[\begin{array}{c}m+1\\ m_1\end{array}\right]}_q{\psi }_{m_1}\left(x\right)-f\left(2,0;0\right){\psi }_{m_1}\left(x\right)\\ & -f\left(1,0;1\right){\psi }_{m_1}\left(x\right)-f\left(0,0;2\right){\psi }_{m_1}\left(x\right).\end{array}

The following is a special case of Theorem 5.2.

Corollary 5.1　 Let n=2\nu +\delta >1. Then

\begin{array}{rl}& \pi \left(\mathcal{L}\left(2\nu +\delta -1,2\nu +\delta -1,\nu -1+\mathrm{m}\mathrm{i}\mathrm{n}\left\{\delta ,1\right\},\Gamma ;2\nu +\delta \right),x\right)\\ & =\sum _{s_1=0}^{\nu }N\left(\nu +{\mathrm{s}}_1+\delta ,{2s}_1+\delta ,s_1,\Gamma ;2\nu +\delta \right){\left(-x\right)}^{\nu -{\mathrm{s}}_1}{\varphi }_v\left(x\right)\gamma \left(x\right),\end{array}

where \gamma \left(x\right)\in \mathbb{Z}\left[x\right] is a monic polynomial.

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