| 高锁刚,薛慧娟,侯 波.基于有限辛空间的一致偏序集和Leonard对[J].数学年刊A辑,2018,39(1):95~112 |
| 基于有限辛空间的一致偏序集和Leonard对 |
| Uniform Posets and Leonard Pairs Based on Symplectic Spaces over Finite Fields |
| Received:January 20, 2015 Revised:December 05, 2015 |
| DOI：10.16205/j.cnki.cama.2018.0010 |
| 中文关键词: 有限域, 辛空间, 一致偏序集, Leonard对 |
| 英文关键词:Finite field, Symplectic space, Uniform poset, Leonard pair |
| 基金项目:本文受到国家自然科学基金 (No.11471097)和河北省自然科学基金(No.A2017403010)的资助. |
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| 中文摘要: |
| 设$\mathbb{F}_{q}$ 为$q$ 个元素的有限域,$q$ 是一个素数的幂.
令$\mathbb{F}_{q}^{(2\nu)}$ 是$\mathbb{F}_{q}$
上的$2\nu$维辛空间, ${\mathcal{M}(m,s;2\nu)}$
表示辛群作用在$\mathbb{F}_{q}^{(2\nu)}$
上的子空间的轨道.$\mathcal{L}{(m,s;2\nu)}$
是${\mathcal{M}(m,s;2\nu)}$ 的子空间生成的集合.
若按照子空间的包含关系来规定$\mathcal{L}{(m,s;2\nu)}$ 的序,
则得一偏序集, 记为$\mathcal{L}_{O}{(m,s;2\nu)}$. 本文,
首先构造了$\mathcal{L}{(m,s;2\nu)}$上的子偏序集$\mathcal{L}_{O}{(m,s;2\nu)}$,
然后证明这个子偏序集是强一致偏序的.
最后利用这个偏序集构造了Leonard对. |
| 英文摘要: |
| Let $\mathbb{F}_q^{(2\nu)}$ be the $2\nu$-dimensional symplectic
space over the finite field $\mathbb{F}_q$, and let
${\mathcal{M}(m,s;2\nu)}$ denote the orbit of subspaces of
$\mathbb{F}_q^{(2\nu)}$ under the symplectic group. Denote by
$\mathcal{L}{(m,s;2\nu)}$ the set of subspaces generated by
${\mathcal{M}(m,s;2\nu)}$. By ordering $\mathcal{L}{(m,s;2\nu)}$ by
ordinary inclusion, the poset denoted $\mathcal{L}_{O}{(m,s;2\nu)}$
is obtained. In this paper, the authors first construct the subposet of
$\mathcal{L}_{O}{(m,s;2\nu)}$. Then it is shown that this subposet is
strongly uniform and construct Leonard pairs from it. |
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