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

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.
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.