基于有限辛空间的一致偏序集和Leonard对

设F_q为q个元素的有限域,q是一个素数的幂.令F_q~((2v))是F_q上的2v维辛空间,M(m,s;2v)表示辛群作用在F_q~((2v))上的子空间的轨道.L(m,s;2v)是M(m,s;2v)的子空间生成的集合.若按照子空间的包含关系来规定L(m,s;2v)的序,则得一偏序集,记为L_O(m,s;2v).本文,首先构造了L(m,s;2v)上的子偏序集L_O(m,s;2v),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.

Uniform Posets and Leonard Pairs Based on Symplectic Spaces over Finite Fields

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.