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.

Let Fq^（2v） be the 2 v-dimensional symplectic space over the finite field Fq, and let M（m, s; 2 v） denote the orbit of subspaces of Fq^（2v） under the symplectic group. Denote by L（m,s;2 v） the set of subspaces generated by M（m,s;2 v）. By ordering L（m,s;2 v） by ordinary inclusion, the poset denoted LO（m,s;2 v） is obtained. In this paper, the authors first construct the subposet of LO（m, s; 2 v）. Then it is shown that this subposet is strongly uniform and construct Leonard pairs from it.