Uniform Posets And Leonard Pairs Based On Symplectic Spaces Over Finite Fields

Let  be a field with q elements, where q is a power of a prime, and let Fq(2ν) be the 2ν-dimensional symplectic space over Fq, where v is a positive integer. In this thesis, we study the applications of symplectic spaces over finite fields to uniform posets and Leonard pairs. Let m, s be nonnegative integers such that 2s≤ m ≤ν+s and let M(m,s;2ν) denote the set of subspaces of Fq(2ν) of type (m, s). Denote L(m,s;2ν) by the set of subspaces which are intersections of subspaces in M(m, s; 2ν and assume the intersection of the empty set of subspaces of Fq(2ν) is Fq(2ν) itself. It is known that if we partially order L(m, s; 2ν) by ordinary inclusion, then L(m, s; 2ν) is a poset denoted Lo(m, s; 2ν). Let Nm,s= min{m - s,2s+1}, where m > 3 and s > 1. We first construct Lo(m, s; 2ν) to be the set of all subspaces of type (m1,s1) in Lo(m,s;2ν) satisfying 0 ≤ 2s1 ≤ m1 ≤ Nm,s. Then we show that Lo1(m, s; 2ν) is a strongly uniform subposet of rank Nm,s of Lo(m, s; 2ν). Finally, we construct Leonard pairs from Lo(m, s; 2ν).The thesis is composed of three chapters and organized as follows:In Chapter 1, we introduce the notions of symplectic space over a finite field, the poset, the uniform poset, Leonard pairs and Leonard system and give some related results.In Chapter 2, we first introduce the notions of the poset Lo(m, s; 2ν) in symplectic space over a finite field and give some related properties. Secondly, we define the subposet Lo(m, s; 2ν) of the poset Lo(m, s; 2ν) and then show the strongly uniform structure on the subposet Lo(m, s; 2ν) in symplectic space over a finite field.In Chapter 3, we construct a Leonard pair from Lo(m,s;2ν).