Suborbits of (m,k)-isotropic subspaces under finite singular classical groups

摘要

Let F_q~(2v+δ+l) be one of the (2v+δ+l)-dimensional singular classical spaces and let G_(2v+δ+l,2v+δ) be the corresponding singular classical group of degree 2v + δ +l. All the (m, k)-isotropic subspaces form an orbit under G_(2v+δ+l,2v+δ). denoted by M(m,k;2v+δ+l, 2v+δ). Let Λ be the set of all the orbitals of (G_(2v+δ+l,2δ+δ),M(m,k;2v+δ+l,2v+δ)). Then (M(m, k; 2v +δ + l,2v + δ), Λ) is a symmetric association scheme. First, we determine all the orbitals and the rank of (G_(2v+δ+1,2v+δ),M(m, k; 2v +δ+ l, 2v+δ)), calculate the length of each suborbit. Next, we compute all the intersection numbers of the symmetric association scheme (M(v + k,k;2v + δ+ l,2v + δ), Λ), where k = 1 or k = l-1. Finally, we construct a family of symmetric graphs with diameter 2 based on M(2,0;4 + δ+l,4 + δ).