Automorphisms Of Generalized Symplectic Graph

Let F_q be a finite field with q elements, where q is a power of a odd prime and K be a2ν×2νnonsingular alternate matrix over F_q. Sp_2ν(F_q,K) is the symplectic group of degree 2νrelative to K over F_q and ? is the set of m?dimensional totally isotropic subspaces of F_q^2ν.Since symplectic group Sp_2ν(F_q,K) acts transitively on the set ?, we get the correspondingassociation scheme X. Every relation∧(r,d) determines a graphΓ^r,d with the set ? as itsvertex set and with the adjacency defined byP～Q (?)rank(PKQ^T) = r,dim(P∩Q) = d.where P～Q means that P and Q are adjacent.In the present paper, we determine the automorphism group ofΓ^1,m-1, that isΓSp_2ν(q,m,K), by a matrix skill and prove the following theorem.Theorem A Let PGSp_2ν(F_q,K) andΓSp_2ν(q,m,K) be the generalized projective sym-plectic group and the generalized symplectic graph separately . Then when 1 < m <ν, theautomorphism group G ofΓSp_2ν(q,m,K) is the product of PGSp_2ν(F_q,K) and Aut(F_q).Basing on the theorem A, we get the following corollary.Corollary A That symplectic group Sp_2ν(F_q,K) acts transitively on the set of V (ΓSp_2ν(q,m,K)) determines an association scheme, denoted by X.Then when 1 < m <ν, the automor-phism group of X is the product of PGSp_2ν(F_q,K) and Aut(F_q).