Association Schemes Based On The Pairs Of Vector Subspaces

Let IF_q be the finite field with q elements,IF_q?n?be the n-dimensional vector space over the finite field IF_q,Mi?0?i?n?be the set of all i-dimensional vector subspaces of IF_q?n?,and GLn?IF_q?denotes the group of all n × n reversible matrices over IF_q with matrix multiplication,which called general linear group of order n over IF_q.Let X_ij ={{A,B}|A?Mj,dim(A?B=0},and here dim?A?B?denotes the dimension of subspace A?B,For any element T ? GLn?IF_q?we define the transformation on X_ij as follow:{A,B} ? {AT,BT}.Thus,we can regard GLn?IF_q?as a transformation group on X_ij ,and it acts transitively on X_ij .The association scheme determined by the group action is recorded as X_ij .In this paper,we determine the association classes of the association scheme X_ij with i = j = ?1,which is X11,and calculate the intersection numbers of association scheme X11.Let X = {?AA,B?|A,B?M1,A?B}.For any element T ? GLn?IF_q?,we define the transformation on X as follow:?A,B???AT,BT?.Thus,we can regard GLn?IF_q?as a transformation group on X,and it acts transitively on X.The association scheme determined by the group action is recorded as X.In this paper,we determine the association classes of this association scheme,and list conditions which indicate two elements in the same orbit.