A Necessary Condition of Tilting Modules Over Path Algebras of Type A_n

Representation theory of algebras is a new branch of algebra which began at the seventies of the last century.Tilting theory is an important tool to study representation theory of algebras.The main purpose of this paper is to study structural characteristics of the tilting module over path algebra of type An in its AR-quiver.AR-quiver analysis and APR-tilting translation was used to proved that:if TA is a tilting A-module over a path algebra of type An,then there are points corresponding indecomposable direct summands of TA in both the highest τ-orbit and the lowest τ-orbit.